Split (1)

train · 2.62k rows

text stringlengths 45 10k
inomial Theorem a formula that can be used to expand any binomial combination a selection of objects in which order does not matter common difference the difference between any two consecutive terms in an arithmetic sequence common ratio the ratio between any two consecutive terms in a geometric sequence complement of an event the set of outcomes in the sample space that are not in the event E diverge a series is said to diverge if the sum is not a real number event any subset of a sample space experiment an activity with an observable result explicit formula a formula that defines each term of a sequence in terms of its position in the sequence finite sequence a function whose domain consists of a finite subset of the positive integers {1, 2, … n} for some positive integer n Fundamental Counting Principle if one event can occur in m ways and a second event can occur in n ways after the first event has occurred, then the two events can occur in m × n ways; also known as the Multiplication Principle geometric sequence a sequence in which the ratio of a term to a previous term is a constant geometric series the sum of the terms in a geometric sequence index of summation in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation infinite sequence a function whose domain is the set of positive integers infinite series the sum of the terms in an infinite sequence lower limit of summation the number used in the explicit formula to find the first term in a series Multiplication Principle if one event can occur in m ways and a second event can occur in n ways after the first event has occurred, then the two events can occur in m × n ways; also known as the Fundamental Counting Principle mutually exclusive events events that have no outcomes in common n factorial the product of all the positive integers from 1 to n nth partial sum the sum of the first n terms of a sequence nth term of a sequence a formula for the general term of a sequence outcomes the possible results of an experiment permutation a selection of objects in which order matters probability a number from 0 to 1 indicating the likelihood of an event probability model a mathematical description of an experiment listing all possible outcomes and their associated probabilities CHAPTER 11 review 1009 recursive formula a formula that defines each term of a sequence using previous term(s ) sample space the set of all possible outcomes of an experiment sequence a function whose domain is a subset of the positive integers series the sum of the terms in a sequence summation notation a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series term a number in a sequence union of two events the event that occurs if either or both events occur upper limit of summation the number used in the explicit formula to find the last term in a series Key equations Formula for a factorial 0! = 1 1! = 1 n!= n(n − 1)(n − 2) ⋯ (2)(1), for n ≥ 2 recursive formula for nth term of an arithmetic sequence an = an −1 + d; n ≥ 2 explicit formula for nth term of an arithmetic sequence recursive formula for nth term of a geometric sequence explicit formula for nth term of a geometric sequence an = a1 + d(n − 1) an = ran − 1, n ≥ 2 an = a1r n −1 sum of the first n terms of an arithmetic series sum of the first n terms of a geometric series sum of an infinite geometric series with −1 < r < 1 number of permutations of n distinct objects taken r at a time number of combinations of n distinct objects taken r at a time number of permutations of n non-distinct objects Binomial Theorem (r + 1)th term of a binomial expansion probability of an event with equally likely outcomes Sn = Sn = Sn = + an) n(a1 _ 2 a1(1 − rn) _________ 1 − r a1 1 P(n, r) = n! _ (n − r)! C(n, r) = n! _ r!(n − r)! n! _ r1!r2! … rk! n (x + y)n = ∑ n _  x n − ky k  k k – 0 n _ r  xn − ryr  n(E) ____ n(S) P(E)= probability of the union of two events P(E ∪ F) = P(E) + P(F) − P(E ∩ F) probability of the union of mutually exclusive events P(E ∪ F) = P(E) + P(F) probability of the complement of an event P(E') = 1 − P(E) 1010 CHAPTER 11 seQuences, proBaBility and counting theory Key Concepts 11.1 Sequences and Their Notations • A sequence is a list of numbers, called terms, written in a specific order. • Explicit formulas define each term of a sequence using the position of the term. See Example 1, Example 2, and Example 3. • An explicit formula for the nth term of a sequence can be written by analyzing the pattern of several terms. See Example 4. • Recursive formulas define each term of a sequence using previous terms. • Recursive formulas must state the initial term, or terms, of a sequence. • A set of terms can be written by using a recursive formula. See Example 5 and Example 6. • A factorial is a mathematical operation that can be defined recursively. • The factorial of n is the product of all integers from 1 to n See Example 7. 11.2 Arithmetic Sequences • An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant. • The constant between two consecutive terms is called the common difference. • The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See Example 1. • The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See Example 2 and Example 3. • A recursive formula for an arithmetic sequence with common difference d is given by an = an − 1 + d, n ≥ 2. See Example 4. • As with any recursive formula, the initial term of the sequence must be given. • An explicit formula for an arithmetic sequence with common difference d is given by an = a1 + d(n − 1). See Example 5. • An explicit formula can be used to find the number of terms in a sequence. See Example 6. • In application problems, we sometimes alter the explicit formula slightly to an = a0 + dn. See Example 7. 11.3 Geometric Sequences • A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant. • The constant ratio between two consecutive terms is called the common ratio. • The common ratio can be found by dividing any term in the sequence by the previous term. See Example 1. • The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See Example 2 and Example 4. • A recursive formula for a geometric sequence with common ratio r is given by an = ran − 1 for n ≥ 2 . • As with any recursive formula, the initial term of the sequence must be given. See Example 3. • An explicit formula for a geometric sequence with common ratio r is given by an = a1r n − 1. See Example 5. • In application problems, we sometimes alter the explicit formula slightly to an = a0r n. See Example 6. 11.4 Series and Their Notations • The sum of the terms in a sequence is called a series. • A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See Example 1. • The sum of the terms in an arithmetic sequence is called an arithmetic series. • The sum of the first n terms of an arithmetic series can be found using a formula. See Example 2 and Example 3. • The sum of the terms in a geometric sequence is called a geometric series. • The sum of the first n terms of a geometric series can be found using a formula. See Example 4 and Example 5. • The sum of an infinite series exists if the series is geometric with −1 < r < 1. CHAPTER 11 review 1011 • If the sum of an infinite series exists, it can be found using a formula. See Example 6, Example 7, and Example 8. • An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See Example 9. 11.5 Counting Principles • If one event can occur in m ways and a second event with no common outcomes can occur in n ways, then the first or second event can occur in m + n ways. See Example 1. • If one event can occur in m ways and a second event can occur in n ways after the first event has occurred, then the two events can occur in m × n ways. See Example 2. • A permutation is an ordering of n objects. • If we have a set of n objects and we want to choose r objects from the set in order, we write P(n, r). • Permutation problems can be solved using the Multiplication Principle or the formula for P(n, r). See Example 3 and Example 4. • A selection of objects where the order does not matter is a combination. • Given n distinct objects, the number of ways to select r objects from the set is C (n, r) and can be found using a formula. See Example 5. • A set containing n distinct objects has 2n subsets. See Example 6. • For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See Example 7. 11.6 Binomial Theorem n _ •  r  is called a binomial coefficient and is equal to C (n, r). See Example 1. • The Binomial Theorem allows us to expand binomials without multiplying. See Example 2. • We can find a given term of a binomial expansion without fully expanding the binomial. See Example 3. 11.7 Probability • Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain. • The probabilities in a probability model must sum to 1. See Example 1. • When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See Example 2. • To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See Example 3. • To fi
nd the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See Example 4. • The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See Example 5. • In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See Example 6. 1012 CHAPTER 11 seQuences, proBaBility and counting theory CHAPTeR 11 ReVIeW exeRCISeS SeQUenCeS AnD THeIR nOTATIOn 1. Write the first four terms of the sequence defined by the recursive formula a1 = 2, an = an − 1 + n. 3. Write the first four terms of the sequence defined by the explicit formula an = 10n + 3. ARITHMeTIC SeQUenCeS 4 _ , 5. Is the sequence 7 39 _ 7 find the common difference. 47 _ , 21 82 _ , 21 , ... arithmetic? If so, 2. Evaluate 6! ________ (5 − 3)!3! . 4. Write the first four terms of the sequence defined by the explicit formula an = n! ________ n(n + 1). 6. Is the sequence 2, 4, 8, 16, ... arithmetic? If so, find the common difference. 7. An arithmetic sequence has the first term a1 = 18 and common difference d = −8. What are the first five terms? 8. An arithmetic sequence has terms a3 = 11.7 and a8 = −14.6. What is the first term? 9. Write a recursive formula for the arithmetic 10. Write a recursive formula for the arithmetic sequence sequence −20, − 10, 0,10,… 11. Write an explicit formula for the arithmetic 37 _ , 24 7 _ , sequence 8 29 _ , 24 15 1, − 0, − , … , and then find the 31st term. 2 2 12. How many terms are in the finite arithmetic sequence 12, 20, 28, … , 172? GeOMeTRIC SeQUenCeS 13. Find the common ratio for the geometric sequence 14. Is the sequence 4, 16, 28, 40, … geometric? If so find 2.5, 5, 10, 20, … the common ratio. If not, explain why. 15. A geometric sequence has terms a7 = 16,384 and a9 = 262,144. What are the first five terms? 16. A geometric sequence has the first term a1 = −3 and 1 _ common ratio r = . What is the 8th term? 2 17. What are the first five terms of the geometric sequence a1 = 3, an = 4 ⋅ an − 1? 18. Write a recursive formula for the geometric 1 _ 27 1 1 _ _ , , sequence 1, 9 3 , … 19. Write an explicit formula for the geometric sequence 20. How many terms are in the finite geometric 1 _ , − − 5 1 _ 15 , − 1 _ 45 , − 1 _ 135 , … 5 5 _ _ , …, − , − sequence −5, − 9 3 5 ______ ? 59,049 SeRIeS AnD THeIR nOTATIOn 21. Use summation notation to write the sum of terms 22. Use summation notation to write the sum that 1 _ m + 5 from m = 0 to m = 5. 2 results from adding the number 13 twenty times. 23. Use the formula for the sum of the first n terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series 2.5, 4, 5.5, … . 24. A ladder has 15 tapered rungs, the lengths of which increase by a common difference. The first rung is 5 inches long, and the last rung is 20 inches long. What is the sum of the lengths of the rungs? CHAPTER 11 review 1013 25. Use the formula for the sum of the first n terms of a 3 __ geometric series to find S9 for the series 12, 6, 3, , … 2 26. The fees for the first three years of a hunting club membership are given in Table 1. If fees continue to rise at the same rate, how much will the total cost be for the first ten years of membership? Year Membership Fees 27. Find the sum of the infinite geometric series ∞ 1 ∑  45 ⋅  − __ 3 k −1 k = 1 . 29. Alejandro deposits $80 of his monthly earnings into an annuity that earns 6.25% annual interest, compounded monthly. How much money will he have saved after 5 years? 1 2 3 $1500 $1950 $2535 Table 1 3 _ of the height of 28. A ball has a bounce-back ratio 5 the previous bounce. Write a series representing the total distance traveled by the ball, assuming it was initially dropped from a height of 5 feet. What is the total distance? (Hint: the total distance the ball travels on each bounce is the sum of the heights of the rise and the fall.) 30. The twins Sarah and Scott both opened retirement accounts on their 21st birthday. Sarah deposits $4,800.00 each year, earning 5.5% annual interest, compounded monthly. Scott deposits $3,600.00 each year, earning 8.5% annual interest, compounded monthly. Which twin will earn the most interest by the time they are 55 years old? How much more? COUnTInG PRInCIPleS 31. How many ways are there to choose a number 32. In a group of 20 musicians, 12 play piano, 7 play from the set { −10, −6, 4, 10, 12, 18, 24, 32} that is divisible by either 4 or 6? trumpet, and 2 play both piano and trumpet. How many musicians play either piano or trumpet? 33. How many ways are there to construct a 4-digit code if numbers can be repeated? 35. Calculate P(18, 4). 37. Calculate C(15, 6). 39. How many subsets does the set {1, 3, 5, … , 99} have? 34. A palette of water color paints has 3 shades of green, 3 shades of blue, 2 shades of red, 2 shades of yellow, and 1 shade of black. How many ways are there to choose one shade of each color? 36. In a group of 5 freshman, 10 sophomores, 3 juniors, and 2 seniors, how many ways can a president, vice president, and treasurer be elected? 38. A coffee shop has 7 Guatemalan roasts, 4 Cuban roasts, and 10 Costa Rican roasts. How many ways can the shop choose 2 Guatemalan, 2 Cuban, and 3 Costa Rican roasts for a coffee tasting event? 40. A day spa charges a basic day rate that includes use of a sauna, pool, and showers. For an extra charge, guests can choose from the following additional services: massage, body scrub, manicure, pedicure, facial, and straight-razor shave. How many ways are there to order additional services at the day spa? 41. How many distinct ways can the word 42. How many distinct rearrangements of the letters of DEADWOOD be arranged? the word DEADWOOD are there if the arrangement must begin and end with the letter D? 1014 CHAPTER 11 seQuences, proBaBility and counting theory BInOMIAl THeOReM 23  . 43. Evaluate the binomial coefficient  _ 8 1 y  44. Use the Binomial Theorem to expand  3x + _ 2 6 . 45. Use the Binomial Theorem to write the first three 46. Find the fourth term of (3a 2 − 2b)11 without fully terms of (2a + b)17. expanding the binomial. PROBABIlITY For the following exercises, assume two die are rolled. 47. Construct a table showing the sample space. 48. What is the probability that a roll includes a 2? 49. What is the probability of rolling a pair? 50. What is the probability that a roll includes a 2 or results in a pair? 51. What is the probability that a roll doesn’t include a 2 52. What is the probability of rolling a 5 or a 6? or result in a pair? 53. What is the probability that a roll includes neither a 5 nor a 6? For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.) 54. What is the percent chance that all the children 55. What is the percent chance that at least one of the attending the party prefer soda? children attending the party prefers milk? 56. What is the percent chance that exactly 3 of the 57. What is the percent chance that exactly 3 of the children attending the party prefer soda? children attending the party prefer milk? CHAPTER 11 practice test 1015 CHAPTeR 11 PRACTICe TeST 1. Write the first four terms of the sequence defined by the recursive formula a = −14, an = 2 + an – 1 ________ . 2 3. Is the sequence 0.3, 1.2, 2.1, 3, … arithmetic? If so find the common difference. 5. Write a recursive formula for the arithmetic 7 _ , − 5, − sequence −2, − 2 13 _ 2 22nd term. , … and then find the 2. Write the first four terms of the sequence defined by the explicit formula an = n2 − n − 1 ________ . n! 4. An arithmetic sequence has the first term a1 = −4 4 _ and common difference d = − . What is the 6th term? 3 6. Write an explicit formula for the arithmetic sequence 15.6, 15, 14.4, 13.8, … and then find the 32nd term. 1 1 _ _ , − 7. Is the sequence − 2, − 1, − , … geometric? If 4 2 so find the common ratio. If not, explain why. 8. What is the 11th term of the geometric sequence − 1.5, − 3, − 6, − 12, … ? 9. Write a recursive formula for the geometric 10. Write an explicit formula for the geometric sequence 1 1 1 _ _ _ , − sequence 1, − , 9 3 4 _ 27 , … 11. Use summation notation to write the sum of terms 5 __ k from k = −3 to k = 15. 3k 2 − 6 12. A community baseball stadium has 10 seats in the first row, 13 seats in the second row, 16 seats in the third row, and so on. There are 56 rows in all. What is the seating capacity of the stadium? 13. Use the formula for the sum of the first n terms of a 14. Find the sum of the infinite geometric series. geometric series to find ∑ 7 −0.2 ⋅ (−5) 15. Rachael deposits $3,600 into a retirement fund each year. The fund earns 7.5% annual interest, compounded monthly. If she opened her account when she was 20 years old, how much will she have by the time she’s 55? How much of that amount was interest earned? 17. A buyer of a new sedan can custom order the car by choosing from 5 different exterior colors, 3 different interior colors, 2 sound systems, 3 motor designs, and either manual or automatic transmission. How many choices does the buyer have? 16. In a competition of 50 professional ballroom dancers, 22 compete in the fox-trot competition, 18 compete in the tango competition, and 6 compete in both the fox-trot and tango competitions. How many dancers compete in the foxtrot or tango competitions? 18. To allocate annual bonuses, a manager must choose his top four employees and rank them first to fourth. In how many ways can he create the “Top-Four” list out of the 32 employees? 19. A rock group needs to choose 3 songs to play at the annual Battle of the Bands. How many ways can they choose their set if have 15 songs to pick from? 20. A self-serve frozen yogurt shop has 8 candy toppings and 4 fruit topp
ings to choose from. How many ways are there to top a frozen yogurt? 21. How many distinct ways can the word EVANESCENCE be arranged if the anagram must end with the letter E? 3 1 y  22. Use the Binomial Theorem to expand  _ _ x − 2 2 5 . 1  23. Find the seventh term of  x2 − __ 2 expanding the binomial. 13 without fully 1016 CHAPTER 11 seQuences, proBaBility and counting theory For the following exercises, use the spinner in Figure 1. 7 1 6 5 2 3 4 Figure 1 24. Construct a probability model showing each 25. What is the probability of landing on an odd possible outcome and its associated probability. (Use the first letter for colors.) number? 26. What is the probability of landing on blue? 27. What is the probability of landing on blue or an odd number? 28. What is the probability of landing on anything other 29. A bowl of candy holds 16 peppermint, 14 than blue or an odd number? butterscotch, and 10 strawberry flavored candies. Suppose a person grabs a handful of 7 candies. What is the percent chance that exactly 3 are butterscotch? (Show calculations and round to the nearest tenth of a percent.) Introduction to Calculus 12 Figure 1 jamaican sprinter Usain Bolt accelerates out of the blocks. (credit: nick Webb) CHAPTeR OUTlIne 12.1 Finding limits: numerical and Graphical Approaches 12.2 Finding limits: Properties of limits 12.3 Continuity 12.4 Derivatives Introduction The eight-time world champion and winner of six Olympic gold medals in sprinting, Usain Bolt has truly earned his nickname as the “fastest man on Earth.” Also known as the “lightning bolt,” he set the track on fire by running at a top speed of 27.79 mph—the fastest time ever recorded by a human runner. Like the fastest land animal, a cheetah, Bolt does not run at his top speed at every instant. How then, do we approximate his speed at any given instant? We will find the answer to this and many related questions in this chapter. 1017 1018 CHAPTER 12 introduction to calculus leARnInG OBjeCTIVeS In this section, you will: • Understand limit notation. • Find a limit using a graph. • Find a limit using a table. 12.1 FInDInG lIMITS: nUMeRICAl AnD GRAPHICAl APPROACHeS Intuitively, we know what a limit is. A car can go only so fast and no faster. A trash can might hold 33 gallons and no more. It is natural for measured amounts to have limits. What, for instance, is the limit to the height of a woman? The tallest woman on record was Jinlian Zeng from China, who was 8 ft 1 in.[36] Is this the limit of the height to which women can grow? Perhaps not, but there is likely a limit that we might describe in inches if we were able to determine what it was. To put it mathematically, the function whose input is a woman and whose output is a measured height in inches has a limit. In this section, we will examine numerical and graphical approaches to identifying limits. Understanding limit notation We have seen how a sequence can have a limit, a value that the sequence of terms moves toward as the number of terms increases. For example, the terms of the sequence 1 1 1 _ _ _ ... , , 1, 4 8 2 gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function f (x) = L, then as the input x gets closer and closer to a, the output y-coordinate gets closer and closer to L. We say that the output “approaches” L. Figure 1 provides a visual representation of the mathematical concept of limit. As the input value x approaches a, the output value f (x) approaches L. f(x) L Output Y approaches L a Input x approaches a x Figure 1 The output (y-coordinate) approaches L as the input (x-coordinate) approaches a. We write the equation of a limit as lim x → a f(x) = L. This notation indicates that as x approaches a both from the left of x = a and the right of x = a, the output value approaches L. Consider the function We can factor the function as shown. f (x) = x 2 − 6x − 7 __________ x − 7 . f (x) =  (x − 7) (x + 1) ____________  x − 7 f (x) = x + 1, x ≠ 7 Cancel like factors in numerator and denominator. Simplify. 36 https://en.wikipedia.org/wiki/Human_height and http://en.wikipedia.org/wiki/List_of_tallest_people SECTION 12.1 Finding limits: numerical and graphical approaches 1019 Notice that x cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition, x ≠ 7, for the simplified function. We can represent the function graphically as shown in Figure 2. y 10 (x) = x2−6x−7 x −7 21 3 4 5 6 7 8 9 10 x −2 −1 −1 −2 Figure 2 Because 7 is not allowed as an input, there is no point at x = 7. What happens at x = 7 is completely different from what happens at points close to x = 7 on either side. The notation f (x) = 8 lim x → 7 indicates that as the input x approaches 7 from either the left or the right, the output approaches 8. The output can get as close to 8 as we like if the input is sufficiently near 7. What happens at x = 7? When x = 7, there is no corresponding output. We write this as f (7) does not exist. This notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as f (x) = x + 1, x ≠ 7. Notice that the limit of a function can exist even when f (x) is not defined at x = a. Much of our subsequent work will be determining limits of functions as x nears a, even though the output at x = a does not exist. the limit of a function A quantity L is the limit of a function f (x) as x approaches a if, as the input values of x approach a (but do not equal a), the corresponding output values of f (x) get closer to L. Note that the value of the limit is not affected by the output value of f (x) at a. Both a and L must be real numbers. We write it as lim x → a f (x) = L Example 1 Understanding the Limit of a Function For the following limit, define a, f (x), and L. lim x → 2 Solution First, we recognize the notation of a limit. If the limit exists, as x approaches a, we write (3x + 5) = 11 We are given lim x → 2 This means that a = 2, f (x) = 3x + 5, and L = 11. (3x + 5) = 11 lim x → a f (x) = L. Analysis Recall that y = 3x + 5 is a line with no breaks. As the input values approach 2, the output values will get close (3x + 5) = 11, which means that as x nears 2 (but is not exactly 2), the to 11. This may be phrased with the equation lim output of the function f (x) = 3x + 5 gets as close as we want to 3(2) + 5, or 11, which is the limit L, as we take values of x sufficiently near 2 but not at x = 2. x → 2 1020 CHAPTER 12 introduction to calculus Try It #1 For the following limit, define a, f (x), and L. (2x2 − 4) = 46 lim x → 5 Understanding Left-Hand Limits and Right-Hand Limits We can approach the input of a function from either side of a value—from the left or the right. Figure 3 shows the values of as described earlier and depicted in Figure 2. f (x) = x + 1, x ≠ 7 Values of x approach 7 from the left (x < 7) 6.99 7.99 6.999 7.999 6.9 7.9 x = 7 7 Undefined x f (x) Values of x approach 7 from the right (x > 7) 7.01 8.01 7.001 8.001 7.1 8.1 Values of output approach the limit, 8 Values of output approach the limit, 8 Figure 3 Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in Figure 3 are 6.9, 6.99, and 6.999. The corresponding outputs are 7.9, 7.99, and 7.999. These values are getting closer to 8. The limit of values of f (x) as x approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function f (x) = x + 1, x ≠ 7 as x approaches 7. Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in Figure 3 are 7.1, 7.01, and 7.001. The corresponding outputs are 8.1, 8.01, and 8.001. These values are getting closer to 8. The limit of values of f (x) as x approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function f (x) = x + 1, x ≠ 7 as x approaches 7. Figure 3 shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input x within the interval 6.9 < x < 7.1 to produce an output value of f (x) within the interval 7.9 < f (x) < 8.1. We also see that we can get output values of f (x) successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values. Figure 4 provides a visual representation of the left- and right-hand limits of the function. From the graph of f (x), we observe the output can get infinitesimally close to L = 8 as x approaches 7 from the left and as x approaches 7 from the right. To indicate the left-hand limit, we write To indicate the right-hand limit, we write x → 7− f (x) = 8. lim x → 7+ f (x) = 8. lim 2 0 −1 −1 −2 f(x) approaches 8 f(x) approaches 8 y = f(x) x approaches 7 from the left 21 approaches 7 from the right x Figure 4 The left- and right-hand limits are the same for this function. SECTION 12.1 Finding limits: numerical and graphical approaches 1021 left- and right-hand limits The left-hand limit of a function f (x) as x approaches a from the left is equal to L, denoted by x → a− f(x) = L. lim The values of f (x) can get as close to the limit L as we like by taking values of x sufficiently close to a such that x < a and x ≠ a. The right-hand limit of a function f (x), as x approaches a from the right, is equal to L, denoted by x → a+ f(x) = L. lim The values of f (x) can get as close to the limit L as we like by taking val
ues of x sufficiently close to a but greater than a. Both a and L are real numbers. Understanding Two-Sided Limits In the previous example, the left-hand limit and right-hand limit as x approaches a are equal. If the left- and right-hand limits are equal, we say that the function f (x) has a two-sided limit as x approaches a. More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist. the two-sided limit of function as x approaches a The limit of a function f (x), as x approaches a, is equal to L, that is, if and only if lim x → a f(x) = L. x → a− f(x) = lim lim In other words, the left-hand limit of a function f (x) as x approaches a is equal to the right-hand limit of the same function as x approaches a. If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist. x → a+ f(x) Finding a limit Using a Graph To visually determine if a limit exists as x approaches a, we observe the graph of the function when x is very near to x = a. In Figure 5 we observe the behavior of the graph on both sides of a. y lim a− x L f (x) = L f (x) = L lim + a x y = f(x) a x < a x > a x Figure 5 To determine if a left-hand limit exists, we observe the branch of the graph to the left of x = a, but near x = a. This is where x < a. We see that the outputs are getting close to some real number L so there is a left-hand limit. To determine if a right-hand limit exists, observe the branch of the graph to the right of x = a, but near x = a. This is where x > a. We see that the outputs are getting close to some real number L, so there is a right-hand limit. If the left-hand limit and the right-hand limit are the same, as they are in Figure 5, then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit. Finally, we can look for an output value for the function f (x) when the input value x is equal to a. The coordinate pair of the point would be (a, f (a)). If such a point exists, then f (a) has a value. If the point does not exist, as in Figure 5, then we say that f (a) does not exist. 1022 CHAPTER 12 introduction to calculus How To… Given a function f (x), use a graph to find the limits and a function value as x approaches a. 1. Examine the graph to determine whether a left-hand limit exists. 2. Examine the graph to determine whether a right-hand limit exists. 3. If the two one-sided limits exist and are equal, then there is a two-sided limit—what we normally call a “limit.” 4. If there is a point at x = a, then f (a) is the corresponding function value. Example 2 Finding a Limit Using a Graph a. Determine the following limits and function value for the function f shown in Figure 6. i. lim x → 2− f (x) ii. lim x → 2+ f (x) iii. lim x → 2 f (x) iv. f (2) b. Determine the following limits and function value for the function f shown in Figure 7. i. lim x → 2− f (x) f (x) ii. lim x → 2+ y iii. lim x → 2 f (x) iv. f (2) 10 3−4 −1 −2 −1 −2 f 21 3 4 5 6 7 8 x f y 10 3−4−5−6 −1 −2 −1 −2 21 3 4 5 6 x Solution a. Looking at Figure 6: Figure 6 Figure 7 i. ii. iii. iv. lim x → 2− f (x) = 8; when x < 2, but infinitesimally close to 2, the output values get close to y = 8. lim x → 2+ f (x) = 3; when x > 2, but infinitesimally close to 2, the output values approach y = 3. f (x) does not exist because lim lim x → 2 f (2) = 3 because the graph of the function f passes through the point (2, f (2)) or (2, 3). f (x) ≠ lim x → 2+ f (x); the left- and right-hand limits are not equal. x → 2− b. Looking at Figure 7: i. ii. iii. iv. lim x → 2− f (x) = 8; when x < 2 but infinitesimally close to 2, the output values approach y = 8. lim x → 2− f (x) = 8; when x > 2 but infinitesimally close to 2, the output values approach y = 8. f (x) = 8 because lim x → 2− lim x → 2 f (2) = 4 because the graph of the function f passes through the point (2, f (2)) or (2, 4). f (x) = 8; the left and right-hand limits are equal. f (x) = lim x → 2+ Try It #2 Using the graph of the function y = f (x) shown in Figure 8, estimate the following limits. a. lim x → 0− f (x) b. lim x → 0+ f (x) c. lim x → 0 f (x) d. lim x → 2− f (x) e. lim x → 2+ f (x) g. lim x → 4− f (x) h. lim x → 4+ f (x) f. lim x → 2 f (x) i. lim x → 4 f (x) y 5 4 3 2 1 −3−4−5−6 −1 −2 −1 −2 −3 f 21 3 4 5 6 x Figure 8 SECTION 12.1 Finding limits: numerical and graphical approaches 1023 Finding a limit Using a Table Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of x approach a from both sides. Then we determine if the output values get closer and closer to some real value, the limit L. Let’s consider an example using the following function:   x3 − 125 _ x − 5 To create the table, we evaluate the function at values close to x = 5. We use some input values less than 5 and some values greater than 5 as in Figure 9. The table values show that when x > 5 but nearing 5, the corresponding output gets close to 75. When x > 5 but nearing 5, the corresponding output also gets close to 75. lim x → 5 x f (x) 4.9 73.51 4.99 4.999 5 5.001 5.01 74.8501 74.985001 Undefined 75.015001 75.1501 5.1 76.51 lim x → 5− f (x) = 75 Figure 9 lim x → 5+ f (x) = 75 Because then lim x → 5− f (x) = 75 = lim x → 5+ f (x), Remember that f (5) does not exist. f (x) = 75. lim x → 5 How To… Given a function f, use a table to find the limit as x approaches a and the value of f (a), if it exists. 1. Choose several input values that approach a from both the left and right. Record them in a table. 2. Evaluate the function at each input value. Record them in the table. 3. Determine if the table values indicate a left-hand limit and a right-hand limit. 4. If the left-hand and right-hand limits exist and are equal, there is a two-sided limit. 5. Replace x with a to find the value of f (a). Example 3 Finding a Limit Using a Table Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit. 5sin(x)   _ 3x Solution We can estimate the value of a limit, if it exists, by evaluating the function at values near x = 0. We cannot find a function value for x = 0 directly because the result would have a denominator equal to 0, and thus would be undefined. lim x → 0 f (x) = 5sin(x) _ 3x We create Figure 10 by choosing several input values close to x = 0, with half of them less than x = 0 and half of them greater than x = 0. Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table. 5 _ The table values indicate that when x < 0 but approaching 0, the corresponding output nears . 3 5 _ When x > 0 but approaching 0, the corresponding output also nears . 3 1024 CHAPTER 12 introduction to calculus x f (x) Because then −0.1 −0.01 −0.001 0 0.001 0.01 0.1 1.66389 1.666639 1.666666 Undefined 1.666666 1.666639 1.66389 lim x → 0− 5 _ f (x) = 3 Figure 10 lim x → 0+ 5 __ f (x) = 3 lim x → 0− 5 _ = lim f (x) = 3 x → 0+ f (x), lim x → 0− 5 _ f (x) = . 3 Q & A… Is it possible to check our answer using a graphing utility? Yes. We previously used a table to find a limit of 75 for the function f (x) = as x approaches 5. To check, we x3 − 125 _ x − 5 graph the function on a viewing window as shown in Figure 11. A graphical check shows both branches of the graph of the function get close to the output 75 as x nears 5. Furthermore, we can use the ‘trace’ feature of a graphing calculator. By approaching x = 5 we may numerically observe the corresponding outputs getting close to 75. y 125 100 L = 75 50 25 f −5 −4 −3 −2 − Figure 11 Try It #3 Numerically estimate the limit of the following function by making a table: lim x → 0 20sin(x)   _ 4x Q & A… Is one method for determining a limit better than the other? No. Both methods have advantages. Graphing allows for quick inspection. Tables can be used when graphical utilities aren’t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph. Example 4 Using a Graphing Utility to Determine a Limit With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as x approaches 0. If the function has a limit as x approaches 0, state it. If not, discuss why there is no limit. π x  f (x) = 3sin  _ SECTION 12.1 Finding limits: numerical and graphical approaches 1025 Solution We can use a graphing utility to investigate the behavior of the graph close to x = 0. Centering around x = 0, we choose two viewing windows such that the second one is zoomed in closer to x = 0 than the first one. The result would resemble Figure 12 for [ − 2, 2] by [ − 3, 3]. y 4 f −2 x 2 −4 Figure 12 The result would resemble Figure 13 for [−0.1, 0.1] by [−3, 3]. y 4 −4 −0.1 x 0.1 Figure 13 even closer to zero, we are even less able to distinguish any limits. The closer we get to 0, the greater the swings in the output values are. That is not the behavior of a function with either a left-hand limit or a right-hand limit. And if there is no left-hand limit or right-hand limit, there certainly is no limit to the function f (x) as x approaches 0. We write lim x → 0− π x   does not exist.  3sin  _ lim x → 0+ π x   does not exist.  3sin  _ lim x → 0 π x   does not exist.  3sin  _ Try It #4 Numerically estimate the following limit: lim x → 0 2 x   .  sin  _ Access these online resources for additional instruction and practice with finding limits. • Introduction to limits (http://openstaxcollege.org/l/introtolimits) • Formal Definition of a limit (http://openstaxcollege.org/l/formaldeflimit) 1026 CHAPTER 12 introduction to calculus 12.1 SeCTIOn exeRCISeS VeRBAl 1. Explain the difference between a value at x = a and the limit as x approaches a. 2. Explain why we say a function does not have a
limit as x approaches a if, as x approaches a, the left-hand limit is not equal to the right-hand limit. GRAPHICAl For the following exercises, estimate the functional values and the limits from the graph of the function f provided in Figure 14. y 6 5 4 3 2 1 −5 −4 −3 −1 −2 −1 −2 −3 −4 −5 −6 f(x) 21 3 4 5 x Figure 14 3. lim x → −2− f (x) 5. lim x → −2 f (x) 7. lim x → −1− f (x) 9. lim x → 1 f (x) 11. lim x → 4− f (x) 13. lim x → 4 f (x) 4. lim x → −2+ f (x) 6. f (−2) 8. lim x → 1+ f (x) 10. f (1) 12. lim x → 4+ f (x) 14. f (4) For the following exercises, draw the graph of a function from the functional values and limits provided. f (x) = 2, lim x → 0+ 15. lim x → 0− f (0) = 4, f (2) = −1, f (−3) does not exist. f (x) = −3, lim x → 2 f (x) = 2, f (x) = 0, lim x → 2+ = −2, lim x → 0 f (x) = 3, 16. lim x → 2− f (2) = 5, f (0) f (x) = 2, lim x → 2+ = −3, lim x → 0 f (x) = 5, f (0) = 1, 17. lim x → 2− f (1) = 0 f (x)= 6, lim x → 6+ f (x) = −1, lim x → 0 f (x) = 5, f (4) = 6, 19. lim x → 4 f (2) = 6 f (x) = 0, lim x → 3+ 18. lim x → 3− f (3) does not exist. f (x) = 5, lim x → 5 f (x) = 0, f (5) = 4, f (x) = 2, lim 20. lim x → −3 x → 1+ f (−3) = 0, f (0) = 0 f (x) = −2, lim x → 3 f (x) = − 4, 21. lim x → π f (π) = √ f (x) = π 2, lim π _ f (x) = , lim 2 x → 1− 2 , f (0) does not exist. x → −π — f (x) = 0, For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as x approaches 0. 22. f (x) = (1 + x) 1 _ x 24. h (x) = (1 + x) 3 _ x 23. g (x) = (1 + x) 2 _ x 25. i(x) = (1 + x) 4 _ x 26. j(x) = (1 + x) 5 _ x 27. Based on the pattern you observed in the exercises above, make a conjecture as to the limit of f (x) = (1 + x) 6 _ x , g (x) = (1 + x) 7 _ x , and h(x) = (1 + x) n _ x . For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. 28. (x) =  \| x \| − 1, x 3, if x ≠ 1 if x = 1 a = 1 29. (x) =  1 _____ x + 1 , (x + 1)2, if x = −2 if x ≠ −2 a = −2 SECTION 12.1 section exercises 1027 nUMeRIC For the following exercises, use numerical evidence to determine whether the limit exists at x = a. If not, describe the behavior of the graph of the function near x = a. Round answers to two decimal places. 30. f (x) = x2 − 4x _ 16 − x 2 ; a = 4 32. f (x) = x 2 − 6x − 7 _ x 2 − 7x ; a = 7 34. f (x) = 1 − x 2 _ x 2 − 3x + 2 ; a = 1 31. f (x 33. f (x) = x 2 − 1 _ x 2 − 3x + 2 ; a = 1 35. f (x) = 10 − 10x2 _ x 2 − 3x + 2 ; a = 1 36. f (x) = x __ 6x 2 − 5x − 6 3 __ ; a = 2 37. f (x) = x __ 4x 2 + 4x + 1 1 __ ; a = − 2 38. f (x For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as x approaches the given value. 39. lim x → 0 7tanx _ 3x 40. lim x → 4 x2 _ x − 4 41. lim x → 0 2sinx _ 4tanx For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. 1 _ x e e 42. lim 46. lim x → 5 e e − 1 __ x 2 43. lim x → 0 44. lim x → 0 \| x \| _ x 47. lim x → −1 1 _ (x + 1)2 48. lim x → 1 1 _ (x − 1)3 45. lim x → − 49. lim 50. Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas  as x approaches 0. Use a graphing utility, if possible, to appear similar: f (x) =  determine the left- and right-hand limits of the functions f (x) and g(x) as x approaches 0. If the functions have a limit as x approaches 0, state it. If not, discuss why there is no limit.  and g(x exTenSIOnS 51. According to the Theory of Relativity, the mass m of a particle depends on its velocity v. That is m = mo _____ √ 1 − v 2 __ c 2 where mo is the mass when the particle is at rest and c is the speed of light. Find the limit of the mass, m, as v approaches c −. 52. Allow the speed of light, c, to be equal to 1.0. If the mass, m, is 1, what occurs to m as v → c? Using the values listed in Table 1, make a conjecture as to what the mass is as v approaches 1.00. v 0.5 0.9 0.95 0.99 0.999 0.99999 Table 1 m 1.15 2.29 3.20 7.09 22.36 223.61 1028 CHAPTER 12 introduction to calculus leARnInG OBjeCTIVeS In this section, you will: • Find the limit of a sum, a difference, and a product. • Find the limit of a polynomial. • Find the limit of a power or a root. • Find the limit of a quotient. 12. 2 FInDInG lIMITS: PROPeRTIeS OF lIMITS Consider the rational function f (x) = x2 − 6x − 7 __________ x − 7 The function can be factored as follows: f (x) =  (x − 7) (x + 1) ____________  x − 7 , which gives us f (x) = x + 1, x ≠ 7. Does this mean the function f is the same as the function g(x) = x + 1? The answer is no. Function f does not have x = 7 in its domain, but g does. Graphically, we observe there is a hole in the graph of f (x) at x = 7, as shown in Figure 1 and no such hole in the graph of g(x), as shown in Figure 2. y 10 1 0 -1 f (x) = x 2 − 6x − 10 x y 10 1 0 -1 g(x 10 x Figure 1 The graph of function f contains a break at x = 7 and is therefore not continuous at x = 7. Figure 2 The graph of function g is continuous. So, do these two different functions also have different limits as x approaches 7? Not necessarily. Remember, in determining a limit of a function as x approaches a, what matters is whether the output approaches a real number as we get close to x = a. The existence of a limit does not depend on what happens when x equals a. Look again at Figure 1 and Figure 2. Notice that in both graphs, as x approaches 7, the output values approach 8. This means Remember that when determining a limit, the concern is what occurs near x = a, not at x = a. In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition. lim x → 7 f (x) = lim x → 7 g(x). SECTION 12.2 Finding limits: properties oF limits 1029 Finding the limit of a Sum, a Difference, and a Product Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits. Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions. properties of limits Let a, k, A, and B represent real numbers, and f and g be functions, such that lim f (x) = A and lim x → a x → a g(x) = B. For limits that exist and are finite, the properties of limits are summarized in Table 1. Constant, k Constant times a function Sum of functions Difference of functions Product of functions Quotient of functions Function raised to an exponent nth root of a function, where n is a positive integer Polynomial function lim x → a k = k lim x → a [k ⋅ f (x)] = k lim x → a f (x) = kA lim x → a [f (x) + g(x)] = lim x → a f (x) + lim x → a g(x) = A + B lim x → a [f (x) − g(x)] = lim x → a f (x) − lim x → a g(x) = A − B lim x → a [f (x) ⋅ g(x)] = lim x → a f (x) ⋅ lim x → a g(x) = A ⋅ B lim x → a f (x) ____ g(x) = lim f (x) A x → a _________ _ = g(x) lim B x → a , B ≠ 0 lim x → a [ f (x)]n =  lim x → ∞ f (x)  n = An, where n is a positive integer n lim x → a √ — n √ f (x) = — lim x → a [ f (x)] = n √ — A lim x → a p(x) = p(a) Table 1 Example 1 Evaluating the Limit of a Function Algebraically Evaluate lim x → 3 (2x + 5). Solution lim x → 3 (2x + 5) = lim x → 3 = 2 lim x → 3 = 2(3) + 5 (2x) + lim x → 3 (x) + lim x → 3 (5) (5) Sum of functions property Constant times a function property Evaluate = 11 Try It #1 Evaluate the following limit: lim x → −12 (−2x + 2). 1030 CHAPTER 12 introduction to calculus Finding the limit of a Polynomial Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit of a polynomial function as x approaches a is equivalent to simply evaluating the function for a. How To… Given a function containing a polynomial, find its limit. 1. Use the properties of limits to break up the polynomial into individual terms. 2. Find the limits of the individual terms. 3. Add the limits together. 4. Alternatively, evaluate the function for a. Example 2 Evaluating the Limit of a Function Algebraically Evaluate lim x → 3 (5x2). Solution Try It #2 Evaluate lim x → 4 (x3 − 5). lim x → 3 (x2) (5x2) = 5 lim x → 3 = 5(32) = 45 Constant times a function property Function raised to an exponent property Example 3 Evaluating the Limit of a Polynomial Algebraically Evaluate lim x → 5 (2x3 − 3x + 1). Solution lim x → 5 (2x3 − 3x + 1) = lim x → 5 = 2 lim x → 5 (2x3) − lim x → 5 (3x) + lim x → 5 (x) + lim x → 5 (1) (1) (x3) − 3 lim x → 5 = 2(53) − 3(5) + 1 Sum of functions Constant times a function Function raised to an exponent = 236 Evaluate Try It #3 Evaluate the following limit: lim x → −1 (x4 − 4x3 + 5). Finding the limit of a Power or a Root When a limit includes a power or a root, we nee
d another property to help us evaluate it. The square of the limit of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots. Example 4 Evaluating a Limit of a Power Evaluate lim x → 2 (3x + 1)5. SECTION 12.2 Finding limits: properties oF limits 1031 Solution We will take the limit of the function as x approaches 2 and raise the result to the 5th power. lim x → 2 (3x + 1)  5 (3x + 1) 5 =  lim x → 2 = (3(2) + 1) 5 = 7 5 = 16,807 Try It #4 Evaluate the following limit: lim x → −4 (10x + 36)3. Q & A… If we can’t directly apply the properties of a limit, for example in lim x → 2 limit of the function as x approaches a? Yes. Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function.  , can we still determine the x2 + 6x + 8  __________ x − 2 Finding the limit of a Quotient Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify. How To… Given the limit of a function in quotient form, use factoring to evaluate it. 1. Factor the numerator and denominator completely. 2. Simplify by dividing any factors common to the numerator and denominator. 3. Evaluate the resulting limit, remembering to use the correct domain. Example 5 Evaluating the Limit of a Quotient by Factoring Evaluate lim x → 2 x2 − 6x + 8  .  __________ x − 2 Solution Factor where possible, and simplify. lim x → 2 x2 − 6x + 8  __________ x − 2 x → 2 (x − 2)(x − 4)    = lim ____________ x − 2  (x − 2) (x − 4)   ____________  x − 2 = lim x → 2 = lim x → 2 (x − 4) = 2 − 4 = −2 Factor the numerator. Cancel the common factors. Evaluate. Analysis When the limit of a rational function cannot be evaluated directly, factored forms of the numerator and denominator may simplify to a result that can be evaluated. Notice, the function f (x) = x2 − 6x + 8 __________ x − 2 is equivalent to the function Notice that the limit exists even though the function is not defined at x = 2. f (x) = x − 4, x ≠ 2. 1032 CHAPTER 12 introduction to calculus Try It #5 Evaluate the following limit: lim x → 7 x2 − 11x + 28  .  ____________ 7 − x Example 6 Evaluating the Limit of a Quotient by Finding the LCD Evaluate lim x → 5 1 1 __ __ − x 5 ______ x − 5  .  Solution Find the LCD for the denominators of the two terms in the numerator, and convert both fractions to have the LCD as their denominator. lim x → 5 1 1 __ __ − x 5 ______ x − 5   = lim x → 5  5x  1 1 __ __  − x 5 __________ 5x(x − 5)  Multiply numerator and denominator by LCD. = lim x → 5 1 1   − 5x  5x  __ __ 5 x _________________ 5x(x − 5)     = lim x → 5 5 − x ________ 5x(x − 5) = lim x → 5 −1(x − 5) _________ 5x(x − 5)   1 _ 5x − = lim x → 5 1 ____ 5(5) = − Apply distributive property. Simplify. Factor the numerator Cancel out like fractions Evaluate for x = 5 = − 1 ___ 25 Analysis When determining the limit of a rational function that has terms added or subtracted in either the numerator or denominator, the first step is to find the common denominator of the added or subtracted terms; then, convert both terms to have that denominator, or simplify the rational function by multiplying numerator and denominator by the least common denominator. Then check to see if the resulting numerator and denominator have any common factors. Try It #6 Evaluate lim x → −5 1 1 __ __ + x . 5 ________ 10 + 2x   How To… Given a limit of a function containing a root, use a conjugate to evaluate.  form, evaluate directly. 1. If the quotient as given is not in indeterminate  0 __ 0 2. Otherwise, rewrite the sum (or difference) of two quotients as a single quotient, using the least common denominator (LCD). 3. If the numerator includes a root, rationalize the numerator; multiply the numerator and denominator by the — b are conjugates. conjugate of the numerator. Recall that a ± √ 4. Simplify. 5. Evaluate the resulting limit. SECTION 12.2 Finding limits: properties oF limits 1033 Example 7 Evaluating a Limit Containing a Root Using a Conjugate —  .  25 − x − 5 √ __ x Evaluate lim x → 0 Solution —  25 − x − 5 √ __ x lim x → 0 — —  = lim  x → 0 25 − x + 5 25 − x − 5 √ √ __ __ ⋅ x 25 − x + 5 √  — = lim x → 0 (25 − x) − 25   __ 25 − x + 5  x  √ — __ = lim  x → 0 25 − x __ = lim   x → 0 25 − x + 5  x  √ — −x = = — −1 __ 25 − 0 + 5 √ −1 _____ 5 + 5 1 ___ 10 = − Multiply numerator and denominator by the conjugate. Multiply:  √ — 25 − x − 5  ⋅  √ — 25 − x + 5  = (25 − x) − 25. Combine like terms. Simplify = −1. −x ___ x Evaluate. Analysis When determining a limit of a function with a root as one of two terms where we cannot evaluate directly, think about multiplying the numerator and denominator by the conjugate of the terms. Try It #7 Evaluate the following limit: lim h → 0 — 16 − h − 4  .  √ ____________ h Example 8 Evaluating the Limit of a Quotient of a Function by Factoring Evaluate lim x → 4  4 − x ________ x − 2 √  . — Solution lim − √  = lim   __ — — — — = lim − √   ___ −  2 − √ x  — = lim 4 Factor. Factor −1 out of the denominator. Simplify. Evaluate. Analysis Multiplying by a conjugate would expand the numerator; look instead for factors in the numerator. Four is a perfect square so that the numerator is in the form and may be factored as a2 − b2 (a + b)(a −b). 1034 CHAPTER 12 introduction to calculus Try It #8 Evaluate the following limit: lim — — How To… Given a quotient with absolute values, evaluate its limit. 1. Try factoring or finding the LCD. 2. If the limit cannot be found, choose several values close to and on either side of the input where the function is undefined. 3. Use the numeric evidence to estimate the limits on both sides. Example 9 Evaluating the Limit of a Quotient with Absolute Values Evaluate lim Solution The function is undefined at x = 7, so we will try values close to 7 from the left and the right. Left-hand limit: \| 6.9 − 7 \| _ = 6.9 − 7 \| 6.99 − 7 \| _ = 6.99 − 7 \| 6.999 − 7 \| _ 6.999 − 7 = − 1 Right-hand limit: \| 7.1 − 7 \| _ = 7.1 − 7 \| 7.01 − 7 \| _ = 7.01 − 7 \| 7.001 − 7 \| _ = 1 7.001 − 7 Since the left- and right-hand limits are not equal, there is no limit. Try It #9 Evaluate lim x → 6 \| Access the following online resource for additional instruction and practice with properties of limits. • Determine a limit Analytically (http://openstaxcollege.org/l/limitanalytic) SECTION 12.2 section exercises 1035 12.2 SeCTIOn exeRCISeS VeRBAl 1. Give an example of a type of function f whose limit, 2. When direct substitution is used to evaluate the limit as x approaches a, is f (a). 3. What does it mean to say the limit of f (x), as x approaches c, is undefined? AlGeBRAIC of a rational function as x approaches a and the 0 _ result is f (a) = , does this mean that the limit of f 0 does not exist? For the following exercises, evaluate the limits algebraically. 4. lim x → 0 (3) 5. lim x → 2 −5x   _ x2 − 1 6. lim x → 2   x2 − 5x + 6 _ x + 2 7. lim x → 3   x2 − 9 _ x − 3 8. lim x → −1   x2 − 2x − 3 _ x + 1   6x2 − 17x + 12 __ 2x − 3 9. lim 3 __ x → 2 10. lim 7 __ x → − 2  8x2 + 18x − 35 __ 2x + 7  11. lim x2 − 9   _ x 2 − 5x + 6 x → 3 14. lim h → 0   (3 + h)3 − 27 __ __ x 18. lim x → 0 15. lim h → 0 19. lim x → 9 12. lim x → −3 x → 3 −12x4 + 108x2  13. lim    x2 + 2x − 3 −7x4 − 21x3 _ __ x − 3   (h + 3)2 − 9   5 − h − √ √ __ __ h h    x __ — 1 + 2x − 1 √ x − x2 √ _ — 1 − √ x 17. lim h → 0 21. lim x → 0  — 5 — — 16. lim h → 0 20. lim x → 1 24. lim x → 2 25. lim x → 2 22. lim 1 __ x → 2 26. lim x → 2 1 __ x2 − 4 _ 2x − 2 − h)3 − 8 __ h     x3 − 64 _ x2 − 16 x2 − 81 _ — 3 − √ x 23. lim x → 4 27. lim x → 4 28. lim x → 4 29. lim 30. lim x → 2   −8 + 6x − x2 __ x − 2 For the following exercise, use the given information to evaluate the limits: lim x → c f (x) = 3, lim x → c g(x) = 5 31. lim x → c  2 f (x) + √ — g(x)  32. lim x → c  3 f (x) + √ — g(x)  33. lim x → c f (x) _ g(x) For the following exercises, evaluate the following limits. 34. lim x → 2 cos(πx) 35. lim x → 2 sin(πx) 36. lim x → 2 π sin  _ x  37. f (x) =  2x2 + 2x + 1, x ≤ 0 x − 3, x > 0 ; lim x → 0+ f (x) 38. f (x) =  2x2 + 2x + 1, x ≤ 0 x − 3, x > 0 ; lim x → 0− f (x) 39. f (x) =  2x2 + 2x + 1, x ≤ 0 x − 3, x > 0 ; lim x → 0 f (x) 40. lim x → 4 — x + 5 − 3 √ ___________ x − 4 41. lim x → 2+  2x − 〚x〛  42. lim x → 2 — x + 7 − 3 √ ___________ x2 − x − 2 43. lim x → 3+ x 2 ______ x 2 − 9 1036 CHAPTER 12 introduction to calculus For the following exercises, find the average rate of change f (x + h) − f (x) _____________ h . 44. f (x) = x + 1 45. f (x) = 2x 2 − 1 46. f (x) = x 2 + 3x + 4 47. f (x) = x 2 + 4x − 100 48. f (x) = 3x 2 + 1 49. f (x) = cos(x) 50. f (x) = 2x 3 − 4x 1 _ 51. f (x) = x 52. f (x) = 1 _ x2 GRAPHICAl 53. f (x) = √ — x 54. Find an equation that could be represented by 55. Find an equation that could be represented by Figure 3. Figure 4. y 5 4 3 2 1 − −1−1 2 −2 −3 −4 −5 −5 −4 −3 21 3 4 5 x −5 −4 −3 y 5 4 3 2 1 − −1−1 2 −2 −3 −4 −5 21 3 4 5 x Figure 3 Figure 4 56. What is the right-hand limit of the function as x 57. What is the left-hand limit of the function as x approaches 0? 21 3 4 5 x −5 −4 −3 −2 approaches 0? −5 −4 −3 −2 y 5 4 3 2 1 −1−1 −2 −3 −4 −5 y 5 4 3 2 1 −1−1 −2 −3 −4 −5 21 3 4 5 x Figure 5 Figure 6 ReAl-WORlD APPlICATIOnS 58. The position function s(t) = −16t 2 + 144t gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval [1, 2] . 60. The amount of money in an account after t years compounded continuously a
t 4.25% interest is given by the formula A = A0 e 0.0425t, where A0 is the initial amount invested. Find the average rate of change of the balance of the account from t = 1 year to t = 2 years if the initial amount invested is $1,000.00. 59. The height of a projectile is given by s(t) = −64t 2 + 192t Find the average rate of change of the height from t = 1 second to t = 1.5 seconds. SECTION 12.3 continuity 1037 leARnInG OBjeCTIVeS In this section, you will: • Determine whether a function is continuous at a number. • Determine the numbers for which a function is discontinuous. • Determine whether a function is continuous. 12. 3 COnTInUITY Arizona is known for its dry heat. On a particular day, the temperature might rise as high as 118° F and drop down only to a brisk 95° F. Figure 1 shows the function T, where the output of T(x) is the temperature in Fahrenheit degrees and the input x is the time of day, using a 24-hour clock on a particular summer day 120 110 100 90 80 0 4 8 12 16 20 24 Time, Hours Since Midnight Figure 1 Temperature as a function of time forms a continuous function. When we analyze this graph, we notice a specific characteristic. There are no breaks in the graph. We could trace the graph without picking up our pencil. This single observation tells us a great deal about the function. In this section, we will investigate functions with and without breaks. Determining Whether a Function Is Continuous at a number Let’s consider a specific example of temperature in terms of date and location, such as June 27, 2013, in Phoenix, AZ. The graph in Figure 1 indicates that, at 2 a.m., the temperature was 96° F. By 2 p.m. the temperature had risen to 116° F, and by 4 p.m. it was 118° F. Sometime between 2 a.m. and 4 p.m., the temperature outside must have been exactly 110.5° F. In fact, any temperature between 96° F and 118° F occurred at some point that day. This means all real numbers in the output between 96° F and 118° F are generated at some point by the function according to the intermediate value theorem, Look again at Figure 1. There are no breaks in the function’s graph for this 24-hour period. At no point did the temperature cease to exist, nor was there a point at which the temperature jumped instantaneously by several degrees. A function that has no holes or breaks in its graph is known as a continuous function. Temperature as a function of time is an example of a continuous function. If temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Let’s create the function D, where D(x) is the output representing cost in dollars for parking x number of hours. See Figure 2. Suppose a parking garage charges $4.00 per hour or fraction of an hour, with a $25 per day maximum charge. Park for two hours and five minutes and the charge is $12. Park an additional hour and the charge is $16. We can never be charged $13, $14, or $15. There are real numbers between 12 and 16 that the function never outputs. There are breaks in the function’s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars. 1038 CHAPTER 12 introduction to calculus s r a l l o D 28 24 20 16 12 89 1 0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hours Parked Figure 2 Parking-garage charges form a discontinuous function. A function that remains level for an interval and then jumps instantaneously to a higher value is called a stepwise function. This function is an example. A function that has any hole or break in its graph is known as a discontinuous function. A stepwise function, such as parking-garage charges as a function of hours parked, is an example of a discontinuous function. So how can we decide if a function is continuous at a particular number? We can check three different conditions. Let’s use the function y = f (x) represented in Figure 3 as an example. y f a Figure 3 x Condition 1 According to Condition 1, the function f (a) defined at x = a must exist. In other words, there is a y-coordinate at x = a as in Figure 4. y f(a) f a Figure 4 x f (x), must exist. This means that at x = a the Condition 2 According to Condition 2, at x = a the limit, written lim x → a left-hand limit must equal the right-hand limit. Notice as the graph of f in Figure 3 approaches x = a from the left and right, the same y-coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at x = a. Condition 3 According to Condition 3, the corresponding y coordinate at x = a fills in the hole in the graph of f. This is written lim x → a f (x) = f (a). Satisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented in Figure 5 so the function is continuous as x = a. SECTION 12.3 continuity 1039 y f(a) f a x Figure 5 All three conditions are satisfied. The function is continuous at x = a. Figure 6 through Figure 9 provide several examples of graphs of functions that are not continuous at x = a and the condition or conditions that fail. y f a x Figure 6 Condition 2 is satisfied. Conditions 1 and 3 both fail. y f(a) f a x Figure 7 Conditions 1 and 2 are both satisfied. Condition 3 fails. y f(a) f a x Figure 8 Condition 1 is satisfied. Conditions 2 and 3 fail. 1040 CHAPTER 12 introduction to calculus y f(a) f a x Figure 9 Conditions 1, 2, and 3 all fail. definition of continuity A function f (x) is continuous at x = a provided all three of the following conditions hold true: • Condition 1: f (a) exists. • Condition 2: lim x → a • Condition 3: lim x → a f (x) exists at x = a. f (x) = f (a). If a function f (x) is not continuous at x = a, the function is discontinuous at x = a. Identifying a Jump Discontinuity Discontinuity can occur in different ways. We saw in the previous section that a function could have a left-hand limit and a right-hand limit even if they are not equal. If the left- and right-hand limits exist but are different, the graph “jumps” at x = a. The function is said to have a jump discontinuity. As an example, look at the graph of the function y = f (x) in Figure 10. Notice as x approaches a how the output approaches different values from the left and from the right. y f(a) f a x Figure 10 Graph of a function with a jump discontinuity. jump discontinuity A function f (x) has a jump discontinuity at x = a if the left- and right-hand limits both exist but are not equal: x → a− f (x) ≠ lim lim x → a+ f (x). Identifying Removable Discontinuity Some functions have a discontinuity, but it is possible to redefine the function at that point to make it continuous. This type of function is said to have a removable discontinuity. Let’s look at the function y = f (x) represented by the graph in Figure 11. The function has a limit. However, there is a hole at x = a. The hole can be filled by extending the domain to include the input x = a and defining the corresponding output of the function at that value as the limit of the function at x = a. SECTION 12.3 continuity 1041 y f a x Figure 11 Graph of function f with a removable discontinuity at x = a. removable discontinuity A function f (x) has a removable discontinuity at x = a if the limit, lim x → a f (x), exists, but either 1. f (a) does not exist or 2. f (a), the value of the function at x = a does not equal the limit, f (a) ≠ lim x → a f (x). Example 1 Identifying Discontinuities Identify all discontinuities for the following functions as either a jump or a removable discontinuity. a. f (x) = x2 − 2x − 15 ___________ x − 5 b. g(x) =  x + 1, x < 2 −x, x ≥ 2 Solution a. Notice that the function is defined everywhere except at x = 5. Thus, f (5) does not exist, Condition 2 is not satisfied. Since Condition 1 is satisfied, the limit as x approaches 5 is 8, and Condition 2 is not satisfied. This means there is a removable discontinuity at x = 5. b. Condition 2 is satisfied because g(2) = − 2. Notice that the function is a piecewise function, and for each piece, the function is defined everywhere on its domain. Let’s examine Condition 1 by determining the left- and right-hand limits as x approaches 2. Left-hand limit: lim Right-hand limit: lim x → 2− (x + 1) = 2 + 1 = 3. The left-hand limit exists. x → 2+ (−x) = − 2. The right-hand limit exists. But x → 2+ f (x). x → 2− f (x) ≠ lim lim So, lim f (x) does not exist, and Condition 2 fails: There is no removable discontinuity. However, since both x → 2 left- and right-hand limits exist but are not equal, the conditions are satisfied for a jump discontinuity at x = 2. Try It #1 Identify all discontinuities for the following functions as either a jump or a removable discontinuity. a. f (x) = x2 − 6x ______ x − 6 — b. g(x) =  √ 2x 1042 CHAPTER 12 introduction to calculus Recognizing Continuous and Discontinuous Real-number Functions Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of f (x) as x approaches a is the same as the value of f (x) when x = a. So lim f (x) = f (a). There are some functions that are continuous everywhere x → a and some that are only continuous where they are defined on their domain because they are not defined for all real numbers. examples of continuous functions The following functions are continuous everywhere: Polynomial functions Exponential functions Sine functions Cosine functions Ex: f (x) = x 4 − 9x 2 Ex: f (x) = 4x + 2 − 5 Ex: f (x) = sin(2x) − 4 π  Ex: f (x) = − cos  x + __ 3 Table 1 The following functions are continuous everywhere they are defined on their domain: Logarithmic functions Ex: f (x) = 2ln(x) , x > 0 Tangent functions Rational functions π __ + kπ, k is an integer Ex: f (x) = tan(x) + 2, x ≠ 2 Ex: f (x) = x 2 − 25 _______ x − 7 , x ≠ 7 Table 2 How To… Given a function f (x), de
termine if the function is continuous at x = a. 1. Check Condition 1: f (a) exists. 2. Check Condition 2: lim x → a 3. Check Condition 3: lim x → a f (x) = f (a). f (x) exists at x = a. 4. If all three conditions are satisfied, the function is continuous at x = a. If any one of the conditions is not satisfied, the function is not continuous at x = a. Example 2 Determining Whether a Piecewise Function is Continuous at a Given Number Determine whether the function f (x) =  x ≤ 3 4x, 8 + x, x > 3 is continuous at a. x = 3 8 __ b. x = 3 Solution To determine if the function f is continuous at x = a, we will determine if the three conditions of continuity are satisfied at x = a. a. Condition 1: Does f (a) exist? f (3) = 4(3) = 12 ⇒ Condition 1 is satisfied. SECTION 12.3 continuity 1043 Condition 2: Does lim x → 3 f (x) exist? To the left of x = 3, f (x) = 4x; to the right of x = 3, f (x) = 8 + x. We need to evaluate the left- and right-hand limits as x approaches 1. • Left-hand limit: lim • Right-hand limit: lim Because lim x → 1− f (x) ≠ lim x → 3− f (x) = lim x → 3+ f (x) = lim x → 1+ f (x), lim x → 1 x → 3− 4(3) = 12 x → 3+ (8 + x) = 8 + 3 = 11 f (x) does not exist. There is no need to proceed further. Condition 2 fails at x = 3. If any of the conditions of continuity are not satisfied at x = 3, the function f (x) is not continuous at x = 3. ⇒ Condition 2 fails. 8 __ b. x = 3 8  exist? Condition 1: Does f  _ 3 Condition 2: Does lim f (x) exist? x → 8 _ 3 32 ⇒ Condition 1 is satisfied. 8 8 _ _ To the left of x = f (x) = 4x; to the right of x = 3 3 8 _ . right-hand limits as x approaches 3 8 _  = 4  f (x) = lim 3 • Left-hand limit: lim 32 x) = 8 + x. We need to evaluate the left- and • Right-hand limit: lim f (x) = lim + 8 + x) = 8 + 3 32 _ 3 f (x) exists, Because lim x → 8 _ 3 8  = lim Condition 3: Is f  _ f (x)? 3 x → 8 _ 3 ⇒ Condition 2 is satisfied. 32 32  = f  _ _ 3 3 lim f (x) x → 8 _ 3 ⇒ Condition 3 is satisfied. 8 8 _ _ Because all three conditions of continuity are satisfied at x = , the function f (x) is continuous at x = . 3 3 Try It #2 Determine whether the function f (x) =  1 _ x ≤ 2 x , 9x − 11.5, x > 2 is continuous at x = 2. Example 3 Determining Whether a Rational Function is Continuous at a Given Number Determine whether the function f (x) = is continuous at x = 5. x2 − 25 _ x − 5 Solution To determine if the function f is continuous at x = 5, we will determine if the three conditions of continuity are satisfied at x = 5. 1044 CHAPTER 12 introduction to calculus Condition 1: f (5) does not exist. ⇒ Condition 1 fails. There is no need to proceed further. Condition 2 fails at x = 5. If any of the conditions of continuity are not satisfied at x = 5, the function f is not continuous at x = 5. Analysis See Figure 12. Notice that for Condition 2 we have lim x → 5 x2 − 25 _______ x − 5 = lim x → 3  (x − 5) (x + 5) ____________  x − 5 = lim x → 5 (x + 5) = 5 + 5 = 10 ⇒ Condition 2 is satisfied. At x = 5, there exists a removable discontinuity. See Figure 12. f y 13 12 11 10 21 3 4 5 6 7 8 Figure 12 x Try It #3 Determine whether the function f (x) = 9 − x2 ______ x2 − 3x is continuous at x = 3. If not, state the type of discontinuity. Determining the Input Values for Which a Function Is Discontinuous Now that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. A piecewise function may have discontinuities at the boundary points of the function as well as within the functions that make it up. To determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that polynomial functions themselves are continuous on the set of real numbers. Any discontinuity would be at the boundary points. So we need to explore the three conditions of continuity at the boundary points of the piecewise function. How To… Given a piecewise function, determine whether it is continuous at the boundary points. 1. For each boundary point a of the piecewise function, determine the left- and right-hand limits as x approaches a, as well as the function value at a. SECTION 12.3 continuity 1045 2. Check each condition for each value to determine if all three conditions are satisfied. 3. Determine whether each value satisfies condition 1: f (a) exists. 4. Determine whether each value satisfies condition 2: lim x → a 5. Determine whether each value satisfies condition 3: lim x → a f (x) exists. f (x) = f (a). 6. If all three conditions are satisfied, the function is continuous at x = a. If any one of the conditions fails, the function is not continuous at x = a. Example 4 Determining the Input Values for Which a Piecewise Function Is Discontinuous Determine whether the function f is discontinuous for any real numbers. f (x) =  x < 2 x + 1, 3, x2 − 11, x ≥ 4 2 ≤ x < 4 Solution The piecewise function is defined by three functions, which are all polynomial functions, f (x) = x + 1 on x < 2, f (x) = 3 on 2 ≤ x < 4, and f (x) = x2 − 5 on x ≥ 4. Polynomial functions are continuous everywhere. Any discontinuities would be at the boundary points, x = 2 and x = 4. At x = 2, let us check the three conditions of continuity. Condition 1: f (2) = 3 ⇒ Condition 1 is satisfied. Condition 2: Because a different function defines the output left and right of x = 2, does x → 2− f (x) = lim lim x → 2+ f (x)? • Left-hand limit: lim • Right-hand limit: lim x → 2− (x + 1+ 3 = 3 x → 2− f (x) = lim x → 2+ f (x) = lim x → 2+ f (x) Because 3 = 3, lim x → 2− f (x) = lim Condition 3: ⇒ Condition 2 is satisfied. f (x) = 3 = f (2) lim x → 2 ⇒ Condition 3 is satisfied. Because all three conditions are satisfied at x = 2, the function f (x) is continuous at x = 2. At x = 4, let us check the three conditions of continuity. Condition 2: Because a different function defines the output left and right of x = 4, does lim x → 4− f (x) = lim x → 4+ f (x)? • Left-hand limit: lim x → 4− f (x) = lim x → 4− 3 = 3 • Right-hand limit: lim x → 4+ f (x) = lim x → 4+ (x2 − 11) = 42 − 11 = 5 Because 3 ≠ 5, lim x → 4− f (x) ≠ lim x → 4+ f (x), so lim x → 4 f (x) does not exist. Because one of the three conditions does not hold at x = 4, the function f (x) is discontinuous at x = 4. ⇒ Condition 2 fails. 1046 CHAPTER 12 introduction to calculus Analysis See Figure 13. At x = 4, there exists a jump discontinuity. Notice that the function is continuous at x = 2. y 10 1−1 −2 −3−4−5 −2 f 21 3 4 5 6 x Figure 13 Graph is continuous at x = 2 but shows a jump discontinuity at x = 4. Try It #4 Determine where the function f (x) = x < 2 πx _ , 4 π _ x , 2πx, x > 6 2 ≤ x ≤ 6 is discontinuous. Determining Whether a Function Is Continuous To determine whether a piecewise function is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous. How To… Given a piecewise function, determine whether it is continuous. 1. Determine whether each component function of the piecewise function is continuous. If there are discontinuities, do they occur within the domain where that component function is applied? 2. For each boundary point x = a of the piecewise function, determine if each of the three conditions hold. Example 5 Determining Whether a Piecewise Function Is Continuous Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity. f (x) =  sin(x), x < 0 x > 0 x 3, Solution The two functions composing this piecewise function are f (x) = sin(x) on x < 0 and f (x) = x 3 on x > 0. The sine function and all polynomial functions are continuous everywhere. Any discontinuities would be at the boundary point, At x = 0, let us check the three conditions of continuity. Condition 1: f (0) does not exist. ⇒ Condition 1 fails. Because all three conditions are not satisfied at x = 0, the function f (x) is discontinuous at x = 0. SECTION 12.3 continuity 1047 Analysis See Figure 14. There exists a removable discontinuity at x = 0; lim x → 0 but f (a) does not exist. f (x) = 0, thus the limit exists and is finite, y f(x) = x3 21 3 4 5 6 x f(x) = sin(x) −5−6 −4 −3 −2 6 5 4 3 2 1 −1−1 −2 −3 −4 −5 −6 Figure 14 Function has removable discontinuity at 0. Access these online resources for additional instruction and practice with continuity. • Continuity at a Point (http://openstaxcollege.org/l/continuitypoint) • Continuity at a Point: Concept Check (http://openstaxcollege.org/l/contconcept) 1048 CHAPTER 12 introduction to calculus 12.3 SeCTIOn exeRCISeS VeRBAl 1. State in your own words what it means for a 2. State in your own words what it means for a function f to be continuous at x = c. function to be continuous on the interval (a, b). AlGeBRAIC For the following exercises, determine why the function f is discontinuous at a given point a on the graph. State which condition fails. 3. f (x) = ln \| . f (x) = ln \| 5x − 2 \| , a = 5 5. f (x) = x2 − 16 _ x + 4 , a = −4 6. f (x) = x2 − 16x _ x , a = 0 7. f (x) =  9. f (x) =  11. f (x) =  13. f (x) =  15. f (x) =  x, x ≠ 3 2x, x = 3 a = 3 1 _____ 2 − x , x ≠ 2 3, x < 1 x, x2 + 2x, x < 1 x, x = 1 −x2, x > 1 a = 1 x2 − 9 ______ x + 3 x − 9, 1 _ x , , x < −3 x = −3 a = −3 x > −3 8. f (x) =  10. f (x) =  12. f (x) =  14. f (x) =   16. f (x) = 5, x ≠ 0 3, x = 0 a = 0 1 _____ x + 6 , x = − 6 x2, x ≠ −6 a = −6 3 − x, x < 1 x, 2x2 x2, 2x + 1, x = − 2 x > − 2 x3, a = −2 x2 − 9 ______ x + 3 x − 9, −6, , x < −3 x = −3 a = 3 x > −3 17. f (x) = x2 − 4 _ x − 2 , a = 2 19. f (x) = x3 − 9x __ x2 + 11x + 24 , a = −3 21. f (x \| 18. f (x) = 25 − x2 __ x2 − 10x + 25 , a = 5 20. f (x) = x3 − 27 _ x2 − 3x , a = 3 22. f (x2 For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous every
where it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous. 23. f (x) = x3 − 2x − 15 27. f (x) = \| x − 2 \| ______ x2 − 2x 24. f (x) = x2 − 2x − 15 ___________ x − 5 28. f (x) = tan(x) + 2 25. f (x) = 2 ⋅ 3x + 4 26. f (x) = −sin(3x) 5 _ 29. f (x) = 2x + x 30. f (x) = log2 (x) SECTION 12.3 section exercises 1049 31. f (x) = ln x2 32. f (x) = e 2x 33. f (x) = √ — x − 4 34. f (x) = sec(x) − 3. 35. f (x) = x2 + sin(x) 36. Determine the values of b and c such that the following function is continuous on the entire real number line. f (x) =  x + 1 x2 + bx + c, GRAPHICAl For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of a, determine which of the three conditions of continuity are satisfied at x = a and which are not. y 37. x = −3 38. x = 2 39. x = 4 For the following exercises, use a graphing utility to graph the function  as in Figure 16. Set the x-axis a short distance before f (x) = sin  and after 0 to illustrate the point of discontinuity. 12π _ x 40. Which conditions for continuity fail at the point of discontinuity? 41. Evaluate f (0). 42. Solve for x if f (x) = 0. 43. What is the domain of f (x)? For the following exercises, consider the function shown in Figure 17. 44. At what x-coordinates is the function discontinuous? 45. What condition of continuity is violated at these points? x 5 10 x Figure 15 y 10 5 0 −5 −10 Figure 16 −10 −5 y 5 4 3 2 1 −5 −4 −3 −1 −2 −1 −2 −3 −4 −5 −6 21 3 4 5 x Figure 17 1050 CHAPTER 12 introduction to calculus 46. Consider the function shown in Figure 18. At what x-coordinates is the function discontinuous? What condition(s) of continuity were violated? 47. Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at x = −7 and x = 1. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 –4 –3 –2 21 3 4 5 x Figure 18 49. Find the limit lim x → 1 f (x) and determine if the following function is continuous at x = 1: f (x 48. The function f (x) = is graphed in Figure 19. x 3 − 1 _ x − 1 It appears to be continuous on the interval [−3, 3], but there is an x-value on that interval at which the function is discontinuous. Determine the value of x at which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph. y x Figure 19 50. The graph of f (x) = is shown in sin(2x) _ x Figure 20. Is the function f (x) continuous at x = 0? Why or why not? y 2.5 2.0 1.5 1.0 0.5 −0.5 −0.5 −1.0 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x Figure 20 SECTION 12.4 derivatives 1051 leARnInG OBjeCTIVeS In this section, you will: • Find the derivative of a function. • Find instantaneous rates of change. • Find an equation of the tangent line to the graph of a function at a point. • Find the instantaneous velocity of a particle. 12. 4 DeRIVATIVeS The average teen in the United States opens a refrigerator door an estimated 25 times per day. Supposedly, this average is up from 10 years ago when the average teenager opened a refrigerator door 20 times per day.[37] It is estimated that a television is on in a home 6.75 hours per day, whereas parents spend an estimated 5.5 minutes per day having a meaningful conversation with their children. These averages, too, are not the same as they were 10 years ago, when the television was on an estimated 6 hours per day in the typical household, and parents spent 12 minutes per day in meaningful conversation with their kids. What do these scenarios have in common? The functions representing them have changed over time. In this section, we will consider methods of computing such changes over time. Finding the Average Rate of Change of a Function The functions describing the examples above involve a change over time. Change divided by time is one example of a rate. The rates of change in the previous examples are each different. In other words, some changed faster than others. If we were to graph the functions, we could compare the rates by determining the slopes of the graphs. A tangent line to a curve is a line that intersects the curve at only a single point but does not cross it there. (The tangent line may intersect the curve at another point away from the point of interest.) If we zoom in on a curve at that point, the curve appears linear, and the slope of the curve at that point is close to the slope of the tangent line at that point. Figure 1 represents the function f (x) = x 3 − 4x. We can see the slope at various points along the curve. • slope at x = −2 is 8 • slope at x = −1 is –1 • slope at x = 2 is 8 y 5 4 3 2 1 −1−1 −2 −3 −4 −5 m = −1 −5 −4 −3 −2 m = 8 f(x) = x3 − 4x 21 3 4 5 x m = 8 Figure 1 Graph showing tangents to curve at –2, –1, and 2. Let’s imagine a point on the curve of function f at x = a as shown in Figure 2. The coordinates of the point are (a, f (a)). Connect this point with a second point on the curve a little to the right of x = a, with an x-value increased by some small real number h. The coordinates of this second point are (a + h, f (a + h)) for some positive-value h. 37 http://www.csun.edu/science/health/docs/tv&health.html Source provided. 1052 CHAPTER 12 introduction to calculus y f(a + h) (a, f(a + h)) f f(a + h) f(a) (a, f(a )) h a a + h x Figure 2 Connecting point a with a point just beyond allows us to measure a slope close to that of a tangent line at x = a. We can calculate the slope of the line connecting the two points (a, f (a)) and (a + h, f (a + h)), called a secant line, by applying the slope formula, slope = change in y __________ change in x . We use the notation msec to represent the slope of the secant line connecting two points. msec = f (a + h) − f (a) _____________ (a + h) − (a) = f (a + h) − f (a) __ a + h − a The slope msec equals the average rate of change between two points (a, f (a)) and (a + h, f (a + h)). msec = f (a + h) − f (a) ______________ h the average rate of change between two points on a curve The average rate of change (AROC) between two points (a, f (a)) and (a + h, f (a + h)) on the curve of f is the slope of the line connecting the two points and is given by AROC = f (a + h) − f (a) _____________ h Example 1 Finding the Average Rate of Change Find the average rate of change connecting the points (2, −6) and (−1, 5). Solution We know the average rate of change connecting two points may be given by AROC = f (a + h) − f (a) _____________ h . If one point is (2, − 6), or (2, f (2)), then f (2) = −6. The value h is the displacement from 2 to −1, which equals −1 − 2 = −3. For the other point, f (a + h) is the y-coordinate at a + h, which is 2 + (−3) or −1, so f (a + h) = f (−1) = 5. AROC = f (a + h) − f (a) _____________ h 5 − (−6) ________ −3 = = 11 ___ −3 = − 11 ___ 3 SECTION 12.4 derivatives 1053 Try It #1 Find the average rate of change connecting the points (−5, 1.5) and ( − 2.5, 9). Understanding the Instantaneous Rate of Change Now that we can find the average rate of change, suppose we make h in Figure 2 smaller and smaller. Then a + h will approach a as h gets smaller, getting closer and closer to 0. Likewise, the second point (a + h, f (a + h)) will approach the first point, (a, f (a)). As a consequence, the connecting line between the two points, called the secant line, will get closer and closer to being a tangent to the function at x = a, and the slope of the secant line will get closer and closer to the slope of the tangent at x = a. See Figure 3. y f Q1 Secant Line P to Q1 P = (a , f(a )) f (a) Secant Line P to Q2 Q2 a x Tangent Line at x = a Figure 3 The connecting line between two points moves closer to being a tangent line at x = a. Because we are looking for the slope of the tangent at x = a, we can think of the measure of the slope of the curve of a function f at a given point as the rate of change at a particular instant. We call this slope the instantaneous rate of change, or the derivative of the function at x = a. Both can be found by finding the limit of the slope of a line connecting the point at x = a with a second point infinitesimally close along the curve. For a function f both the instantaneous rate of change of the function and the derivative of the function at x = a are written as f ′(a), and we can define them as a two-sided limit that has the same value whether approached from the left or the right. The expression by which the limit is found is known as the difference quotient. f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h definition of instantaneous rate of change and derivative The derivative, or instantaneous rate of change, of a function f at x = a, is given by f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h The expression is called the difference quotient. f (a + h) − f (a) _____________ h We use the difference quotient to evaluate the limit of the rate of change of the function as h approaches 0. Derivatives: Interpretations and Notation The derivative of a function can be interpreted in different ways. It can be observed as the behavior of a graph of the function or calculated as a numerical rate of change of the function. • The derivative of a function f (x) at a point x = a is the slope of the tangent line to the curve f (x) at x = a. The derivative of f (x) at x = a is written f ′(a). • The derivative f ′(a) measures how the curve changes at the point (a, f (a)). • The derivative f ′(a) may be thought of as the instantaneous rate of change of the function f (x) at x = a. • If a function measures distance as a function of time, then the derivative measures the instantaneous velocity at time t = a. 1054 CHAPTER 12 introduction to calculus notations for the derivative The equation of the derivative of a function f (x) is written as y′ = f ′(x), where y = f (x). The notation f ′(x) is read as “ f prime of x.” Alternate notations for the derivative include the following: f ′(x) = y′ = = = f (x) = Df (x) dy ___
dx df ___ dx d ___ dx The expression f ′(x) is now a function of x ; this function gives the slope of the curve y = f (x) at any value of x. The derivative of a function f (x) at a point x = a is denoted f ′(a). How To… Given a function f, find the derivative by applying the definition of the derivative. 1. Calculate f (a + h). 2. Calculate f (a). 3. Substitute and simplify f (a + h) − f (a) _____________ h . 4. Evaluate the limit if it exists: f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h . Example 2 Finding the Derivative of a Polynomial Function Find the derivative of the function f (x) = x 2 − 3x + 5 at x = a. Solution We have: f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h Substitute f (a + h) = (a + h)2 − 3(a + h) + 5 and f (a) = a2 − 3a + 5. Definition of a derivative (a + h)(a + h) − 3(a + h) + 5 − (a2 − 3a + 5) ______________________________________ h a2 + 2ah + h2 − 3a − 3h + 5 − a2 + 3a − 5 ____________________________________ h a2 + 2ah + h2 −3a − 3h + 5 −a2 + 3a −5 __________________________________ h Simplify. f ′(a) = lim h → 0 = lim h → 0 = lim h → 0 = lim h → 0 = lim h → 0 2ah + h2 − 3h ____________ h  h (2a + h − 3) ____________  h Factor out an h. Evaluate the limit. Evaluate to remove parentheses. = 2a + 0 − 3 = 2a − 3 Try It #2 Find the derivative of the function f (x) = 3x2 + 7x at x = a. Finding Derivatives of Rational Functions To find the derivative of a rational function, we will sometimes simplify the expression using algebraic techniques we have already learned. SECTION 12.4 derivatives 1055 Example 3 Finding the Derivative of a Rational Function Find the derivative of the function f (x) = at x = a. 3 + x _____ 2 − x Solution f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h 3 + (a + h) −  __________ 2 − (a + h) ____________________ h 3 + a  _____ 2 − a = lim h → 0 (2 − (a + h))(2 − a)  3 + (a + h) −  __________ 2 − (a + h) 3 + a _____ 2 − a   _______________________________________ (2 − (a + h))(2 − a)(h) = lim h → 0 Substitute f (a + h) and f (a). Multiply numerator and denominator by (2 − (a + h))(2 − a). (2 − (a + h))(2 − a)  3 + (a + h) ___________ (2 − (a + h)) (2 − (a + h))(2 − a)(h)  − (2 − (a + h))  (2 − a)  3 + a  _____  2 − a ________________________________________________________ Distribute. = lim h → 0 = lim h → 0 6 − 3a + 2a − a2 + 2h − ah − 6 + 3a + 3h − 2a + a2 + ah _________________________________________________ (2 − (a + h))(2 − a)(h) Multiply. = lim h → 0 5  h ____________________ (2 − (a + h))(2 − a)(  h ) = lim h → 0 5 _________________ (2 − (a + h))(2 − a) = 5 _________________ = (2 − (a + 0))(2 − a) 5 ____________ (2 − a)(2 − a) = 5 _______ (2 − a)2 Combine like terms. Cancel like factors. Evaluate the limit. Try It #3 Find the derivative of the function f (x) = 10x + 11 ________ 5x + 4 at x = a. Finding Derivatives of Functions with Roots To find derivatives of functions with roots, we use the methods we have learned to find limits of functions with roots, including multiplying by a conjugate. Example 4 Finding the Derivative of a Function with a Root Find the derivative of the function f (x) = 4 √ x at x = 36. — Solution We have f ′(a) = lim h → 0 = lim h → 0 f (a + h) − f (a) _____________ h — a + h − 4 √ 4 √ _______________ h — a Substitute f (a + h) and f (a) Multiply the numerator and denominator by the conjugate: — a + h + 4 √ 4 √ __ 1056 CHAPTER 12 introduction to calculus — — a + h − 4 √ 4 √  ______________ f ′(a) = lim √ __  16(a + h) − 16a  __ = lim h → 0 h4  √ a + h + 4 √ — — Multiply.  ⋅   a  16a + 16h −   16a   __ = lim — h → 0 a  h4  √ a + h + 4 √ — Distribute and combine like terms. Simplify. Evaluate the limit by letting h = 0. Evaluate the derivative at x = 36. __  = lim  — 16h — = lim h →0 16 __ — = 16 ′(36) = 2 _ — √ 36 2 _ = 6 1 _ = 3 Try It #4 Find the derivative of the function f (x) = 9 √ — x at x = 9. Finding Instantaneous Rates of Change Many applications of the derivative involve determining the rate of change at a given instant of a function with the independent variable time—which is why the term instantaneous is used. Consider the height of a ball tossed upward with an initial velocity of 64 feet per second, given by s(t) = −16t 2 + 64t + 6, where t is measured in seconds and s(t) is measured in feet. We know the path is that of a parabola. The derivative will tell us how the height is changing at any given point in time. The height of the ball is shown in Figure 4 as a function of time. In physics, we call this the “s-t graph.” s(t) 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 s −1−2−3−4−5 −5 1 2 3 4 5 6 7 Figure 4 t SECTION 12.4 derivatives 1057 Example 5 Finding the Instantaneous Rate of Change Using the function above, s(t) = −16t2 + 64t + 6, what is the instantaneous velocity of the ball at 1 second and 3 seconds into its flight? Solution The velocity at t = 1 and t = 3 is the instantaneous rate of change of distance per time, or velocity. Notice that the initial height is 6 feet. To find the instantaneous velocity, we find the derivative and evaluate it at t = 1 and t = 3: f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h = lim h → 0 −16(t + h)2 + 64(t + h) + 6 − (−16t 2 + 64t + 6) ________________________________________ h Substitute s(t + h) and s(t). = lim h → 0 −16t 2 − 32ht − h 2 + 64t + 64h + 6 + 16t 2 − 64t − 6 ___________________________________________ h Distribute. = lim h → 0 = lim h → 0 −32ht − h 2 + 64h _______________ h h( − 32t − h + 64) ________________ h −32t − h + 64 = lim h → 0 s′(t) = − 32t + 64 For any value of t, s′(t) tells us the velocity at that value of t. Evaluate t = 1 and t = 3. Simplify. Factor the numerator. Cancel out the common factor h. Evaluate the limit by letting h = 0. s′(1) = −32(1) + 64 = 32 s′(3) = −32(3) + 64 = −32 The velocity of the ball after 1 second is 32 feet per second, as it is on the way up. The velocity of the ball after 3 seconds is −32 feet per second, as it is on the way down. Try It #5 The position of the ball is given by s(t) = −16t 2 + 64t + 6. What is its velocity 2 seconds into flight? Using Graphs to Find Instantaneous Rates of Change We can estimate an instantaneous rate of change at x = a by observing the slope of the curve of the function f (x) at x = a. We do this by drawing a line tangent to the function at x = a and finding its slope. How To… Given a graph of a function f (x), find the instantaneous rate of change of the function at x = a. 1. Locate x = a on the graph of the function f (x). 2. Draw a tangent line, a line that goes through x = a at a and at no other point in that section of the curve. Extend the line far enough to calculate its slope as change in y __________ . change in x Example 6 Estimating the Derivative at a Point on the Graph of a Function From the graph of the function y = f (x) presented in Figure 5, estimate each of the following: f (0); f (2); f ′(0); f ′(2) 1058 CHAPTER 12 introduction to calculus y 5 4 3 2 1 −1−1 −2 −3 −4 −5 −5 −4 −3 −2 f 21 3 4 5 x Figure 5 Solution To find the functional value, f (a), find the y-coordinate at x = a. To find the derivative at x = a, f ′(a), draw a tangent line at x = a, and estimate the slope of that tangent line. See Figure 6. y m = 4 (2, 1) 3 5 4 (0, 1) 21 x 5 4 3 2 1 −1−1 −2 −3 −4 −5 m = 0 −5 −4 −3 −2 f Figure 6 a. f (0) is the y-coordinate at x = 0. The point has coordinates (0, 1), thus f (0) = 1. b. f (2) is the y-coordinate at x = 2. The point has coordinates (2, 1), thus f (2) = 1. c. f ′(0) is found by estimating the slope of the tangent line to the curve at x = 0. The tangent line to the curve at x = 0 appears horizontal. Horizontal lines have a slope of 0, thus f ′(0) = 0. d. f ′(2) is found by estimating the slope of the tangent line to the curve at x = 2. Observe the path of the tangent line to the curve at x = 2. As the x value moves one unit to the right, the y value moves up four units to another point on the line. Thus, the slope is 4, so f ′(2) = 4. Try It #6 Using the graph of the function f (x) = x3 − 3x shown in Figure 7, estimate: f (1), f ′(1), f (0), and f ′(0). y 7 6 5 4 3 2 1 −1−1 −2 −3 −4 −5 −5 −4 −3 −2 21 3 4 5 x Figure 7 SECTION 12.4 derivatives 1059 Using Instantaneous Rates of Change to Solve Real-World Problems Another way to interpret an instantaneous rate of change at x = a is to observe the function in a real-world context. The unit for the derivative of a function f (x) is output units __ input unit Such a unit shows by how many units the output changes for each one-unit change of input. The instantaneous rate of change at a given instant shows the same thing: the units of change of output per one-unit change of input. One example of an instantaneous rate of change is a marginal cost. For example, suppose the production cost for a company to produce x items is given by C(x), in thousands of dollars. The derivative function tells us how the cost is changing for any value of x in the domain of the function. In other words, C′ (x) is interpreted as a marginal cost, the additional cost in thousands of dollars of producing one more item when x items have been produced. For example, C′ (11) is the approximate additional cost in thousands of dollars of producing the 12th item after 11 items have been produced. C′ (11) = 2.50 means that when 11 items have been produced, producing the 12th item would increase the total cost by approximately $2,500.00. Example 7 Finding a Marginal Cost The cost in dollars of producing x laptop computers is f (x) = x2 − 100x. At the point where 200 computers have been produced, what is the approximate cost of producing the 201st unit? Solution If f (x) = x 2 − 100x describes the cost of producing x computers, f ′(x) will describe the marginal cost. We need to find the derivative. For purposes of calculating the derivative, we can use the following functions: f (a + b) = (x + h)2 − 100(x + h) f (a) = a2 − 100a f ′(x) = f (a + h) − f (a) _____________ h = (x + h)2 − 100(x + h) − (x2 − 100
x) ______________________________ h = x2 + 2xh + h2 − 100x − 100h − x2 + 100x __________________________________ h = 2xh + h2 − 100h ______________ h = h(2x + h − 100) ______________ h = 2x + h − 100 = 2x − 100 f ′(x) = 2x − 100 f ′(200) = 2(200) − 100 = 300 Formula for a derivative Substitute f (a + h) and f (a). Multiply polynomials, distribute. Collect like terms. Factor and cancel like terms. Simplify. Evaluate when h = 0. Formula for marginal cost Evaluate for 200 units. The marginal cost of producing the 201st unit will be approximately $300. Example 8 Interpreting a Derivative in Context A car leaves an intersection. The distance it travels in miles is given by the function f (t), where t represents hours. Explain the following notations: f (0) = 0 ; f ′(1) = 60; f (1) = 70; f (2.5) = 150 Solution First we need to evaluate the function f (t) and the derivative of the function f ′(t), and distinguish between the two. When we evaluate the function f (t), we are finding the distance the car has traveled in t hours. When we evaluate the derivative f ′(t), we are finding the speed of the car after t hours. 1060 CHAPTER 12 introduction to calculus a. f (0) = 0 means that in zero hours, the car has traveled zero miles. b. f ′(1) = 60 means that one hour into the trip, the car is traveling 60 miles per hour. c. f (1) = 70 means that one hour into the trip, the car has traveled 70 miles. At some point during the first hour, then, the car must have been traveling faster than it was at the 1-hour mark. d. f (2.5) = 150 means that two hours and thirty minutes into the trip, the car has traveled 150 miles. Try It #7 A runner runs along a straight east-west road. The function f (t) gives how many feet eastward of her starting point she is after t seconds. Interpret each of the following as it relates to the runner. f (0) = 0; f (10) = 150; f ′(10) = 15; f ′(20) = −10; f (40) = −100 Finding Points Where a Function’s Derivative Does Not Exist To understand where a function’s derivative does not exist, we need to recall what normally happens when a function f (x) has a derivative at x = a. Suppose we use a graphing utility to zoom in on x = a . If the function f (x) is differentiable, that is, if it is a function that can be differentiated, then the closer one zooms in, the more closely the graph approaches a straight line. This characteristic is called linearity. Look at the graph in Figure 8. The closer we zoom in on the point, the more linear the curve appears. y Graph appears linear x Figure 8 We might presume the same thing would happen with any continuous function, but that is not so. The function f (x) = \| x \| , for example, is continuous at x = 0, but not differentiable at x = 0. As we zoom in close to 0 in Figure 9, the graph does not approach a straight line. No matter how close we zoom in, the graph maintains its sharp corner. y 0.1 −0.1 0 x 0.1 −0.1 Figure 9 Graph of the function f (x) = \|x\|, with x-axis from –0.1 to 0.1 and y-axis from –0.1 to 0.1. We zoom in closer by narrowing the range to produce Figure 10 and continue to observe the same shape. This graph does not appear linear at x = 0. SECTION 12.4 derivatives 1061 y 0.001 −0.001 0 x 0.001 −0.001 Figure 10 Graph of the function f (x) = \|x\|, with x-axis from –0.001 to 0.001 and y-axis from –0.001 to 0.001. What are the characteristics of a graph that is not differentiable at a point? Here are some examples in which function f (x) is not differentiable at x = a. In Figure 11, we see the graph of f (x) =  x 2, 8 − x, x ≤ 2 x > 2 . Notice that, as x approaches 2 from the left, the left-hand limit may be observed to be 4, while as x approaches 2 from the right, the right-hand limit may be observed to be 6. We see that it has a discontinuity at x = 2. y 10 8 6 4 2 −1−2−3−4−5 −2 1 2 3 4 5 x Figure 11 The graph of f (x) has a discontinuity at x = 2. In Figure 12, we see the graph of f (x) = \| x \| . We see that the graph has a corner point at x = 0. y 5 4 3 2 1 −1−1 −2 −3−4−5 −2 1 2 3 4 5 x Figure 12 The graph of f (x) = \|x\| has a corner point at x = 0 . In Figure 13, we see that the graph of f (x) = x 2 _ 3 has a cusp at x = 0. A cusp has a unique feature. Moving away from the cusp, both the left-hand and right-hand limits approach either infinity or negative infinity. Notice the tangent lines as x approaches 0 from both the left and the right appear to get increasingly steeper, but one has a negative slope, the other has a positive slope Figure 13 The graph of f (x) = x 2 _ has a cusp at x = 0. 3 1062 CHAPTER 12 introduction to calculus In Figure 14, we see that the graph of f (x) = x 1 _ 3 has a vertical tangent at x = 0. Recall that vertical tangents are vertical lines, so where a vertical tangent exists, the slope of the line is undefined. This is why the derivative, which measures the slope, does not exist there5 −4 −3 −2 4 3 2 1 −1−1 −2 −3 −4 Figure 14 The graph of f (x) = x 1 _ has a vertical tangent at x = 0. 3 differentiability A function f (x) is differentiable at x = a if the derivative exists at x = a, which means that f ′(a) exists. There are four cases for which a function f (x) is not differentiable at a point x = a. 1. When there is a discontinuity at x = a. 2. When there is a corner point at x = a. 3. When there is a cusp at x = a. 4. Any other time when there is a vertical tangent at x = a. Example 9 Determining Where a Function Is Continuous and Differentiable from a Graph Using Figure 15, determine where the function is a. continuous b. discontinuous c. differentiable d. not differentiable At the points where the graph is discontinuous or not differentiable, state why. y 5 4 3 2 1 −1−1 −2 −3 −4 −5 −5 −4 −3 −2 1 2 3 4 5 x f Figure 15 Solution The graph of f (x) is continuous on (−∞, −2) ∪ (−2, 1) ∪ (1, ∞). The graph of f (x) has a removable discontinuity at x = −2 and a jump discontinuity at x = 1. See Figure 16. y 6 5 4 3 2 1 −1 −1 −2 −3 −4 −5 −6 −5−6 −4 −3 −2 1 2 3 4 5 6 x (−∞, −2) (−2, 1) (1, ∞) Figure 16 Three intervals where the function is continuous SECTION 12.4 derivatives 1063 The graph of f (x) is differentiable on (−∞, −2) ∪ (−2, −1) ∪ (−1, 1) ∪ (1, 2) ∪ (2, ∞). The graph of f (x) is not differentiable at x = −2 because it is a point of discontinuity, at x = −1 because of a sharp corner, at x = 1 because it is a point of discontinuity, and at x = 2 because of a sharp corner. See Figure 17. Sharp corner Discontinuity −5 −4 −3 −2 Sharp corner y 5 4 3 2 1 −1 −1 −2 −3 −4 −5 1 2 3 4 5 x f Discontinuity (−∞, −2) (−2, −1) (−1, 1)(1, 2) (2, ∞) Figure 17 Five intervals where the function is differentiable Try It #8 Determine where the function y = f (x) shown in Figure 18 is continuous and differentiable from the graph1−1 −2 −3 −4 −5 −4 −3 −2 y = f (x) 1 2 3 4 5 x Figure 18 Finding an equation of a line Tangent to the Graph of a Function The equation of a tangent line to a curve of the function f (x) at x = a is derived from the point-slope form of a line, y = m(x − x1) + y1. The slope of the line is the slope of the curve at x = a and is therefore equal to f ′(a), the derivative of f (x) at x = a. The coordinate pair of the point on the line at x = a is (a, f (a)). If we substitute into the point-slope form, we have m = f ′(a) y1 = f (a) x1 = a y = m(x − x1) + y1 ↑ f (a) ↑ f ′(a) ↑ a The equation of the tangent line is y = f ′(a)(x − a) + f (a) the equation of a line tangent to a curve of the function f The equation of a line tangent to the curve of a function f at a point x = a is y = f ′(a)(x − a) + f (a) 1064 CHAPTER 12 introduction to calculus How To… Given a function f, find the equation of a line tangent to the function at x = a. 1. Find the derivative of f (x) at x = a using f ′(a) = lim h → 0 2. Evaluate the function at x = a. This is f (a). 3. Substitute (a, f (a))and f ′(a) into y = f ′(a)(x − a) + f (a). 4. Write the equation of the tangent line in the form y = mx + b. f (a + h) − f (a) _____________ h . Example 10 Finding the Equation of a Line Tangent to a Function at a Point Find the equation of a line tangent to the curve f (x) = x 2 − 4x at x = 3. Solution Using: f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h Substitute f (a + h) = (a + h)2 − 4(a + h) and f (a) = a2 − 4a. f ′(a) = lim h → 0 (a + h)(a + h) − 4(a + h) − (a2 − 4a) _______________________________ h a2 + 2ah + h2 − 4a − 4h − a2 + 4a _____________________________ h  a2 + 2ah + h2  −4a − 4h  − a2  + 4a ____________________________ h = lim h → 0 = lim h → 0 = lim h → 0 = lim h → 0 2ah + h2 − 4h ____________ h  h (2a + h − 4) ____________  h = 2a + 0 − 4 f ′(a) = 2a − 4 f ′(3) = 2(3) − 4 = 2 Equation of tangent line at x = 3: y = f ′(a)(x − a) + f (a) y = f ′(3)(x − 3) + f (3) y = 2(x − 3) + (−3) y = 2x − 9 Remove parentheses. Combine like terms. Factor out h. Evaluate the limit. Analysis We can use a graphing utility to graph the function and the tangent line. In so doing, we can observe the point of tangency at x = 3 as shown in Figure 19. y 5 4 3 2 1 f (x) = x 2 − 4x −5 −4 −3 −2 −1−1 −2 −3 −4 −5 y = 2x − 9 1 2 3 4 5 6 7 x Figure 19 Graph confirms the point of tangency at x = 3. SECTION 12.4 derivatives 1065 Try It #9 Find the equation of a tangent line to the curve of the function f (x) = 5x2 − x + 4 at x = 2. Finding the Instantaneous Speed of a Particle If a function measures position versus time, the derivative measures displacement versus time, or the speed of the object. A change in speed or direction relative to a change in time is known as velocity. The velocity at a given instant is known as instantaneous velocity. In trying to find the speed or velocity of an object at a given instant, we seem to encounter a contradiction. We normally define speed as the distance traveled divided by the elapsed time. But in an instant, no distance is traveled, and no time elapses. How will we divide zero by zero? The use of a derivative solves this problem. A derivative allows us to say that even while the object’s velocity is constantly changing, it has
a certain velocity at a given instant. That means that if the object traveled at that exact velocity for a unit of time, it would travel the specified distance. instantaneous velocity Let the function s(t) represent the position of an object at time t. The instantaneous velocity or velocity of the object at time t = a is given by s′(a) = lim h → 0 s(a + h) − s(a) _____________ h Example 11 Finding the Instantaneous Velocity A ball is tossed upward from a height of 200 feet with an initial velocity of 36 ft/sec. If the height of the ball in feet after t seconds is given by s(t) = −16t 2 + 36t + 200, find the instantaneous velocity of the ball at t = 2. Solution First, we must find the derivative s′ (t) . Then we evaluate the derivative at t = 2, using s(a + h) = − 16(a + h)2 + 36(a + h) + 200 and s(a) = − 16a2 + 36a + 200. s′(a) = lim h → 0 s(a + h) − s(a) _____________ h = lim h → 0 −16(a + h)2 + 36(a + h) + 200 − (−16a2 + 36a + 200) ______________________________________________ h −16(a2 + 2ah + h2) + 36(a + h) + 200 − (−16a2 + 36a + 200) ____________________________________________________ h −16a2 − 32ah − 16h2 + 36a + 36h + 200 + 16a2 − 36a − 200 ___________________________________________________ h  − 36a  −16a2 − 32ah − 16h2  +36a + 36h  __________________________________________________ + 200  + 16a2  − 200 h = lim h → 0 = lim h → 0 = lim h → 0 = lim h → 0 = lim h → 0 −32ah − 16h2 + 36h _________________ h h(−32a − 16h + 36) _________________ h = lim h → 0 ( −32a − 16h + 36) = −32a − 16 ⋅ 0 + 36 s′(a) = −32a + 36 s′(2) = −32(2) + 36 = −28 Analysis This result means that at time t = 2 seconds, the ball is dropping at a rate of 28 ft/sec. 1066 CHAPTER 12 introduction to calculus Try It #10 A fireworks rocket is shot upward out of a pit 12 ft below the ground at a velocity of 60 ft/sec. Its height in feet after t seconds is given by s = −16t2 + 60t − 12. What is its instantaneous velocity after 4 seconds? Access these online resources for additional instruction and practice with derivatives. • estimate the Derivative (http://openstaxcollege.org/l/estimatederiv) • estimate the Derivative ex. 4 (http://openstaxcollege.org/l/estimatederiv4) SECTION 12.4 section exercises 1067 2. What is the difference between the average rate of change of a function on the interval [x, x + h] and the derivative of the function at x? 4. Explain the concept of the slope of a curve at point x. 12.4 SeCTIOn exeRCISeS VeRBAl 1. How is the slope of a linear function similar to the derivative? 3. A car traveled 110 miles during the time period from 2:00 P.M. to 4:00 P.M. What was the car's average velocity? At exactly 2:30 P.M., the speed of the car registered exactly 62 miles per hour. What is another name for the speed of the car at 2:30 P.M.? Why does this speed differ from the average velocity? 5. Suppose water is flowing into a tank at an average rate of 45 gallons per minute. Translate this statement into the language of mathematics. AlGeBRAIC For the following exercises, use the definition of derivative lim h → 0 function. f (x + h) − f (x) _____________ h to calculate the derivative of each 6. f (x) = 3x − 4 7. f (x) = −2x + 1 8. f (x) = x2 − 2x + 1 9. f (x) = 2x2 + x − 3 10. f (x) = 2x2 + 5 14. f (x) = √ — 1 + 3x 11. f (x) = −1 _____ x − 2 15. f (x) = 3x3 − x2 + 2x + 5 12. f (x) = 2 + x _____ 1 − x 16. f (x) = 5 13. f (x) = 5 − 2x ______ 3 + 2x 17. f (x) = 5π For the following exercises, find the average rate of change between the two points. 19. (4, −3) and (−2, −1) 18. (−2, 0) and (−4, 5) 20. (0, 5) and (6, 5) 21. (7, −2) and (7, 10) For the following polynomial functions, find the derivatives. 22. f (x) = x3 + 1 23. f (x) = −3x2 − 7x + 6 24. f (x) = 7x2 25. f (x) = 3x3 + 2x2 + x − 26 For the following functions, find the equation of the tangent line to the curve at the given point x on the curve. 26. f (x) = 2x2 − 3x x = 3 27. f (x) = x3 + 1 x = 2 — 28. f (x) = √ x x = 9 For the following exercise, find k such that the given line is tangent to the graph of the function. 29. f (x) = x2 − kx, y = 4x − 9 GRAPHICAl For the following exercises, consider the graph of the function f and determine where the function is continuous/ discontinuous and differentiable/not differentiable. 30. y 31. y −5 −4 −3 −2 5 4 3 2 1 −1−1 −2 −3 −4 −5 x 21 3 4 5 f(x) −5 −4 −3 −2 5 4 3 2 1 −1−1 −2 −3 −4 −5 21 3 4 5 x f (x) 1068 CHAPTER 12 introduction to calculus 32. y 33. −5 −4 −3 −2 6 5 4 3 2 1 −1−1 −2 −3 −4 −5 −6 21 3 4 5 6 x f (x) −5−x) −2−3−4 −1 −1 −2 −3 −4 −5 −6 For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated. f (x5−6 −2−3−4 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 34. f (−1) 40. f ′(0) 35. f (0) 41. f ′(1) 36. f (1) 42. f ′(2) Figure 20 37. f (2) 43. f ′(3) 44. Sketch the function based on the information below: f ′(x) = 2x, f (2) = 4 TeCHnOlOGY 38. f (3) 39. f ′(−1) 45. Numerically evaluate the derivative. Explore the behavior of the graph of f (x) = x 2 around x = 1 by graphing the function on the following domains: [0.9, 1.1] , [0.99, 1.01] , [0.999, 1.001], and [0.9999, 1.0001] . We can use the feature on our calculator that automatically sets Ymin and Ymax to the Xmin and Xmax values we preset. (On some of the commonly used graphing calculators, this feature may be called ZOOM FIT or ZOOM AUTO). By examining the corresponding range values for this viewing window, approximate how the curve changes at x = 1, that is, approximate the derivative at x = 1. ReAl-WORlD APPlICATIOnS For the following exercises, explain the notation in words. The volume f (t) of a tank of gasoline, in gallons, t minutes after noon. 46. f (0) = 600 47. f ′(30) = −20 50. f (240) = 500 49. f ′(200) = 30 48. f (30) = 0 SECTION 12.4 section exercises 1069 For the following exercises, explain the functions in words. The height, s, of a projectile after t seconds is given by s(t) = −16t 2 + 80t. 51. s(2) = 96 55. s(0) = 0, s(5) = 0. 54. s′(3) = −16 52. s′(2) = 16 53. s(3) = 96 4 __ For the following exercises, the volume V of a sphere with respect to its radius r is given by V = πr 3. 3 56. Find the average rate of change of V as r changes 57. Find the instantaneous rate of change of V when from 1 cm to 2 cm. r = 3 cm. For the following exercises, the revenue generated by selling x items is given by R(x) = 2x 2 + 10x. 58. Find the average change of the revenue function as 59. Find R′(10) and interpret. x changes from x = 10 to x = 20. 60. Find R′(15) and interpret. Compare R′(15) to R′(10), and explain the difference. For the following exercises, the cost of producing x cellphones is described by the function C(x) = x 2 − 4x + 1000. 61. Find the average rate of change in the total cost as 62. Find the approximate marginal cost, when x changes from x = 10 to x = 15. 15 cellphones have been produced, of producing the 16th cellphone. 63. Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21st cellphone. exTenSIOn For the following exercises, use the definition for the derivative at a point x = a, lim x → a of the functions. f (x) − f (a) __________ x − a , to find the derivative 1 __ 64. f (x) = x2 65. f (x) = 5x 2 − x + 4 66. f (x) = −x 2 + 4x + 7 67. f (x) = −4 ______ 3 − x2 1070 CHAPTER 12 introduction to calculus CHAPTeR 12 ReVIeW Key Terms average rate of change the slope of the line connecting the two points (a, f (a)) and (a + h, f (a + h)) on the curve of f (x); it is given by AROC = f (a + h) − f (a) _____________ h . continuous function a function that has no holes or breaks in its graph derivative the slope of a function at a given point; denoted f ′(a), at a point x = a it is f ′(a) = lim h → 0 providing the limit exists. f (a + h) − f (a) _____________ , h differentiable a function f (x) for which the derivative exists at x = a. In other words, if f ′(a) exists. discontinuous function a function that is not continuous at x = a instantaneous rate of change the slope of a function at a given point; at x = a it is given by f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h instantaneous velocity the change in speed or direction at a given instant; a function s(t) represents the position of an object at time t, and the instantaneous velocity or velocity of the object at time t = a is given by s′(a) = lim h → 0 s(a + h) − s(a) _____________ h . jump discontinuity a point of discontinuity in a function f (x) at x = a where both the left and right-hand limits exist, but lim x → a− f (x) ≠ lim x → a+ f (x) left-hand limit the limit of values of f (x) as x approaches a from the left, denoted lim x → a− f (x) = L. The values of f (x) can get as close to the limit L as we like by taking values of x sufficiently close to a such that x < a and x ≠ a. Both a and L are real numbers. limit when it exists, the value, L, that the output of a function f (x) approaches as the input x gets closer and closer to a but does not equal a. The value of the output, f (x), can get as close to L as we choose to make it by using input values of x f (x) = L. sufficiently near to x = a, but not necessarily at x = a. Both a and L are real numbers, and L is denoted lim x → a properties of limits a collection of theorems for finding limits of functions by performing mathematical operations on the limits removable discontinuity a point of discontinuity in a function f (x) where the function is discontinuous, but can be redefined to make it continuous right-hand limit the limit of values of f (x) as x approaches a from the right, denoted lim f (x) = L. The values of f (x) can get as close to the limit L as we like by taking values of x sufficiently close to a where x > a, and x ≠ a. Both a and L are real numbers. x → a+ secant line a line that intersects two points on a curve tangent line a line that intersects a curve at a single point two-sided limit the limit of a function f (x), as x approaches a, is equal to L, that is, lim x → a f (x) = L if and on
ly if x → a− f (x) = lim lim x → a+ f (x). Key equations average rate of change AROC = f (a + h) − f (a) _____________ h derivative of a function f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h CHAPTER 12 review 1071 Key Concepts 12.1 Finding Limits: Numerical and Graphical Approaches • A function has a limit if the output values approach some value L as the input values approach some quantity a. See Example 1. • A shorthand notation is used to describe the limit of a function according to the form lim x → a f (x) = L, which indicates that as x approaches a, both from the left of x = a and the right of x = a, the output value gets close to L. • A function has a left-hand limit if f (x) approaches L as x approaches a where x < a. A function has a right-hand limit if f (x) approaches L as x approaches a where x > a. • A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit. • A graph provides a visual method of determining the limit of a function. • If the function has a limit as x approaches a, the branches of the graph will approach the same y- coordinate near x = a from the left and the right. See Example 2. • A table can be used to determine if a function has a limit. The table should show input values that approach a from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See Example 3. • A graphing utility can also be used to find a limit. See Example 4. 12.2 Finding Limits: Properties of Limits • The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves. See Example 1. • The limit of a polynomial function can be found by finding the sum of the limits of the individual terms. See Example 2 and Example 3. • The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See Example 4. • The limit of the root of a function equals the corresponding root of the limit of the function. • One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify. See Example 5. • Another method of finding the limit of a complex fraction is to find the LCD. See Example 6. • A limit containing a function containing a root may be evaluated using a conjugate. See Example 7. • The limits of some functions expressed as quotients can be found by factoring. See Example 8. • One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful. See Example 9. 12.3 Continuity • A continuous function can be represented by a graph without holes or breaks. • A function whose graph has holes is a discontinuous function. • A function is continuous at a particular number if three conditions are met: • Condition 1: f (a) exists. • Condition 2: lim x → a • Condition 3: lim x → a f (x) exists at x = a. f (x) = f (a). • A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.” • A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See Example 1. 1072 CHAPTER 12 introduction to calculus • Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See Example 2 and Example 3. • For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries. See Example 4 and Example 5. 12.4 Derivatives • The slope of the secant line connecting two points is the average rate of change of the function between those points. See Example 1. • The derivative, or instantaneous rate of change, is a measure of the slope of the curve of a function at a given point, or the slope of the line tangent to the curve at that point. See Example 2, Example 3, and Example 4. • The difference quotient is the quotient in the formula for the instantaneous rate of change: • Instantaneous rates of change can be used to find solutions to many real-world problems. See Example 5. f (a + h) − f (a) __ h • The instantaneous rate of change can be found by observing the slope of a function at a point on a graph by drawing a line tangent to the function at that point. See Example 6. • Instantaneous rates of change can be interpreted to describe real-world situations. See Example 7 and Example 8. • Some functions are not differentiable at a point or points. See Example 9. • The point-slope form of a line can be used to find the equation of a line tangent to the curve of a function. See Example 10. • Velocity is a change in position relative to time. Instantaneous velocity describes the velocity of an object at a given instant. Average velocity describes the velocity maintained over an interval of time. • Using the derivative makes it possible to calculate instantaneous velocity even though there is no elapsed time. See Example 11. CHAPTER 12 review 1073 CHAPTeR 12 ReVIeW exeRCISeS FInDInG lIMITS: A nUMeRICAl AnD GRAPHICAl APPROACH For the following exercises, use Figure 1. y 10 8 6 4 2 42 6 8 10 x −10 −2−4−6−8 −2 −4 −6 −8 −10 Figure 1 1. lim x → −1+ f (x) 2. lim x → −1− f (x) 3. lim x → −1 f (x) 4. lim x → 3 f (x) 5. At what values of x is the function discontinuous? What condition of continuity is violated? 6. Using Table 1, estimate lim x → 0 f (x). x −0.1 −0.01 −0.001 0 0.001 0.01 0.1 0.15 f (x) 2.875 2.92 2.998 Undefined 2.9987 2.865 2.78145 2.678 Table 1 For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x approaches a. If the function has limit as x approaches a, state it. If not, discuss why there is no limit. 8. f (x) =  1 _____ , x + 1 (x + 1)2, if x = −2 a = −2 if x ≠ −2 7. f (x) =  9. f (x) =  \| x \| − 1, x3, if x ≠ 1 if , if x < 1 if x > 1 a = 1 FInDInG lIMITS: PROPeRTIeS OF lIMITS For the following exercises, find the limits if lim x → c f (x) = −3 and lim x → c g(x) = 5. 10. lim x → c ( f (x) + g(x)) 12. lim x → c ( f (x) ⋅ g(x)) 14. lim x → 0− f (x), f (x) =  3x2 + 2x + 1 x > 0 x < 0 5x + 3 11. lim x → c f (x) ____ g(x) 13. lim x → 0+ f (x), f (x) =  3x2 + 2x + 1 x > 0 x < 0 5x + 3 15. lim x → 3+ (3x − 〚x〛) 1074 CHAPTER 12 introduction to calculus For the following exercises, evaluate the limits using algebraic techniques. 17. lim x → 25  x 2 − 625 _ x − 5 √  — 18. lim x → 1 −x 2 − 9x   ________ x 16. lim h → 0 (h + 6)2 − 36   ___________ h 20. lim x → − ______ 3 + x COnTInUITY 19. lim x → 4 —   7 − √ 12x + 1 _____________ x − 4 For the following exercises, use numerical evidence to determine whether the limit exists at x = a. If not, describe the behavior of the graph of the function at x = a. 21. f (x) = −2 _____ x − 4 ; a = 4 22. f (x) = −2 _______ (x − 4)2 ; a = 4 23. f (x) = −x _________ 24. f (x) = 6x 2 + 23x + 20 _____________ 4x 2 − 25 5 ___ ; a = − 2 25. f (x) = — x − 3 √ _______ 9 − x ; a = 9 For the following exercises, determine where the given function f (x) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities. 26. f (x) = x 2 − 2x − 15 30. f (x) = 1 __ x 2 − x _______ 2 − x DeRIVATIVeS 27. f (x) = x 2 − 2x − 15 ___________ x − 5 28. f (x) = x 2 − 2x __________ x 2 − 4x + 4 29. f (x) = x 3 − 125 _____________ 2x2 − 12x + 10 31. f (x) = x + 2 ___________ x 2 − 3x − 10 32. f (x) = x + 2 ______ x 3 + 8 For the following exercises, find the average rate of change f (x + h) − f (x) _____________ h . 33. f (x) = 3x + 2 34. f (x) = 5 37. f (x) = e2x 35. f (x) = 1 _____ x + 1 36. f (x) = ln(x) For the following exercises, find the derivative of the function. 38. f (x) = 4x − 6 39. f (x) = 5x 2 − 3x 40. Find the equation of the tangent line to the graph of f (x) at the indicated x value. f (x) = −x 3 + 4x; x = 2. For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable. 41. f (x) = x _ \| x \| 42. Given that the volume of a right circular cone 1 __ is V = πr 2h and that a given cone has a fixed 3 height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of π. CHAPTER 12 practice test 1075 CHAPTeR 12 PRACTICe TeST For the following exercises, use the graph of f in Figure 1. y 5 4 3 2 1 21 3 4 5 x −1−2−3−4−5 −1 −2 −3 −4 −5 Figure 1 1. f (1) 2. lim x → −1+ f (x) 3. lim x → −1− f (x) 4. lim x → −1 f (x) 5. lim x → −2 f (x) 6. At what values of x is f discontinuous? What property of continuity is violated? For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. 7. f (x) =  1 __ − 3, x x3 + 1, if x ≤ 2 if x > 2 a = 2 8. f (x) =  x3 + 1, 3x2 − 1, − √ if x < 1 if , if x > 1 For the following exercises, evaluate each limit using algebraic techniques. 9. lim x → − _______ 10 + 2x 10. lim h → 0 —   h2 + 25 − 5 √ ____________ h2 11. lim h → 0  1 __ − h  1 ______ h2 + h For the following exercises, determine whether or not the given function f is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail. 12. f (x) = √ — x 2 − 4 13. f (x) = x3 − 4x 2 − 9x + 36 ________________ x 3 − 3x 2 + 2x − 6 For the following exercises, use the definition of a derivative to find the derivative of the given function at x = a. 14.
f (x) = 3 ______ 5 + 2x 15. f (x) = 3 _ — √ x 16. f (x) = 2x 2 + 9x 17. For the graph in Figure 2, determine where the function is continuous/discontinuous and differentiable/not differentiable. y 5 4 3 2 1 x 21 3 4 5 f(x) −1−2−3−4−5 –1 –2 –3 –4 –5 Figure 2 1076 CHAPTER 12 introduction to calculus For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable. 18. f (x) = \|x − 2\| − \|x + 2\| 19. f (x) = 2 ______ 1 + e 2 _ x For the following exercises, explain the notation in words when the height of a projectile in feet, s, is a function of time t in seconds after launch and is given by the function s(t). 20. s(0) 21. s(2) 22. s′(2) 23. s(2) − s(1) _________ 2 − 1 24. s(t) = 0 For the following exercises, use technology to evaluate the limit. 25. lim x → 0 sin(x) _____ 3x 28. Evaluate the limit by hand. 26. lim x → 0 tan2(x) ______ 2x 27. lim x → 0 sin(x)(1 − cos(x)) _______________ 2x 2 f (x), where f (x) =  lim x → 1 4x − At what value(s) of x is the function discontinuous? For the following exercises, consider the function whose graph appears in Figure 3. y 5 4 3 2 1 −1−2−3−4−5 −1 −2 −3 −4 −5 29. Find the average rate of change of the function from x = 1 to x = 3. 30. Find all values of x at which f ′(x) = 0. 21 3 4 5 x 31. Find all values of x at which f ′(x) does not exist. 32. Find an equation of the tangent line to the graph of f the indicated point: f (x) = 3x2 − 2x − 6, x = − 2 Figure 3 For the following exercises, use the function f (x) = x(1 − x) 2 _ 5 . 33. Graph the function f (x) = x(1 − x) 2 _ 5 by entering f (x) = x( (1 − x) 2 ) 1 _ 5 and then by entering 2 f (x) = x  (1 − x) 1 _ 5  . 34. Explore the behavior of the graph of f (x) around x = 1 by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at x = 1. For the following exercises, find the derivative of each of the functions using the definition: lim f (x + h) − f (x) _____________ h h → 0 35. f (x) = 2x − 8 36. f (x) = 4x2 − 7 40. f (x) = −x3 + 1 41. f (x) = x2 + x3 1 __ 37. f (x) = x − x2 2 — 42. f (x) = √ x − 1 38. f (x) = 1 _ x + 2 39. f (x) = 3 _ x − 1 Basic Functions and Identities A1 Graphs of the Parent Functions Identity f (x) Square f (x) Square Root f (x) y = x –4 –3 –2 4 3 2 1 –1 –1 –2 –3 –4 y = x 2 21 3 4 x –4 –3 –2 4 3 2 1 –1 –1 –2 –3 –4 y = 21 3 4 x –4 –3 –2 4 3 2 1 –1 –1 –2 –3 –4 21 3 4 x Domain: (−∞, ∞) Range: (−∞, ∞) Cubic f (x) Domain: (−∞, ∞) Range: [0, ∞) Figure A1 Cube Root f (x) Domain: [0, ∞) Range: [0, ∞) Reciprocal f (x) y = x 3 –4 –3 –2 4 3 2 1 –1 –1 –2 –3 –4 3 y = 21 3 4 x –4 –3 –2 4 3 2 1 –1 –1 –2 –3 –4 y = 1 x 21 3 4 x –4 –3 –2 4 3 2 1 –1 –1 –2 –3 –4 21 3 4 x Domain: (−∞, ∞) Range: (−∞, ∞) Absolute Value y = x –4 –3 –2 f (x) 4 3 2 1 –1 –1 –2 –3 –4 Domain: (−∞, ∞) Range: (−∞, ∞) Figure A2 Exponential f (x) 4 3 2 1 –1 –1 –2 –3 –4 Domain: (−∞, 0) ∪ (0, ∞) Range: (−∞, 0) ∪ (0, ∞) Natural Logarithm f (x) 4 3 2 1 –1 –1 –2 –3 –4 21 3 4 x y = e x y = ln(x) 21 3 4 x –4 –3 –2 21 3 4 x –4 –3 –2 Domain: (−∞, ∞) Range: [0, ∞) Domain: (−∞, ∞) Range: [0, ∞) Figure A3 Domain: (0, ∞) Range: (−∞, ∞) A-1 A-2 A2 Graphs of the Trigonometric Functions Sine y y = sin x Cosine y y = cos x Tangent y y = tan x 4 3 2 1 π –1 2 –2 –3 –4 2π 3π 2 π π 2 π 3π 2 2π x 2π 3π 2 π 4 3 2 1 π –1 2 –2 –3 –4 π 2 π 3π 2 2π x 2π 3π 2 π 4 3 2 1 π –1 2 –2 –3 –4 π 2 π 3π 2 2π x Domain: (−∞, ∞) Range: (−1, 1) Domain: (−∞, ∞) Range: (−1, 1) Figure A4 π Domain: x ≠ k where k is an odd integer 2 Range: (−∞, −1] ∪ [1, ∞) Cosecant y y = csc x Secant y y = sec x Cotangent y y = cot x 8 6 4 2 π –2 2 –4 –6 –8 2π 3π 2 π π 2 π 3π 2 2π x 2π 3π 2 π 8 6 4 2 π –2 2 –4 –6 –8 π 2 π 3π 2 2π x 2π 3π 2 π 20 15 10 5 π –5 2 –10 –15 –20 π 2 π 3π 2 2π x Domain: x ≠ πk where k is an integer Range: (−∞, −1] ∪ [1, ∞) π Domain: x ≠ k where k is an odd integer 2 Range: (−∞, −1] ∪ [1, ∞) Domain: x ≠ πk where k is an integer Range: (−∞, ∞) Inverse Sine y y = sin−1 x Figure A5 Inverse Cosine y y = cos−1 x Inverse Tangent y y = tan− 3π 8 π 4 π 8 π 8 π 4 3π 3π 2 π 2 π 4 π 4 π 2 3π 3π 8 π 4 π 8 π 8 π 4 3π Domain: [−1, 1] Range: Domain: [−1, 1] Range: [0, π) Figure A6 Inverse Cosecant y y = csc−1 x Inverse Secant y y = sec−1 x Domain: (−∞, ∞) Range: Inverse Cotangent y y = cot−1 x –4 –3 –2 –1 3π 2 π π 2 π 2 π 3π 2 1 2 3 4 x –4 –3 –2 –1 3π 2 π π 2 π 2 π 3π 2 1 2 3 4 x –4 –3 –2 –1 3π 2 π π 2 π 2 π 3π 2 1 2 3 4 x Domain: (−∞, −1] ∪ [1, ∞) Range: Domain: (−∞, −1] ∪ [1, ∞) Range: Domain: (−∞, ∞) Range: Figure A7 APPENDIX A-3 A3 Trigonometric Identities Identities Pythagorean Identities Even-odd Identities Equations sin2 θ + cos2 θ = 1 1 + tan2 θ = sec2 θ 1 + cot2 θ = csc2 θ cos(−θ) = cos θ sec(−θ) = sec θ sin(−θ) = −sin θ tan(−θ) = −tan θ csc(−θ) = −csc θ cot(−θ) = −cot θ Cofunction identities Fundamental Identities Sum and Difference Identities Double-Angle Formulas π _ sin θ = cos  − θ  2 π _ cos θ = sin  − θ  2 π _ tan θ = cot  − θ  2 π _ cot θ = tan  − θ  2 π _ sec θ = csc  − θ  2 π _ csc θ = sec  − θ  2 sin θ _ cos θ 1 _ cos θ 1 _ sin θ 1 _ tan θ tan θ = cot θ = csc θ = sec θ = = cos θ _ sin θ cos (α + β) = cos α cos β − sin α sin β cos (α − β) = cos α cos β + sin α sin β sin (α + β) = sin α cos β + cos α sin β sin (α − β) = sin α cos β − cos α sin β tan (α + β) = tan α + tan β __ 1 − tan α tan β tan (α − β) = tan α − tan β __ 1 + tan α tan β sin(2θ) = 2sin θ cos θ cos(2θ) = cos2 θ − sin2 θ cos(2θ) = 1 − 2sin2 θ cos(2θ) = 2cos2 θ − 1 tan(2θ) = 2tan θ ________ 1 − tan2 θ Table A1 APPENDIX A-4 Identities Equations Half-Angle formulas Reduction Formulas Product-to-Sum Formulas Sum-to-Product Formulas Law of Sines Law of Cosines α _ sin 2 α _ cos 2 α _ tan 2 = ± √ = ± √ = ± √ _________ 1 − cos α _______ 2 _________ 1 + cos α _______ 2 _________ 1 − cos α _______ 1 + cos α = sin α _______ 1 − cos α = 1 − cos α _______ sin α sin2θ = 1 − cos(2θ) _________ 2 cos2θ = 1 + cos(2θ) _________ 2 tan2θ = 1 − cos(2θ) _________ 1 + cos(2θ) 1 __  cos  α − β  + cos  α + β   cos α cos β = 2 1 __  sin  α + β  + sin  α − β   sin α cos β = 2 1 __  cos  α − β  − cos  α + β   sin α sin β = 2 1 __  sin  α + β  − sin  α − β   cos α sin β = 2 sin α + sin β = 2 sin  α + β _____ 2  cos  α − β  _____ 2 sin α − sin β = 2 sin  α − β _____ 2  cos  α + β  _____ 2 cos α − cos β = −2 sin  α + β _____ 2  sin  α − β  _____ 2 cos α + cos β = 2cos  α + β _____ 2  cos  α − β  _____ 2 sin α _ a sin a _ α = = sin β _ b sin b _ β = = sin γ _ c sin c _ γ a2 = b2 + c2 − 2bccos α b2 = a2 + c2 − 2accos β c2 = a2 + b2 − 2aacos γ Table A1 APPENDIX Try It Answers Chapter 1 Section 1.1 — 7. g(1) = 8 4. g(5) = 1 (Note: If two players had been tied for, say, 4th 5. m = 8 8. x = 0 or x = 2 1. a. Yes b. Yes place, then the name would not have been a function of rank.) 2. w = f (d) 3. Yes 3 x ___ 6. y = f (x) = √ 2 9. a. Yes, because each bank account has a single balance at any given time. b. No, because several bank account numbers may have the same balance. c. No, because the same output may 10. a. Yes, letter grade correspond to more than one input. is a function of percent grade. b. No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade. 12. No, because it does not pass the horizontal line test. 11. Yes 5. a. Values that are less than or equal to –2, or Section 1.2 1 1 __ __ 1. {−5, 0, 5, 10, 15} 2. (−∞, ∞) 3.  −∞, , ∞   ∪  2 2 5 , ∞  4.  − __ 2 values that are greater than or equal to –1 and less than 3; b. { x \|x ≤ –2 or –1 ≤ x < 3 } c. (–∞, –2] ∪ [–1, 3) 6. Domain = [1950, 2002]; Range = [47,000,000, 89,000,000] 7. Domain: (−∞, 2]; Range: (−∞, 0] 8. y 21 1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 Section 1.3 1. $2.84 − $2.31 ___________ = 5 years $0.53 _____ 5 years = $0.106 per year. 1 __ 2. 2 4. The local maximum appears to occur at (−1, 28), 3. a + 7 and the local minimum occurs at (5, −80). The function is increasing on (−∞, −1) ∪ (5, ∞) and decreasing on (−1, 5). f (x) (−1, 28) 40 20 –4 –3 –2 –1 –20 –40 –60 –80 –100 –120 321 4 5 6 7 8 x (5, −80) Section 1.4 1. a. (fg)(x) = f (x)g(x) = (x − 1)(x 2 − 1f − g)(x) = f (x) − g(x) = (x − 1) − (x2 − 1) = x − x 2 b. No, the functions are not the same. 2. A gravitational force is still a force, so a(G(r)) makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G(a(F)) does not make sense. 3. f ( g (1)) = f (3) = 3 and g ( f (4)) = g (1) = 3 4. g ( f (2)) = g (5) = 3 5. a. 8; b. 20 7. Possible answer: g(x) = √ 6. [−4, 0) ∪ (0, ∞) 4 + x2 ; h(x ____ 3 − x Section 1.5 1. b(t) = h (t) + 10 = −4.9t 2 + 30t + 10 2. y 5 4 3 2 1 –1 –1 –2 g(x) f(x) 21 3 4 5 –5 –4 –3 –2 The graphs of f (x) and g (x) are shown here. The transformation is a horizontal shift. The function is shifted to the left by 2 units. x 3. h(x) 4. g(x) = 1 ____ x − 1 + 1 10 h(x) \|x – 2\|+ 4 8 6 4 2 –6 –5 –4 –3 –2 –1 –2 21 3 4 5 6 x 5. a. y 3 2 1 –1 –1 –2 –3 –4 –3 –2 b. 21 3 4 x y 4 3 2 1 –4 –3 –2 –1 –1 –2 21 3 4 6. a. g(x) = −f (x) b. h(x) = f (−x) x 4 x −2 0 g(x) −5 −10 −15 −20 4 2 x −2 h(x) 15 0 10 2 5 unknown Notice: h(x) = f (−x) looks the same as f (x). 7. y f (x) = x2 h(x) = f (− x)= (− x)2 –5 –4 –3 –2 5 4 3 2 1 0 –1–1 –2 –3 –4 –5 21 3 4 5 x g(x) = −f (x) = −x2 9. 8. even 10. g(x) = 3x − 2 2 6 4 9 12 15 x 8 g(x) 0 1 __ x  so using the square root function we get 11. g(x) = f  ___ 3 1 g(x) = √ __ x 3 B-1 B-2 Section 1.6 4. x = −1 or x = 2 1. \| x − 2 \| ≤ 3 2. Using the variable p for passing, \| p − 80 \| ≤ 20 3. f (x) = −\| x + 2 \| + 3 5. f (0) = 1, so the graph intersects the vertical axis at (0, 1). f (x) = 0 when x = −5 and x = 1 so the graph intersects the horizontal axis at (−5, 0) and (1, 0). 6. 4 ≤ x ≤ 8 7. k ≤ 1 or
k ≥ 7; in interval notation, this would be (−∞, 1] ∪ [7, ∞). Section 1.7 4. The domain of function 3. Yes 2. Yes 1. h(2) = 6 f −1 is ( −∞, −2) and the range of function f −1 is (1, ∞). 5. a. f (60) = 50. In 60 minutes, 50 miles are traveled. b. f −1(60) = 70. To travel 60 miles, it will take 70 minutes. 8. f −1(x) = (2 − x)2; 6. a. 3 b. 5.6 domain of f : [0, ∞); domain of f −1: (−∞, 2] 9. 7. x = 3y + 5 y y=x f –1(x) 21 3 4 5 76 (x) –3 –2 –1–1 –2 –3 Chapter 2 Section 2.1 4 − 3 1 _ −2 _ 0 − 2 1,868 − 1,442 __ 2,012 − 2,009 1. m = 2. m = = 1 __ ; decreasing because m < 0. = − 2 426 _ 3 = 142 people per year = 3. y − 2 = −2(x + 2); y = −2x − 2 4. y − 0 = −3(x − 0); y = −3x 6. H(x) = 0.5x + 12.5 5. y = −7x + 3 Section 2.2 1. y 2. Possible answers include (−3, 7), (−6, 9), or (−9, 11) (0, 6) (4, 3) (8, 0) 8 10 x 42 6 –10 –8 –6 –4 3. –10 –8 y y = 2x + 4 y = x 4. (16, 0) 5. a. f (x) = 2x; b. g(x) = − 1 __ x 2 6. y = − 1 __ x + 6 3 42 6 8 10 x 10 8 6 4 2 –2 –2 –4 –6 –8 –10 10 8 6 4 2 –6 –4 –2 –2 –4 –6 –8 –10 y = 2x 7. a. (0, 5) b. (5, 0) c. Slope –1 d. Neither parallel nor perpendicular e. Decreasing function function, perform a vertical flip (over the t-axis) and shift up 5 units. f. Given the identity Section 2.3 1. C(x) = 0.25x + 25,000; The y-intercept is (0, 25,000). If the company does not produce a single doughnut, they still incur a 2. a. 41,100 b. 2020 cost of $25,000. 3. 21.15 miles Section 2.4 1. 54° F 2. 150.871 billion gallons; extrapolation Chapter 3 Section 3.1 2. — — 6 −24 = 0 + 2i √ 1. √ 3. (3 − 4i) − (2 − 5i) = 1 − 9i 4. −8 − 24i 6. 102 − 29i 5. 18 + i 7. − 3 __ 17 5 __ + i 17 i 5 4 3 2 1 –1–1 –2 –3 –4 –5 –5 –4 –3 –2 21 3 4 5 r (x + 4)2 + 7. To make the shot, h(−7.5) would Section 3.2 1. The path passes through the origin and has vertex at (−4, 7), so (h)x = − 7 __ 16 need to be about 4 but h(−7.5) ≈ 1.64; he doesn’t make it. 2. g(x) = x 2 − 6x + 13 in general form; g(x) = (x − 3)2 + 4 3. The domain is all real numbers. The in standard form 8 __ , or  range is f (x) ≥ 11 5. 3 seconds; 256 feet; 7 seconds 4. y-intercept at (0, 13), No x-intercepts , ∞  . 8 __ 11 3. The degree is 6. The leading term is −x 6. Section 3.3 1. f (x) is a power function because it can be written as f (x) = 8x 5. The other functions are not power functions. 2. As x approaches positive or negative infinity, f (x) decreases without bound: as x → ±∞, f (x) → −∞ because of the negative coefficient. The leading coefficient is −1. 4. As x → ∞, f (x) → −∞; as x → −∞, f (x) → −∞. It has the shape of an even degree power 5. The leading term is function with a negative coefficient. 0.2x 3, so it is a degree 3 polynomial. As x approaches positive infinity, f (x) increases without bound; as x approaches negative 6. y-intercept (0, 0); infinity, f (x) decreases without bound. x-intercepts (0, 0), (−2, 0), and (5, 0) 7. There are at most 8. The end 12 x-intercepts and at most 11 turning points. behavior indicates an odd-degree polynomial function; there are 3 x-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead 9. The x-intercepts are (2, 0), coefficient must be negative. (−1, 0), and (5, 0), the y-intercept is (0, 2), and the graph has at most 2 turning points. Section 3.4 1. y-intercept (0, 0); x-intercepts (0, 0), (–5, 0), (2, 0), and (3, 0) 2. The graph has a zero of –5 with multiplicity 3, a zero of –1 with multiplicity 2, and a zero of 3 with multiplicity 4. TRY IT ANSWERS B-3 y 6 5 4 3 2 1 –2 –1 –2 –3 –4 –5 –10 –8 –6 –4 642 8 10 x 1 _ Horizontal asymptote at y = . 2 Vertical asymptotes at x = 1 4 and x = 3. y-intercept at  0, _  . 3 x-intercepts at (2, 0) and (–2, 0). (–2, 0) is a zero with multiplicity 2, and the graph bounces off the x-axis at this point. (2, 0) is a single zero and the graph crosses the axis at this point. 3. y 10 –4 –3 –2 –1 1 2 x –10 –20 –30 –40 –50 8. 4. Because f is a polynomial function and since f (1) is negative and f (2) is positive, there is at least one real zero between x = 1 and x = 2. 5. f (x) = − 1 __ (x − 2)3(x + 1)2(x − 4) 6. The minimum occurs 8 at approximately the point (0, −6.5), and the maximum occurs at approximately the point (3.5, 7). Section 3.5 1. 4x2 − 8x + 15 − 2. 3x 3 − 3x 2 + 21x − 150 + 78 _____ 4x + 5 1,090 _____ x + 7 and f ( f −1 (x)) = f (3x − 5) = Section 3.8 1. f −1( f(x)) = f −1  2. f −1(x ______ 2 4. f −1(x) =  − 5 = (x − 5) + 5 = x x + 5 _____ 3  = 3  x + 5 _____ 3 (3x − 5) + 5 __________ = 3 3. f −1(x) = √ x − 1 — 3x __ 3 = x , x ≥ 0 5. f −1(x) = 2x + 3 ______ x − 1 3. x = 20 Section 3.9 1. 128 ____ 3 9 __ 2. 2 Chapter 4 Section 4.1 — 4. (0, 129) and (2, 236); N(t) = 129(1.3526)t 3. About 1.548 billion people; by the year 2031, 1. g(x) = 0.875x and j(x) = 1095.6−2x represent exponential functions. 2. 5.5556 India's population will exceed China's by about 0.001 billion, or 1 million people. — 2 )x ; Answers may vary due to 5. f (x) = 2(1.5)x 6. f (x) = √ 2 ( √ round-off error. the answer should be very close to 1.4142(1.4142)x. 7. y ≈ 12 · 1.85x 9. $13,693 8. About $3,644,675.88 10. e−0.5 ≈ 0.60653 11. $3,659,823.44 12. 3.77E-26(Thi s is calculator notation for the number written as 3.77 × 10−26 in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.) Section 4.2 1. f(x) f(x) = 4x 6 5 4 3 2 1 (–1, 0.25) –5 –4 –3 –2 –1–1 –2 (1, 4) (0, 1) 21 3 4 5 x 2. f(x) (–1, 3.25) 10 8 6 4 2 –5 –4 –3 –2 –1–2 –4 f(x) = 2x – 1 + 3 (1, 4) (0, 3.5) 21 3 4 y = 3 x 5 The domain is ( −∞, ∞); the range is (0, ∞); the horizontal asymptote is y = 0. The domain is ( −∞, ∞); the range is (3, ∞); the horizontal asymptote is y = 3. 3. x ≈ −1.608 3. 3x2 − 4x + 1 Section 3.6 1. f (−3) = −412 2. The zeros are 2, −2, and −4. 1 __ 3. There are no rational zeros. 4. The zeros are −4, , and 1. 2 5. f (x) = − 1 x 3 + 5 __ __ 2 2 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros. 7. 3 meters by 4 meters by 7 meters 6. There must be 4, 2, or x 2 − 2x + 10 Section 3.7 1. End behavior: as x → ±∞, f (x) → 0; Local behavior: as x → 0, f (x) → ∞ (there are no x- or y-intercepts). 2. y The function and the asymptotes are shifted 3 units right and 4 units down. As x → 3, f (x) → ∞, and as x → ±∞, f (x) → −4. The function is f (x) = ______ (x − 3)2 − 4. 1 –5 –4 –3 –2 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 x y = –4 x = 3 3. 4. The domain is all real numbers except x = 1 and x = 5. 12 __ 11 5. Removable discontinuity at x = 5. Vertical asymptotes: x = 0, 6. Vertical asymptotes at x = 2 and x = –3; horizontal x = 1. asymptote at y = 4. squared function, we find the rational form. 7. For the transformed reciprocal f(x) = 1 ______ (x − 3)2 − 4 = 1 − 4(x − 3)2 ___________ (x − 3)2 = 1 − 4(x2 − 6x + 9) _______________ (x − 3)(x − 3) = −4x2 + 24x − 35 ______________ x2 − 6x + 9 Because the numerator is the same degree as the denominator we know that as x → ±∞, f (x) → −4; so y = –4 is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is x = 3, because as x → 3, f (x) → ∞. We then set the numerator equal to 0 and find the x-intercepts are at (2.5, 0) and (3.5, 0). Finally, we evaluate the function at 0 and find the y-intercept to be at  0, −35  . ____ 9 TRY IT ANSWERS B-4 4. f(x) 1 f(x) = (4)x 2 (1, 2) The domain is ( −∞, ∞); the range is (0, ∞); the horizontal asymptote is y = 0. (0, 0.5) y = 0 3 21 5 4 x 5 4 3 2 1 –1–1 –2 (–1, 0.125) –3 –5 –2 –4 5. g(x) = 1.25–x (–1, 1.25) g(x) 5 4 3 2 1 (0, 1) (1, 0.8) –10 –8 –6 –4 –2–1 42 6 8 10 y = 0 x The domain is ( −∞, ∞); the range is (0, ∞); the horizontal asymptote is y = 0. 1 __ 6. f(x) = − ex − 2; the domain is ( −∞, ∞); the range is 3 (− ∞, 2); the horizontal asymptote is y = 2. Section 4.3 1. a. log10(1,000,000) = 6 is equivalent to 106 = 1,000,000 b. log5(25) = 2 is equivalent to 52 = 25 2. a. 32 = 9 is equivalent to log3(9) = 2 b. 53 = 125 is equivalent to log5(125) = 3 1 1 is equivalent to log2  _ _  = −1 c. 2−1 = 2 2 1 121 = 121 1 _ 3. log121(11) = (recalling that √ 2 1 4. log2  _  = −5 5. log(1,000,000) = 6 6. log(123) ≈ 2.0899 32 8. It is not 7. The difference in magnitudes was about 3.929. possible to take the logarithm of a negative number in the set of real numbers. _ = 11) 2 — Section 4.4 1. (2, ∞) 3. 2. (5, ∞) f(x) 4 3 2 1 –2–1 –2 –3 –10 –8 –6 –4 4. x = −4 (−1, 1) y 5 4 3 2 1 –6 –5 –4 –3 –2 –1 (−3, 0) –1 –2 –3 –4 –5 , 11 5 (1, 0) 42 6 8 f (x) = log (x) x = 0 10 1 5 x = 0 f (x) = log3(x + 4) y = log3(x) 4 5 6 (3, 1) x 321 (1, 0) The domain is (0, ∞), the range is (−∞, ∞), and the vertical asymptote is x = 0. x The domain is (−4, ∞), the range (−∞, ∞), and the asymptote x = –4. The domain is (0, ∞), the range is (−∞, ∞), and the vertical asymptote is x = 0. 5. y f(x) = log2(x) + 2 x = 0 (0.5, 1) (0.25, 0) (2, 1) (1, 0) y = log2(x) x 1 f(x) = log4(x) 2 (16, 1) x The domain is (0, ∞), the range is (−∞, ∞), and the vertical asymptote is x = 0. The domain is (2, ∞), the range is (−∞, ∞), and the vertical asymptote is x = 2. x The domain is (−∞, 0), the range is (−∞, ∞), and the vertical asymptote is x = 0. 9. x ≈ 3.049 11. f (x) = 2ln(x + 3) − 1 10. x = 1 x 6. y x = 0 y = log4(x) (4, 1) (1, 0) 7. 8 10 –10 –8 –6 –4 –2–1 –2 –3 –4 –5 y 4 3 2 1 –10 –8 – – 46 –2–1 –2 42 x = 0 6 8 10 Section 4.5 3. 2ln(x) 6. 2log(x) + 3log(y) − 4log(z) 1. logb(2) + logb(2) + logb(2) + logb(k) = 3logb(2) + logb(k) 2. log3(x + 3) − log3(x − 1) − log3(x − 2) 4. −2ln(x) 5. log3(16) 2 1 __ __ ln(x − 1) + ln(2x + 1) − ln(x + 3) − ln(x − 3) 7. ln(x) 8. 2 3 3 ⋅ 5 5 ____ __ 9. log   ; can also be written log   by reducing the 4 ⋅ 6 8  fraction to lowest terms. 10. log  x 5(x − 1)3 √ ___________ (7x − 1) (2x + 3)4 ; this answer could also be written log  11. log ln(8) ____ ln(0.5) x12(x + 5)4 ________ 12. The pH increases by about 0.301. 4.6051 _____ 1.6094 ln(100)
_____ ≈ ln(5) = 2.861 x3(x + 5)  _______ . (2x + 3) 13. 14. — 4 Section 4.6 1. x = −2 2. x = −1 2 4. The equation has no solution. 11 11 __ __  or ln  6. t = 2ln   3 3 7. t = ln  1 1 __ _  = − ln(2) — 2 2 √ 10. x = e5 − 1 11. x ≈ 9.97 ln(0.8) ______ ln(0.5) 1 __ 3. x = 2 5. x = ln(3) _ 2 _  ln  3 8. x = ln(2) 9. x = e4 12. x = 1 or x = −1 13. t = 703,800,000 × years ≈ 226,572, 993 years. Section 4.7 1. f (t) = A0 e −0.0000000087t 2. Less than 230 years; 229.3157 to be exact 3. f (t) = A0 e  ln(2) 5. 895 cases on day 15 6. Exponential. y = 2e 0.5x 4. 6.026 hours 7. y = 3e (ln 0.5)x ____ 3  t TRY IT ANSWERS B-5 Section 4.8 1. a. The exponential regression model that fits these data is y = 522.88585984(1.19645256)x. b. If spending continues at this rate, the graduate’s credit card debt will be $4,499.38 after one year. 2. a. The logarithmic regression model that fits these data is y = 141.91242949 + 10.45366573ln(x) b. If sales continue at this rate, about 171,000 games will be sold in the year 2015. 3. a. The logistic regression model that fits these data is _____________________ 1 + 6.113686306e−0.3852149008x . b. If the population continues to grow at this rate, there will be about 25,634 seals in c. To the nearest whole number, the carrying capacity 2020. is 25,657. 25.65665979 y = Section 5.4 1. 7 ____ 25 2. sin(t) = 33 __ 65 sec(t) = 65 __ 56 2 π √ 3. sin  __ ____  = 2 4 π — sec  __ 2  = √ 4 — Chapter 5 Section 5.1 1. y 240° Section 5.2 2. 3π ___ 2 5. α = 150° 215π ____ 18 8. x 3. −135° 4. 6. β = 60° 7π _ 10 7. 7π ___ 6 = 37.525 units 9. 1.88 10. − rad/s 3π ___ 2 11. 1,655 kilometers per hour cos(t) = 56 __ 65 csc(t) = 65 __ 33 2 π √ cos  __ ____  = 2 4 π — csc  __ 2  = √ 4 — tan(t) = 33 __ 56 cot(t) = 56 __ 33 π tan  __  = 1 4 π cot  __  = 1 4 π — __ . 3 ; missing angle is 6 4. 2 5. Adjacent = 10; opposite = 10 √ 6. About 52 ft. Chapter 6 Section 6.1 π __ 1 __ 1. 6π 3. 2. compressed ; right 2 2 1 __ 5. Midline: y = 0; Amplitude : \|A\| = 4. 2 units up ; 2 2π C _ __ = 6π; Phase shift: = π 6. f (x) = sin(x) + 2 Period : P = ∣ B ∣ B π π __ __  + 4 or x − 7. Two possibilities: y = 4sin  5 5 π __ y = −4sin  x + 5 8. 4π ___  + 4 5 y 1 g(x) = –0.8 cos (2x) –π – π 2 x π π 2 Midline: y = 0; Amplitude : \|A\| = 0.8; Period : P = = π; Phase shift : = 0 or 2π ___ \|B\| C __ B none — 2 √ ____ 1. cos(t) = − 2 3. sin(t) = − 7 __ 25 6. a. cos(315°) = π 1 _ ___  = − sin  − 6 2 , sin(t) = — 2 √ ____ 2 — 2 √ ____ 2 7.  1 __ , − 2 , sin(315°) = − 3 √  ____ 2 — 2. cos(π) = −1, sin(π) = 0 — 2 √ π ___ ____  = b. cos  − 2 6 π _ 5. 3 — 3 √ ____ , 2 4. Approximately 0.866025403 — Section 5.3 2 1. sin t = − √ ____ 2 — 2 , sec t = √ 3 √ π __ ____ = 2. sin 2 3 π _ = 2 sec 3 3. sin  − — — — 2 2 7π ___  = √ ____ 2 4 7π ___ — sec  −  = √ 4 3 4. − √ 5. −2 7. cot t = − 8 __ 17 17 __ 15 8. sin t = −1 csc t = sec t = Undefined — 2 9. sec t = √ cot t = 1 10. ≈ −2.414 –1 y 2 1 π g(x) = –2 cos 3 –7 –5 –3 –1 1 3 5 7 9. 10. tan t = −1 cot t = −1 π __ = √ tan 3 π __ = cot 3 tan  − — 3 — 3 √ ____ 3 7π ___  = 1 4 7π ___  = 1 4 cot  − tan t = − 15 __ 8 –2π –π — — — 2 2 cos t = √ ____ 2 csc t = − √ π __ 1 __ = cos 2 3 3 π __ 2 √ _____ = csc 3 3 2 7π ___ √ ____  = cos  − 4 2 7π ___ — csc  −  = √ 4 6. sin t sin t = 2 — 15 __ 17 cot t = − 8 __ 15 –1 –2 y 7 5 3 1 –1 –3 –5 –7 f(x) = 7 cos x π 2π x x + Midline : y = 0; π 6 Amplitude : \|A\| = 2; = 6 ; Period : P = 2π ___ \|B\| C __ B 1 __ = − 2 Phase shift : x 7 cos t = 0 csc t = −1 2 csc t = √ — tan t = Undefined cot t = 0 tan t = 1 11. y 3cos(x) − 4 –1 –2 –3 –5 –6 –7 π 2π 3π 4π x y = −4 y = 3 cos x − 4 TRY IT ANSWERS B-6 Section 6.2 1. y y = 3 tan 6 πx 5 3 1 2. It would be reflected across the line y = −1, becoming an increasing function. –7 –5 –1 1 5 7 x 3. g(x) = 4tan(2x) x = −9 x = −3 x = 3 x = 9 –5 4. y 6 4 2 f (x) = −2.5sec (0.4x) This is a vertical reflection of the preceding graph because A is negative. –π π 2π 3π 4π x –2 –4 –6 5. y f(x) = −6sec (4x + 24 –8 –12 –16 –20 –24 y 6. f (x) = 0.5csc (2x) 6 4 2 –2 –4 –6 7. π 4 3π (x) = 2 sec π x 2 + 1 f (x) = 2 cos –5 –3 –2 –4 –6 Section 6.3 1. arccos(0.8776) ≈ 0.5 Chapter 7 Section 7.1 1. csc θ cos θ tan θ =  = sin θ  _ cos θ  cos θ  1 _ sin θ cos θ sin θ   ____ ____ cos θ sin θ sin θcos θ ________ sin θcos θ = = 1 = cot θ ____ csc θ cos θ ____ sin θ _ = 1 ____ sin θ sin2 θ − 1 ______________ = tan θ sin θ − tan θ 2. 3. cos θ ____ ⋅ sin θ sin θ ____ 1 = cos θ (sin θ + 1)(sin θ − 1) ________________ = tan θ(sin θ − 1) sin θ + 1 _______ tan θ 4. This is a difference of squares formula: 25 − 9sin2 θ = (5 − 3sin θ)(5 + 3sin θ). 5. cos θ _______ 1 + sin θ  1 − sin θ  = _______ 1 − sin θ cos θ(1 − sin θ) ____________ 1 − sin2 θ cos θ(1 − sin θ) ____________ cos2 θ = = 1 − sin θ _______ cos θ Section 7.2 — — — — 6 6 3. 2 – √ √ _________ 4 2 + √ √ _________ 4 2. 1. 5. tan(π − θ) = tan(π) − tan θ ____________ 1 + tan(π)tan θ = 0 − tan θ __________ 1 + 0 ∙ tan θ — 1 − √ _______ 1 + √ 3 4. cos  3 — 5π ___  14 = −tan θ Section 7.3 1. cos(2α) = 7 __ 32 2. cos4 θ − sin4 θ = (cos2 θ + sin2 θ)(cos2 θ − sin2 θ) = cos(2θ) 3. cos(2θ)cos θ = (cos2 θ − sin2 θ)cos θ = cos3 θ − cos θsin2 θ 4. 10cos4 x = 10(cos2 x)2 1 + cos(2x) 2  _________ Substitute reduction formula for cos2 x. 2 = = = 10  10 __ 4 10 __ 4 10 __ 4 30 __ 8 15 __ 4 = = = [1 + 2cos(2x) + cos2(2x)] + + + cos(2x) + cos(2x) + 10 __ 2 10 __ 2 10  __ 4 10 __ 8 10 __ 8 5 __ + 5cos(2x) + cos(4x) 4 + 5cos(2x) + cos(4x) 10 __ 8 1 + cos2(2x)  __________ 2 Substitute reduction formula for cos2 x. cos(4x) 5 __ __ __ b. − 2. a. − c. π d. 3 4 2 Section 7.4 3. 1.9823 or 113.578° 4. sin−1(0.6) = 36.87° = 0.6435 radians π _ 5. ; 8 2π _ 6. 9 3π _ 4 12 _ 13 7. 8. 9. — 2 4 √ _ 9 4x _ 16x2 + 1 √ — 1 1 __ __ (sin 2x + sin 2y) 3. (cos 6θ + cos 2θ) 2. 1. 2 2 — −2 − √ ________ 4 3 4. 2 sin(2θ)cos(θ) TRY IT ANSWERS 5. tan θ cot θ − cos2 θ =  cos θ _ sin θ  − cos2 θ   sin θ _ cos θ = 1− cos2 θ = sin2 θ Section 7.5 1. x = 7π _ , 6 11π _ 6 π _ 3. θ ≈ 1.7722 ± 2πk and ± πk 2. 3 3π π _ _ θ ≈ 4.5110 ± 2πk 4. cos θ = −1, θ = π 5. , 2 2 2π _ , 3 4π _ , 3 Section 7.6 2 _ 1. The amplitude is 3, and the period is . 3 2. y x 3sin(3x) y = 3 sin (3x) –5 –4 π 6 π 3 π 2 x 2π 2π _ 3 0 3 0 −3 0 3 2 1 –1 –2 –3 3. y 40 32 24 16 8 π __ 12 t  + 32; The y = 8sin  temperature reaches freezing at noon and at midnight. 4 8 12 16 20 24 t 2 __ 4. initial displacement = 6, damping constant = −6, frequency = π 5. y = 10e −0.5t cos(πt) 6. y = 5cos(6πt) Chapter 8 Section 8.1 1. α = 98°, a = 34.6; β = 39°, b = 22; γ = 43°, c = 23.8 2. Solution 1 α = 80°, a = 120; β ≈ 83.2°, b = 121; γ ≈ 16.8°, c ≈ 35.2 Solution 2 α' = 80°, a' = 120; β' ≈ 96.8°, b' = 121; γ' ≈ 3.2°, c' ≈ 6.8 3. β ≈ 5.7°, γ ≈ 94.3°, c ≈ 101.3 5. About 8.2 square feet 6. 161.9 yd. 4. Two Section 8.2 1. a ≈ 14.9, β ≈ 23.8°, γ ≈ 126.2° 2. α ≈ 27.7°, β ≈ 40.5°, γ ≈ 111.8° 3. Area = 552 square feet 4. About 8.15 square feet — 3. (x, y) =  √ 3 1  ____ _ , − 2 2 in the standard form for a circle, x2 + (y − 1)2 = 1 4. r = √ — 3 5. x2 + y2 = 2y or, B-7 Section 8.4 1. The equation fails the symmetry test with respect to the π _ line θ = and with respect to the pole. It passes the polar axis 2 symmetry test. π _  , and the maximum value is (3, 0). polar axis. The zero is  θ, 2 3. 4. The graph is a rose curve, n even 2. Tests will reveal symmetry about the 4 –3 –2 –1 1 2 3 4 –1 –2 –3 –4 y (−4, 2) (0, 2) (4, 2) (0, 0) 2 = 8y x x y = −2 1 2 3 4 5 6 –3 –3 –2 –1 –1 –2 –3 –4 –5 –6 5. The graph is a rose curve, n odd (y + 1)2 = −4(x − 8) y 6. x (9, 1) (9, −1) (9, −3) y = −1 (8, −1) x = 7 Section 8.5 1. (−4, 8) y2 = −16x y (−4, 0) (0, 0) x (−4, −8) x = 4 — 2 — 50 = 5 √ 3. ∣ z ∣ = √ 2. 13 π π  + isin  4. z = 3  cos   + isin  5. z = 2  cos    6 6 3 − 2i 6. z = 2 √ — 7. z1z2 = −4 √ — 3 ; z1 _ = − z2 — √ 3 ____ 2 3 _ + i 2 8. z0 = 2(cos(30°) + isin(30°)), z1 = 2(cos(120°) + isin(120°)) z2 = 2(cos(210°) + isin(210°)), z3 = 2(cos(300°) + isin(300°)) Section 8.6 1. t x(t) −1 0 1 2 −4 −3 −2 −1 y (−4, 2) (0, 2) (4, 2) (0, 0) x 2 = 8y x y = −2 y(t) 2 4 6 8 Section 8.3 1. (−15, 0) (−12, 0) y (0, 9) (0, −9) 2. y (15, 0) x (12, 0) (3, 6) (−5, −4) (3, −4) x (11, −4) (3, −14) 2. x(t) = t3 − 2t, y(t) = t 4. y = ln  √ — x  5. _______ 1 3. y = 5 − √ __ x − 3 2 y2 x2 _ _ 6. y = x2 4 9 = 1 + TRY IT ANSWERS B-8 Section 8.7 1. y y = 8 2. (−2, 3) (x + 2)2 = −20(y − 3) (−2, −2) (−12, −2) x = −2 x (8, −2) y 0.5 0.25 −0.75 −0.5 −0.25 0.25 0.5 0.75 x −0.25 −0.5 3. –5 –4 –3 –2 y 5 4 3 2 1 –1–1 –2 –3 –4 –5 1 2 3 4 5 The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations. Section 8.8 y 1. 5 4 3 2 1 1– –1 1 2 3 4 5 6 Chapter 9 2. 3u = 〈15, 12〉 〈3, 5〉 3. u = 8i − 11j 4. v = √ — 34 cos(59°)i + √ — 34 sin(59°)j x — Magnitude = √ 34 5 θ = tan−1  _  = 59.04° 3 Section 9.1 1. Not a solution 2. The solution to the system is the ordered pair(−5, 3). y (−4, 0) –4 –5 –3 –2 7 6 5 4 3 2 1 –1–1 –2 –3 –4 –5 –6 –7 (0, 7) (0, 0) 2 1 (4, 0) 3 4 5 − (0, −7) 3. (−2, −5) 4. (−6, −2) 5. (10, −4) 6. No solution. It is an inconsistent system. 7. The system is dependent so there are infinite solutions of the form (x, 2x + 5). 8. 700 children, 950 adults Section 9.2 1. (1, −1, 1) 2. No solution solutions of the form (x, 4x − 11, −5x + 18) 3. Infinite number of Section 9.3 1 1 _ _  and (2, 8) 2. (−1, 3) 3. {(1, 3),(1, −3),(−1, 3),(−1, −3)} 1.  − , 2 2 4. y 10 8 6 4 2 (4, 2 + 2 ) (4, 2) (10, 2) (−2, −2) –10 –8 –4 –6 (0, 2) –2–2 –4 –6 4 10 6 8 2 (4, 2 − 2 ) (8, 2) 1. Section 9. + x2 − 2x + 3 2 _ x − 2 4. 2. 6 _ − x − 1 2x + 1 __ (x2 − 2x + 3)2 5 _ (x − 1)2 3. 3 _ + x − 1 2x − 4 _ x2 + 1 Section 9.5 2 6  +  1. A + B =  1 0 −3 1 −2 −8  2. −2B =  −4 −6 − + (−4) −4 6 + (−23 0 + 5 −3 + 3 2. x − y + z = 5 2x − y + 3z = 1 y + z = −9 3. (2, 1) 5. (1, 1, 1) Section 9.6  −3 1.  4 11 4 2 3  4.  5 __ 2 − 5 __ 2 17 __  Section 9.7 1. AB =  1 4   −3 −1 −3 1 6. $150,000 at 7%, $750,000 at 8%, $600,000 at 10% −4 1 1(−3)
+ 4(1) −1(−3) + −3(13(1) + −4(−1)  =  1(1) + 1(−1) =  1 0  1 0 3. A−1 =  1 2 4 −3 −5 3  1 2 6 1(−4) + 4(1)  −1(−4) + −3(1) −3(4) + −4(−3)  1(4) + 1(−3) 4. X =  4 38 58  BA =  −3 1 −4 1   1 4 −3 −1 3 __ 5 2. A−1 =  − 2 __ 5 1 __ 5  1 __ 5 Section 9.8 1. (3, −7) 2. −10 3 3.  −2, _ , 5 12 _  5 Chapter 10 Section 10.1 2. (x − 1)2 _ + 16 (y − 3)2 _ 4 = 1 y = 1 1. x2 + y2 _ 16 3. Center: (0, 0); Vertices: (±6, 0); Co-vertices: (0, ±2); Foci: (±4 √ — 2 , 0) (−6, 0) –5 –4 –3 –2 4 3 2 1 –1–1 –2 –3 –4 (0, 2) (0, 0) 2 1 3 4 5 (0, −2) (6, 0) x2 _ 16 + y2 _ 49 = 1; 4. Standard form: Center: (0, 0); Vertices: (0, ±7); Co-vertices: (±4, 0) Foci: (0, ± √ 33 ) — (−4, 0) –4 –5 –3 –2 y (0, 7) (0, 0) 1 2 (4, 0) 3 4 5 − (0, −7) 7 6 5 4 3 2 1 –1–1 –2 –3 –4 –5 –6 –7 TRY IT ANSWERS B-9 (y + 1)2 = −4(x − 8) y y = −1 (8, −1) x = 7 y x (9, 1) (9, −1) (9, −3) y = 8 5. Center: (4, 2); Vertices: (−2, 2) and (10, 2); Co-vertices:  4, 2 − 2 √ and  4, 2 + 2 √ 5  ; Foci: (0, 2) and (8, 2) 5  — — y 10 8 6 4 2 (−2, −2) –10 –8 –6 –4 (0, 2) –2–2 –4 –6 (4, 2 + 2 ) (4, 2) (10, 2) 4 10 6 2 8 (4, 2 − 2 ) (8, 2) 3. x2 = 14y 4. Vertex: (8, −1); Axis of symmetry: y = −1; Focus: (9, −1); Directrix: x = 7; Endpoints of the latus rectum: (9, −3) and (9, 1). 6. (x − 3)2 _ 4 + (y + 1)2 _ 16 = 1; Center: (3, −1); Vertices: (3, −5) and (3, 3); Co-vertices: (1, −1) and (5, −1); Foci:  3, −1 − 2 √ and  3, −1 + 2 √ — 3  7. a. x2 _ 57,600 + y2 _ 25,600 = 1; b. The — 3  5. Vertex: (−2, 3); people are standing 358 feet apart. Section 10.2 1. Vertices: (±3, 0); Foci:  ± √ — 34 , 0  3. (y − 3)2 _ 25 + (x − 1)2 _ 144 = 1 4. Vertices: (±12, 0); Co-vertices: (0, ±9); Foci: (±15, 0); 3 _ Asymptotes: y = ± x 4 (−15, 0) (−12, 0) Axis of symmetry: x = −2; Focus: (−2, −2); Directrix: y = 8; Endpoints of the latus rectum: (−12, −2) and (8, −2). (−2, 3) (x + 2)2 = −20(y − 3) (−2, −2) (−12, −2) x = −2 x (8, −2) 2. − = 1 y2 _ 4 x2 _ 16 6. a. y2 = 1,280x b. The depth of the cooker is 500 mm. y (0, 9) (0, −9) (15, 0) x (12, 0) Section 10.4 1. a. hyperbola b. ellipse 3. a. hyperbola b. ellipse Section 10.5 2. x' 2 _ 4 + y' 2 _ 1 = 1 5. Center: (3, −4); Vertices: (3, −14) and (3, 6); Co-vertices: (−5, −4) and (11, −4); Foci:  3, −4 − 2 √ and  3, −4 + 2 √ 5 _ (x − 3) − 4 41  ; Asymptotes: y = ± 4 — — 41  1 _ 1. ellipse; e = ; x = −2 3 y 2. 0.5 0.25 3. r = 1 _ 1 − cos θ 4. 4 − 8x + 3x2 − y2 = 0 y (3, 6) (−5, −4) (3, −4) x (11, −4) (3, −14) −0.75 −0.5 −0.25 0.25 0.5 0.75 x −0.25 −0.5 Chapter 11 6. The sides of the tower can be modeled by the hyperbolic equation. x2 _ 400 − y2 _ 3600 = 1 or x2 _ 202 − y2 _ 602 = 1. Section 10.3 1. Focus: (−4, 0); Directrix: x = 4; Endpoints of the latus rectum: (−4, ±8) y (−4, 8) y2 = −16x (−4, 0) (0, 0) 2. Focus: (0, 2); Directrix: y = −2; Endpoints of the latus rectum: (±4, 2) (−4, −8) x = 4 y (−4, 2) (0, 2) (4, 2) (0, 0) 2 = 8y x y = −2 x x 2. The first five 4. an = (−1)n + 1 9n 3. The first six terms are 3n _ 4n 17 _  6 Section 11.1 1. The first five terms are {1, 6, 11, 16, 21}. 5 3 terms are  −2, 2, − _ _  . , 1, − 8 2 5. an = − {2, 5, 54, 10, 250, 15}. 5 6. an = e n − 3 7. {2, 5, 11, 23, 47} 8.  0, 1, 1, 1, 2, 3, _ , 2 3 9. The first five terms are  1, _ , 4, 15, 72  . 2 Section 11.2 1. The sequence is arithmetic. The common difference is −2. 2. The sequence is not arithmetic because 3 − 1 ≠ 6 − 3. 3. {1, 6, 11, 16, 21} for n ≥ 2 sequence. her 42 minutes. 6. an = 53 − 3n 7. There are 11 terms in the 8. The formula is Tn = 10 + 4n, and it will take 5. a1 = 25; an = an − 1 + 12, 4. a2 = 2 TRY IT ANSWERS Section 12.3 1. a. Removeable discontinuity at x = 6 b. Jump discontinuity at x = 4 3. No, the function is not continuous at x = 3. There exists a removable discontinuity at x = 3. 4. x = 6 2. Yes Section 12.4 1. 3 3. f '(a) = −15 _ (5a + 4) 2 6. −2, 0, 0, −3 2. f '(a) = 6a + 7 d. After 20 seconds, she is moving 3 _ 4. 2 7. a. After zero seconds, she b. After 10 seconds, she has traveled c. After 10 seconds, she is moving eastward 5. 0 has traveled 0 feet. 150 feet east. at a rate of 15 ft/sec. westward at a rate of 10 ft/sec. 100 feet westward of her starting point. of f is continuous on (−∞, 1)∪(1, 3)∪(3, ∞). The graph of f is discontinuous at x = 1 and x = 3. The graph offis differentiable on(−∞, 1)∪(1, 3)∪(3, ∞). The graph of f is not differentiable at 10. −68 ft/sec, it is x = 1 and x = 3. dropping back to Earth at a rate of 68 ft/s. e. After 40 seconds, she is 8. The graph 9. y = 19x − 16 B-10 Section 11.3 1. The sequence is not geometric because 10 _ 5 ≠ 15 _ . 10 1 _ 2. The sequence is geometric. The common ratio is . 5 2 2 2 _ _ _ 3.  18, 6, 2,  an − 1 for n ≥ 2 4. a1 = 2; an = , 3 9 3 5. a6 = 16,384 6. an = −(−3)n − 1 7. a. Pn = 293 ∙ 1.026an b. The number of hits will be about 333. 3. 328 2. 26.4 Section 11.4 1. 38 6. ≈ 2,000.00 7. 9,840 is defined. It is geometric. is defined. 12. 3 15. $92,408.18 5. $2,025 4. −280 8. $275,513.31 10. The sum of the infinite series 9. The sum 11. The sum of the infinite series is defined. 13. The series is not geometric. 14. − 3 _ 11 Section 11.5 1. 7 4. 60 8. C(10, 3) = 120 2. There are 60 possible breakfast specials. 6. P(7, 7) = 5,040 5. 12 9. 64 sundaes 10. 840 3. 120 7. P(7, 5) = 2,520 Section 11.6 1. a. 35 b. 330 2. a. x 5 − 5x 4y + 10x 3y2 − 10x2y3 + 5xy 4 − y 5 b. 8x 3 + 60x 2y + 150xy 2 + 125y 3 3. −10,206x 4y 5 7 _ 13 3. 2 _ 2. 3 5 _ 5. 6. a. 6 1 _ 91 4. 2 _ 13 5 _ 91 b. c. 86 _ 91 Section 11.7 1. Outcome Probability Roll of 1 Roll of 2 Roll of 3 Roll of 4 Roll of 5 Roll of 6 Chapter 12 Section 12.1 1. a = 5, f (x) = 2x2 − 4, and L = 46 2. a. 0 b. 2 c. Does not exist d. −2 e. 0 i. 4 20sin(x)  = 5  __ 3. lim 4x x → 0 −0.01 −0.001 −0.1 f. Does not exist g. 4 h. 4 0.001 0.01 0.1 x 0 f (x) 4.9916708 4.9999167 4.9999992 Error 4.9999992 4.9999167 4.9916708 20sin(x)  __ 4x x → 0+  lim 5 5 20sin(x)  x → 0−  __ lim 4x 4. Does not exist Section 12.2 1. 26 2. 59 1 _ 7. − 8 8. 2 √ 3. 10 4. −64 5. −3 6. − 1 _ 50 — 3 9. −1 TRY IT ANSWERS Odd Answers ChapteR 1 Section 1.1 81. The range for this viewing window is [−1,000,000, 1,000,000]. y 1. A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first 3. When a vertical line intersects the graph of a coordinate. relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be 5. When a only one output if the relation is to be a function. horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input. 7. Function 13. Function 15. Function 21. Function 27. f (−3) = −11, f (2) = −1, f (−a) = −2a − 5, −f (a) = −2a + 5, f (a + h) = 2a + 2h − 5 f (−a) = √ 9. Function 11. Function 17. Function 23. Function 5 + 5, f (2) = 5, 2 − a − 5, f (a + h) = 19. Function 25. Not a function 2 + a + 5, −f (a) = − √ 29. f (−3 31. f (−3) = 2, f (2) = −2, √ f (−a) = ∣ −a − 1 ∣ − ∣ −a + 1 ∣ , −f (aa + h ∣ 33. g(x) − g(a, x ≠ a 35. a. f (−2) = 14 b. x = 3 37. a. f (5) = 10 b. x = 4 or −1 2 __ 39. a. r = 6 − t 3 43. Function 49. Function 41. Not a function b. f (−3) = 8 c. t = 6 47. Function 45. Function 53. a. f (0) = 1 b. f (x) = −3, x = −2 or 2 51. Function 55. Not a function, not one-to-one 57. One-to-one function 59. Function, not one-to-one 65. Not a function 69. f (−2) = 14; f (−1) = 11; f (0) = 8; f (1) = 5; f (2) = 2 71. f (−2) = 4; f (−1) = 4.414; f (0) = 4.732; f (1) = 5; f (2) = 5.236 67. f (x) = 1, x = 2 61. Function 63. Function 1 1 __ __ 73. f (−2) = ; f (0) = 1; f (1) = 3; f (2) = 9 75. 20 ; f (−1) = 9 3 77. The range for this viewing window is [0, 100]. 79. The range for this viewing window is [−0.001, 0.001]. y 100 80 60 40 20 –5 –20 –40 –60 –80 –100 –10 x 5 10 –0.1 y 0.001 0.0008 0.0006 0.0004 0.0002 –0.05 –0.0002 –0.0004 –0.0006 –0.0008 –0.001 0.05 0.1 x 83. The range for this viewing window is [0, 10]. y 10 8 6 4 2 x –20 20 40 60 80 100 x 50 100 10.105 8.105 6.105 4.105 2.105 –50 –2.105 –4.105 –6.105 –8.105 –10.105 –100 85. The range for this viewing window is [−0.1, 0.1]. y 87. The range for this viewing window is [−100, 100]. 0.1 0.08 0.06 0.04 0.02 –0.0005 –0.02 –0.04 –0.06 –0.08 –0.1 –0.0001 0.0005 0.0001 x –10.105 y 100 80 60 40 20 –5.105 –20 –40 –60 –80 –100 5.105 10.105 x 89. a. g(5000) = 50 b. The number of cubic yards of dirt 91. a. The required for a garden of 100 square feet is 1. height of the rocket above ground after 1 second is 200 ft. b. The height of the rocket above ground after 2 seconds is 350 ft. Section 1.2 — 3 √ 1. The domain of a function depends upon what values of the independent variable make the function undefined or imaginary. — 3. There is no restriction on x for f (x) = x because you can take the cube root of any real number. So the domain is all real numbers, (−∞, ∞). When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x-values are restricted for f (x) = √ domain is [0, ∞). function over its corresponding domain. Use the same scale for the x-axis and y-axis for each graph. Indicate included endpoints with a solid circle and excluded endpoints with an open circle. Use an arrow to indicate −∞ or ∞. Combine the graphs to find the graph of the piecewise function. x to nonnegative numbers and the 5. Graph each formula of the piecewise 9. (−∞, 3]  ∪  − 1 15.  −∞, − 1 , ∞  _ _ 2 2 17. (−∞, −11)∪(−11, 2)∪(2, ∞) 19. (−∞, −3)∪(−3, 5)∪(5, ∞) 21. (−∞, 5) 25. (−∞, −9)∪(−9, 9)∪(9, ∞) 27. domain: (2, 8], range: [6, 8) 29. domain: [−4, 4], range: [0, 2] 31. domain: [−5, 3), range: [0, 2] 33. domain: (−∞, 1], range: [0, ∞) 11. (−∞, ∞) 13. (−∞, ∞) 7. (−∞, ∞) 23. [6, ∞) C-1 C-2 1 , 6  , range:  −6, − 1 1 35. domain:  −6 37. domain: [−3, ∞), range is [0, ∞) 39. domain:(−∞, ∞) 41. domain: (−∞, ∞) y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –5 –4 –3 –2 21 3 4 5 x –2 –1 y 5 4 3 2 1 –1 –2 –3 –4 –5 1 2
x 43. domain: (−∞, ∞) 45. domain: (−∞, ∞) y 5 4 3 2 1 –1 –2 –3 –4 –5 –2 –1 1 2 x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 x 47. f (−3) = 1; f (−2) = 0; f (−1) = 0; f (0) = 0 49. f (−1) = −4; f (0) = 6; f (2) = 20; f (4) = 34 51. f (−1) = −5; f (0) = 3; f (2) = 3; f (4) = 16 53. (−∞, 1)∪(1, ∞) 55. y y 104 96 88 80 72 64 56 48 40 32 24 16 8 –8 –0.5 –0.4 –0.3 –0.2 –0.1 104 96 88 80 72 64 56 48 40 32 24 16 8 x 0.1 –0.1 –8 0.1 0.2 0.3 0.4 0.5 x The viewing window: [−0.5, −0.1] has a range: [4, 100]. The viewing window: [0.1, 0.5] has a range: [4, 100]. 59. Many answers; one function is f (x) = 1 _______ 57. [0, 8] . — x − 2 √ 61. a. The fixed cost is $500. b. The cost of making 25 items is $750. c. The domain is [0, 100] and the range is [500, 1500]. Section 1.3 1. Yes, the average rate of change of all linear functions is constant. 3. The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region in an open interval. −1 _________ 11. 13(13 + h) 4 _ 7. 3 19. Increasing on (−∞, −2.5)∪(1, ∞) and decreasing 5. 4(b + 1) 13. 3h2 + 9h + 9 15. 4x + 2h − 3 9. 4x + 2h 7. 3 on (−2.5, 1) on (1, 3)∪(4, ∞) minimum: (3, 50) 21. Increasing on (−∞, 1)∪(3, 4) and decreasing 23. Local maximum: (−3, 50) and local 33. ≈ −0.167 b. −1,250 people per year 25. absolute maximum at approximately (7, 150) and absolute minimum at approximately (−7.5, −220) 27. a. −3,000 people per year 29. −4 31. 27 (3, −22), decreasing on (−∞, 3), increasing on (3, ∞) 37. local minimum: (−2, −2), decreasing on (−3, −2), increasing on (−2, ∞) minima: (−3.25, −47) and (2.1, −32), decreasing on (−∞, −3.25) and (−0.5, 2.1), increasing on (−3.25, −0.5) and (2.1, ∞) 43. b = 5 41. A 47. ≈ −0.6 milligrams per day 39. local maximum: (−0.5, 6), local 45. ≈ 2.7 gallons per minute 35. local minimum: Section 1.4 1. Find the numbers that make the function in the denominator g equal to zero, and check for any other domain restrictions on f and g, such as an even-indexed root or zeros in the denominator. 3. Yes, sample answer: Let f (x) = x + 1 and g(x) = x − 1. Then f (g(x)) = f (x − 1) = (x − 1)+ 1 = x and g ( f (x)) = g (x + 1) = (x + 1)− 1 = x so f ∘ g = g ∘ f. 5. (f + g)(x) = 2x + 6; domain: (−∞, ∞) (f − g)(x) = 2x 2 + 2x − 6; domain: (−∞, ∞) (fg)(x) = −x 4 − 2x 3 + 6x 2 + 12x; domain: (−∞, ∞) f  g  (x) = _ 7. (f + g)(x) = ; domain: (−∞, 0)∪(0, ∞) 6 )∪(− √ 6 )∪( √ 6 , ∞) 6 , √ — — — — x 2 + 2x ______ 6 − x 2 ; domain: (−∞, − √ 4x3 + 8x2 + 1 ___________ 2x 4x3 + 8x2 − 1 ___________ 2x ; domain: (−∞, 0)∪(0, ∞) (f − g)(x) = — — (fg)(x) = x + 2; domain: (−∞, 0)∪(0, ∞) f  g  (x) = 4x 3 + 8x 2; domain: (−∞, 0)∪(0, ∞) _ 9. (f + g)(x) = 3x 2 + √ x − 5 ; domain: [5, ∞) (f − g)(x) = 3x 2 − √ (fg)(x) = 3x 2 √ f  3x2 g  (x) = _ _ x − 5 √ b. f ( g(x)) = 18x 2 − 60x + 51 d. ( g ∘ g)(x) = 9x − 20 e. ( f ∘ f )(−2) = 163 x − 5 ; domain: [5, ∞) x − 5 ; domain: [5, ∞) ; domain: (5, ∞) — — 11. a. f (g(2)) = 3 c. g( f (x)) = 6x 2 − 2 13. f ( g(x)) = √ — x2 + 3 + 2 ; g( f (x)) = ( f (x)) = 15. f ( g(x)) = x x __ , x ≠ 0; g( f (x)) = 2x − 4, x ≠ 4 17. f ( g(x)) = 2 1 _ (x + 3)2 + 1 3 — 19. f ( g(h(x))) = 21. a. (g ∘ f )(x) = − 1 __  b.  −∞, 2 _ 2 − 4x √ 23. a. (0, 2)∪(2, ∞) except x = −2 b. (0, ∞) c. (0, ∞) 25. (1, ∞) 27. Many solutions; one possible answer: f (x) = x3; g(x(x) = (x + 2)2 29. Many solutions; one possible answer: f (x) = 3 x ; g(x) = √ 31. Many solutions; one possible answer: f (x) = — 33. Many solutions; one possible answer: f (x) = 4 √ 35. Many solutions; one possible answer: f (x) = √ 37. Many solutions; one possible answer: f (x) = 39. Many solutions; one possible answer: f (x) = x 3; g(x) = — 3 √ x ; g(x _____ 2x − 3 3x − 2 ______ x + 5 x ; g(x) = 2x + 6 x ; g(x) = — ODD ANSWERS — 41. Many solutions; one possible answer: f (x) = √ 51. 2 43. 2 55. 4 63. 2 67. 11 45. 5 57. 4 69. 0 1 _ 75. f (g(0)) = , g(f (0)) = 5 5 47. 4 59. 9 71. 7 49. 0 61. 4 73. f (g(0)) = 27, g(f (0)) = −94 x ; g(x) = 53. 1 65. 3 77. f (g(x)) = 18x2 + 60x + 51 2x − 1 _ 3x + 4 87. (f ∘ g )(6) = 6; (g ∘ f )(6) = 6 85. False 81. ( f ∘ g)(x) = 2, (g ∘ f )(x) = 2 79. g ∘ g(x) = 9x + 20 83. (−∞, ∞) 89. ( f ∘ g )(11) = 11; (g ∘ f )(11) = 11 93. A(t) = π  25 √ square inches 97. a. N(T(t)) = 575t 2 + 65t − 31.25 b. ≈ 3.38 hours t + 2  2 and A(2) = π  25 √ 95. A(5) = 121π square units 91. C — 4  2 = 2,500π — Section 1.5 1. A horizontal shift results when a constant is added to or subtracted from the input. A vertical shift results when a constant 3. A horizontal is added to or subtracted from the output. compression results when a constant greater than 1 multiplies the input. A vertical compression results when a constant between 0 5. For a function f, substitute and 1 multiplies the output. (−x) for (x) in f (x) and simplify. If the resulting function is the same as the original function, f (−x) = f (x), then the function is even. If the resulting function is the opposite of the original function, f (−x) = −f (x), then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even. 1 _ 9. g(x) = (x + 4)2 + 2 7. g(x) = ∣ x − 1 ∣ − 3 11. The graph of f (x + 43) is a horizontal shift to the left 43 units of the graph of f. 13. The graph of f (x − 4) is a horizontal shift to the right 4 units 15. The graph of f (x) + 8 is a vertical shift of the graph of f. up 8 units of the graph of f. 19. The graph of vertical shift down 7 units of the graph of f. f (x + 4) − 1 is a horizontal shift to the left 4 units and a vertical shift 21. Decreasing on (−∞, −3) and down 1 unit of the graph of f. increasing on (−3, ∞) 25. 23. Decreasing on (0, ∞) y 17. The graph of f (x) − 7 is a y 27. 10 1 0 –1 –2 –3 –4 h 21 3 4 x –5 –4 –3 –2 10 1 –1 –2 –3 –4 f x 21 3 –9 –8 –7 –6 –5 –4 –3 –2 C-3 k 21 3 4 5 6 29. –6 –5 –4 –3 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12 –13 –14 — x x — x + 3 − 1 31. g(x) = f (x − 1), h(x) = f (x) + 1 33. f (x) = ∣ x − 3 ∣ − 2 35. f (x) = √ 37. f (x) = (x − 2)2 39. f (x) = ∣ x + 3 ∣ − 2 41. f (x) = − √ 43. f (x) = −(x + 1)2 + 2 45. f (x) = √ 47. Even 51. Even of g is a vertical reflection (across the x-axis) of the graph 55. The graph of g is of f. a vertical stretch by a factor of 4 of the graph of f. −x + 1 49. Odd 53. The graph — 1 _ 57. The graph of g is a horizontal compression by a factor of 5 59. The graph of g is a horizontal stretch of the graph of f. by a factor of 3 of the graph of f. horizontal reflection across the y-axis and a vertical stretch by a 61. The graph of g is a factor of 3 of the graph of f. 65. g(x) = 1 _ 3(x + 2)2 − 3 63. g(x) = ∣ −4x ∣ 67. g(x) = 1 _ (x − 5)2 + 1 2 69. This is a parabola shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units. y 71. This is an absolute value function stretched vertically by a factor of 2, shifted 4 units to the right, reflected across the horizontal axis, and then shifted 3 units up. y 10 1 –1 –2 –3 –4 –5 –5 –4 –3 –2 g –6 –5 –4 –3 –2 21 3 4 5 x h 21 1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 73. This is a cubic function compressed vertically by a 1 _ factor of . 2 y 75. The graph of the function is stretched horizontally by a factor of 3 and then shifted downward by 3 units. y –5 –4 –3 –1 –1 –2 –3 –4 –5 –6 –7 –8 m 21 3 4 5 x –6 –5 –4 –3 –2 p 21 1 –1 –2 –3 –4 –5 –6 –7 –8 ODD ANSWERS C-4 77. The graph of — f (x) = √ x is shifted right 4 units and then reflected across the y-axis. y –8 –7 –6 –5 –4 –3 –2 81. a –8 –7 –6 –5 –4 –3 –2 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 x Section 1.6 79. 41. 43. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 g 21 3 4 5 6 7 8 x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 x –6 –5 –4 –3 –2 g 21 1 –1 –2 –3 –4 –5 –6 –7 –8 45. 471 –1 –2 –3 –4 –5 –8 –7 –6 –5 –4 –3 –2 21 3 4 x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –1 –1 –2 –3 –4 –5 21 3 4 5 6 x 21 3 4 5 x 1. Isolate the absolute value term so that the equation is of the form ∣ A ∣ = B. Form one equation by setting the expression inside the absolute value symbol, A, equal to the expression on the other side of the equation, B. Form a second equation by setting A equal to the opposite of the expression on the other side of the equation, −B. Solve each equation for the variable. 3. The graph of the absolute value function does not cross the x-axis, so the graph is either completely above or completely below the 5. First determine the boundary points by finding x-axis. the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a test value in each interval to 1 7. ∣ x + 4 ∣ = _ 2 13.  − 9 13  _ _ 4, 4 5 7 19.   _ _ , 2 2 determine which values satisfy the inequality. 9. ∣ f (x) − 8 ∣ < 0.03 29 10 20 15 21. No solution 23. {−57, 27} 25. (0, −8); (−6, 0) and (4, 0) 29. ( −∞, − 8)∪(12, ∞) 27. (0, −7); no x-intercepts.  ∪[6, ∞) 35.  −∞, − 8 , 4  33.  −∞, − 8 31.  − 4  ∪[16, ∞) _ _ _ 3 3 3 11 17.  _ 5 11. {1, 11} , , 37. y 39. y 49. 51. 321 4 5 x –7 –6 –5 –4 –3 –2 y –1 –1 –2 –3 –4 –5 –6 –7 21 1 –1 –2 –3 –4 –5 –6 –5 –4 –3 –2 53. range: [0, 20] y 55. y f –100 –75 –50 2 1.5 1 0.5 –25 –0.5 –1 –1.5 –2 25 50 75 100 x 20 18 16 14 12 10 8 6 4 2 21 3 4 5 x –4 –3 –2 –1 –1 –2 59. There is no value for a that will keep the 57. (−∞, ∞) function from having a y-intercept. The absolute value function always crosses the y-intercept when x = 0. 61. ∣ p − 0.08 ∣ ≤ 0.015 63. ∣ x − 5.0 ∣ ≤ 0.01 (–1, 2) (3, 2) (0, 1) (2, 1) (1, 0) x (–2, 3) (2, 3) SeCtion 1.7 (–1, 2) (1, 2) (0, 1) x 1. Each output of a function must have exactly one input for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that y-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no y-values repeat and the function is one-to-one. ODD ANSWERS 1 _ x is its own inverse. 3. Yes. For example, f (x) = 7. f −1(x) = x − 3 9. f −1(x) = 2 − x 13. Doma
in of f (x): [−7,∞); f −1 (x) = √ 5. y = f −1(x) 11. f −1(x) = − 2x _ x − 1 x − 7 — 15. Domain of f (x): [0, ∞); f −1 (x) = √ 17. f ( g(x)) = x and g( f (x)) = x 21. One-to-one 23. Not one-to-one — x + 5 19. One-to-one 25. 3 27. 2 33. 6 37. 0 39. 1 31. [2, 10] 35. −4 41. x f −1(x) 1 3 4 6 7 9 12 13 16 14 5 _ 45. f −1 (x) = (x − 32) 9 d _ ; t(180) = 50 47. t(d) = 180 _ 50 . The time for the car to travel 180 miles is 3.6 hours. 29. y 10 8 6 4 2 –2 –2 –4 f –1 f 42 6 8 10 x –10 –8 –6 –4 1 _ 43. f −1 (x) = (1 –1 –2 –3 –4 –5 –5 –4 –3 –2 f f –1 21 3 4 5 x Chapter 1 Review exercises 1. Function 3. Not a function 5. f (−3) = −27; f (2) = −2; f (−a) = −2a2 − 3a; −f (a) = 2a2 −3a; f (a + h) = −2a2 − 4ah − 2h2 + 3a + 3h 7. One-to-one 11. Function 9. Function 13. –5 –4 –3 –2 23. –5 –4 –3 –2 y 3 2 1 –1 –1 –2 –3 y 3 2 1 –1 –1 –2 –3 21 3 4 5 21 3 4 5 x x 15. 2 19. 17. −1.8 or 1.8 −64 + 80a − 16a2 __ −1 + a = −16a + 64; a ≠ 1 25. 31 27. Increasing on (2, ∞), decreasing on (−∞, 2) 29. Increasing on (−3, 1), constant on (−∞, −3) and (1, ∞) 31. Local minimum: (−2, −3); local maximum: (1, 3) 33. Absolute maximum: 10 35. ( f ∘ g )(x) = 17 − 18x, ( g ∘ f )(x) = −7 −18x 37. ( f ∘ g )(x) = √ ______ )(x) = 1 _ — x + 2 √ 39. (f ∘ g )(x) = = 1 + x _____ 1 + 4x  ∪  − 1 ; Domain:  −∞, − 1 , 0  ∪(0, ∞) _ _ 4 4 1 ____ 1 __ x + 1 _ 1 ____ 1 __ x + 4 C-5 x x 2x −1 _ 3x + 4 and 41. ( f ∘ g )(x) = 1 _ ; Domain: (0, ∞) — x √ 43. Many solutions; one possible answer: g(x) = f (x) = √ — x . 45 476 –5 –4 –3 –2 –1 –1 –2 21 3 4 5 6 x –6 –5 –4 –3 –2 21 3 4 5 6 –1 –1 –2 21 3 4 5 x 49. –5 –4 –3 –2 53. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 y 5 4 3 2 1 –6 –5 –4 –3 –2 –1 –1 21 3 4 5 6 x 67. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –4 –3 –2 21 3 4 5 76 x 321 4 5 6 7 8 51. y –4 –3 –2 2 –1 –2 –4 –6 –8 –10 –12 –14 –16 –18 –20 –22 –24 55. f (x) = ∣ x − 3 ∣ 57. Even 59. Odd 61. Even 1 ∣ x + 2 ∣ + 1 _ 63. f (x) = 2 65. f (x) = −3 ∣ x − 3 ∣ + 3 69. {−22, 14} 71.  − 5 , 3  _ 3 73. f −1(x) = 75. f −1(x) = √ x − 9 _ 10 — x − 1 –5 –4 –3 –2 5 4 3 2 1 –1 –1 –2 –3 –4 –5 1 2 3 4 5 x Chapter 1 practice test 1. Relation is a function parabola and the graph fails the horizontal line test. — 9. −2(a + b) + 1; b ≠ a 7. 2a2 − a 11. √ 2 3. −16 5. The graph is a 21. (−∞, −2)∪(−2, 6)∪(6, ∞) 77. The function is one-to-one. y 79. 5 ODD ANSWERS C-6 13. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 –5 –4 –3 –2 21 3 4 5 6 x ChapteR 2 Section 2.1 15. Even 19. {−7, 10} 17. Odd 21. f −1(x) = x + 5 _ 3 23. (−∞, −1.1) and (1.1, ∞) 25. (1.1, −0.9) 29. f (x) =  \|x\| if x ≤ 2 if x > 2 3 31. x = 2 33. Yes x − 11 _ 2 35. f −1(x) = − or 27. f (2) = 2 11 − x _ 2 9. No 11. No 1. Terry starts at an elevation of 3,000 feet and descends 70 feet per second 3. 3 miles per hour 7. Yes 17. Decreasing 5. d(t) = 100 − 10t 13. No 15. Increasing 21. Increasing 29. 4 1 _ _ 27. − 5 3 1 __ x + 35. y = − 33. y = 2x + 3 3 19. Decreasing 23. Decreasing 22 __ 3 25. 3 1 7 __ __ 31 39. − 5 4 __ __ 37. y = 5 4 45. y = 3 49. Linear, f(x) = 5x − 5 53. Linear, f (x) = 10x − 24 2 __ x + 1 41. y = 3 43. y = −2x + 3 47. Linear, g(x) = −3x + 5 51. Linear, g(x) = − 25 __ 2 55. f (x) = −58x + 17.3 x + 6 642 8 10 x 57. y 30,000 25,000 20,000 15,000 10,000 5,000 −10 −4−6−8 −2 −5,000 −10,000 −15,000 −20,000 −25,000 −30,000 61. y 30 20 10 –0.1 –0.05 0.05 0.1 x –10 –20 –30 59. a. a = 11,900, b = 1001.1 b. q(p) = 1000p − 100 x − 65. x = a 63. x = − 16 __ 3 d ad ____ ____ 67. y = c − a c − a 69. $45 per training session 71. The rate of change is 0.1. For every additional minute talked, the monthly charge increases by $0.1 or 10 cents. The initial value is 24. When there are no minutes talked, initially the charge is $24. 73. The slope is −400. this means for every year between 1960 and 1989, the population dropped by 400 per year in the city. 75. C Section 2.2 1. The slopes are equal; y-intercepts are not equal. 3. The point of intersection is (a, a). This is because for the horizontal line, all of the y-coordinates are a and for the vertical line, all of the x coordinates are a. The point of intersection is on both lines and therefore will have these two characteristics. 7. Neither 11. Parallel 5. First, find the slope of the linear function. Then take the negative reciprocal of the slope; this is the slope of the perpendicular line. Substitute the slope of the perpendicular line and the coordinate of the given point into the equation y = mx + b and solve for b. Then write the equation of the line in the form y = mx + b by substituting in m and b. 9. Perpendicular 1 15.  __ , 0  , (0, 1) 5 19. Line 1: m = 8, Line 2: m = −6, neither 21. Line 1: m = − 1 _ , Line 2: m = 2, perpendicular 2 23. Line 1: m = −2, Line 2: m = −2, parallel 25. g(x) = 3x − 3 5 31.  − 17  _ _ , 5 3 27. p(t) = − 1 __ t + 2 29. (−2, 1) 3 35. C 13. (−2, 0), (0, 4) 17. (8, 0), (0, 28) 33. F 37. A 39. –6 –5 –4 –3 –2 43. –6 –5 –4 –3 –2 47. –6 –5 –4 –3 –2 51. –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 321 4 5 6 x 321 4 5 6 x 41. 321 4 5 6 x –6 –5 –4 –3 –2 45. 321 4 5 6 x –6 –5 –4 –3 –2 49. y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 321 4 5 6 x –6 –5 –4 –3 –2 321 4 5 6 x –1 –1 –2 –3 –4 –5 –6 53. 321 4 5 6 x –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 321 4 5 6 x ODD ANSWERS C-7 57. 321 4 5 6 x –6 –5 –4 –3 –2 55. –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 321 4 5 6 x 9. No 150 125 100 75 50 0 0 1.6 3.2 4.8 6.4 8 59. a. g(x) = 0.75x − 5.5 b. 0.75 c. (0, −5.5) 11. No 61. y = 3 63. x = −3 65. no point of intersection 67. (2, 7) 69. (−10, −5) 71. y = 100x − 98 73. x < 1999 _ 201 , x > 1999 _ 201 75. Greater than 3,000 texts 250 200 150 100 50 0 Section 2.3 19. W(t) = 0.5t + 7.5 7. 20.012 square units 25. C(t) = 12,025 − 205t 5. 6 square 11. 64,170 23. At age 5.8 27. (58.7, 0) In 58.7 1. Determine the independent variable. This is the variable upon 3. To determine the initial value, which the output depends. find the output when the input is equal to zero. 9. 2,300 units. 13. P(t) = 2500t + 75,000 15. (−30, 0) 30 years before the start of this model, the town has no citizens. (0, 75,000) Initially, 17. Ten years after the the town had a population of 75,000. 21. (−15, 0) The model began. x-intercept is not a plausible set of data for this model because it means the baby weighed 0 pounds 15 months prior to birth. (0, 7.5) the baby weighed 7.5 pounds at birth. months years, the number of people afflicted with the common cold would be zero (0, 12,025) Initially, 12,025 people were afflicted 29. 2063 with the common cold 33. In 2070, the company’s profits will be zero 35. y = 30t − 300 37. (10, 0) In the year 1990, the company’s profits were zero 39. Hawaii 43. $105,620 45. a. 696 people b. 4 years c. 174 people per year d. 305 people e. P(t) = 305 + 174t f. 2,219 people 47. a. C(x) = 0.15x + 10 b. The flat monthly fee is $10 and there is a $0.15 fee for each additional minute used c. $113.05 49. a. P(t) = 190t + 4,360 b. 6,640 moose cubic feet c. During the year 2017 133 minutes 57. More than $66,666.67 in sales 55. More than $42,857.14 worth of jewelry 51. a. R(t)= −2.1t + 16 b. 5.5 billion 41. During the year 1933 31. y = −2t + 180 53. More than Section 2.4 1. When our model no longer applies, after some value in the 3. We predict a value domain, the model itself doesn’t hold. 5. The closer the outside the domain and range of the data. number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data. 7. 61.966 years 0 2.5 5 7.5 10 12.5 13. Interpolation, about 60° F 15. C 17. B 68 64 60 56 52 48 44 0 10 19. 8 6 4 2 0 21. 10 8 6 4 2 0 0 15 20 25 30 35 0 2 4 6 8 10 0 2 4 6 8 10 29. y = −1.981x + 60.197; r = −0.998 23. Yes, trend appears linear; during 2016 25. y = 1.640x + 13.800, r = 0.987 27. y = −0.962x + 26.86, r = −0.965 31. y = 0.121x − 38.841, r = 0.998 (5, −20), (6, −22), (9, −28) sells 18,980 units, its profits will be zero dollars 37. y = 0.00587x + 1985.41 41. y = −10.75x + 742.50 35. (189.8, 0) If the company 39. y = 20.25x − 671.5 33. (−2, −6), (1, −12), ODD ANSWERS C-8 Chapter 2 Review exercises ChapteR 3 0 2 4 6 8 10 x 49. 3i 51. 0 53. 5 − 5i 3. Increasing 5. y = −3x + 26 1. Yes 9. y = 2x − 2 11. Not linear 13. Parallel 15. (−9, 0); (0, −7) 17. Line 1: m −2, Line 2: m = −2, parallel 19. y = −0.2x + 21 21. 7. 3 y 23. More than 250 25. 118,000 27. y = −300x + 11,500 29. a. 800 b. 100 students per year c. P(t) = 100t + 1700 31. 18,500 33. y = $91, 625 –6 –5 –4 –3 –2 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 321 4 5 6 x 35. Extrapolation y 37. y 120 100 80 60 40 20 ,900 6,800 6,700 6,600 6,500 6,400 6,300 6,200 6,100 6,000 5,900 5,800 5,700 5,600 0 1985 1990 1995 2000 x 2005 2010 Year 39. Midway through 2023 41. y = −1.294x + 49.412; r = −0.974 43. Early in 2027 45. 7, 660 Chapter 2 practice test 1. Yes 3. Increasing 5. y = −1.5x − 6 7. y = −2x − 1 9. No 15. y = −0.25x + 12 13. (−7, 0); (0, −2) 11. perpendicular 21. 165,000 19. 150 23. y = 875x + 10,625 25. a. 375 b. dropped an average of 46.875, or about 47 people per year c. y = −46.875t + 1250 17. Slope = −1 and y-intercept = 3 –2 –1–1 –2 –3 –4 321 4 5 6 7 8 9 x 29. Early in 2018 31. y = 0.00455x + 1979.5 33. r = 0.999 27. y 35 30 25 20 15 10 5 0 0 2 4 6 8 10 12 x Section 3.1 1. Add the real parts together and the imaginary parts together. 3. i times i equals −1, which is not imaginary. (answers vary) 23 __ 5. −8 + 2i 7. 14 + 7i 9. − 29 i 11. 2 real and 0 nonreal 15 __ 29 + 13. −2−3−4−5 i 5 4 3 2 1 −1−1 −2 −3 −4 −5 15. 321 4 5 r −2−3−4−5 i 5 4 3 2 1 −1−1 −2 −3 −4 −5 321 4 5 r 17. 8 − i 19. −11 + 4i 21. 2 − 5i 25. −16 + 32i 27. −4 − 7i 29. 25 23. 6 + 15i 2 __ 31. 2 − i 3 — 2 __ + 33. 4 − 6i 35. 5 11 __ i 5 41. 1 43. −1 45. 128i — 3 39. 1 + i √ 37. 15i 6 471 + 2 2 9 9 __ __ 55. −2i − 57. i 2 2 Section 3.2 37  _ 12 − 37 _ 12 17. Minimum is − 1. When written in that form, the vertex can be easily identified. 3. I
f a = 0 then the function becomes a linear function. 5. If possible, we can use factoring. Otherwise, we can use the 7. g(x) = (x + 1)2 − 4; vertex: (−1, −4) 33  _ 4 quadratic formula. 2 5 − 33 5 ; vertex:  −  9. f (x)=  11. k(x) = 3(x − 1)2 − 12; vertex: (1, −12) 2 13. f (x) = 3  x − 5 5 ; vertex:   _ _ , − 6 6 17 5 5 _ _ _ 15. Minimum is − ; axis of symmetry: x = and occurs at 2 2 2 17 ; axis of symmetry: x = − 1 1 _ _ _ and occurs at − 8 8 16 7 _ 19. Minimum is − and occurs at −3; axis of symmetry: x = −3 2 21. Domain: (−∞, ∞); range: [2, ∞) range: [−5, ∞) 27.  2i √ 31. {2 + i, 2 − i} 2  33. {2 + 3i, 2 − 3i} 35. {5 + i, 5 − i} 3 3 39.  − 1 i 37, − 3 41  43 47. f (x) = x 2 + 1 45. f (x) = x 2 − 4x + 4 297 6 ___ __ 49 49 25. Domain: (−∞, ∞); range: [−12, ∞) 51. f (x) = −x 2 + 1 29.  3i √ 49. f (x) = 2 , −2i √ 3 , −3i √ 60 __ 49  23. Domain: (−∞, ∞); ODD ANSWERS C-9 51. y-intercept: (0, 0); x-intercepts: (0, 0) and (2, 0); as x → −∞, f (x ) → ∞, as x → ∞, f (x) → ∞ 53. y-intercept: (0, 0); x-intercepts: (0, 0), (5, 0), (7, 0); as x → −∞, f (x ) → −∞, as x → ∞, f (x) → ∞ 53. Vertex: (1, −1), axis of symmetry: x = 1, intercepts: (0, 0), (2, 0) 49 5 55. Vertex:   , axis of _ _ , − 4 2 5 _ symmetry: x = , intercepts: 2 (6, 0), (−1, 0) y = f (x1−1 −2 −3 −4 −5 −6 −2−3−4−5−6 321 4 5 6 x −10 −5 y 15 12 9 6 3 0 −3 −6 −9 −12 −15 y = f (x) 5 10 x −5−6 −4 y 5 4 3 2 1 −3 −2 −1−1 −2 −3 −4 −5 y 200 160 120 80 40 −1−2 −40 −80 −120 −160 −200 21 3 4 5 6 x 21 3 4 5 6 7 8 9 10 x 93. $10.70 59. y-intercept: (0, 0); x-intercepts: (−3, 0), (0, 0), (5, 0); as x → −∞, f (x ) → −∞, as x → ∞, f (x) → ∞ 39 5 57. Vertex:   , axis of _ _ , − 4 8 5 _ symmetry: x = , intercept: 4 (0, −8) −10 −5 y 12 8 4 −4 −8 −12 −16 −20 −24 5 10 x y = f (x) 59. f (x) = x2 − 4x + 1 61. f (x) = −2x 2 + 8x − 1 1 7 __ __ 63. f (x) = x 2 − 3x + 2 2 65. f (x) = x 2 + 1 67. f (x) = 2 − x 2 69. f (x) = 2x 2 71. The graph is shifted up or down (a vertical shift). 73. 50 feet 75. Domain: (−∞, ∞); range: [−2, ∞) 77. Domain: (−∞, ∞); range: (−∞, 11] 81. f (x) = 3x 2 − 9 83. f (x) = 5x 2 − 77 85. 50 feet by 50 feet 89. 6 and −6; product is −36 91. 2909.56 meters 87. 125 feet by 62.5 feet 79. f (x) = 2x 2 − 1 Section 3.3 13. Degree: 2, coefficient: −2 15. Degree: 4, 1. The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function. 3. As x decreases without bound, so does f (x). As x increases without bound, so does f (x). 5. The polynomial function is of even degree and leading coefficient is negative. 7. Power function 9. Neither 11. Neither coefficient: −2 17. As x → ∞, f (x) → ∞, as x → −∞, f (x) → ∞ 19. As x → −∞, f (x) → −∞, as x → ∞, f (x) → −∞ 21. As x → −∞, f (x) → −∞, as x → ∞, f (x) → −∞ 23. As x → ∞, f (x) → ∞, as x → −∞, f (x) → −∞ 25. y-intercept is (0, 12), t-intercepts are (1, 0), (−2, 0), and (3, 0) 27. y-intercept is (0, −16), x-intercepts are (2, 0), and (−2, 0) 29. y-intercept is (0, 0), x-intercepts are (0, 0), (4, 0), and (−2, 0) 31. 3 33. 5 least possible degree: 3 degree: 2 45. Yes, 0 turning points, least possible degree: 1 47. As x → −∞, f (x ) → ∞, as x → ∞, f (x) → ∞ 37. 5 39. Yes, 2 turning points, 41. Yes, 1 turning point, least possible 49. As x → −∞, f (x ) → ∞, as x → ∞, f (x) → −∞ 43. Yes, 0 turning points, least possible degree: 1 35. 3 x 10 100 −10 −100 f (x) 9,500 99,950,000 9,500 99,950,000 x 10 100 −10 −100 f (x) −504 −941,094 1,716 1,061,106 55. y-intercept: (0, 0); x-intercepts: (−4, 0), (0, 0), (4, 0); as x → −∞, f (x ) → −∞, as x → ∞, f (x) → ∞ 57. y-intercept: (0, −81); x-intercepts: (−3, 0), and (3, 0); as x → −∞, f (x ) → ∞, as x → ∞, f (x) → ∞ y 500 400 300 200 100 −10 −2−4−6−8 −100 −200 −300 −400 −500 42 6 8 10 x y 100 80 60 40 20 21 3 4 5 6 x −1−2−3−4−5−6 −20 −40 −60 −80 −100 y 50 40 30 20 10 −1−2−3−4−5 −10 −20 −30 −40 −50 21 3 4 5 6 x 61. f (x ) = x 2 − 4 63. f (x ) = x 3 − 4x 2 + 4x 65. f (x ) = x 4 + 1 67. V(m ) = 8m 3 + 36m 2 + 54m + 27 69. V(x ) = 4x 3 − 32x 2 + 64x Section 3.4 9. (3, 0), (−1, 0), (0, 0) 7. (−2, 0), (3, 0), (−5, 0) 3. If we evaluate the function at a and at b and 1. The x-intercept is where the graph of the function crosses the x-axis, and the zero of the function is the input value for which f (x) = 0. the sign of the function value changes, then we know a zero exists 5. There will be a factor raised to an even between a and b. power. 11. (0, 0), (−5, 0), (2, 0) 15. (2, 0), (−2, 0), (−1, 0) 19. (1, 0), (−1, 0) 23. (0, 0), (1, 0), (−1, 0), (2, 0), (−2, 0) 25. f (2) = −10, f (4) = 28; sign change confirms 27. f (1) = 3, f (3) = −77; sign change confirms 29. f (0.01) = 1.000001, f (0.1) = −7.999; sign change confirms 31. 0 with multiplicity 2, − 3 _ multiplicity 5, 4 multiplicity 2 2 33. 0 with multiplicity 2, −2 with multiplicity 2 35. − 2 _ with multiplicity 5, 5 with multiplicity 2 3 13. (0, 0), (−5, 0), (4, 0) 17. (−2, 0), (2, 0),  1 _ 3 , 0  3 , 0  ,  − √ 21. (0, 0),  √ 2 , 0  — — ODD ANSWERS C-10 37. 0 with multiplicity 4, 2 with multiplicity 1, −1 with multiplicity 1 3 _ 39. with multiplicity 2, 0 with multiplicity 3 2 41. 0 with 2 _ with multiplicity 2 multiplicity 6, 3 43. x-intercept: (1, 0) with multiplicity 2, (−4, 0) with multiplicity 1; y-intercept: (0, 4); as x → −∞, g (x) → −∞, as x → ∞, g (x) → ∞ 45. x-intercept: (3, 0) with multiplicity 3, (2, 0) with multiplicity 2; y-intercept: (0, −108); as x → −∞, k (x) → −∞, as x → ∞, k (x) → ∞ k(x) g(x) 20 16 12 8 4 −1−2−3−4−5 −4 − 24 12 −1−2−3−4−5−6 −12 −24 −36 −48 −60 −72 −84 −96 −108 −120 47. x-intercepts: (0, 0), (−2, 0), (4, 0) with multiplicity 1; y-intercept: (0, 0); as x → −∞, n (x) → ∞, as x → ∞, n (x) → −∞ n(x) 75 60 45 30 15 49. f (x) = − 2 _ (x − 3)(x + 1)(x + 3) 9 1 _ 51. f (x) = (x + 2)2(x − 3) 4 53. −4, −2, 1, 3 with multiplicity 1 55. −2, 3 each with multiplicity 2 57. f (x) = − 2 _ (x + 2)(x − 1)(x − 3) 3 1 _ 59. f (x) = (x − 3)2(x − 1)2(x + 3) 3 61. f (x) = −15(x − 1)2(x − 3)3 1 2 3 4 5 x 63. f (x) = −2(x + 3)(x + 2)(x − 1) −1−2−3−4−5 −15 −30 −45 −60 −75 41. Yes, 35. x 3 − 9x 2 + 27x − 27 39. Yes, (x − 2)(3x 3 − 5) 43. No 33. x 3 − 6x 2 + 12x − 8 37. 2x 3 − 2x + 2 (x − 2)(4x 3 + 8x 2 + x + 2) 45. (x − 1)(x 2 + 2x + 4) 49. Quotient: 4x 2 + 8x + 16, remainder: −1 51. Quotient 53. Quotient is x 3 − 2x 2 + is 3x 2 + 3x + 5, remainder: 0 4x − 8, remainder: −6 55 57 47. (x − 5)(x 2 + x + 1) 61. 1 + 59 63. x 2 + ix − 1 + 1 − i _ x − i 65. 2x 2 + 3 67. 2x + 3 69. x + 2 71. x − 3 73. 3x 2 − 2 Section 3.6 1. The theorem can be used to evaluate a polynomial. 3. Rational zeros can be expressed as fractions whereas real zeros include irrational numbers. can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again. 5. Polynomial functions 7. −106 9. 0 11. 255 13. −1 21. − 5 _ , √ 2 1 _ 15. −2, 1, 2 — 6 , − √ — 6 17. −2 3 _ 23. 2, −4, − 2 — 5 19. −3 25. 4, −4, −5 — 5 3 _ 31. 2 33. 2, 3, −1, −2 1 _ 29. , 2 1 _ 27. 5, −3, −3 , − 35. 2 2 5 39. − 3 1 _ _ 37. −1, −1, √ 41. 2, 3 + 2i, 3 − 2i , − 4 2 43. − 2 , 1 + 2i, 1 − 2i 45. − 1 _ _ , 1 + 4i, 1 − 4i 2 3 49. 1 positive, 0 negative 47. 1 positive, 1 negative 5 , − √ — — f (x) 5 4 3 2 1 f (x) 5 4 3 2 1 −3 −2 −1−1 −2 −3 −4 −5 21 3 4 5 6 x 21 3 4 5 6 x −1−2−3−4−5−6 −1 −2 −3 −4 −5 65. f (x) = − 3 _ (2x − 1)2(x − 6)(x + 2) 2 67. Local max: (−0.58, −0.62); local min: (0.58, −1.38) 69. Global min: (−0.63, −0.47) 71. Global min: (0.75, −1.11) 73. f (x) = (x − 500)2(x + 200) 75. f (x) = 4x 3 − 36x 2 + 80x −5−6 −4 77. f (x) = 4x 3 − 36x 2 + 60x + 100 1 _ π (9x 3 + 45x 2 + 72x + 36) 79. f (x) = Section 3.5 1. The binomial is a factor of the polynomial. 3. x + 6 + , quotient: x + 6, remainder: 5 5 _ x − 1 5. 3x + 2, quotient: 3x + 2, remainder: 0 9. 2x − 7 + x − 5, remainder: 0 6 _ 3x + 1 remainder 16 11. x − 2 + 7. x − 5, quotient: 16 _ , quotient: 2x − 7, x + 2 , quotient: x − 2, remainder: 6 13. 2x 2 − 3x + 5, quotient: 2x 2 − 3x + 5, remainder: 0 15. 2x 2 + 2x + 1 + 10 _ x − 4 17. 2x 2 − 7x + 1 − 2 _ 2x + 1 51. 0 positive, 3 negative 53. 2 positive, 2 negative f (x) −14 −12 −10 −8 −6 −4 100 80 60 40 20 −2−20 −40 −60 −80 −100 42 6 8 10 x f (x) 40 32 24 16 8 21 3 4 5 6 x −1−2−3−4−5−6 −8 −16 −24 −32 −40 19. 3x 2 − 11x + 34 − 23. 4x 2 − 21x + 84 − 106 _ x + 3 323 _ x + 4 21. x 2 + 5x + 1 25. x 2 − 14x + 49 27. 3x 2 + x + 29. x 3 − 3x + 1 31 3x − 1 ODD ANSWERS 55. 2 positive, 2 negative f (x) 15 12 9 6 3 21 3 4 5 6 x −1−2−3−4−5−6 −3 −6 −9 −12 −15 , ±1, ±5, ± 5 57. ± 1 _ _ 2 2 59. ±1 61. 1, 3 2 , − 3 1 _ _ 63. 2, 4 2 4 _ 67. f (x) = (x 3 + x 2 − x − 1) 9 69. f (x) = − 1 __ (4x 3 − x) 5 71. 8 by 4 by 6 inches 5 _ 65. 4 73. 5.5 by 4.5 by 3.5 inches 77. Radius: 6 meters; height: 2 meters meters, height: 4.5 meters 75. 8 by 5 by 3 inches 79. Radius: 2.5 Section 3.7 3. The numerator and denominator 5. Yes. The numerator of the 1. The rational function will be represented by a quotient of polynomial functions. must have a common factor. formula of the functions would have only complex roots and/or factors common to both the numerator and denominator. 7. All reals except x = −1, 1 9. All reals except x = −1, 1, −2, 2 11. Vertical asymptote: x = − 2 _ ; horizontal asymptote: y = 0; 5 domain: all reals except x = − 2 _ 5 x = 4, −9; horizontal asymptote: y = 0; domain: all reals except x = 4, −9 15. Vertical asymptotes: x = 0, 4, −4; horizontal asymptote: y = 0; domain: all reals except x = 0, 4, −4 17. Vertical asymptotes: x = −5; horizontal asymptote: y = 0; domain: all reals except x = 5, −5 13. Vertical asymptotes: 1 2 _ _ 19. Vertical asymptote: x = ; horizontal asymptote: y = − ; 3 3 1 _ domain: all reals except x = 21. None 3 1 23. x-intercepts: none, y-intercept:  0,  _ 4 − + 25. Local behavior: x → − 1 , f (x) → −∞, x → − 1 _ _ , f (x) → ∞ 2 2 1 _ End behavior: x → ±∞, f (x) → 2 27. Local behavior: x → 6+, f (x) → −∞, x → 6−, f (x) → ∞ End behavior: x → ±∞, f (x) → −2 − + 1 1 _ _ 29. Local behavior: x → − , f (x) → −∞, , f (x) → ∞, x) → ∞, x → − 5
_ _ , f (x) → −∞ 2 2 1 _ End behavior: x → ±∞, f (x) → 3 33. y = 2x 31. y = 2x + 4 35. Vertical asymptote at x = 0, horizontal asymptote at y = 2 y 10 8 6 4 2 y = 2 −10 −4−6−8 −2−2 −4 −6 −8 −10 42 6 8 10 x x = 0 37. Vertical asymptote at x = 2, horizontal asymptote at y = 0 y 10 8 6 4 2 −10 −4−6−8 y = 0 −2−2 −4 −6 −8 −10 x 8 10 42 6 x = 2 39. Vertical asymptote at x = −4; horizontal asymptote 3 3  , 0  ,  0, − at y = 2;  _ _ 4 2 p(x) C-11 41. Vertical asymptote at x = 2; horizontal asymptote at y = 0; (0, 1) s(x) 10 8 6 4 2 −2−2 −4 −6 −8 −10 −10 −4−6−8 x = −4 y = 2 x 42 6 8 10 12 10 10 −4−6−8 −2−1 −2 x = 2 y = 0 42 6 8 10 x 43. Vertical asymptote 4 _ at x = −4, ; horizontal 3 asymptote at y = 1; (5, 0), 1 , 0  ,  0,  − _ 3 5  _ 16 f(x) 12 10 8 6 4 2 −2−2 −4 −6 −8 −10 −12 −12 −4−6−8−10 x = −4 x = 4 3 y = 1 42 6 8 10 12 x 47. Vertical asymptote at x = 4; slant asymptote at 1 y = 2x + 9; (−1, 0),  , 0  , _ 2 1   0, _ 4 h(x) 50 40 30 20 10 −50 −40 −30 −20 −10 −10 −20 −30 −40 −50 y = 2x + 9 x = 4 10 20 30 40 50 45. Vertical asymptote at x = −1; horizontal asymptote at y = 1; (−3, 0), (0, 3) a(x) x =−1 15 12 9 6 3 −15 −6−9−12 −3−3 −6 −9 −12 −15 y = 1 15 x 63 9 12 49. Vertical asymptote at x = −2, 4; horizontal asymptote at y = 1; 15 (1, 0), (5, 0), (−3, 0),  0, −  _ 16 w(x) x = 4 x = −2 12 10 8 6 4 2 −12 −4−6−8−10 x −2−2 −4 −6 −8 −10 y = 1 42 6 8 10 12 x 51. f (x) = 50 53. f (x) = 7 x2 − x − 2 ________ x2 − 25 x2 + 2x − 24 __ x2 + 9x + 20 1 _ 55. f (x) = ⋅ 2 x2 − 4x + 4 __ x + 1 57. f (x) = 4 x − 3 __ x2 − x − 12 59. f (x) = −9 x − 2 __ x 2 − 9 1 _ 61. f (x 63. f (x) = −6 (x − 1)2 ____________ (x + 3)(x − 2)2 65. Vertical asymptote at x = 2; horizontal asymptote at y = 0 x y x y 2.01 100 10 2.001 1,000 100 0.125 0.0102 2.0001 10,000 1,000 0.001 1.99 1.999 −100 −1,000 10,000 100,000 0.0001 0.00001 ODD ANSWERS –10 –8 –6 –4 10 8 6 4 2 –2–2 –4 –6 –8 –10 42 6 8 10 x –10 –8 –6 –4 10 8 6 4 2 –2–2 –4 –6 –8 –10 42 6 8 10 x 21 3 4 5 x −1−2−3−4−5 −1 −2 −3 45. [−4, 2) ∪ [5, ∞) y 5 4 3 2 1 16 14 12 8 4 84 12 14 16 −16 −14 −4−8−12 −4 −8 −12 −14 −16 47. (−2, 0), (4, 2), (22, 3) y C-12 67. Vertical asymptote at x = −4; horizontal asymptote at y = 2 x y x y −4.1 82 10 −4.01 802 100 1.4286 1.9331 −4.001 −3.99 −3.999 −7998 −798 8,002 1,000 1.992 10,000 100,000 1.9992 1.999992 69. Vertical asymptote at x = −1; horizontal asymptote at y = 1 −0.999 998,001 −1.01 10,201 −0.99 9,801 −0.9 81 −1.1 121 x y x y 10 0.82645 100 0.9803 1,000 0.998 10,000 0.9998 100,000 3 , ∞  71.  _ 2 f(x) 73. (−∞, 1) ∪ (4, ∞) f(x) 75. (2, 4) 77. (2, 5) 79. (−1, 1) 81. C(t) = 8 + 2t _ 300 + 20t 83. After about 6.12 hours 87. radius 2.52 meters 85. 2 by 2 by 5 feet Section 3.8 1. It can be too difficult or impossible to solve for x in terms of y. 3. We will need a restriction on the domain of the answer. 7. f −1(x) = √ x + 3 −1 x + 4 — — 9. f −1(x) = − √ 5. f −1 (x) = √ ______ 13. f −1(x) = 11. f −1(x) = √ — 9 − x 15. f −1(x) = √ 3 — 4 − x 17. f −1(x) = , [0, ∞) 19. f −1(x) = (x − 9)2 + 4 __________ 4 , [9, ∞) √ x 2 − 1 ______ 2 x − 9 _____  2 7x − 3 ______ 27. f −1(x) = 1 − x 3 23. f −1(x) = 2 − 8x ______ x 5x − 4 ______ 4x + 3 31. f −1(x) = √ — x + 6 + 3 21. f −1(x) =  25. f −1(x) = 29. f −1(x) = √ 33. f −1(x — 10 8 6 4 2 −10 −2−4−6−8 −2 −4 −6 −8 −10 42 6 8 10 x 35. f −1(x) = √ y — x + 4 12 10 8 6 4 2 42 6 8 10 12 x −12 −2−4−6−8−10 −2 −4 −6 −8 37. f −1(x) = 3 √ y — 1 − x 39. f −1 (x2−1 −1 −2 −3 −4 −5 −3−4−5 1 2 3 4 5 x 15 12 9 6 3 −15 −3−6−9−12 −3 −6 −9 −12 −15 63 9 12 15 x __ 41. f −1 (x) = √ 1 _ x 43. [−2, 1) ∪ [3, ∞) y x x x f (x) 10 8 6 4 2 −10 −6−8 −4−2 −2 −4 −6 −8 −10 42 6 8 10 x −3−4−5 40 32 24 16 8 −2 –1 −8 −16 −24 −32 −40 1 2 3 4 5 49. (−4, 0), (0, 1), (10, 2) y 51. (−3, −1), (1, 0), (7, 1) y 40 32 24 16 8 −2 –1 −8 −16 −24 −32 −40 −3−4−5 1 2 3 4 5 x −3−4−5 40 32 24 16 8 −2 –1 −8 −16 −24 −32 −40 59. r(V) = , 5.53 seconds b2 + 4x √ _ 55. f −1(x) = , ≈ 3.63 feet 61. n(C) = — 3V _ 4π ___ V _ 6π 53. f −1(x) = − b _ + 2 ________ 57. t(h) = √ 200 − h _ 4.9 √ 63. r(V) = √ Section 3.9 1. The graph will have the appearance of a power function. 5. y = 5x 2 3. No. Multiple variables may jointly vary. 18 _ x 2 , ≈ 3.99 m 65. r(V) = √ 100C − 25 _ 0.6 − C ___ V _ 4π 9. y = 6x 4 13. y = 11. y = 17. y = 10xzw 19. y = 10x √ — z 81 _ x 4 , ≈ 1.99 inches , 250 mL 23. y = 40 25. y = 256 xz _ — w t2 √ 31. y = 27 27. y = 6 29. y = 6 33. y = 3 15. y = 7. y = 10x 3 20 _ — 3 x √ xz _ w 21. y = 4 ODD ANSWERS 35. y = 18 37. y = 90 3 _ 41. y = x 2 4 y 39. y = 81 _ 2 1 __ 43. y = √ 3 — x y 75 60 45 30 15 −10 −2−4−6−8 −15 −30 −45 −60 −75 4 __ 45. y = x2 y 10 8 6 4 2 −10 −2−4−6−8 −2 −4 −6 −8 −10 42 6 8 10 x 10 8 6 4 2 −25 −20 −15 −5−10 −2 −4 −6 −8 −10 5 10 15 20 25 x 47. ≈ 1.89 years 49. ≈ 0.61 years 51. 3 seconds 53. 48 inches 55. ≈ 49.75 pounds 57. ≈ 33.33 amperes 59. ≈ 2.88 42 6 8 10 x Chapter 3 Review exercises 1. 2 − 2i 3. 24 + 3i 7. f (x) = (x − 2)2 −9; vertex: (2, −9); intercepts: (−1, 0), (5, 0), (0, −5) f(x) −2−3−4−5 10 8 6 4 2 −1−2 −4 −6 −8 −10 321 4 5 6 7 x 5. {2 + i, 2 − i} 3 _ 25 9. f (x) = (x + 2)2 + 3 11. 300 meters by 150 meters, the longer side parallel to the river 13. Yes; degree: 5, leading coefficient: 4 15. Yes; degree: 4; leading coefficient: 1 17. As x → −∞, f (x) → −∞, as x → ∞, f (x) → ∞ 1 _ 19. −3 with multiplicity 2, − with multiplicity 1, −1 with 2 21. 4 with multiplicity 1 1 _ 23. with 2 multiplicity 3 multiplicity 1, 3 with multiplicity 3 25. x 2 + 4 with remainder 12 27. x 2 − 5x + 20 − 61 _____ x + 3 29. 2x 2 − 2x − 3, so factored form is (x + 4)(2x 2 − 2x − 3) 1 31.  −2, 4, − 1 33.  1, 3, 4,   _ _ 2 2 35. 2 or 0 positive, 1 negative 37. Intercepts: (−2, 0),  0, − 2  , _ 5 asymptotes: x = 5 and y = 1 y 39. Intercepts: (3, 0), (−3, 0), 27  0,  ; asymptotes: x = 1, −2 _ 2 and y = 3 25 20 15 10 5 y = 1 −25 −20 −15 −5−10 −5 −10 −15 −20 −25 5 10 15 20 25 30 x x = 5 y 40 32 24 16 8 −15 −12 −3−6−9 −8 −16 −24 −32 −40 x = −2 y = 3 x 3 6 9 12 15 x = 1 41. y = x − 2 43. f −1(x) = √ — x + 2 45. f −1(x) = √ — x + 11 − 3 C-13 47. f −1(x) = , x ≥ −3 49. y = 64 51. y = 72 (x + 3)2 − 5 __ 4 53. ≈ 148.5 pounds Chapter 3 practice test 1. 20 − 10i 3. {2 + 3i, 2 − 3i} 5. As x → −∞, f (x) → −∞, as x → ∞, f (x) → ∞ 7. f (x) = (x + 1)2 − 9, vertex: (−1, −9), intercepts: (2, 0), (−4, 0)(0, −8) y 9. 60,000 square feet 11. 0 with multiplicity 4, 3 with multiplicity 2 13. 2x 2 − 4x + 11 − 26 _ x + 2 15. 2x 2 − x − 4, so factored form is (x + 3)(2x 2 − x − 4) 100 80 60 40 20 –4 0 –2 –20 –40 –60 –80 –100 –10 –8 –6 42 6 8 10 x g(x) = 2x3 – 50x — 15 −1 ± i √ __ 2 17. − 1 _ (has multi plicity 2), 2 21. f(x) = 2(2x − 1)3(x + 3) 19. −2 (multiplicity 3), ±i 23. Intercepts: (−4, 0),  0, − 4  ; _ 3 asymptotes: x = 3, −1 and y = 0 25. y = x + 4 — 27. f −1(x) = x + 4 _ 3 3 √ 31. 4 seconds y 10 8 6 4 2 −2 −4 −6 −8 −10 −10 −2−4−6−8 x = −1 29. y = 18 y = 0 2 4 6 8 10 x x = 3 ChapteR 4 Section 4.1 5. Exponential; the population 7. Not exponential; 1. Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original. 3. When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account. Thus, the annual percentage rate does not necessarily correspond to the real interest earned, which is the very definition of nominal. decreases by a proportional rate. the charge decreases by a constant amount each visit, so the statement represents a linear function. 11. After 20 years forest A will have 43 more trees than forest B. 13. Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but because forest B actually grows at a faster rate, the population will eventually become larger than forest A and will remain that way as long as the population growth models hold. Some factors that might influence the long-term validity of the exponential growth model are drought, an epidemic that culls the population, and other environmental and biological factors. 9. Forest B ODD ANSWERS C-14 − 3 – 5 23. Linear x – 5 ≈ 2.93(0.699)x 15. Exponential growth; the growth factor, 1.06, is greater than 1. 17. Exponential decay; the decay factor, 0.97, is between 0 and 1. 19. f (x) = 2000(0.1)x 1 1  21. f (x) =  __ __   6 6 27. Linear 25. Neither −nt r 33. P = A(t) ⋅  1 + _ n  39. Continuous growth; the growth rate is greater than 0. 41. Continuous decay; the growth rate is less than 0. 43. $669.42 49. f (3) ≈ 483.8146 55. y ≈ 0.2 ⋅ 1.95x 45. f (−1) = −4 47. f (−1) ≈ −0.2707 31. $13,268.58 37. 4% 29. $10,250 35. $4,569.10 53. y ≈ 18 ⋅ 1.025x 51. y = 3 ⋅ 5x 57. APY = A(t) − a _ a 365(1) a  1 + r ___  365 ___ a − a = − 1; n 365 365 − 1 − 1  r _  365 ___  365 ___ = a r I(n) =  1 + _ n  59. Let f be the exponential decay function f (x such that b > 1. Then for some number n > 0, f (x 61. 47,622 foxes 67. $82,247.78; $449.75 63. 1.39%; $155,368.09 x = a(e−n)x = a(e)−nx. = a(b−1)x = a ((en)−1 ) 65. $35,838.76 Section 4.2 1. An asymptote is a line that the graph of a function approaches, as x either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely 3. g(x) = 4(3)−x; y-intercept: (0, 4); large or extremely small. domain: all real numbers; range: all real numbers greater than 0. 5. g(x) = −10x + 7; y-intercept: (0, 6); domain: all real numbers; range: all real numbers less than 7. 1 7. g(x) = 2   _ 4 range: all real numbers greater than 0. ; y-intercept: (0, 2); domain: all real numbers; x 9. y-intercept: (0, −2) y 11. 5 4 3 2 1 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 g(x) = −2(0.25)x x 21 3 4 5 13. B 23. g(−x) = −2(0.25)−x 17. E 15. A 19. D 25. y 5 5 4 3 2 1 f(x) = 1 2(4)x x −5 −4 5 3 21 4 −f(x) = −1 2 (4)x −5 −4 −3 −2 −1−1 −2 −3 −4 −5 y g(x) = 3(2)x h(x) = 3(4)x 4( )x f (x) = 3 1 5 4 3 2 1 21 3 4 5 x −5 −4 −3 −2 −1−1 −2 −3 −4 −5 213 −2 −1−1 −2 −3 −4 −5 −f (x) = 4(2)x −
2 21 3 4 5 x f (x) = −4(2)x + 2 27. Horizontal asymptote: h(x) = 3; domain: all real numbers; range: all real numbers strictly greater than 3. h(x1−2−3−4−5 21 3 4 5 x 29. As x → ∞, f (x) → −∞; as x → −∞, f (x) → −1 31. As x → ∞, f (x) → 2; as x → −∞, f (x) → ∞ 33. f (x) = 4x − 3 35. f (x) = 4x − 5 37. f (x) = 4−x 39. y = −2x + 3 41. y = −2(3)x + 7 43. g(6) ≈ 800.3 49. x ≈ −0.222 47. x ≈ −2.953 45. h(−7) = −58 x 51. The graph of g(x) =  1  _ is the reflection about the y-axis b of the graph of f (x) = bx; for any real number b > 0 and function x 1  f (x) = bx, the graph of  _ is the reflection about the y-axis, f (−x). b 53. The graphs of g(x) and h(x) are the same and are a horizontal shift to the right of the graph of f (x). For any real number n, real number b > 0, and function f (x) = bx, the graph of  1 bn  bx is the _ horizontal shift f (x − n). Section 4.3 (x), the 9. x y = 64 1. A logarithm is an exponent. Specifically, it is the exponent to which a base b is raised to produce a given value. In the expressions given, the base b has the same value. The exponent, y, in the expression by can also be written as the logarithm, logbx, and the value of x is the result of raising b to the power of y. 3. Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation b y = x, and then properties of exponents can be applied to solve for x. 5. The natural logarithm is a special case of the logarithm with base b in that the natural log always has base e. Rather than notating the natural logarithm as loge notation used is ln(x). 7. ac = b 15. e n = w 21. logn(103) = 4 1 _ 27. x = 8 35. x = e2 45. 4 55. ≈ 2.708 defined value for x = 0. To verify, suppose x = 0 is in the domain of the function f (x) = log(x). Then there is some number n such that n = log(0). Rewriting as an exponential equation gives: 10n = 0, which is impossible since no such real number n exists. Therefore, x = 0 is not the domain of the function f (x) = log(x). 61. Yes. Suppose there exists a real number, x such that ln(x) = 2. Rewriting as an exponential equation gives x = e2, which is a real number (x = e2 ≈ 7.389056099). To verify, let x = e 2. Then, by 19. log19 (y) = x  = x 57. ≈ 0.151 59. No, the function has no 25. ln(h) = k 1 _ 216 17. logc(k) = d 23. logy  47. −3 49. −12 51. 0 53. 10 39 _ 100 31. x = 3 1 _ 43. 2 13. 13a = 142 41. 14.125 11. 15b = a 29. x = 27 39. 1.06 33. x = 37. 32 definition, ln(x) = ln(e2) = 2. is undefined. 65. 2 63. No; ln(1) = 0, so ln(e1.725) _ ln(1) ODD ANSWERS Section 4.4 1. Since the functions are inverses, their graphs are mirror images about the line y = x. So for every point (a, b) on the graph of a logarithmic function, there is a corresponding point (b, a) on 3. Shifting the the graph of its inverse exponential function. function right or left and reflecting the function about the y-axis 5. No. A horizontal asymptote would will affect its domain. suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers. 1 7. Domain:  −∞,  ; range: (−∞, ∞) _ 2 9. Domain:  − 17 , ∞  ; range: (−∞, ∞ ) _ 4 11. Domain: (5, ∞); vertical asymptote: x = 5 , ∞  ; vertical asymptote: x = − 1 13. Domain:  − 1 _ _ 3 3 15. Domain: (−3, ∞); vertical asymptote: x = −3 17. Domain:  3 3 , ∞  ; vertical asymptote: x = _ _ ; end behavior: 7 7 3 as x →   _ 7 19. Domain: (−3, ∞); vertical asymptote: x = −3; end behavior: as x → −3+, f (x) → −∞ and as x → ∞, f (x) → ∞ 21. Domain: (1, ∞); range: (−∞, ∞); vertical asymptote: x = 1; 5 x-intercept:  , 0  ; y-intercept: DNE _ 4 23. Domain: (−∞, 0); range: (−∞, ∞); vertical asymptote: x = 0; x-intercept: (−e2, 0); y-intercept: DNE 25. Domain: (0, ∞); range: (−∞, ∞) vertical asymptote: x = 0; x-intercept: (e3, 0); y-intercept: DNE 27. B 35. 5 4 3 2 1 , f (x) → −∞ and as x → ∞, f (x) → ∞ 33. C 37. f(x) = log(x) 29. C 31. B x f (x) = e 391−1 −2 −3 −4 −5 41. −5 −4 −3 −2 −1−1 −2 −3 −4 −5 21 3 4 5 876 9 10 x −5 −4 g(x) = log (x) l 2 43. f(x) 5 4 3 2 1 21 3 4 5 x −3−4−5−6−7−8 −3 −2 −1−1 −2 −3 −4 −5 x 4 5 21 3 g(x) = ln(x) y 5 4 3 2 1 −2 −1−1 −2 −3 −4 −5 x 21 47. f (x) = log2(−(x − 1)) 49. f (x) = 3log4(x + 2) 51. x = 2 53. x ≈ 2.303 55. x ≈ −0.472 45. g(x) 5 4 3 2 1 21 3 4 x −3−4−5−6 −2 −1−1 −2 −3 −4 −5 C-15 57. The graphs of f (x) = log (x) and g(x) = −log 2(x) appear to be 1 _ 2 the same; conjecture: for any positive base b ≠ 1, logb(x) = −log 1 _ b (x). 59. Recall that the argument of a logarithmic function must be positive, so we determine where > 0. From the graph of x + 2 _ x − 4 the function f (x) = , note that the graph lies above the x + 2 _ x − 4 x-axis on the interval (−∞, −2) and again to the right of the vertical asymptote, that is (4, ∞). Therefore, the domain is (−∞, −2)∪(4, ∞). Section 4.5 1. Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, logb(x 1 3. logb (2) + logb (7) + logb(x) + logb(y) 5. logb(13) − logb(17) 7. −kln(4) 11. logb(4) 13. logb(7) 15. 15log(x) + 13log(y) − 19log(z) 8 3 _ _ log(x) + log(x) − 2log(y) 19. log(y) 21. ln(2x7) 17. 3 2 1 _ n logb(x). n ) = 9. ln(7xy) 14 _ 3 _ 23. log  xz3  _ — y √ 1 _ 27. log11(5) = b 25. log7(15) = 6 29. log11  _ 11 35. ≈ 0.93913 ln(15) _ ln(7)  = or 37. ≈ −2.23266 33. ≈ 2.81359 39. x = 4, By the quotient rule: log6(x + 2) − log6(x − 3) = log6  Rewriting as an exponential equation and solving for x: 61 = x + 2 _____ x − 3  = 1 31. 3 − 6 x + 2 _____ x − 3 x + 2 _____ x − 3 6(x − 3) x + 2 _____ _______ − (x − 3) x − 3 x + 2 − 6x + 18 _____________ _____ x − 3 x = 4 Checking, we find that log6(4 + 2) − log6 (4 − 3) = log6(6) − log6(1) is defined, so x = 4. 41. Let b and n be positive integers greater than 1. Then, by the change-of-base formula, logb(n) = logn(n) _ logn(b) = 1 _ . logn(b) Section 4.6 1. Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use 3. The one-to-one property properties of logarithms to solve. can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base. 1 _ 5. x = − 3 11. x = 10 13. No solution 17. k = − ln(38) _ 3 19. x = 6 _ 9. b = 7. n = −1 5 17 15. p = log   − 7 _ 8 38 ln   − 8 _ 3 __ 9 21. x = ln(12) ODD ANSWERS C-16 23. x = 25. No solution 27. x = ln(3) y 67. About 5 years 3 ln   − 3 _ 5 __ 8 1 _ 100 29. 10−2 = 31. n = 49 33. k = 1 _ 36 35. x = 9 − e _ 8 37. n = 1 43. x = ± 10 _ 3 51. x = 9 y 39. No solution 41. No solution 45. x = 10 47. x = 0 3 _ 49. x = 4 53. x = ≈ 2.5 e2 _ 3 321 4 5 6 7 8 9 10 11 12 x 1 −1 −2 −3 −4 −5 −6 −7 y 5 4 3 2 1 −2 −1−1 −2 −3 −4 −5 21 3 4 5 x e + 10 _ 4 ≈ 3.2 57. x = y 321 4 5 6 7 8 9 10 x 1 −1−1 −2 −3 −4 −5 −6 x –1 61. x = 11 _ 5 ≈ 2.2 y 55. x = −5 y 5 4 3 2 1 −2−3−4−5−6−7−8 −1 −1 −2 −3 59. No solution y 3 2 1 71. ≈ 2.078 69. ≈ 0.567 73. ≈ 2.2401 75. ≈ −44655.7143 77. About 5.83 1 _ k ) 79. t = ln (  y ______ A  81. t = ln (  1 _ − T − Ts k  ______ T0 − Ts ) (5, 20,000) 25,000 20,000 15,000 10,000 5,000 0 1 2 3 4 5 6 x Section 4.7 1. Half-life is a measure of decay and is thus associated with exponential decay models. The half-life of a substance or quantity is the amount of time it takes for half of the initial amount of that 3. Doubling time is a measure substance or quantity to decay. of growth and is thus associated with exponential growth models. The doubling time of a substance or quantity is the amount of time it takes for the initial amount of that substance or quantity 5. An order of magnitude is the nearest to double in size. power of ten by which a quantity exponentially grows. It is also an approximate position on a logarithmic scale; Sample response: Orders of magnitude are useful when making comparisons between numbers that differ by a great amount. For example, the mass of Saturn is 95 times greater than the mass of Earth. This is the same as saying that the mass of Saturn is about 102 times, or 2 orders of magnitude greater, than the mass of Earth. 7. f (0) ≈ 16.7; the amount initially present is about 16.7 units. 9. 150 13. Logarithmic 11. Exponential; f (x) = 1.2x 15. Logarithmic 5 4 3 2 1 −3 −2 −1−1 −2 −3 −4 −5 f(x) 21 3 4 5 x 10 10 x y 13 12 11 10 9 0 4 5 6 7 8 9 10 11 12 13 x 21 3 4 5 6 7 8 9 10 x −5 −4 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 63. x = y 101 _ 11 ≈ 9.2 6 5 4 3 2 1 −2 −1 −2 42 6 8 10 12 14 x 65. About $27,710.24 y 35,000 30,000 25,000 20,000 15,000 10,000 5,000 −1 0 −1 (20, 27710.24) f(x) = 6500e0.0725x 2 4 6 8 10 12 14 16 18 20 22 24 x 17. P(t) 1,000 900 800 700 600 500 400 300 200 100 −20 −18 −16 −14 −12 −4−6−8−10 −2 −100 42 6 8 10 12 14 16 18 20 t ODD ANSWERS 19. About 1.4 years 23. Four half-lives; 8.18 minutes 21. About 7.3 years __ __ 3 _ 2 10 3M 2 _ 25. M = 3 27. Let y = b x for some non-negative real number b such that b ≠ 1. Then, ln (y) = ln (b x) ln (y) = x ln (b) e ln(y) = e xln(b) y = e xln(b) log  S _  S0 M = log  S _  S0 2 =  S _  S0 S010 3M 2 = S 29. A = 125e (−0.3567t); A ≈ 43mg 33. f (t) = 250e −0.00914t; half-life: about 76 minutes 35. r ≈ − 0.0667; hourly decay rate: about 6.67% 37. f (t) = 1350 e 0.034657359t; after 3 hours; P (180) ≈ 691,200 39. f (t) = 256 e (0.068110t); doubling time: about 10 minutes 41. About 88minutes 43. T(t) = 90 e (−0.008377t) + 75, where t is in minutes 45. About 113 minutes 47. log10x = 1.5; x ≈ 31.623 49. MMS Magnitude: ≈ 5.82 31. About 60 days 51. N(3) ≈ 71 53. C Section 4.8 1. Logistic models are best used for situations that have limited values. For example,
populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic 3. Regression analysis is model best describes populations. the process of finding an equation that best fits a given set of data points. To perform a regression analysis on a graphing utility, first list the given points using the STAT then EDIT menu. Next graph the scatter plot using the STAT PLOT feature. The shape of the data points on the scatter graph can help determine which regression feature to use. Once this is determined, select the appropriate regression analysis command from the STAT then CALC menu. 5. The y-intercept on the graph of a logistic equation corresponds to the initial population for the population model. 7. C 13. p ≈ 2.67 19. About 6.8 months. 21. 11. P (0) = 22; 175 15. y-intercept: (0, 15) 17. 4 koi 9. B y 23. About 38 wolves 25. About 8.7 years 27. f (x) = 776.682 (1.426)x 29. y C-17 43. f (10) ≈ 2.3 45. When f (x) = 8, x ≈ 0.82 47. f (x) = 25.081 __ 1 + 3.182e−0.545x 49. About 25 41. y 10 51. y 140 130 120 110 100 90 80 70 60 50 40 30 20 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 53. 0 y 140 130 120 110 100 90 80 70 60 50 40 30 20 10 10 11 12 13 14 15 16 17 18 x x 55. When f (x) = 68, x ≈ 4.9 g (x) = 4.035510; the regression curves are symmetrical about y = x, so it appears that they are inverse functions. 57. f (x) = 1.034341(1.281204) x; 59. f −1(x) = c ln (a) − ln  _ − 1  x __ b 600 550 500 450 400 350 300 250 200 150 100 50 5 10 15 20 x 0 y 31. 600 550 500 450 400 350 300 250 200 150 100 ,000 6,000 5,000 4,000 3,000 2,000 1,000 0 1 2 3 4 5 6 7 x 33. f (x) = 731.92e −0.3038x 35. When f (x) = 250, x ≈ 3.6 37. y = 5.063 + 1.934 log(x) 39. y 10 Chapter 4 Review exercises 5. $42,888.18 1. Exponential decay; the growth factor, 0.825, is between 0 and 1. 3. y = 0.25(3) x 7. Continuous decay; the growth rate is negative y 9. Domain: all real numbers; range: all real numbers strictly greater than zero; y-intercept: (0, 3.5) 10 1−2−3−4−5−6−7−8 −1 1 2 3 4 x ODD ANSWERS 19. x = ln(1000) _______ + 5 ln(16) __ 3 ≈ 2.497 21. a = ln (4) + 8 ________ 10 23. No solution 25. x = ln(9) — 27. x = ± 3 √ 3 ____ 2 29. f (t) = 112e−0.019792t; half-life: about 35 days 31. T(t) = 36 e−0.025131t + 35; T(60) ≈ 43° F 33. Logarithmic C-18 11. g(x) = 7(6.5)−x; y-intercept: (0, 7); domain: all real numbers; range: all real numbers greater than 0. 2 _ 15. loga b = − 5 21. ln(e−0.8648) = −0.8648 23. 19. log(0.000001) = −6 13. 17x = 4,913 17. x = 4 g(x) 1 2 3 4 5 x 1 −1−2−3−4−5 −1 −2 −3 −4 −5 −6 −7 −8 25. Domain: x > −5; vertical asymptote: x = −5; end behavior: as x → −5+, f (x) → −∞ and as x → ∞, f (x) → ∞ 27. log 8(65xy) z 29. ln  _ xy  31. logy(12) 33. ln(2) + ln(b) + ln(b + 1) − ln(b − 1) __ 2 35. log 7  v 3w 6 _ — 3 u √  5 _ 37. x = 39. x = −3 41. No solution 43. No solution 3 45. x = ln(11) 51. About 5.45 years 47. a = e4 − 3 53. f −1(x) = 9 __ 49. x = ± 5 24x − 1 — 3 √ 55. f (t) = 300(0.83)t ; f (24) ≈ 3.43 g 57. About 45 minutes y 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 59. About 8.5 days 61. Exponential y 100 90 80 70 60 50 40 30 20 10 10 11 x y 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 63. y = 4(0.2)x; y = 4e–1.609438x 65. About 7.2 days 67. Logarithmic y = 16.68718 − 9.71860ln(x 10 11 x 35. Exponential; y = 15.10062(1.24621)x y 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 10 x 37. Logistic; y = 18.41659 __ 1 + 7.54644 e−0.68375x y 20 19 18 17 16 15 14 13 12 11 10 .5 1 1.5 2 2.5 3 3.5 4 x Chapter 4 practice test 1. About 13 dolphins 5. y-intercept: (0, 5) 3. $1,947 y f(−x) = 5(0.5)−x f(x) = 5(0.5)− 49 7. 8.5a = 614.125 9. x = 11. ln(0.716) ≈ − 0.334 13. Domain: x < 3; vertical asymptote: x = 3; end behavior: as x → 3−, f (x) → −∞ and as x → −∞, f (x) → ∞ −1−2−3−4−5 −1 1 2 3 4 5 x 15. log t(12) 17. 3ln(y) + 2ln(z) + ln(x − 4 ChapteR 5 Section 5.1 1. Terminal side Vertex Initial side 3. Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction. ODD ANSWERS 5. Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time. 7. 9. Section 5.2 1. The unit circle is a circle of radius 1 centered at the origin. 3. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t, formed by the terminal side of the angle t and the horizontal axis. C-19 11. 13. 15. 17. 240° 19. 4π ___ 3 21. 2π ___ 3 35. −3π radians ≈ 12.72 cm2 27. 20° 23. 7π ___ ≈ 11.00 in2 2 81π ____ 25. 20 π __ 29. 60° 31. −75° 33. radians 2 radians 39. 37. π radians 25π ___ 9 43. 47. 104.7198 cm2 5π ___ 6 5.02π _____ 3 41. 21π ___ 10 49. 0.7697 in2 51. 250° 53. 320° ≈ 6.60 meters 45. ≈ 5.26 miles ≈ 8.73 centimeters 55. 4π ___ 3 57. 8π ___ 9 59. 1320 rad/min 210.085 RPM 61. 7 in/s, 4.77 RPM, 28.65 deg/s 63. 1,809,557.37 mm/min = 30.16 m/s 67. 120° 73. 11.5 inches 69. 794 miles per hour 71. 2,234 miles per hour 65. 5.76 miles — √ 3 _ 2 1 __ 13. 2 5. The sine values are equal. 7. I 9. IV 11. 15 __ 33. 8 17. 0 19. −1 21. 23. 60° , cos (135° ) = − π __ 29. 3 25. 80° 27. 45° 39. 60°, Quadrant II, sin(120°) = 37. 45° , Quadrant II, sin(135°) = 35. 60°, Quadrant IV, sin(300°) = − 1 __ , cos (300 __ 31 __ _ , cos (120 __ ____ 41. 30°, Quadrant II, sin(150°) = , cos (150 7π 7π 1 , cos  , Quadrant III, sin  ___ __ ___ __  = −  = − _ 43 3π 2 , cos  ___ ____  = − _ 4 2 2 — 1 2π √ 3 , cos  __ ___ ____  = − 3 2 2 — √ 7π 2 , cos  ___ ____  = 4 2 2π π , Quadrant II, sin  ___ __  = 47. 3 3 π 7π , Quadrant IV, sin  ___ __  = − 49. 4 4 — √ 15 ____ 3  4 1 __ , cos t = − 57. (−2.778, 15.757) 59. [−1, 1] 61. sin t = 2 — √ 3 ____ 2 3π π , Quadrant II, sin  ___ __  = 45. 4 4 55.  −10, 10 √ — √ 77 ____ 9 65. sin t = 63. sin t = − — √ 2 ____ 2 , cos t = − 53. − 51. — 67. sin t = − — √ 2 ____ 2 — √ 2 ____ 2 — √ 2 ____ 2 — √ 2 ____ 2 71. sin t = −0.596, cos t = 0.803 75. sin t = − 1 _ , cos t = 2 , cos t = 79. sin t = 1, cos t = 0 85. −0.7071 — √ 6 _ 4 87. −0.1392 — √ 2 _ 4 95. 93. − — √ 3 _ 2 81. −0.1736 — √ 2 _ 4 69. sin t = 0, cos t = −1 — √ 3 ____ 2 1 __ 73. sin t = , cos t = 2 77. sin t = 0.761, cos t = −0.649 83. 0.9511 89. −0.7660 91. — √ 2 _ 4 97. 99. 0 101. (0, −1) — √ 3 ____ 2 1 __ , cos t = − 2 103. 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds Section 5.3 π _ 1. Yes, when the reference angle is and the terminal side of the 4 π _ angle is in quadrants I or III. Thus, at x = , 4 5π _ 4 , the sine and cosine values are equal. in for y in the Pythagorean Theorem x 2+ y take the negative solution. cotangent will repeat every π units. 3. Substitute the sine of the angle 2 = 1. Solve for x and 5. The outputs of tangent and — — 3 7. 9. √ — 2 √ 3 _____ 3 19. − 2 √ √ 3 3 ____ ____ 3 3 — 27. −2 29. − √ 3 ____ 3 17. — 11. √ — 2 13. 1 15. 2 21. √ — 3 23. − √ — 2 25. −1 31. 2 33. 35. −2 37. −1 — √ 3 ____ 3 ODD ANSWERS C-20 — — , sec t = − 3, csc t = − 3 √ 39. sin t = − 2 √ 2 2 ____ ____ , 4 3 2 , cot t = tan t = 2 √ 41. sec t = 2, csc t = — — 2 √ 3 ____ , 3 tan t = √ 3 , cot t = — 43. − 45. 3.1 — √ 2 ____ 4 — √ 3 ____ 3 — √ 2 ____ 2 — √ 2 ____ 2 — √ 2 ____ 2 47. 1.4 49. sin t = , cos t = , tan t = 1, cot t = 1, sec t = √ 51. sin t = − — t = √ 3 , cot t = , sec t = − 2, csc t = − — 2 — 2 , csc t = √ — √ 3 ____ 3 57. 1.414 55. −2.414 53. −0.228 61. 1.556 63. sin(t) ≈ 0.79 65. csc(t) ≈ 1.16 69. Even 71. 75. 7.73 inches sin t _ cos t = tan t 73. 13.77 hours, period: 1000π — √ 1 3 __ ____ , cos t = − , tan 2 2 — 2 √ 3 ____ 3 59. 1.540 67. Even Section 5.4 1. Opposite side Hypotenuse Adjacent side 5. For example, the sine of 3. The tangent of an angle is the ratio of the opposite side to the adjacent side. an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement. π _ 7. 6 π _ 9. 4 11. b = — 20 √ 3 ______ 3 , c = — 40 √ 3 ______ 3 13. a = 10,000, c = 10,000.5 15. b = 17. — 5 √ 29 _____ 29 5 __ 19. 2 21. — √ 29 ____ 2 — 5 √ 3 ____ 3 , c = 23. — 10 √ 3 _____ 3 — 5 √ 41 _____ 41 29. c = 14, b = 7 √ — 3 33. b = 9.9970, c = 12.2041 27. 5 __ 25. 4 — √ 41 ____ 4 31. a = 15, b = 15 35. a = 2.0838, b = 11.8177 39. a = 46.6790, b = 17.9184 43. 188.3159 49. 1,060.09 ft. 55. 368.7633 ft. 45. 200.6737 51. 27.372 ft. 37. a = 55.9808, c = 57.9555 41. a = 16.4662, c = 16.8341 47. 498.3471 ft. 53. 22.6506 ft. Chapter 5 Review exercises 1. 45° 3. − 7π ___ 5. 10.385 meters 7. 60° 9. 6 11. 13. 2π ___ 11 15. 1,036.73 miles per hour 17. 19. −1 23. − 31. √ — √ 2 ____ 2 — 2 25. [−1, 1] 33. 0.6 35. 37. Sine, cosecant, tangent, cotangent π __ 21. 4 29. √ — 2 — √ 3 ____ 2 27. 1 — √ 2 ____ 2 or − — √ 2 ____ 2 — √ 3 ____ 3 39. 47. a = 4, b = 4 41. 0 43. b = 8, c = 10 45. 49. 14.0954 ft. — 11 √ 157 _______ 157 Chapter 5 practice test 1. 150° 7. 3. 6.283 centimeters 5. 15° 2π _ 75 13. [–1, 1] 11. − 9. 3.351 feet per second, radians per second — √ 3 ____ 2 — 3 — √ 3 ____ 2 — √ 3 _ 3 π __ 21 _____ __ 23. a = 2 2 17. 15. √ 19. ChapteR 6 Section 6.1 1. The sine and cosine functions have the property that f (x + P) = f (x) for a certain P. This means that the function 3. The absolute values repeat for every P units on the x-axis. value of the constant A (amplitude) increases the total range and the constant D (vertical shift) shifts the graph vertically. 5. At the point where the terminal side of t intersects the unit circle, you can determine that the sin t equals the y-coordinate of the point. 2 _ 7. Amplitude: ; period: 2π; 3 midline: y = 0; maximum: 2 _ occurs at x = 0; y = 3 minimum: y = − 2 _ occurs 3 at x = π; for one period, the graph starts at 0 and ends at 2π. f(x) 9. Amplitude: 4; period: 2π; midline: y = 0; maximum: π _ y = 4 occurs at x = ; 2 minimum: y = −4 occurs at x = ; for one period, the graph starts at 0 and ends at 2π. f(x) 3π _
2 1 2 3 1 3 −π 0 π− 3 1− 2 2− 3 −1 π 2 π 3π 2 2π x −2π 3π− 2 −π 4 3 2 1 0 π− 2 −1 −2 −3 −4 π 2 π 3π 2 2π x −2π − 3π 2 11. Amplitude: 1; period: π; midline: y = 0; maximum: y = 1 occurs at x = π; minimum: y = −1 occurs at π _ x = ; for one period, the 2 graph starts at 0 and ends at π. f(t) 1 13. Amplitude: 4; period: 2; midline: y = 0; maximum: y = 4 occurs at x = 0; minimum: y = −4 occurs at x = 1; for one period, the graph starts at 0 and ends at π. f(t) 4 3 2 1 −π − 3π 4 π− 0 π−2 4 −1 π 4 π 2 3π 4 π t 0.5 1 1.5 2 t 0 −2−1.5−1−0.5 −1 −2 −3 −4 ODD ANSWERS π _ 15. Amplitude: 3; period: ; 4 midline: y = 5; maximum: y = 8 occurs at x = 0.12; minimum: y = 2 occurs at x = 0.516; horizontal shift: −4; vertical translation: 5; for one period, the graph starts at 0 π _ . and ends at 16 π 8 3π 16 π 4 x π− 4 3π− 16 π− 8 0 π− 16 −1 −2 −3 2π _ ; 5 17. Amplitude: 5; period: midline: y = −2; maximum: y = 3 occurs at x = 0.08; minimum: y = −7 occurs at x = 0.71; phase shift: −4; vertical translation: −2; for one period, the graph starts at 2π _ . 0 and ends at 5 y 2π− 3π−5 10 π 10 π 5 3π 10 2π 5 x 5 4 3 2 1 π− 5 0 π− 10 −1 −2 −3 −4 −5 −6 −7 19. Amplitude: 1; period: 2π ; midline: y = 1; maximum: y = 2 occurs at t = 2.09; minimum: y = 0 occurs at t = 5.24; phase shift: π _ − ; vertical translation: 1; for one period, the graph starts at 0 and 3 ends at 2π . f(t) 43. A linear function is added to a periodic sine function. The graph does not have an amplitude because as the linear function increases without bound the combined function h(x) = x + sin x will increase without bound as well. The graph is bounded between the graphs of y = x + 1 and y = x − 1 because sine oscillates between −1 and 1. 45. There is no amplitude because the function is not bounded. f (x) 5 4 3 2 1 C-21 h(t 3π 2 2π t −2π 3π− 0 π−2 −π 2 −1 −2 −3 −4 −5 −6 47. The graph is symmetric with respect to the y-axis and there is no amplitude because the function’s bounds decrease as \|x\| grows. There appears to be a horizontal asymptote at y = 0. π 2 π 3π 2π 2 x 3π−− 2π 2 −π −2 π −1 −2 −3 −4 −5 y 5 4 3 2 1 π 2π 3π 4π 5π x −4π 0 −5π −3π−2π−π −1 −2 −3 −4 −5 5π−−2π 4π−3 2π−3 −π 3 5 4 3 2 1 0 π− 3 −1 −2 −3 −4 −5 −3π 8π− 7π−3 3 π 3 2π 3 π 4π 3 5π 3 2π 7π 3 8π 3 3π t Section 6.2 21. Amplitude: 1; period: 4π; midline: y = 0; maximum: y = 1 occurs at t = 11.52; minimum: y = −1 occurs at t = 5.24; phase shift: − 10π _ ; vertical shift: 0; for 3 one period, the graph starts at 0 and ends at 4π. f(t) 6 5 4 3 2 1 −4π −2π 0 −1 −2 −3 −4 −5 −6 1. Since y = csc x is the reciprocal function of y = sin x, you can plot the reciprocal of the coordinates on the graph of y = sin x to obtain the y-coordinates of y = csc x. The x-intercepts of the graph y = sin x are the vertical asymptotes for the graph of y = csc x. 3. Answers will vary. Using the unit circle, one can show that tan (x + π) = tan x. 5. The period is the same: 2π 7. IV 9. III 11. Period: 8; horizontal shift: 1 unit to the left 17. −cot x cos x − sin x 13. 1.5 15. 5 2π 4π 6π 8π t 19. Stretching factor: 2; π _ period: ; asymptotes: 4 π 1  _ _ + πk  + 8, where x = 4 2 k is an integer f (x) 8 6 4 2 25. Amplitude: 2, midline: y = 3; 2π x  + 3 _ 5 23. Amplitude: 2, midline: y = −3; period: 4; equation: π f (x) = 2sin  _ x  −3 2 period: 5; equation: f (x) = −2cos  27. Amplitude: 4, midline: y = 0; period: 2; equation: π f (x) = −4cos  π  x − _   2 period: 2; equation: f (x) = 2cos(πx) + 1 π π 33. sin  _ _  = 1 35. 2 2 with respect to the origin. 31. 0, π 37. f (x) = sin x is symmetric 29. Amplitude: 2, midline: y = 1; π _ 39. , 3 5π _ 3 41. Maximum: 1 at x = 0; minimum: −1 at x = π x π 16 π 8 3π 16 π 4 3π− π− 4 16 π− π− 16 8 −2 −4 −6 −8 21. Stretching factor: 6; period: 6; asymptotes: x = 3k, where k is an integer 23. Stretching factor: 1; period: π; asymptotes: x = πk, where k is an integer m(x) 16 12 8 4 0 −4 −8 −12 −16 −6 −3 3 6 x p(x) 8 6 4 2 π 4 π 2 3π 4 π x −p 3π− 4 0 π− π− 4 2 −4 −6 −8 ODD ANSWERS C-22 25. Stretching factor: 1; period: π; asymptotes: π _ x = + πk, where k is an 4 integer f(x) 16 12 8 4 π 4 π 2 3π 4 π 5π 4 x −π 3π− 4 0 2 π− π− 4 −8 −12 −16 27. Stretching factor: 2; period: 2π; asymptotes: x = πk, where k is an integer π 37. y =  tan 39. f (x) = csc(2x) 41. f (x) = csc(4x) 43. f (x) = 2csc x 1 _ 45. f (x) = tan(100πx) 2 f (x) 47. 8 6 4 2 5π− 12 π 2 π 3π 2 2π x π− 4 π− π− 6 3 0 π− 12 −2 −4 −6 −8 π 12 π 6 π 4 π 3 5π 12 x 8 6 4 2 f(x) 16 12 8 4 −2π − 3π 2 −π 0 π− 2 −8 −12 −16 π 4 π 2 3π 4 π x 49. f(x) −π 8 6 4 2 3π− 4 2 0 π− π− 4 −2 −4 −6 −8 period: ; asymptotes: x = π 2π _ _ 5 10 where k is an integer k, π 4 π 2 3π 4 π x −π 3π− 4 2 0 π− π− 4 −2 −4 −6 −8 29. Stretching factor: 4; 31. Stretching factor: 7; period: ; asymptotes: x = π 2π _ _ k, 3 6 where k is an integer f (x) 16 12 8 4 π 6 π 3 π 2 2π 3 x 2π 4 −8 −12 −16 f(x) 51. 14 7 0 π− 10 −7 π 10 π 5 3π 10 2π 5 x 2π− 3π−5 π−10 5 −14 33. Stretching factor: 2; period: 2π ; asymptotes: x = − π _ + πk, 4 where k is an integer −2000π −1500π −1000π f (x500π 0 −1 −2 −3 −4 −5 −6 −7 −8 500π 1000π 1500π 2000π x 3π 2 − 5π 4 − −π − 3π 4 f(x1 −2 −3 −4 −5 −6 −7 −8 −9 π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 x 9π 4 − −2π − 7π 4 53. y 4 3 2 1 π 2 π 3π 2 5π 2π 2 x 5π− 2 −2π 3π− 2 −π 0 π− 2 −1 −2 −3 −4 ; period: 2π ; asymptotes: x = π 7 _ _ + πk, 35. Stretching factor: 5 4 where k is an integer f (x) π 55. a. f(x) 7π −4 5π −4 − 7 6 5 4 3 2 1 3π −4 0 π 4 −1 −2 −3 −4 −5 −6 −7 π 4 3π 4 5π 4 7π 4 9π 4 x 16 14 12 10 − 2 π− 3 π− −2 6 −4 −6 −8 c. x = − π π _ _ and x = ; the distance 2 2 grows without bound as ∣ x ∣ π _ approaches –i.e., at right angles 2 to the line representing due north, the boat would be so far away, the fisherman could not see it d. 3; when x = − π _ , the boat is 3 km away 3 π _ e. 1.73; when x = , the boat is about 6 1.73 km away f. 1.5 km; when x = 0 ODD ANSWERS 3. Amplitude: 3; period: is 2π; midline: y = 0; no asymptotes C-23 f (x) 4 3 2 1 13π− −4π 3 11π− 3 10π− 3 −3π 8π− 3 7π− 3 −2π 5π− 3 4π− 3 −π 2π− 3 π− −1 3 −2 −3 −4 π 3 2π 3 π 4π 3 5π 3 2π 7π 3 8π 3 3π 10π 3 11π 3 x 5. Amplitude: 3; period: is 2π; midline: y = −4; no asymptotes f(x) π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 −2π 9π 4 x 7π− 4 3π− 2 5π− 4 −π 3π− 4 1 π− π− 4 −1 2 −2 −3 −4 −5 −6 −7 7. Amplitude: 6; period: is 2π _ ; 3 midline: y = −1; no asymptotes f(x1 −2 −3 −4 −5 −6 −7 2π π − − 9 3 π 9 2π 9 π 3 4π 9 5π 9 2π 3 x − 2π 3 5π −9 4π 9 − 9. Stretching factor: none; period: π; midline: y = −4; π _ asymptotes: x = + πk, 2 where k is an integer f(x) 10 8 6 4 2 x π 2 π −π π− 2 −2 −4 −6 −8 −10 11. Stretching factor: 3; π _ ; midline: y = −2; period: 4 π π _ _ + asymptotes: x = k, 4 8 where k is an integer f(x) 13. Amplitude: none; period: 2π; no phase shift; asymptotes: π _ x = k, where k is an odd integer 2 f(x) 5 4 3 2 1 0 π− 2 −1 −2 −3 −4 −5 −2π 3π− 2 −π π 2 π 3π 2 2π x π 16 π 8 3π 16 π 4 x 10 16 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 3π π − −4 − 16 π 8 57. a. h(x) = 2 tan  b. f (x) π _ 120 x  100 90 80 70 60 50 40 30 20 10 0 −5 −10 −20 5 10 15 20 25 30 35 40 45 50 55 60 x d. As x approaches c. h(0) = 0: after 0 seconds, the rocket is 0 mi above the ground; h(30) = 2: after 30 seconds, the rockets is 2 mi high; 60 seconds, the values of h(x) grow increasingly large. As x → 60 the model breaks down, since it assumes that the angle of elevation continues to increase with x. In fact, the angle is bounded at 90 degrees. Section 6.3 1. The function y = sin x is one-to-one on  − π π  ; thus, this _ _ , 2 2 interval is the range of the inverse function of y = sin x, f (x) = sin−1 x. The function y = cos x is one-to-one on [0, π]; thus, this interval is the range of the inverse function of π _ y = cos x, f (x) = cos−1 x. 3. is the radian measure of an 6 π angle between − π _ _ 5. In order for whose sine is 0.5. and 2 2 any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval  − π π  so that it is one-to-one and possesses an _ _ , 2 2 7. True. The angle, θ1 that equals arccos (−x), inverse. x > 0, will be a second quadrant angle with reference angle, θ2, where θ2 equals arccos x, x > 0. Since θ2 is the reference angle for θ1, θ2 = π − θ1 and arccos(−x) = π − arccos x 3π _ 17. 1.98 11. 4 21. 1.41 29. −0.71 radians 9. − π _ 6 19. 0.93 27. 0.71 radians 4 _ 35. 33. 5 13. − π π _ _ 15. 3 3 23. 0.56 radians 25. 0 31. − π _ radians 4 — x 2 − 1 √ _ x 39. 41. 5 _ 13 37. x + 0.5 __ − __ − x 2 + 2x √ 2x + 1 √ _ x + 1 — 45. 43. 49. Domain: [−1, 1]; range: [0, π] y — 2x + 1 √ _ x 47. t 51. x = 0 53. 0.395 radians 55. 1.11 radians 57. 1.25 radians 59. 0.405 radians 61. No. The angle the ladder makes with the horizontal is 60 degrees. π −1 1 x −π Chapter 6 Review exercises 1. Amplitude: 3; period: is 2π; midline: y = 3; no asymptotes f (x) 6 5 4 3 2 1 π− 2 −2 −3 −4 −5 −6 −2π 3π− 2 −π x π 2 π 3π 2 2π ODD ANSWERS C-24 15. Amplitude: none; period: no phase shift; asymptotes: π _ x = k, where k is an integer 5 f(x) 2π _ ; 5 17. Amplitude: none; period: 4π ; no phase shift; asymptotes: x = 2π k, where k is an integer 5. Amplitude: 1; period: 2π ; midline: y = 1 7π− 3 −2π 5π− 3 4π− 3 f(x) 2 1.5 1 0.5 −π π− 2π− 3 3 −0.5 −1 −1.5 −2 π 3 2π 3 π 4π 3 5π 3 x f(x) 5 4 3 2 1 π 10 π 5 3π 10 x π 2π 3π 4π x −4π 0 −3π −π −2π −1 −2 −3 −4 −5 7. Amplitude: 3; period: 6π ; midline: y = 0 f (x) 3π 10 − π 5 − − 10 8 6 4 2 π 10 0 −2 −4 −6 −8 −10 21. Amplitude: 8,000; 19. Largest: 20,000; smallest: 4,000 period: 10; phase shift: 0 23. In 2007, the predicted population is 4,413. In 2010, the population will be 11,924. π _ 33. 3 π _ 31. 4 27. 10 seconds 25. 5 in. 35. No solution π _ 29. 6 12 _ 37. 5 39. The graphs are not symmetrical with respect to the line y = x. They are symmetrical with respect to the y-axis 3π 2 2π x −2π 3π− 2 −π 0 π− 2 −1 −2 −3 −4 −5 5π− 6 2π− 3 π− 2 π− 3 41. The graphs appear to be identical. y 1 0.8 0.6 0.4 0.2 −0.2 0 −0.2 −0.4
−0.6 −0.8 −1 −1 −0.6 0.2 0.6 1 x −1 −0.6 y 1 0.8 0.6 0.4 0.2 0 −0.2 −0.2 −0.4 −0.6 −0.8 −1 0.2 0.6 1 x 9π− 4 −2π 7π− 4 3π− 2 5π− 4 −π 3π− 4 4 3 2 1 7π− −3π 2 5π− 2 −2π 3π− 2 −π π − 2 −1 −2 −3 −4 π 2 π 3π 2 2π 5π 2 3π 7π 2 4π 9π 2 5π 2 11π 6π 2 13π 7π 2 15π 8π 2 17π x 9. Amplitude: none; period: π; midline: y = 0; asymptotes: + πk, where k is some x = 2π _ 3 integer 11. Amplitude: none; period: midline: y = 0; asymptotes: π _ x = k, where k is some integer 3 2π _ ; 3 f (x) f(x) 8 6 4 2 π− 6 −2 −4 −6 −8 π 6 π 3 π 2 2π 3 5π 6 π 7π 6 x 2π− 3 π− 2 π− 3 20 15 10 5 π− 6 −5 −10 −15 −20 π 6 π 3 π 2 2π 3 x 13. Amplitude: none; period: 2π; midline: y = −3 π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 x f (x) 6 5 4 3 2 1 π− π− 4 −1 2 −2 −3 −4 −5 −6 −7 −8 −9 −10 15. Amplitude; 2; period: 2; midline: y = 0; f (x) = 2sin(π(x − 1)) 17. Amplitude; 1; period: 12; phase shift: −6; midline: y = −3 19. D(t) = 68 − 12sin  π _ 21. Period: ; horizontal 6 shift: −7 23. f (x) = sec(πx); period: 2; phase shift: 0 25. 4 π _ 12 x  Chapter 6 practice test 1. Amplitude: 0.5; period: 2π ; midline: y = 0 3. Amplitude: 5; period: 2π ; midline: y = 0 y y 0.5 0.25 0 π− −π 2 −0.25 −0.5 −2π 3π− 2 π 2 π 3π 2 2π x −2π 3π− 2 −π 6 5 4 3 2 1 0 π− 2 −1 −2 −3 −4 −5 −6 27. The views are different because the period of the wave is Over a bigger domain, there will be more cycles of the graph. n(x) n(x) 1 _ . 25 π 2 π 3π 2 2π x 0.02 0.01 −0.2 −0.01 −0.02 0.2 0.4 0.6 0.8 1 1.2 1.4 0.02 0.01 x −0.2 −0.01 −0.02 0.2 0.4 0.6 0.8 1 x ODD ANSWERS 3 _ 29. 5 2π π π π 5π  ,   ,  , π  ,  _ _ _ _ _ 31 33. f (x) = 2 cos  12  x) 5 4 3 2 1 π 24 π 12 π 8 π 6 5π 12 π 4 x π− 4 5π− 24 π− 6 π− π− 8 12 π− 24 −1 −2 −3 −4 −5 3π _ , 2 4π  ,  _ 3 5π  ,  _ 3 7π _ , 6 35. This graph is periodic with a period of 2π. 11π _ 6 , 2π  y 4 3.5 3 2.5 2 1.5 1 0.5 π 2 π 3π 2 2π x −2π 3π− 2 −π π− 2 −0.5 −1 −1.5 −2 π _ 37. 3 π _ 39. 2 x + 1 _ x 41. √ — 1 − (1 − 2x)2 1 — 43. _ 1 + x 4 √ 49. 0.07 radians 45. csc t = 47. False ChapteR 7 Section 7.1 1. All three functions, F, G, and H, are even. This is because F(−x) = sin(−x)sin(−x) = (−sin x)(−sin x) = sin 2 x = F(x), G(−x) = cos(−x)cos(−x) = cos x cos x = cos 2 x = G(x) and H(−x) = tan(−x)tan(−x) = (−tan x)(−tan x) = tan 2 x = H(x). 1 _ 3. When cos t = 0, then sec t = , which is undefined. 0 9. csc t 13. sec 2 x 7. sec x 5. sin x 15. sin 2 x + 1 17. 21. tan x 11. −1 1 _ cot x 19. 1 _ sin x + 1 27. ± — 1 − sin 2 x √ ___________ sin x 23. −4sec x tan x 25. ± √ _________ 1 _____ cot 2 x 29. Answers will vary. Sample proof: cos x − cos 3x = cos x(1 − cos 2 x) = cos x sin 2 x 31. Answers will vary. Sample proof: 1 + sin 2 x _ = cos 2 x = sec 2 x + tan 2 x sin 2 x _ cos 2 x 1 _ cos 2 x + = tan 2 x + 1 + tan 2 x = 1 + 2tan 2 x 33. Answers will vary. Sample proof: cos 2 x − tan 2 x = 1 − sin 2 x −(sec 2 x − 1) = 1 − sin 2 x − sec 2 x + 1 = 2 − sin 2 x − sec 2 x 37. False 35. False 39. Proved with negative and Pythagorean identities 41. True 3 sin 2 θ + 4 cos 2 θ = 3 sin 2 θ + 3 cos 2 θ + cos 2 θ = 3(sin 2 θ + cos 2 θ)+ cos 2 θ = 3 + cos 2 θ Section 7.2 1. The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those π _ − x. Then angles measures x, the second angle measures 2 π − x  . The same holds for the other cofunction sin x = cos  _ 2 identities. The key is that the angles are complementary. 3. sin(−x) = −sin x, so sin x is odd. cos(−x) = cos(0 − x ) = cos x, so cos x is even. C-25 9. −2 − √ — 3 7. — — 6 − √ √ 2 _________ 4 — 3 √ _ 2 sin x 1 _ 13. − cos x − 2 x  19. tan  ___ 10 2 − √ 6 √ _________ 4 23. — — y 25. sin x 1 0.8 0.6 0.4 0.2 −π π− 2 −0.4 −0.6 −0.8 −1 −2π 3π− 2 π 2 π 3π 2 2π x 5. — — 2 + √ √ 6 _________ 4 — 2 √ _ 2 11. − sin x − cos x — 2 √ _ 2 15. csc θ 17. cot x 21. sin(a − b = 15 — —  2 √ 2 _ 3 cos(a + b 15 = — —  2 √ 2 _ 3 π − x  27. cot  _ 6 −2π 5π− 3 4π− 3 −π 2π− 3 π + x  29. cot  _ 4 y 10 8 6 4 2 π− 3 −2 −4 −6 −8 −10 π 3 3π 3 π 4π 3 5π 3 2π x y 10 8 6 4 2 π− 2 π− 4 −4 −6 −8 −10 π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 2π x −2π 7π− 4 3π− 2 5π− 4 −π 3π− 4 31. (sin x + cos x) — √ 2 _ 2 y 1 0.8 0.6 0.4 0.2 π− π− 4 2 −0.4 −0.6 −0.8 −1 −2π 7π− 4 3π− 2 5π− 4 −π 3π− 4 π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 2π x 33. They are the same. g(x) = sin(9x) − cos(3x) sin(6x) 35. They are the different, try 37. They are the same. 39. They are the different, try g(θ) = 41. They are different, try g(x) = 2tan θ _ 1 − tan 2 θ tan x − tan(2x) __ 1 + tan x tan(2x) , or 0.9659 43. − — 3 − 1 √ _______ or − 0.2588 2 2 √ — π  = 47. tan  45. — 2 √ 2 π  tan x + tan  _ 4 _________________ π  1 − tan x tan  _ 4 tan x + 1 tan x + 1 _ __ = 1 − tan x 1 − tan x(1) = ODD ANSWERS C-26 49. cos(a + b) _ = cos a cos b cos a cos b _ − cos a cos b sin a sin b _ cos a cos b = 1 − tan a tan b 51. cos (x + h) − cos x __ = h cos x(cos h − 1) − sin x sin h ___ h cos x cos h − sin x sin h − cos x ___ h cos h − 1 _ = cos x h − sin x = sin h _ h 53. True expand the right hand side. 55. True. Note that sin(α + β) = sin(π − γ) and Section 7.3 1. Use the Pythagorean identities and isolate the squared term. 3. 1 − cos x sin x ________ ________ , 1 + cos x sin x — 1 − cos x and √ √ — 1 + cos x , respectively. , multiplying the top and bottom by 5. a. — 3 √ 7 _ 32 c. — 3 √ 7 _ 31 7. a. — √ 1 3 __ ____ c. − √ b. − 2 2 — 3 b. 31 _ 32 — 5 2 √ _ 5 9. cos θ = − , sin θ = sec θ = − , cot θ = − − √ __ 3 2 15. — 5 √ _ 5 1 _ , tan θ = − , csc θ = √ 2 — 5 , π  11. 2sin  __ 2 — — 2 − √ √ __ 2 13. 2 17. 2 + √ — 3 19. −1 − √ — 2 21. a. 23. a. b. − 3 __ c. − 2 — 2 √ 13 _ 13 — 3 √ 13 _ 13 — 6 √ √ 10 _ _ 4 4 — 13 13 2 √ 3 √ 2 _ _ _ 29. cos(74°) 31. cos(18x) 33. 3sin(10x) , , 3 13 13 — √ 15 _ 3 119 _ 169 120 _ 169 120 _ 119 25. b. c. , − , − — — 27. 35. −2sin(−x)cos(−x) = −2(−sin(x)cos(x)) = sin(2x) 37. tan 2(θ ) = sin(2θ) __ 1 + cos(2θ) 2sin(θ )cos(θ ) __ 2cos 2(θ ) 1 + cos(12x) __ 2 2sin(θ )cos(θ ) __ 1 + cos 2(θ ) − sin 2(θ ) tan 2(θ ) = sin(θ ) _ cos(θ ) 3 + cos(12x) − 4cos(6x) ___ 8 39. 41. tan 2(θ ) = tan 2(θ ) = cot(θ )tan 2(θ ) = tan(θ ) 43. 2 + cos(2x) − 2cos(4x) − cos(6x) ____ 32 45. 3 + cos(4x) − 4cos(2x) ___ 3 + cos(4x) + 4cos(2x) 47. 1 − cos(4x) __ 8 49. 3 + cos(4x) − 4cos(2x) ___ 4(cos(2x) + 1) 53. 4sin x cos x (cos 2 x − sin 2 x) 51. (1 + cos(4x)) sin x __ 2 55. 2tan x _ = 1 + tan 2 x 2sin x _ cos x _ sin 2 x _ 1 + cos 2 x = 2sin x _ cos x __ cos 2 x + sin 2 x __ cos 2 x = cos 2 x _ 1 2sin x _ cos x . sin(2x) _ cos(2x) = tan(2x) 57. 2sin x cos x __ = 2cos 2 x − 1 = 2sin x cos x = sin(2x) 59. sin(x + 2x) = sin x cos(2x) + sin(2x)cos x = sin x(cos 2 x − sin 2 x) + 2sin x cos x cos x = sin x cos 2 x − sin 3 x + 2sin x cos 2 x = 3sin x cos 2 x − sin 3 x 61. 1 + cos(2t) __ sin(2t) − cos t = 1 + 2cos 2 t − 1 __ 2sin t cos t − cos t 2cos 2 t __ cos t(2sin t − 1) = = 2cos t _ 2sin t − 1 63. (cos 2 (4x) − sin 2(4x) − sin(8x))(cos 2(4x) − sin 2(4x) + sin(8x)) = (cos(8x) − sin(8x))(cos(8x) + sin(8x)) = cos 2(8x) − sin 2(8x) = cos(16x) Section 7.4 1. Substitute α into cosine and β into sine and evaluate. 3. Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a sin(3x) + sin x __ cos x product are easier to solve. For example: = 1. When converting the numerator to a product the equation — — — 3 − 1) 5. 8(cos(5x) − cos(27x)) 2sin(2x)cos x __ = 1. cos x becomes: 1 _ 7. sin(2x) + sin(8x) (cos(6x) − cos(4x)) 9. 2 13. 2cos(7x) 11. 2cos(5t)cos t 15. 2cos(6x)cos(3x) 1 1 1 3 − 2) ( √ 3 ) (1 + √ ( √ _ _ _ 17. 19. 21. 4 4 4 1 _ 23. cos(80°) − cos(120°) (sin(221°) + sin(205°)) 25. 2 — 27. √ 2 cos(31°) 31. 2sin(−1.5°)cos(0.5°) 33. 2sin(7x) − 2sin x = 2sin(4x + 3x) − 2sin(4x − 3x) = = 2(sin(4x)cos(3x) + sin(3x)cos(4x)) − 2(sin(4x)cos(3x) − sin(3x)cos(4x)) = 2sin(4x)cos(3x) + 2sin(3x)cos(4x)) − 2sin(4x)cos(3x) + 2sin(3x)cos(4x)) = 4sin(3x)cos(4x) 35. sin x + sin(3x) = 2sin  −2x ____ 2 = 2(2sin x cos x)cos x = 4sin x cos 2 x  = 2sin(2x)cos x 29. 2cos(66.5°)sin(34.5°)  cos  4x ___ 2 37. 2tan x cos(3x) = = 2sin x cos(3x) __ cos x 2(0.5 (sin (4x) − sin (2x))) ___ cos x 1 _ cos x (sin(4x) − sin(2x)) = = sec x (sin(4x) − sin(2x)) 45. It is an identity. 39. 2cos(35°)cos(23°), 1.5081 41. −2sin(33°)sin(11°), − 0.2078 1 __ (cos(99°) − cos(71°)), −0.2410 43. 2 47. It is not an identity, but 2cos 3 x is. 49. tan(3t) 55. Start with cos x + cos y. Make a substitution and let x = α + β and let y = α − β, so cos x + cos y becomes cos(α + β) + cos(α − β) = 53. −sin(14x) 51. 2cos(2x) = cos α cos β − sin α sin β + cos α cosβ + sin α sin β = 2cos α cos β Since x = α + β and y = α − β, we can solve for α and β in terms of x and y and substitute in for 2cos α cos β and get x − y 2cos   . _ 2 x + y _ 2  cos  ODD ANSWERS 57. cos(3x) + cos x __ cos(3x) − cos x = 2cos(2x) cos x __ −2sin(2x) sin x = −cot(2x) cot x 59. cos(2y) − cos(4y) __ = sin(2y) + sin(4y) −2sin(3y) sin(−y) __ 2sin(3y) cos y = 2sin(3y) sin(y) __ 2sin(3y) cos y = tan y 61. cos x − cos(3x) = −2sin(2x)sin(−x) = 2(2sin x cos x)sin x π − t  = 63. tan  _ 4 = 4sin2 x cos x π  − tan t tan  _ 4 __ = π  tan(t) 1 + tan  _ 4 1 − tan t ________ 1 + tan t Section 7.5 1. There will not always be solutions to trigonometric function equations. For a basic example, cos(x) = −5. 3. If the sine or cosine function has a coefficient of one, isolate the term on one side of the equals sign. If the number it is set equal to has an absolute value less than or equal to one, the equation has solutions, otherwise it does not. If the sine or cosine does not have a coefficient equal to one, still isolate the term but then divide both sides of the equation by the leading coefficient. Then, if the number it is set equal to has an absolute value greater than one, the equation has no solution. 7π _ 4 29π _ 18 19. 7. 37 ___ 6 17. 15. 29. 0, π π _ 9. , 4 5π _ 4 π _ , 18 3π _ , 4 7π _ , 6 13π ____ , 12 , 5π ____ , 4 25π _ , 18 29 ___ , 6 3π π _ _ 11. , 4 4 17π _ , 18 25 ___ , 6 π 5π 2π _ _ _ 5. , 4 3 3 π 7π 11π 13π 5π _ _ _ _ _ 13. , , , 4 4 18 18 6 3π 19π 11π 5π 17 21π 13 5 1 __
_ ____ ____ ___ ___ ____ ___ __ __ 21. , , , , , , , , 12 12 12 12 6 12 6 6 6 π π 5π 5π 5π 3π π __ __ ___ ___ ___ ___ __ 23. 0, 25. 27. , π, , π 11π 7π 1 1 ___ ____ __ __ 31. π − sin−1  −  ,  , 2π + sin− 2π π 1 1 1   ,  sin−1    ,  sin−1    ,  sin− − + 33. 3 10 3 3 10 3 10 3 9 9 9 5π 4π 1 1 1   ,   ,  sin−1   sin−1    ,  sin− 10 3 10 3 10 2 2 2  , π + sin−1   , π − sin−1  37. θ = sin−1   , _ _ _ 35. 0 3 3 3 4π π 5π π 3π 2 2π − sin−1  ___ __ ___ __ ___  _ 41. 0, 39. , π, , , 3 3 3 6 6 2 43. There are no solutions. 1 45. cos−1  __  1 − √ 3 1 47. tan−1  _  √ 2 1 π + tan−1  _  √ 2 49. There are no solutions. 2π 3π 5π 7π 4π π _ _ _ _ _ _ 53. 0, 55 3π  , , π − sin−1   , 57. sin−   , 2π − cos−1  − 59. cos− 5π  ,  , 2π − cos−1  − , cos−1  − _ _ _ _ 61. 4 4 3 3 3 3 2 2   , 2π − cos−1   , cos−1  − 63. cos−1   , 2π − cos−   , 2π − cos−1  __  1 − √ 3 1 29 − 5   , π + tan−1  _  − √ 2 1 29 − 5   , 2π + tan−1  _  − √ 2 29 − 5   51. There are no solutions. 7   29 − -27 5π _ 3 π _ 65. 0, , π, 2 π 1 1  ,  , 2π − cos−1  − , cos−1  − _ _ _ 67. 4 4 3 71. π + tan−1(−2), 3π _ 2 69. There are no solutions. 3 3 __ π + tan−1  −  , 2π + tan−1(−2), 2π + tan−1  −  _ 2 2 73. 2πk + 0.2734, 2πk + 2.8682 75. πk − 0.3277 77. 0.6694, 1.8287, 3.8110, 4.9703 79. 1.0472, 3.1416, 5.2360 3π 1 1 81. 0.5326, 1.7648, 3.6742, 4.9064 83. sin−1   ,  , π − sin−1  _ _ _ 4 4 2 π 3π π _ _ _ 85. 87. There are no solutions. 89. 0, , π, , 2 2 2 3π _ 2 91. There are no solutions. 93. 7.2° 95. 5.7° 97. 82.4° 99. 31.0° 103. 59.0° 101. 88.7° 105. 36.9° Section 7.6 1. Physical behavior should be periodic, or cyclical. cumulative rainfall is always increasing, a sinusoidal function would not be ideal here. 7. 5sin(2x) + 2 π x  − 1 5. y = −3cos  _ 6 11. y = tan  9. y = 4−6cos  xπ ___  2 3. Since xπ ___  8 13. Answers will vary. Sample answer: This function could model the average monthly temperatures for a city in the northern hemisphere. 15. Answers will vary. Sample answer: This function could model the population of an invasive fish species in thousands over the next 80 years. 8 16 24 32 40 48 56 64 72 80 x 60 40 20 y 0 y 100 80 60 40 20 0 2 4 6 8 10 12 x 21. 2:49 23. 4:30 27. Floods: April 16 to 17. 75 °F 19. 8 a.m. 25. From June 15 to November 16 July 15. Drought: October 16 to January 15. 1 _ 29. Amplitude: 8, period: , frequency: 3 Hz 3 1 _ 31. Amplitude: 4, period: 4, frequency: Hz 4 t  + 800 + 160 π 33. P(t) = −19cos  _ _ t 12 6 π 35. P(t) = −33cos  _ t  + 900(1.07)t 6 37. D(t) = 10(0.85)t cos(36πt) 39. D(t) = 17(0.9145)tcos(28πt) 41. 6 years 45. Spring 2 comes to rest first after 8.0 seconds. π 49. y = 6(4)x + 5sin  _  2 x π 53. y = 3 (2)xcos  _ x  + 1 2 π 51. y = 4(−2)x + 8sin  _ x  2 47. 234.3 miles, at 72.2° 43. 15.4 seconds Chapter 7 Review exercises 1. sin−1  7π ___ , 6 3. — — — 3 3 3 √ √ √  , 2π − sin−1   , π + sin−1   , π − sin−1  _ _ _ 3 3 3 11π 1 1  7. 1  , π − sin−1  5. sin−1  ____ _ _ 4 4 6 — √ 2 ____ 13. 2 9. Yes — 3 11. −2 − √ — 3 √  _ 3 ODD ANSWERS C-28 15. cos(4x) − cos(3x)cosx = cos(2x + 2x) − cos(x + 2x)cos x = cos(2x)cos(2x) − sin(2x)sin(2x) − cos x cos(2x)cos x + sin x sin(2x) cos x = (cos 2 x − sin 2 x) 2 − 4cos 2 x sin 2 x − cos 2 x(cos 2 x − sin 2 x) + sin x (2)sin x cos x cos x = (cos 2 x − sin 2 x)2 − 4cos 2 x sin 2 x − cos 2 x(cos 2 x − sin 2 x) + 2 sin 2 x cos 2 x = cos 4 x − 2cos 2 x sin 2 x + sin 4 x − 4cos 2 x sin 2 x − cos 4 x + cos 2 x sin 2 x + 2 sin 2 x cos 2 x = sin 4 x − 4 cos 2 x sin 2 x + cos 2 x sin 2 x = sin 2 x (sin 2 x + cos 2 x) − 4 cos 2 x sin 2 x = sin 2 x − 4 cos 2 x sin 2 x 5 x  17. tan  _ 8 24 _ 25 — 10 27. cot x cos(2x) = cot x (1 − 2sin 2 x , 10 — √ 3 _ 3 21. − 7 _ , 25 25. 19. 23. √ 24 _ 7 — , − 2  — (2)sin 2 x = cot x − cos x _ sin x = −2sin x cos x + cot x = −sin (2x) + cot x 29. 10sin x − 5sin(3x) + sin(5x) ___ 8(cos(2x) + 1) 1 __ (sin(6x) + sin(12x)) 35. 2 , π 39. 43. 7π ___ 4 3π ___ , 4 5π π ___ __ 41. 0, , 6 6 47. 0.2527, 2.8889, 4.7124 51. 3sin  1 _ 57. Amplitude: 3, period: 2, frequency: Hz 2 59. C(t) = 20sin(2πt) + 100(1.4427)t xπ  − 2 _ 2 53. 71.6 33. − 31. 2 9 13 x  x  cos  37. 2sin  _ _ 2 2 3π ___ 2 45. No solution 49. 1.3694, 1.9106, 4.3726, 4.9137 π t  55. P(t) = 950 − 450sin  _ 6 Chapter 7 practice test 1. 1 3 11. 2cos(3x)cos(5x) — — 3 2 − √ 5. − √ 1 13. x = cos−1  __  5 7. 0, π π _ 9. , 2 ,− 3 ,− 4 3 __ __ __ 15. 4 5 5 3π _ 2 17. tan3 x−tan x sec2 x = tan x (tan2 x − sec2 x) = tan x (tan2 x − (1 + tan2 x)) = tan x (tan2 x − 1 − tan2 x) = −tan x = tan(−x) = tan(−x) 19. sin(2x) _ − sin x cos(2x) _ cos x = 2sin xcos x _ sin x − 2 cos2 x−1 _ cos x 1 _ cos x = 2cos x−2cos x + = 1 _ cos x = sec x = sec x 1 __ 60 , frequency: 60 Hz 1 __ 21. Amplitude: , period 4 23. Amplitude: 8, fast period: 1 ___ 500 , slow frequency: 10 Hz period: 1 __ 10 cos (4πt), 31 second , fast frequency: 500 Hz, slow 25. D(t) = 20 (0.9086)t ChapteR 8 Section 8.1 11. b ≈ 3.78 7. β = 72°, a ≈ 12.0, 5. A triangle with two 9. γ = 20°, b ≈ 4.5, c ≈ 1.6 1. The altitude extends from any vertex to the opposite side or to 3. When the line containing the opposite side at a 90° angle. the known values are the side opposite the missing angle and another side and its opposite angle. given sides and a non-included angle. b ≈ 19.9 13. c ≈ 13.70 15. One triangle, α ≈ 50.3°, β ≈ 16.7°, a ≈ 26.7 17. Two triangles, γ ≈ 54.3°, β ≈ 90.7°, b ≈ 20.9 or γ′ ≈ 125.7°, 19. Two triangles, β ≈ 75.7°, γ ≈ 61.3°, β′ ≈ 19.3°, b′ ≈ 6.9 b ≈ 9.9 or β′ ≈ 18.3°, γ′ ≈ 118.7°, b′ ≈ 3.2 21. Two triangles, α ≈ 143.2°, β ≈ 26.8°, a ≈ 17.3 or α′ ≈ 16.8°, β′ ≈ 153.2°, a′ ≈ 8.3 23. No triangle possible 27. 8.6 37. 29.7° 43. A ≈ 39.4, C ≈ 47.6, BC ≈ 20.7 49. 430.2 51. 10.1 57. L ≈ 49.7, N ≈ 56.1, LN ≈ 5.8 61. The distance from the satellite to station A is approximately 1,716 miles. The satellite is approximately 1,706 miles above the ground. 63. 2.6 ft 65. 5.6 km 67. 371 ft 69. 5,936 ft 71. 24.1 ft 73. 19,056 ft 2 75. 445,624 square miles 77. 8.65 ft 2 25. A ≈ 47.8° or A′ ≈ 132.2° 33. 12.2 47. 42.0 55. AB ≈ 2.8 39. x = 76.9°or x = 103.1° 53. AD ≈ 13.8 59. 51.4 feet 41. 110.6° 29. 370.9 45. 57.1 31. 12.3 35. 16.0 Section 8.2 9. 34.7 7. 11.3 11. 26.7 17. 95.5° 19. 26.9° 13. 257.4 21. B ≈ 45.9°, 1. Two sides and the angle opposite the missing side. 3. s is the semi-perimeter, which is half the perimeter of the triangle. 5. The Law of Cosines must be used for any oblique (non-right) triangle. 15. Not possible C ≈ 99.1°, a ≈ 6.4 25. A ≈ 37.8°, B ≈ 43.8°, C ≈ 98.4° 27. 177.56 in 2 29. 0.04 m 2 39. 70.7° 31. 0.91 yd 2 35. 29.1 49. 1.41 41. 77.4° 51. 0.14 59. 7.62 65. 99.9 ft 67. 37.3 miles 69. 2,371 miles 71. 43. 25.0 53. 18.3 61. 85.1 23. A ≈ 20.6°, B ≈ 38.4°, c ≈ 51.1 37. 0.5 47. 43.52 63. 24.0 km 55. 48.98 45. 9.3 33. 3.0 57. 528 6 5 BO 9.18 PH 150.28 x DC 20.78 73. 599.8 miles 75. 65.4 cm 2 77. 468 ft 2 Section 8.3 1. For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations. 3. Determine θ for the point, then move r units from the pole to plot the point. If r is negative, move r units from the pole in the opposite direction but along the same angle. The point is a distance of r away from the origin at an angle of θ from the π  has a positive angle but a 5. The point  −3, __ 2 and π __ negative radius and is plotted by moving to an angle of 2 polar axis. ODD ANSWERS then moving 3 units in the negative direction. This places the π  has a point 3 units down the negative y-axis. The point  3, − _ 2 negative angle and a positive radius and is plotted by first moving π __ to an angle of − and then moving 3 units down, which is the 2 positive direction for a negative angle. The point is also 3 units down the negative y-axis. 7. (−5, 0) 9.  − 13.  √ — 34 , 5.253  — 3 3 √ _____ 2 3  __ , − 2 15.  8 √ π —  __ 2 , 4 11. (2 √ — 5 , 0.464) 17. r = 4csc θ 19. r = 21. r = 3cos θ 23. r = 3sin θ _ cos (2θ) — sin θ _ 2cos 4 θ 3 √ 9sin θ _ cos 2 θ 25. r = 29. x 2 + y 2 = 4x or 27. r = √ __________ 1 __________ 9cos θ sin θ y 2 ___ 4 = 1; circle (x − 2)2 _______ 4 + 61. r = 3cosθ 63. x 2 + y 2 = 16 y C-29 1 2 3 4 5 652 −2 −4 −6 −2 −2 −4 −6 −8 −4−6−8 2 4 6 8 x 67. x 2 + (y + 5)2 = 25 y 10 8 6 4 2 2 4 6 8 10 x −10 −4−6−8 −2 −2 −4 −6 −8 −10 31. 3y + x = 6; line 33. y = 3; line 37. x 2 + y 2 = 4; circle 3π  41.  3, ___ 4 45. 43. (5, π) 47. 39. x − 5y = 3; line 35. xy = 4; hyperbola −4−6−8 69. (1.618, −1.176) 71. (10.630, 131.186°) 73. (2, 3.14) or (2, π) 75. A vertical line with a units left of the y-axis. 77. A horizontal line with a units below the x-axis. 79. 81. p u = 4 49. 51. 83. 53. 55. r = 6 __ 5cos θ − sin θ 2 4 6 8 10 Section 8.4 57. r = 2sin θ 59. r = 2 _ cos . Symmetry with respect to the polar axis is similar to symmetry about the x-axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the is similar to symmetry about the y-axis. π __ line θ = 2 3. Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, π __ limaçon, lemniscate, etc., then plot points at θ = 0, , π and 2 5. The shape of the polar graph is and sketch the graph. determined by whether or not it includes a sine, a cosine, and 7. Symmetric with respect to constants in the equation. 9. Symmetric with respect to the polar axis, the polar axis π __ symmetric with respect to the line θ = 2 , symmetric with respect 3π ___ , 2 to the pole 15. Symmetric with respect to the pole 11. No symmetry 13. No symmetry ODD ANSWERS C-30 17. Circle (θ from 0 to 2π) 19. Cardioid (θ from 0 to 2π) 41. Archimedes’ spiral 43. Archimedes’ spiral (θ from 0 to 3π) (θ from 0 to 3π 1
0 10 20 30 21. Cardioid 23. One-loop/dimpled limaçon 45. (θ from 0 to 8) 47. (θ from −π to π) (θ from 0 to 2π) (θ from 0 to 2π 11 0.5 1 1.5 2 0.5 1 1.5 2 25. One-loop/dimpled limaçon (θ from 0 to 2π) 27. Inner loop/two-loop limaçon 49. (θ from 0 to 2π) 51. (θ from 0 to 3π) 1 2 3 2 4 6 8 10 1 3 5 7 9 1 2 3 4 53. (θ from 0 to 2π) 29. Inner loop/two-loop limaçon 31. Inner loop/two-loop limaçon (θ from 0 to 2π) (θ from 0 to 2π 12 1 3 5 7 9 11 33. Lemniscate (θ from −π to π) 35. Lemniscate (θ from −π to π) 1 2 3 4 1 2 3 37. Rose curve 39. Rose curve (θ from 0 to 2π) 1 2 3 4 1 2 3 4 55. They are both spirals, but not quite the same. 57. Both graphs are curves with 2 loops. The equation with a coefficient of θ has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to 4π to get a better picture. 59. When the width of the domain is increased, more petals of 61. The graphs are three-petal, rose the flower are visible. curves. The larger the coefficient, the greater the curve’s distance from the pole. coefficient, the tighter the spiral. 5π 3 π 3   ,  67.  ___ __ __ __ , , 3 2 3 2 — — 4 4  ,  71.  √ √ π 8 8 ____ __ ____ , , 4 2 2 63. The graphs are spirals. The smaller the 5π π   ,  4, 65.  4, ___ __ 3 3 3π π  , (0, 2π)  , (0, π) ,  0, 69.  0, ___ __ 2 2  and at θ = 7π 5π ___ ___ 4 4 since r is squared 3π ___ , 4 Section 8.5 — −1 1. a is the real part, b is the imaginary part, and i = √ 3. Polar form converts the real and imaginary part of the complex number in polar form using x = r cos θ and y = r sin θ. 5. zn = rn(cos (nθ) + i sin (nθ)) It is used to simplify polar form when a number has been raised to a power. — 9. √ 14.45 — 5 cis(333.4°) 13. 4 √ 11. √ 7. 5 √ — 38 2 — — 17. π  15. 2cis  __ 6 — 3 √ 3 ____ 21. −1.5 − i 2 7 √ 3 _____ 2 7 __ + i 2 — 3 cis(198°) 23. 4 √ 19. −2 √ — 3 − 2i 3 __ 25. cis(180°) 4 ODD ANSWERS 31. 5cis(80°) — 29. 7cis(70°) 37. 9cis(240°) 17π  ____ 24 35. 125cis(135°) 3 cis  27. 5 √ π 33. 5cis   __ 3 3π 39. cis   ___ 41. 3cis(80°), 3cis(200°), 3cis(320°) 4 8π 2π  , 2 4 cis   , 2 4 cis  3 3 ___ ___ 43. 2 √ √ 9 9 15π 7π  2 cis   , 2 √ 2 cis  ____ ___ 8 8 49. 14π  ____ 9 4 cis  45. 2 √ Imaginary Imaginary 47 Real −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 1 2 3 4 5 6 Real 51. Imaginary 53. Imaginary 2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 1 2 3 4 5 6 Real −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 1 2 3 4 5 6 Real 55. Plot of 1 − 4i in the complex plane (1 along the real axis, −4 along the imaginary axis). 59. −2 + 3.46i 61. −4.33 − 2.50i 57. 3.61e−0.59i Section 8.6 1 − y _____ 5 7. y = −2 + 2x 3. Choose one equation to 1. A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, x = f (t) and y = f (t). solve for t, substitute into the other equation and simplify. 5. Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions. ______ x − 1 x 9. y = 3 √  or y = 1 − 5ln  _____ _ 11. x = 2e 2 2 y − 3 y 13. x = 4log   y 3  15. x =  __ _ _ 17  +  19.   __ __ __ x 23. y = x 2 + 2x + 1 = 1 21  25. y =  _____ 2 31.  x(t) = t y(t) = 2sin t + 1 35.  39.  33.  37.  x(t) = 4 cos t y(t) = 6 sin t x(t) = 4 + 2t y(t) = 1 − 3t x(t) = √ y(t) = √ 41.  27. y = −3x + 14 x(t) = √ y(t) = t 10 cos t — 10 sin t 29. y = x + 3 3 − 2 ; Ellipse ; Circle t + 2t — 2 — x(t) = − 1 + 4t y(t) = − 2t 43. Yes, at t = 2 45. 1 t x −3 y 1 2 0 7 3 5 17 C-31 47. Answers may vary: x(t) = t − 1 y(t) = t 2 x(t) = t + 1 y(t) = (t + 2)2 49. Answers may vary: ,  and   x(t) = t y(t) = t 2 − 4t + 4 and  x(t) = t + 2 y(t) = t 2 Section 8.7 3. The arrows show the orientation, the direction 1. Plotting points with the orientation arrow and a graphing calculator of motion according to increasing values of t. parametric equations show the different vertical and horizontal motions over time. 5. The 7. y 9. 10 2−3−4−5−6 −1 −1 −2 11. y 6 5 4 3 2 1 −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 −2−3−4−5−6 − 13. −2.5 −1.5 y 6 5 4 3 2 1 −1 −2 −3 −4 −5 −6 y 6 5 4 3 2 1 −0.5 −1 −2 −3 −4 −5 −t from −1 to 5) 0.5 1.5 2.5 x 1 2 3 4 5 6 x 15. y 17. 6 5 4 3 2 1 −3−4 −1−2 −1 −2 −3 −4 −5 −t from −5 to 5) −3−4−5−6 −1−2 −1 −2 −3 −4 −5 −6 19. y 0 −0.5 −1.0 −1.5 −2.0 −2.5 −3.0 0.5 1.0 1.5 (t from 0 to 360) 2.0 x 2.5 3.0 21. y 30 25 20 15 10 5 (t from 0 to 360) −2−3−4−5−6 −1 1 2 3 4 5 6 x ODD ANSWERS C-32 23. 253 −2 −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 27. y 29. 100 80 60 40 20 −1000 −20 −40 −60 −80 −100 −120 −140 1000 2000 3000 4000 5000 x −15 −10 (t from −1 to 5) y 3.0 2.5 2.0 1.5 1.0 0.5 −1 −0.5 −1.0 −1.5 −2.0 −2.5 −3.0 y 30 25 20 15 10 5 −5 −5 −10 −15 −20 −25 −30 31. y 33. y (t from 0 to 1000) −1000 −800 −600 −400 35 30 25 20 15 10 5 −200 −5 −10 −15 −20 −25 x 200 35. −3 −2 −1 y 3 2 1 −1 −2 −3 1 x 2 (t from −π to 0) 3 51. y 1.5 1 0.5 (t from 0 to 2π) 1 2 3 x −1.5 −1 −0.5 0.5 11.5 x −0.5 −1 −15. 57. (cont.) y 3 2.5 2 1.5 1 0.5 −2−3−4−5 −1 −0.5 −1 −1.5 −2 −2.5 −3 59. (t from 0 to 2π 20 15 10 5 (t from −4π to 6π) (t from −5 to 5) 5 10 15 x 6 5 4 3 2 1 −2−3−4−5−6−7 −1 −1 −2 −3 −4 −5 −6 π (t from − to ) 2 x π 2 1 2 3 4 5 37. y 6 4 2 −2 −1.5 −1 −0.5 0.5 1 x −5 −10 (t from−π to 0) 59. (cont.) −6 −4 −2 2 4 6 x −2 −4 −6 y 1 0.5 (t from −4π to 6π) −10 −5 5 10 15 20 x 53. a = 4, b = 3, c = 6, d = 1 55. a = 4, b = 2, c = 3, d = 3 57. y (t from 0 to 2π) x 0.5 1 6 5 4 3 2 1 −0.5 −1 −2 −3 −4 −5 −6 (t from 0 to 2π2 −1.5 −1 y 3 2.5 2 1.5 1 0.5 −2−3−4−5 −1 −0.5 −1 −1.5 −2 −2.5 −3 y 1 0.5 (t from −4π to 6π) −10 −5 5 10 15 20 x −0.5 −1 −1.5 −2 2 61. The y-intercept changes. x x 63. y(x) = −16  + 20  _ _   15 15 65.  x(t) = 64cos(52°) y(t) = −16t2 + 64tsin(52°) 67. Approximately 3.2 seconds 69. 1.6 seconds 39. There will be 100 back-and-forth motions. opposite of the x(t) equation. 43. The parabola opens up. 41. Take the 45.  x(t) = 5 cos t y(t) = 5 sin t 47. y 1.5 1 0.5 49. y 1.5 1 0.5 (t from 0 to 2π) 71. y 73. (t from 0 to 2π) y 6 4 2 (t from 0 to 2π) −1.5 −1 −0.5 0.5 11.5 x −1.5 −1 −0.5 0.5 11.5 x −10 −5 −0.5 −1 −1.5 −0.5 −1 −1.5 5 10 x −6 −4 −2 2 4 6 x −2 −4 −6 −0.5 −1 −1.5 −2 12 10 8 6 4 2 −2 −4 −6 −8 −10 −12 ODD ANSWERS Section 8.8 Chapter 8 Review exercises C-33 3. They are unit 1. Lowercase, bold letter, usually u, v, w vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1. 5. The first number always represents the coefficient of the i, and the second represents the j. 11. Equal 19. u + v = 〈−5, 5〉, u − v = 〈−1, 3〉, 2u − 3v = 〈0, 5〉 — 23. − 2 √ 29 _ 29 229 15 √ _ j 229 — 29 i + 5 √ _ j 29 — — 10 10 7. 〈7, − 5〉 15. −7i − 3j 9. Not equal 17. −6i − 2j 229 2 √ _ 229 21. −10i − 4j 13. Equal 27. − 25. − i + — — 3. C = 120°, a = 23.1, c = 34.1 1. Not possible 5. Distance of the plane from point A: 2.2 km, elevation of the plane: 1.6 km 7. B = 71.0°, C = 55.0°, a = 12.8 9. 40.6 km 11. 13. (0, 2) 15. (9.8489, 203.96°) 17. r = 8 19. x 2 + y 2 = 7x 29. \|v\| = 7.810, θ = 39.806° 33. −6 35. −12 37. y 3v 31. \|v\| = 7.211, θ = 236.310° 21 23. Symmetric with respect to π __ the line θ = 2 v v1 2−3−4−5−6−7 −1 −1 −2 −3 −4 −5 −6 −7 x x x 25. (θ from 0 to 2π) 27. (θ from 0 to 2π) 39. 41. u + v u − v 2u u − v u + v 2u 43. 45. — — 2 j 49. v = −7i + 3j 2 i + 3 √ — 3 j 47. 〈4, 1〉 51. 3 √ 53. i − √ 55. a. 58.7; b. 12.5 57. x = 7.13 pounds, y = 3.63 pounds 59. x = 2.87 pounds, y = 4.10 pounds 61. 4.635 miles, 17.764° N of E 63. 17 miles, 10.071 miles 65. Distance: 2.868, Direction: 86.474° North of West, or 3.526° West of North 67. 4.924°, 659 km/hr 73. 21.801°, relative to the car’s forward direction 75. Parallel: 16.28, perpendicular: 47.28 pounds 77. 19.35 pounds, 51.65° from the horizontal 79. 5.1583 pounds, 75.8° from the horizontal 69. 4.424° 71. (0.081, 8.602 __ 33. 2.3 + 1.9i 31. cis  − 29. 5  3 3π 4π 37. 3cis   39. 25cis   ___ ___ 2 3 43. π  35. 60cis  __ 2 3π  , 5cis  41. 5cis  7π ___ ___  4 4 1 __ y = 1 45. x 2 + 2 Imaginary 6 5 4 3 2 1 47.  x(t) = −2 + 6t y(t) = 3 + 4t 1 2 3 4 5 6 Real −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 49. y = −2x 5 (t from −1 to 1) 0.5 1.0 1.5 2.0 x y 20 −20 −40 −60 −80 −100 51. a. x(t) = (80 cos (40°))t y(t) = − 16t 2 + (80 sin (40°))t + 4  b. The ball is 14 feet high and 184 feet from where it was launched. c. 3.3 seconds 53. Not equal 10 3 √ _ 10 59. Magnitude: 3 √ 61. 16 55. 4i 10 √ _ j 10 — 2 , Direction: 225° 57. − i, − — — ODD ANSWERS C-34 63. u − v u + v 3v Chapter 8 practice test 1. α = 67.1°, γ = 44.9°, a = 20.9 3. 1,712 miles 5.  1, √ 7. y = −3 (θ from 0 to 2π) 9. y 6 5 4 3 2 1 −2−3−4−5−6 −1 −1 −2 −3 −4 −5 − 11. √ — 106 + 5 __ 13. − 2 — 2 cis(198°) — 5 √ 3 _____ i 2 — 2 cis(18°), 2 √ 15. 4cis (21°) 19. y = 2(x − 1)2 17. 2 √ 21. y 6 5 4 3 2 1 23. −4i − 15j 25. — 2 √ 13 ______ 13 i + — 3 √ 13 ______ j 13 (θ from 0 to 2π) 1 2 3 4 5 6 x −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 ChapteR 9 Section 9.1 1. No, you can either have zero, one, or infinitely many. Examine 3. This means there is no realistic break-even point. graphs. By the time the company produces one unit they are already making profit. y ), graphically, or by addition. 13. (−3, 1) 2) 15.  − 3 , 0  _ 5 21. (6, −6) 5. You can solve by substitution (isolating x or 11. (−1, 72 132  19.  _ _ 5 5 25. No solutions exist. 7. Yes 9. Yes , 31. (−4, 4) 17. No solutions exist 1 1 ___ __  23.  − , 10 2 x + 3   29.  x, 2 1 __ __ 27  35.   33.  _ _ _ 37. ( x, 2(7x − 6)) , 8 2 6 4 5 __ __ 39.  −  41. Consistent with one solution , 3 6 43. Consistent with one solution 45. Dependent with infinitely many solutions 47. (−3.08, 4.91) 49. (−1.52, 2.29) 51  55 EC − BF _ AE − BD DC − AF _ , BD − AE 59. (1,250, 100,000) 63. 24,000 67. 56 men, 74 women 53.  − 57. They never turn a profit. 61. The numbers are 7.5 and 20.5. 65. 790 sophomores, 805 freshman 69. 10 gallons of 10% solution, 15 gallons of 60% solution 71. Swan Peak: $750,000, Riverside: $350,000 the first account, $10,500 in
the second account tops: 45, Low-tops: 15 more information. 73. $12,500 in 75. High- 77. Infinitely many solutions. We need — 3  Section 9.2 1. No, there can be only one, zero, or infinitely many solutions. 3. Not necessarily. There could be zero, one, or infinitely many solutions. For example, (0, 0, 0) is not a solution to the system below, but that does not mean that it has no solution. 2x + 3y − 6z = 1 −4x − 6y + 12z = −2 x + 2y + 5z = 10 5. Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable. 7. No 9. Yes 11. (−1, 4, 2) 13.  − 85 ____ , 107 312 ____ , 107 191 ____  107 1 , 0  15.  1, _ 2 17. (4, −6, 1) 19.  x, 65 − 16x _______ , 27 23. No solutions exist 25. (0, 0, 0) 29. (7, 20, 16) 31. (−6, 2, 1) 17 ___ 13 28 + x 45 ______ ___  , −2  21.  − , 13 27 3 1 4 __ __ __  27.  , − , − 7 7 7 33. (5, 12, 15)  39.  4 1 1 __ __ __ , , 5 5 2 45. (1, 1, 1) 51. 24, 36, 48 49. (6, −1, 0) 37. (10, 10, 10) 35. (−5, −5, −5)  41.  4 2 1 __ __ __ 43. (2, 0, 0) , , 5 5 2  47.  28 23 128 ____ ____ ____ , , 557 557 557 53. 70 grandparents, 140 parents, 190 children 55. Your share was $19.95, Sarah’s share was $40, and your other roommate’s share was $22.05. more information. 61. The BMW was $49,636, the Jeep was $42,636, and adults the Toyota was $47,727. 63. $400,000 in the account that pays 3% interest, $500,000 in the account that pays 4% interest, and 65. The United $100,000 in the account that pays 2% interest. States consumed 26.3%, Japan 7.1%, and China 6.4% of the world’s 67. Saudi Arabia imported 16.8%, Canada imported oil. 15.1%, and Mexico 15.0% 18.6%, and mammals were 17.1% of endangered species 57. There are infinitely many solutions; we need 59. 500 students, 225 children, and 450 69. Birds were 19.3%, fish were Section 9.3 1. A nonlinear system could be representative of two circles that overlap and intersect in two locations, hence two solutions. A nonlinear system could be representative of a parabola and a circle, where the vertex of the parabola meets the circle and the branches 3. No. There also intersect the circle, hence three solutions. does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region. ODD ANSWERS 5. Choose any number between each solution and plug into C(x) and R(x). If C(x) < R(x), then there is profit. — — 3 √ 3 √ 2 2 _____ _____ , 2 2 — √ 62 _____ 8 9.  − 13.  1 __ 4 — —   ,  √ √ 199 199 398 398 ____ ______ ____ ______ , , __ , 4 — 3 √ 2 _____ 2 , − —  3 √ 2 _____ 2 —  √ 62 _____ 8 17. (0, 2), (1, 3) 1 __ 5 − 1) , (1 − √ 2 — __________ 5 )  ,  √ 1 — __ ( √ 2 1 __ 5 − 1) , (1 − √ 2 — 5 7. (0, −3), (3, 0) 11. (−3, 0), (3, 0) 15.  − __________ 19.  − √ 1 — __ ( √ 2 )  21. (5, 0) 29. No solutions exist 23. (0, 0) 25. (3, 0) 27. No solutions exist 31.  − 33. (2, 0) — √ 2 ____ , − 2 —  ,  − √ 2 ____ 2 35. (− √ — √ 2 ____ , 2 —  ,  √ 2 ____ 2 7 , −3), (− √ — — — √ 2 ____ , − 2 7 , 3), ( √ —  √ 2 ____ 2 — √ 2 ____ , 2 —  ,  √ 2 ____ 2 7 , −3), ( √ — — 7 , 3) ___________ 1 — _ ( √ 2 1 73 −5) , _ (7 − √ 2 37.  − √ ___________ 1 — __ ( √ 2 1 __ 73 − 5) , (7 − √ 2 — 73 )   √ — 73 )  39. y 10 8 6 4 2 −2−3−4−5 −1 −2 −4 −6 −8 −10 41. 45. 1 2 3 4 5 x y 10 2−3−4−5 −1 −1 −2 −3 −4 − 10 x 1 2 3 4 5 x −10 −4−6−8 −2 −2 −4 −6 −8 −10 y 5 4 3 2 1 −2−3−4−5 −1 −1 −2 −3 −4 −5 2 4 6 8 x −4−6−8 −2 −2 −4 −6 −8 −10 70 ____ 383 ____ 70 ____ 383 , −2 √ ___ 49.  −2 √ ____  , 35 ___ 29 ___  ,  −2 √ 35 , 2 √ ___ 29 ___ ____  ,  2 √ 35 70 , −2 √ ___ ____ 29 383 , 2 √ ___  2 √ ____ 35  ___ 29 70 ____ 383 51. No solution exists 53. x = 0, y > 0 and 0 < x < 1, √ 57. 2–20 computers — 1 _ x x < y < 55. 12,288 43. 47. Section 9.4 C-35 1. No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example, cannot be 1 _ x 2 + 1 7. − + + + − + + 15 19. 11. 9. 23. − 17. 13. 21. 25. − 1 _____ x + 5 5 _____ x − 8 4 ______ 4x − 1 3 _____ x − 2 5 _____ x − 2 9 _____ x + 2 5 ________ + 2(x + 3) 9 ________ + 5(x + 2) 1 _____ x − 2 5 _______ 2(x − 3) 11 ________ 5(x − 3) 2 _______ (x − 2) 2 1 _____ x − 7 decomposed because the denominator cannot be factored. 3. Graph both sides and ensure they are equal. 5. If we choose x = −1, then the B-term disappears, letting us immediately know that A = 3. We could alternatively plug in 5 _ x = − giving us a B-value of −2. 3 8 _____ x + 3 3 ______ 5x − 2 3 _____ x + 2 8 _____ x − 3 6 ______ 4x + 5 3 4 ________ __ + − 27. 2(x + 1) x 2 4 ___ __ + 29. x 2 x x + 1 _________ + x 2 + x + 3 2x − 1 __________ + x 2 + 6x + 1 2 _________ + x 2 −3x + 9 1 __ + 43. x x + 1 _____ x + 2 1 ___ 8x 3 ________ (4x + 5) 2 7 ________ 2(x + 1) 2 7 _________ 2(3x + 2) 2 4 − 3x __________ + x 2 + 3x + 8 1 _________ + x 2 + x + 1 1 ___________ + 4x 2 + 6x + 9 3 ______ 3x + 2 3 _____ x + 2 2 _____ x + 3 3 _____ x + 3 1 _____ x − 1 4 _____ x − 1 1 ___________ − x 2 + 3x + 25 4x __ x 2 − 6x + 36 2 _______ (x − 7) 2 x + 6 ______ x 2 + 1 1 _____ x + 6 41. − 49. 45. 33. 37. 47. 51. 39. 35. 31. + − + − − − + 1 ______ 2x − 3 4x + 3 _______ (x 2 + 1) 2 3x _____________ (x 2 + 3x + 25) 2 2x + 3 _______ (x + 2) 2 x ________ + 8(x 2 + 4) 9 ___ x 2 + 16 _____ x − 1 53. − − 55. − 57. − 16 ___ x 1 _____ x + 1 5 _____ x − 2 5 ___ 4x 59. − − 2 _______ + (x + 1) 2 3 _________ + 10(x + 2) 5 ________ + 2(x + 2) − 10 − x _________ 2(x 2 + 4) 2 7 _______ (x − 1) 2 5 _______ (x + 1)3 7 _____ x + 8 11 ________ + 2(x + 4) − 7 _________ 10(x − 8) 5 _______ 4(x − 4) Section 9.5 1. No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a 2×2 matrix and the  has no sum. second is a 2×3 matrix. 3. Yes, if the dimensions of A are m × n and the dimensions of B are n × m, both products will be defined 5. Not necessarily. To find AB, we multiply the first row of A by the first column of B to get the first entry of AB. To find BA, we multiply the first row of B by the first column of A to get the first entry of BA. Thus, if those are unequal, then the matrix multiplication does not commute. ODD ANSWERS C-36 7.  11 19 15 94 17 67  9.  −4 2  8 1 11. Undefined; dimensions do not match 13.  9 27 63 36 0 192  15.  −64 −12 −28 −72 −360 −20 −12 −116  17.  21.  1,800 1,200 1,300 800 1,400 600 700 400 2,100 60 41 2 −16 120 −216   23.  19.  20 102  28 28 −68 24 136 −54 −12 64 −57 30 128  25. Undefined; dimensions do not match.  29.  −8 41 −3 40 −15 −14 4 27 42 27.  31.  33. Undefined; inner dimensions do not match. −350 1,050 350 350  −840 650 −530 330 360 250 −10 900 110  35.  39.  43.  47.  51.  55.  37.  1,400 700  −1,400 700 490,000 0 41.   0 490,000 45.   49.  53.  −4 29 21  −27 −3 1 1 −18 −9 −198 505 369 −72 126 91 2 24 −4.5 12 32 −9 −8 64 61    332,500 927,500 −227,500 87,500 −2 3 4  −7 9 −7 −3 −2 −2  −28 59 46 −4 16 7 0 1.6 9 −1   0.5 3 0.5 2 1 2 10 7 10  57.   , n even,  , n odd. 59. Bn =    Section 9.6 1. Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar. 3. No, there are numerous correct methods of using row operations on a matrix. Two possible ways are the following: (1) Interchange rows 1 and 2. Then R 2 = R 2 − 9R 1. (2) R 2 = R 1 −9R 2. Then divide row 1 by 9. 5. No. A matrix with 0 entries for an entire row would have either zero or infinitely many solutions. 7.  0 16 9 −1 \| 4 2  9.  1 5 8 12 3 0 3 4 9 \| 19 4 −7  11. −2x + 5y = 5 6x − 18y = 26 15. 4x + 5y − 2z = 12 y + 58z = 2 3x + 2y = 13 13. −x − 9y + 4z = 53 8x + 5y + 7z = 80 17. No solutions 19. (−1, −2) 8x + 7y − 3z = −5 21. (6, 7) 23. (3, 2) 25.   27.  x, 1 4 1 ___ __ __ , 5 15 2 5 196 ____ ___  31.  , − 33. (31, −42, 87) 39 13 15 15 18 ___ ___ ___  , 13 13 13 x __ 41.  x, − , −1  2 47. (1, 2, 3) (5x + 1)  29. (3, 4)  35.  9 1 21 __ ___ ___ , , 8 20 40 3 1 y  39.  x, y, _ _ − x − 2 2 45. (8, 1, −2) 10 __ , z  7 53. 860 red velvet, 1,340 chocolate 51. No solutions exist. 55. 4% for account 1, 6% for account 2 59. Banana was 3%, pumpkin was 7%, and rocky road was 2% 61. 100 almonds, 200 cashews, 600 pistachios 49.  −4z + 17 __ 7 43. (125, −25, 0) 57. $126 37.  , 3z − , − Section 9.7 1. If A−1 is the inverse of A, then AA−1 = I, the identity matrix. Since A is also the inverse of A−1, A−1 A = I. You can also check by proving this for a 2 × 2 matrix. 3. No, because ad and bc are both 0, so ad − bc = 0, which requires us to divide by 0 in the  . The inverse is formula. found with the following calculation: 1 0 0 1 0 1 0 −1 1  __________ −1 0 0(0) − 1(1)  = I 1 0 5. Yes. Consider the matrix   =   . 9. AB = BA =   = I 13. 1 0 17. There is no inverse A−1 = 21. 15. 1 ___ 17 . AB = BA =  11. AB = BA =  −2 7 1   ___ 9 3 69 −5 5 −3  20 −3 12 1 −1 4 18 60 −168  −56 −140 448 40 80 −280 2 __ 33.  − , − 3 39.  − 35 __ 34 229 _  , − 690 25.   31.  5 1 __ __ , − 3 2 37. (5, 0, −1) 41.  568 ___ 345 13 _ 138  , − 23. 1 ____ 209    ___ −1 3 29 4  __ 19. 7 47 −57 69 10 19 −12 −24 38 −13  0.5 1.5 1 −0.5  27. (−5, 6) 29. (2, 0) , − , −  35.  7, 1 1 11 __ __ ___  , 6 5 2 77 97 __ __  17 34 43.  −  8 37 ___ ___ , 15 30 2 1 −1 −1  0 1 1 −1 0 −1 1 1 0 1 −1 1   45.  10 ____ 123  2 __ , −1, 5 1 __ 47. 2 3 2 1 −7 18 −53 32 10 24 −36 21 9 −9 46 −16 −5  1 ___ 39 49. ODD ANSWERS 511 −1 −1 −1 −1 1  53. Infinite solutions 55. 50% oranges, 25% bananas, 20% apples 57. 10 straw hats, 50 beanies, 40 cowboy hats 59. Tom ate 6, Joe ate 3, and Albert ate 3 61. 124 oranges, 10 lemons, 8 pomegranates Section 9.8 51
. x − 3z = 7 y + 2z = − 5 with infinite solutions C-37 53.  −2 2 1 7 2 −8 5 0 19 −10 22 3 \|  55.  1 0 3 −1 4 0 0 1 2 \| 12 0 −7  59. No solutions exist 57. No solutions exist  63. No inverse exists 1 2 7  __ 61. 6 1 8 67. (−1, 0.2, 0.3) 1 __  77. ( x, 5x + 3) 71. 0 73. 6 75.  6, 2 69. 17% oranges, 34% bananas, 39% apples  79.  0, 0, − 1 __ 2 65. (−20, 40) 1. A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product—even if it does end up being 0. 7. 7 9. −4 11. 0 19. 224 3. The inverse does not exist. 13. −7, 990.7 5. −2 15. 3 29. (2, 5) 37. (−1, 0, 3) 21. 15 23. −17.03 25. (1, 1) 1 __  31.  −1, − 3 , 1, 2  39.  1 __ 2 45. 24 33. (15, 12) 41. (2, 1, 4) 17. −1  27.  1 1 __ __ , 3 2 35. (1, 3, 2) 49. Yes; 18, 38 55. 120 children, 1,080 adult 47. 1 53. $7,000 in first account, $3,000 in 43. Infinite solutions 51. Yes; 33, 36, 37 second account yellow, 6 gal blue 61. Strawberries 18%, oranges 9%, kiwi 10% first movie, 230 for the second movie, 312 for the third movie 65. 20–29: 2,100, 30–39: 2,600, 40–49: 825 400 cranberries, 300 cashews 59. 13 green tomatoes, 17 red tomatoes 57. 4 gal 63. 100 for the 67. 300 almonds, Chapter 9 Review exercises 3. (−2, 3) 1. No 9. (300, 60) 5. (4, −1) 11. (10, −10, 10) 17.  x, 7. No solutions exist 13. No solutions exist 14x 8x ____ ___  , 5 5 23. No solution 25. No solution 19. 11, 17, 33 15. (−1, − 2, 3) 21. (2, − 3), (3, 2) 272−3−4−5 −1 −1 −2 −3 −4 −5 29. y 5 4 3 2 1 −2−3 −1 −1 −2 −3 −4 −5 1 2 3 4 5 6 7 x 31. − 10 33. 7 _ x + 5 − 15 _ (x + 5)2 35. 37. + −4x + 1 3 __ _ x − 5 x 2 + 5x + 25 −16 8  −4 −12 39.  43. Undefined; inner dimensions do not match x − 4 _ x 2 − 2 + 5x + 3 _ (x2 − 2)2 41. Undefined; dimensions do not match 45.  113 28 10 44 81 −41 84 98 −42  47.  −127 −74 176 −2 11 40 28 77 38  49. Undefined; inner dimensions do not match Chapter 9 practice test , − 3. No solutions exist 1. Yes 13x 16x ____ ____ 7.  x,  5 5 — — 17 ), (−2 √ 9. (−2 √ 2 , − √ 112−3−4−5 −1 −1 −2 −3 −4 − — 17 ), (2 √ — 2 , − √ — 17 ), (2 √ — 2 , √ — 17 ) − 13. 17 51 −8 11 5 ______ 3x + 1 15.  17.  2x + 3 ________ (3x + 1)2  12 −20  −15 30 1 __ 19. − 8 14 −2 13 −2 3 −6 1 −5 12 21.  \| 25. (100, 90) 27.  140 −1 11 23. No solutions exist.  , 0  29. 32 or more cell phones per day 1 ____ 100 ChapteR 10 Section 10.1 1. An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant. 3. This special case would be a circle. 5. It is symmetric about the x-axis, y-axis, and the origin. 7. Yes; + x 2 ___ 32 = 1 y 2 ___ 22 9. Yes; x 2 ______ 2   1 __ 2 = 1; endpoints of major axis: (0, 7) and (0, −7); y 2 ______ 2   1 __ 3 = 1 + x 2 ___ 22 y 2 ___ 72 — — + 5 ), (0, −3 √ 11. endpoints of minor axis: (2, 0) and (−2, 0); foci: (0, 3 √ 13; endpoints of major axis: (1, 0) and (−1, 0); (1)2 + 2   1 __ 3  ; foci:  1 1  ,  0, − endpoints of minor axis:  0, _ _ 3 3 (x − 2)2  − _ = 1; endpoints of major + 72 axis: (9, 4), (−5, 4); endpoints of minor axis: (2, 9), (2, − 1); foci: (2 + 2 √ (y − 4)2 _ 52 ), (2 − 2 √ 15. 6 , 4) — — — — ODD ANSWERS C-38 (x + 5)2 _ + 22 (y − 7)2 _ 32 = 1; endpoints of major axis: (−5, 10), 17. (−5, 4); endpoints of minor axis: (−3, 7), (−7, 7); foci: (−5, 7 + √ (x − 1)2 (−5, 7 − √ _ = 1; endpoints of + 32 major axis: (4, 4), (−2, 4); endpoints of minor axis: (1, 6), (1, 2); 5 , 4) 21. foci: (1 + √ (y − 4)2 _ 22 5 , 4), (1 − √ 19 ), (x − 3) ), (3 − 3 √ (y − 5)); endpoints of = 1; — — 2 ); foci: (7, 5), (−1, 5) endpoints of major axis: (3 + 3 √ minor axis: (3, 5 + √ 2 ), (3, 5 − √ — — (x + 5)2 _______ 52 (y − 2)2 _______ 22 — — + 25. 21 , 2) = 1; endpoints of major axis: (0, 2), (−10, 2); 23. endpoints of minor axis: (−5, 4), (−5, 0); foci: (−5 + √ (x + 3)2 (−5 − √ _______ 52 major axis (2, −4), (−8, −4); endpoints of minor axis (−3, −2), 21 , −4), (−3 − √ (−3, − 6); foci: (−3 + √ 27. Foci: (−3, −1 + √ 11 ), (−3, −1 − √ 31. Foci: (−10, 30), (−10, −30) 21 , 2), = 1; endpoints of 21 , −4). 11 ) 29. Focus: (0, 0) (y + 4)2 _______ 22 + — — — — 33. Center: (0, 0); vertices: (4, 0), (−4, 0), (0, 3), (0, −3); 7 , 0), (− √ foci: ( √ y 7 , 0) — — 1 __ 35. Center (0, 0); vertices,  ,  0, − 1 1 1 __ __ __  ;  − , − foci  0, 4 √ 4 √ 2 2 _____ _____ 63 63 5 4 3 2 1 −2−3−4−5 −1 −1 −2 −3 −4 −5 1 2 3 4 5 x y .3 .2 .1 .1 .2 .3 x −.3 −.2 −.1 −.1 −.2 −.3 37. Center (−3, 3); vertices (0, 3), (−6, 3), (−3, 0), (−3, 6); focus: (−3, 3). Note that this ellipse is a circle. The circle has only one focus, which coincides with the center. y 10 7.5 5 2.5 2.5 5 7.5 10 x −10 −7.5 −5 −2.5 −2.5 −5 −7.5 −10 41. Center: (−4, 5); vertices: (−2, 5), (−6, 5), (−4, 6), (−4, 4); 3 , 5) foci: (−4 + √ y 3 , 5), (−4 − √ — — 10 7.5 5 2.5 2.5 5 7.5 x −10 −5−7.5 −2.5 −2.5 −5 −7.5 39. Center: (1, 1); vertices: (5, 1), (−3, 1), (1, 3), (1, −1); foci: 3 ) (1, 1 + 4 √ 3 ), (12−3−4−5 −1 −1 −2 −3 −4 −5 43. Center: (−2, 1); vertices: (0, 1), (−4, 1), (−2, 5), (−2, −3); 3 ) foci: (−2, 1 + 2 √ 3 ), (−2, 1 − 2 √ — — y .75 .5 .25 .25 .5 .75 x −.75 −.5 −.25 −.25 −.5 −.75 45. Center: (−2, − 2); vertices: (0, − 2), (−4, −2), (−2, 0), (−2, −4); focus: (−2, −2) y 2 1 x 1 2 −2−3−4−5 −1 −1 −2 −3 −4 −5 = 1 47. 49. 51. 53. 55. + y 2 x 2 ___ ___ 29 25 (x − 4)2 _______ 25 (x + 3)2 _______ 16 + y 2 x 2 ___ ___ 81 9 (x + 2)2 _______ 4 + + (y − 2)2 _______ 1 (y − 4)2 _______ 4 = 1 = 1 = 1 + (y − 2)2 _______ 9 = 1 — 5 π square units 59. Area = 2 √ x 2 ____ 4h 2 = 1, distance: 17.32 feet y 2 _ + 1 __ h 2 4 63. = 1 57. Area = 12π square units 61. Area = 9π square units + 65. x 2 ____ 400 y 2 ____ 144 67. Approximately 51.96 feet Section 10.2 — — − − 9. Yes x 2 ___ 52 7. Yes = 1 11. segment joining the foci. 1. A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant. 3. The foci must lie on the transverse axis and be in the interior of 5. The center must be the midpoint of the line the hyperbola. y 2 x 2 _ _ 32 = 1 62 − y 2 ___ = 1; vertices: (5, 0), (−5, 0); 62 y 2 x 2 ___ ___ 52 42 6 6 61 , 0), (− √ foci: ( √ 61 , 0); asymptotes: y = _ _ x — 85 ), 92 = 1; vertices: (0, 2), (0, − 2); foci: (0, √ _ _ 22 − 2 2 85 ); asymptotes: y = _ _ x, y = − x 9 9 (y − 2)2 _______ = 1; vertices: (4, 2), (−2, 2); foci: (6, 2), 42 4 4 __ __ (x − 1) + 2, y = − (−4, 2); asymptotes: y = 3 3 (y + 7)2 _______ 72 = 1; vertices: (9, −7), (−5, −7); foci: (x − 1)2 _______ 32 (x − 1) + 2 (0, − √ 15. 13. − — 2 , −7); asymptotes: y = x − 9, y = −x − 5 — − 17. (x − 2)2 _______ 72 2 , −7), (2 − 7 √ (2 + 7 √ (y − 3)2 _______ 32 19. − — — (x + 3)2 _______ 32 (−3 + 3 √ (y − 4)2 _ − 22 (3, 4 −2 √ 21. — — 2 , 3), (−3 − 3 √ (x − 3)2 _ 42 1 __ 5 ); asymptotes: y = 2 = 1; vertices: (0, 3), (−6, 3); foci: 2 , 3); asymptotes: y = x + 6, y = −x = 1; vertices: (3, 6), (3, 2); foci: (3, 4 + 2 √ — 5 ), 1 __ (x − 3) + 4, y = − 2 (x − 3) + 4 − 23. (y + 5)2 _______ 72 foci: (−1, −5 + 7 √ (x + 1)2 _______ 702 101 ), (−1, −5 − 7 √ — = 1; vertices: (−1, 2), (−1, − 12); — 101 ); asymptotes: − 25. y = 1 ___ 10 (x + 1) − 5 (x + 1) − 5, y = − 1 ___ 10 (x + 3)2 _______ 52 (−3 + √ 2 2 2 __ __ __ y = − (x − 3) − 4 (x − 3) − 4, y = − (x + 3) + 4 27. y = 5 5 5 2 29 , 4); asymptotes: y = __ 5 = 1; vertices: (2, 4), (−8, 4); foci: (y − 4)2 _______ 22 29 , 4), (−3 − √ (x + 3) + 4, — — 3 __ 29. y = 4 3 __ (x − 1) + 1, y = − 4 (x − 1) + 1 ODD ANSWERS 31. y Vertex (−7, 0) 10 Vertex (7, 0) Focus (−8.06, 0) −15 −10 −20 −5 33. 5 5 −5 −10 y Focus (8.06, 0) x 10 15 20 10 5 Focus (0, 5.83) Vertex (0, 3) −20 −15 −10 −5 5 x 10 Vertex (0, −3) 20 15 35. −24 −32 Vertex (4, −2) −16 −8 −5 −10 y 16 8 −8 −16 Focus (0, −5.83) Focus (4, 0.83) 8 16 Vertex (4, −8) 24 32 37. Vertex (3, 0) −24 Focus (4, −10.83) y 24 16 8 Focus (3, 7.24) Vertex (3, 6) −32 −24 −16 −8 8 16 24 32 −8 −16 −24 y Focus (3, −1.24) Vertex (−1, −2) 5 Focus (9.1, −2) 39. −20 −15 −10 −5 5 10 15 20 −5 Focus (−1.1, −2) −10 Vertex (9, −2) y 10 5 41. Vertex (−4, −4) −10 −5 43. Vertex (2, −4) 10 5 x Focus (−9.54, −4) −5 −10 Focus (7.54, −4) −16 −8 = 1 45. 47. − y 2 x 2 ___ ___ 9 16 (x − 6)2 _______ 25 − (y − 1)2 _______ 11 = 1 x x x y 24 16 8 Vertex (5, 15) Focus (5, 15.05) 8 x 16 Focus (5, −5.05) Vertex (5, −5) −8 −16 −24 C-39 49. (x − 4)2 _______ 25 − (y − 2)2 _______ 1 = 1 − = 1 53. y 2 ___ 9 (x + 1)2 _______ 9 — 57. y(x(x) = −3 √ x 2 + 1 y — 10 8 6 4 2 2 4 6 8 10 x −10 −8−6 −2−4 −2 −4 −6 −8 −10 61. − x 2 ___ 25 y 2 ___ 25 = 1 y − 51. y 2 ___ 16 (x + 3)2 _______ 25 − = 1 x 2 ___ 25 (y + 3)2 _______ 25 = 1 55. 59. y(x) = 1 + 2 √ y(x) = 1 − 2 √ — x 2 + 4x + 5 — x 2 + 4x + 5 , y 10 8 6 4 2 2 4 6 8 10 x −10 −8−6 −2−4 −2 −4 −6 −8 −10 63. x 2 ____ 100 − y 2 ___ = 1 25 y Fountain 15 10 5 16 8 Fountain −24 −16 −8 8 16 24 x −15 −10 −5 5 10 15 x −5 −10 −15 −8 −16 67. 69. − = 1 (x − 1)2 _______ 0.25 (x − 3)2 _______ 4 − y 2 ____ 0.75 y 2 ___ 5 = 1 65. x 2 ____ 400 − = 1 y 2 ____ 225 y Fountain 24 16 8 −40 −32 −24 −8−16 −8 −16 −24 8 16 24 32 40 x Section 10.3 1. A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix. 3. The graph will open down. 5. The distance between the focus and directrix will increase. 7. Yes y = 4(1)x 2 1 1 x, V: (0, 0); F:  _ _ 11. y 2 = 32 8 1 y, V: (0, 0); F:  0, − _ 13. x 2 = − 4 1 1 x, V: (0, 0); F:  _ _ 144 36 9. Yes (y − 3)2 = 4(2)(x − 2) 1 , 0  ; d: x = − _ 32 1 1  ; d: y = _ _ 16 16 1 , 0  ; d: x = − _ 144 15. y 2 = 17. (x − 1)2 = 4(y − 1), V: (1, 1); F: (1, 2); d: y = 0 7 5 19. (y − 4)2 = 2(x + 3), V: (−3, 4); F:  − , 4  ; d: x = − _ _ 2 2 21. (x + 4)2 = 24(y + 1), V: (−4, −1); F: (−4, 5); d: y = −7 23. (y − 3)2 = −12(x + 1), V: (−1, 3); F: (−4, 3); d: x = 2 14 4  ; d: y = − (y + 3), V: (5, −3); F:  5, − _ _ 25. (x − 5)2 = 5 5 9 11  ; d: y = 27. (x − 2)2 = −2(y − 5), V: (2, 5); F:  2, _ _ 2 2 16 14 4 , 1  ; d: x = (x − 5), V: (5, 1); F:  _ _ _ 29. (y − 1)2 = 3 3 3 16 _ 5 ODD ANSWERS C-40 31. x = −2 y 8 6 4 2 −8−6 −2−4 −2 −4 −6 −8 33. y Focu
s (2, 0) 2 4 6 8 x 24 16 8 Focus (0, 9) −32 −24 −16 −8 8 16 24 32 y = −9 −8 −16 −24 37. −10 −7.5 −5 −2.5 y 2.5 x = 25 6 2.5 x 5 35. y 7.5 5 2.5 x − 5 3 7 3 −10 −7.5 −5 −2.5 x 2.5 39. −2.5 y 5 y = 0 −5 −20 −10 −15 Focus (−4, −2) −5 −10 −15 −20 41. x 5 x = −4 −10 −5 −2.5 −5 −7.5 −10 5 10 15 Focus (0, −5) 25 6 y 10 5 −5 −10 −15 43. y 15 10 5 −5 −10 −15 −5 x = 2 5 10 20 15 Focus (8, −1) x — — 3 )2 = −4 √ 45. x 2 = −16y 47. (y − 2)2 = 4 √ 49. (y + √ 51. x 2 = y 1 __ 53. (y − 2)2 = 4 3 )2 = 4 √ 55. (y − √ 57. y 2 = −8x 59. (y + 1)2 = 12(x + 3) 61. (0, 1) (x + 2) — — 2 (x − 2) — 2 (x − √ — 2 ) 5 (x + √ — 2 ) 63. At the point 2.25 feet above the vertex 67. x 2 = −125(y − 20), height is 7.2 feet 65. 0.5625 feet 69. 0.2304 feet Section 10.4 5. It gives the angle of 1. The xy term causes a rotation of the graph to occur. 3. The conic section is a hyperbola. rotation of the axes in order to eliminate the xy term. 7. AB = 0, parabola 9. AB = −4 < 0, hyperbola 11. AB = 6 > 0, ellipse 15. B 2 − 4AC = 0, parabola 19. 7x′2 + 9y′2 − 4 = 0 23. θ = 60°, 11x′2 − y′2 + √ 25. θ = 150°, 21x′2 + 9y′2 + 4x′ − 4 √ 27. θ ≈ 36.9°, 125x′2 + 6x′ − 42y′ + 10 = 0 29. θ = 45°, 3x′2 − y′2 − √ 3 x′ + y′ − 4 = 0 3 y′ − 6 = 0 2 y′ + 1 = 0 2 x′ + √ 13. B 2 − 4AC = 0, parabola — — — — 17. B 2 − 4AC = −96 < 0, ellipse 21. 3x′2 + 2x′y′ − 5y′2 + 1 = 0 31. — 2 √ ____ 2 1 __ (x′ + y′ ) = (x′ − y′ )2 2 y 10 5 x −10 −5 5 10 x −5 −10 35. (x′ + y′ )2 ________ − 2 y (x′ − y′ )2 ________ 2 = 1 5 2.5 −7.5 −5 −2.5 2.5 5 7.5 −2.5 −5 39. θ = 45º (3, 3) y 8 4 −8 −4 4 8 x x x −4 −8 (−3, −3) y θ = 30º 43. 4 3 2 1 33. (x′ − y′ )2 ________ 8 + (x′ + y′ )2 ________ 2−3−4−5 −1 −1 −2 −3 37. — √ 3 ____ 2 1  __ x′ + 2 y y′ = 1 __ x′ − 2 — 3 √ y′ − 1  ____ 2 2 10 5 −5 −10 5 10 15 20 x 41. θ = 45º y 8 4 (0, 2) −8 −4 4 8 x −4 −8 y (0, −2) 45. θ = 30º x −1.4 −1 −0.6 1.5 1 0.5 −0.2 −0.5 −1 −1.5 (0, 1) 0.2 0.6 1 1.4 x (0, −1) 49. y 8 4 −4 −8 −4 (0, 0) 4 θ = 63º 8 12 16 x 53. θ = 60° y 3 2 1 x' 1 2 3 4 5 x y' −2−3−4−5 −1 −1 −2 −3 57. − 59. k = 2 x x x −2−3−4 −1−1 −2 1 2 3 4 −4 47. θ = 37º y 4 (0, 0) −8 −4 4 8 −4 −8 51. θ = 45° y y' 3 2 1 x' −2−3−4−5 −1 −1 −2 −3 55. θ ≈ 36.9° y y' 3 2 1 −2−3−4−5 −1 −1 −2 −3 1 2 3 4 5 x' 1 2 3 4 5 ODD ANSWERS Section 10.5 Chapter 10 Review exercises C-41 11. Parabola 1. If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a 3. The directrix will be parallel to the polar axis. hyperbola. 5. One of the foci will be located at the origin. 7. Parabola with 3 _ e = 1 and directrix units below the pole. 4 9. Hyperbola 5 _ with e = 2 and directrix units above the pole. 2 3 _ with e = 1 and directrix 10 units to the right of the pole. 2 _ 13. Ellipse with e = and directrix 2 units to the right of the pole. 7 5 11 _ _ 15. Hyperbola with e = and directrix 5 3 7 8 _ _ 17. Hyperbola with e = units to the right of and directrix 7 8 units above the pole. 19. 25x 2 + 16y 2 − 12y − 4 = 0 the pole. 21. 21x 2 − 4y 2 − 30x + 9 = 0 25. 25x 2 − 96y 2 − 110y − 25 = 0 27. 3x 2 + 4y 2 − 2x − 1 = 0 29. 5x 2 + 9y 2 − 24x − 36 = 0 31. 23. 64y 2 = 48x + 9 33. y y — — 1. + + x 2 ___ 52 39 ) 3. 39 ), (0, − √ (x + 3)2 _ 12 y 2 ___ 82 (0, −8); foci: (0, √ = 1; center: (0, 0); vertices: (5, 0), (−5, 0), (0, 8), (y − 2)2 _ 32 2 ), (−3, 2); (−2, 2), (−4, 2), (−3, 5), (−3, −1); (−3, 2 + 2 √ (−3, 2 − 2 √ 2 ) 5. Center: (0, 0); vertices: (6, 0), (−6, 0), (0, 3), (0, −3); 3 , 0), (−3 √ foci: (3 √ 7. Center: (−2, −2); vertices: (2, −2), (−6, −2), (−2, 6), (−2, −10); foci: (−2, − 2 + 4 √ 3 ) (−2, − 2 − 4 √ 3 , 0 ), y 7.5 5 2.5 −10 −7.5 −5 −2.5 −2.5 −5 −7.5 y 15 10 5 2.5 5 7.5 10 x −20 −15 −10 −5 −5 −10 −15 5 10 15 20 x Vertex (−5, 0) 5 2.5 −7.5 −5 −2.5 −2.5 Vertex (−1.67, −2.89) 35. Vertex (−1.67, 2.89) Focus (0, 0) x 5 7.5 Vertex 5 3 y 15 10 5 16 Vertex (0, 10) 14 12 10 8 6 4 2 9. + = 1 y 2 x 2 ___ ___ 25 16 (y + 1)2 (x − 4)2 _ _ − 13. 42 62 (4, −5); foci: (4, −1 + 2 √ (y + 3)2 _ 3  2  2 √ (x − 2)2 _______ 22 15. − — 11. Approximately 35.71 feet = 1; center: (4, −1); vertices: (4, 3), — 13 ), (4, −1 − 2 √ — 13 ) = 1; center: (2, −3); vertices: (4, −3), 2 4 6 8 10 x (0, −3); foci: (6, −3), (−2, −3) 17. y 19. −10 −4−6−8 Focus (0, 0) −2 −2 −4 −6 37. y 3 2 1 Focus (0, 0) Vertex (−1, 8) Focus (−1, 8.28) 20 15 10 5 −2−3−4−5 −1 −1 −2 −3 x 1 2 3 4 5 Vertex (0, −1) y = −2 Vertex (−1, −6) −20 −15 −10 −5 −5 −10 −15 −20 5 10 15 20 x Focus (−1, −6.28) y Focus (0, 9) Vertex (0, 6.46) x 5 10 15 20 Focus (0, −3) Vertex (0, −0.46) −20 −15 −10 20 15 10 5 −5 −5 −10 −15 −20 −20 −15 −10 −5 Focus (0, 0) 15 10 5 x Vertex (−8, 0) −5 −10 39. Focus (0, 0) −16 −14 −12 −10 −8−6 −15 y 14 12 10 8 6 4 2 −2−4 −2 −4 −6 −8 −10 −12 −14 41. Vertex (0, 6) Vertex (−2.68, 2.4) y 7.5 5 2.5 2 4 6 x −7.5 −5 −2.5 −2.5 Vertex (2.68, 2.4) Focus (0, 0) 2.5 5 7.5 43. r = 4 _________ 5 + cos θ x = 5 45. r = 51. r = 4 _ 1 + 2sin θ 12 _ 2 + 3sin θ 47. r = 49. r = 53. r = 55. r = 1 _ 1 + cos θ 15 _ 4 − 3cos θ 59. r = ± 7 __ 8 − 28cos θ 3 _ 3 − 3cos θ 57. r = ± 2 __ 1 + sin θ cos θ √ — 2 __ 4cos θ + 3sin θ (x − 5)2 _ 1 1 _ 23. (x + 2)2 = (y − 1); 2 (y − 7)2 _ − = 1 21. 3 9 7  ; directrix: y = vertex: (−2, 1); focus:  −2, _ _ 8 8 7 25. (x + 5)2 = (y + 2); vertex: (−5, −2); focus:  −5, −  ; _ 4 9 _ directrix: y = − 4 27. x = −25 8 x y 3 2 1 29. y x = 5 4 10 5 x x 5 10 −2−3−4−5 −1 1 2 3 4 5 −15 −10 −5 1 4 −5 −1 −2 −3 Vertex (−3, 1)  (y − 1) 31. (x − 2)2 =  1 __ 2 33. B 2 − 4AC = 0, parabola 35. B 2 − 4AC = − 31 < 0, ellipse 37. θ = 45°, x′ 2 + 3y′ 2 − 12 = 0 −15 −10 1 2 ODD ANSWERS C-42 39. θ = 45° y (−2, 0) −2−3−4 −1 4 3 2 1 −1 −2 −3 −4 41. Hyperbola with e = 5 and directrix 2 units to the left of the pole. 3 1 _ _ 43. Ellipse with e = and directrix 4 3 unit above the pole. (2, 0) 1 2 3 4 x 45. y Focus (4, 0) 7.5 5 2.5 −10 −7.5 −5 y = −3 −2.5 −2.5 −5 −7.5 2.5 5 7.5 10 Vertex 49. r = 3 _ 1 + cos θ 47. x Focus (0, 0) y 24 16 8 −32 −24 −16 −8 8 16 24 32 −8 −16 −24 Vertex (10, 0) x Chapter 10 practice test = 1; center: (0, 0); vertices: (3, 0), (−3, 0), (0, 2), — 1. + x 2 ___ 32 y 2 ___ 22 (0, −2); foci: ( √ 5 , 0) 3. Center: (3, 2); vertices: (11, 2), (−5, 2), (3, 8), (3, −4); foci: 7 , 2) (3 + 2 √ 5 , 0), (− √ — — — 7 , 2), (3 − 2 √ y 15 10 5 −15 −10 −5 −5 −10 −15 x 5 10 15 5. (x − 1)2 _______ 36 + (y −2)2 ______ 27 = 1 7. y 2 x 2 _ _ 92 = 1; center: (0, 0); 72 − — vertices (7, 0), (−7, 0); foci: ( √ 130 , 0); 130 , 0), (− √ 9 __ asymptotes: y = ± x 7 — 9. Center: (3, −3); vertices: (8, −3), (−2, −3); foci: (3 + √ (3 − √ 1 26 , −3); asymptotes: y = ± _ (x − 3) − 3 5 — y Vertex (−2, −3) 5 Center (3, −3) Focus — 26 , −3), −20 −15 −10 −5 5 10 15 20 x Focus −5 −10 Vertex (8, −3) − 11. (x − 1)2 (y − 3)2 _______ _______ 8 1 11 ___ (2, −1); focus:  2, − 12 15. y = 1 1 _ 13. (x − 2)2 = (y + 1); vertex: 3  ; directrix: y = − 13 ___ 12 17. Approximately 8.48 feet 19. Parabola; θ ≈ 63.4° Vertex (−3, 4) Focus (−5, 4) −20 −15 −10 −5 15 10 5 −5 −10 x 5 10 x = −1 21. x′ 2 − 4x′ + 3y2, 3 3 _ 23. Hyperbola with e = , 2 5 _ units to the and directrix 6 right of the pole. 1 2 3 4 5 6 x −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 25. y 0.8 0.4 1 y = 2   1 0, 4 Vertex −1.6 −1.2 −0.8 −0.4 0.4 0.8 1.2 1.6 x Focus (0, 0) −0.4 −0.8 −1.2 −1.6 −2 ChapteR 11 Section 11.1 3. Yes, both sets go on 1. A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers. indefinitely, so they are both infinite sequences. 5. A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out 13 ∙ 12 ∙ 11 ∙ 10 ∙ . 16 _ 5 7. First four terms: −8, − 9. First four terms: , − 4, − 16 _ 3 1 _ , 2, 2 11. First four terms: 1.25, −5, 20, −80 8 1 _ _ , 4 27 9 16 1 4 _ _ _ _ 13. First four terms 17. , , , 7 5 3 19. −0.6, −3, −15, −20, −375, −80, −9375, −320 4 _ − , 4, −20, 100 5 , 31, 44, 59 25 _ 11 16 _ 9 , 15. First four terms: n − 1 2n _ 2n 2n − 1 _ n or −1, 1, −9, 23. an = 21. an = n 2 + 3 1  25. an =  − _ 2 27. First five terms: 3, −9, 27, −81, 243 29. First five terms: 531,441 9 81 2187 , 4 4 4 8 2 16 35. a 1 = −8, an = an − 1 + n 891 _ 5 14 _ 5 37. a 1 = 35, an = an − 1 + 3 4 _ , 2, 10, 12 , 5 39. 720 41. 665,280 33. 2, 10, 12, 27 _ , 11 31. 1 _ 24 , 3 1 2 _ _ _ 43. First four terms: 1, , , 2 3 2 6 24 _ _ 45. First four terms: −1, 2, , 5 11 ODD ANSWERS 47. an 6 5 4 3 2 1 −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 51. an 36 30 24 18 12 6 −2−3−4−5 −1 −6 −12 −18 −24 −30 −36 (1, 0) 1 2 3 4 5 6 n 49. an 6 5 4 3 2 1 −2−3−4−5−6 −1 −1 −2 −3 −4 −5 −6 (1, 2) (2, 1) (4, 1) n 1 2 3 4 5 6 (5, 0) (3, 0) (5, 30) (4, 20) (3, 12) (2, 6) 2 3 4 5 6 7 1 (1, 2) n 53. an = 2n − 2 55. a 1 = 6, an = 2an − 1 − 5 57. First five terms: 3188 716 29 _____ ____ ___ , , , 999 333 37 152 ____ , 111 13724 ______ 2997 59. First five terms: 2, 3, 5, 17, 65537 61. a 10 = 7,257,600 63. First six terms: 0.042, 0.146, 0.875, 2.385, 4.708 65. First four terms: 5.975, 32.765, 185.743, 1057.25, 6023.521 67. If an = −421 is a term in the sequence, then solving the equation −421 = −6 − 8n for n will yield a non-negative integer. However, if −421 = −6 − 8n, then n = 51.875 so an = −421 is not a term in the sequence. 69. a 1 = 1, a 2 = 0, an = an − 1 − an − 2 71. (n + 2)! _______ = (n − 1)! (n + 2) · (n + 1) · (n) · (n − 1) · ... · 3 · 2 · 1 ___________________________________ (n − 1) · ... · 3 · 2 · 1 = n(n + 1)(n + 2) = n 3 + 3n 2 + 2n Section 11.2 1. A sequence where each successive term of the sequence 3. We find increases (or decreases) by a constant value. whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common 5. Both
arithmetic sequences and linear functions difference. have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers. 7. The common 21. a 1 = 5 9. The sequence is not arithmetic because 1 _ difference is 2 8 4 2 _ _ _ 13. 0, −5, −10, −15, −20 16 − 4 ≠ 64 − 16. 11. 0, , 2, , 3 3 3 19. a 1 = 2 17. a 6 = 41 15. a 4 = 19 23. a 1 = 6 25. a 21 = −13.5 27. −19, −20.4, −21.8, −23.2, −24.6 29. a 1 = 17; an = an − 1 + 9; n ≥ 2 31. a 1 = 12; an = an − 1 + 5; n ≥ 2 1 1 __ __ 33. a 1 = 8.9; an = an − 1 + 1.4; n ≥ 2 35. a 1 = ; n ≥ 2 ; an = an − 1 + 5 4 1 __ ; n ≥ 2 39. a 1 = 4; an = an − 1 + 7; a 14 = 95 ; an = an − 1 − 37. a 1 = 6 43. an = 1 + 2n 41. First five terms: 20, 16, 12, 8, 4 45. an = −105 + 100n 1 1 _ _ 51. an = n − 3 3 55. There are 6 terms in the sequence. 57. The graph does not represent an arithmetic sequence. 53. There are 10 terms in the sequence. 49. an = 13.1 + 2.7n 47. an = 1.8n 13 ___ 12 59. an 10 5 −0.5 −5 −10 −15 −20 −25 −30 −35 65. (1, 9) (2, −1) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 (3, −11) n (4, −21) (5, −31) an 10 2−3−4−5−6 −1 −1 (4, 7) (4, 7.5) (3, 6.5) (2, 6) (1, 5.5) 1 2 3 4 5 6 n C-43 61. 1, 4, 7, 10, 13, 16, 19 63. an 14 13 12 11 10 2−3−4−5−6 −1 −1 (5, 13) (4, 10) (3, 7) (2, 4) (1, 1) 1 2 3 4 5 6 n 67. Answers will vary. Examples: an = 20.6n and an = 2 + 20.4n 69. a 11 = −17a + 38b 71. The sequence begins to have negative values at the 13th 1 _ term, a 13 = − 3 sequence is arithmetic. Example: recursive formula: a 1 = 3, an = an − 1 − 3. First 4 terms: 3, 0, −3, −6; a 31 = −87 73. Answers will vary. Check to see that the Section 11.3 1. A sequence in which the ratio between any two consecutive 3. Divide each term in a sequence by terms is constant. the preceding term. If the resulting quotients are equal, then 5. Both geometric sequences the sequence is geometric. and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an 7. The common ratio exponential function must be positive. is −2 1 _ 11. The sequence is geometric. The common ratio is − . 2 13. The sequence is geometric. The common ratio is 5. 9. The sequence is geometric. The common ratio is 2. 1 _ 15. 5, 1, , 5 1 _ 125 1 _ , 25 16 _ 27 19. a 4 = − 21. a 7 = − 1 _ 25. a = −32, an = an − 1 2 3 1 _ _ , an = 29. a 1 = an − 1 5 6 3 3 _ _ 33. 12, −6, 3, − , 4 2 17. 800, 400, 200, 100, 50 23. 7, 1.4, 0.28, 0.056, 0.0112 2 _ 729 27. a 1 = 10, an = −0.3 an − 1 31. a 1 = , an = −4an − 1 1 _ 512 n − 1 35. an = 3n − 1 4 __  39. an = −  5 1 ________ 43. a 12 = 177, 147 37. an = 0.8 ∙ (−5)n − 1 1 __ 41. an = 3 ∙  −  3 45. There are 12 terms in the sequence. 47. The graph does not represent a geometric sequence. 49. n − 1 an 60 48 36 24 12 −0.5 −1 (5, 48) (4, 24) (3, 12) (1, 3) (2, 6) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n 51. Answers will vary. Examples: a 1 = 800, an = 0.5an − 1 and a 1 = 12.5, an = 4an − 1 53. a 5 = 256b 55. The sequence exceeds 100 at the 14th term, a 14 ≈ 107. ODD ANSWERS C-44 32 ___ is the first non-integer value 3 57. a 4 = − 59. Answers will vary. Example: explicit formula with a decimal common ratio: an = 400 ∙ 0.5n − 1; first 4 terms: 400, 200, 100, 50; a 8 = 3.125 Section 11.4 3. A geometric series is the sum of the terms in a 1. An nth partial sum is the sum of the first n terms of a sequence. geometric sequence. equal payments that earn a constant compounded interest. 11. ∑ 7. ∑ 5. An annuity is a series of regular 13. S 5 = 8k + 2 5n 20 4 25 _ 2 n = 0 15. S 13 = 57.2 k = 1 7 8 ∙ 0.5k − 1 k = 1 5 9. ∑ 4 17 __ = 1 _ 1 − 3 k = 1 5 19. S 5 = 121 _ 9 ≈ 13.44 21. S 11 = 64(1 − 0.211) __ = 1 − 0.2 23. The series is defined. S = 25. The series is defined. S = ≈ 80 781,249,984 __ 9,765,625 2 _ 1 − 0.8 −1 __________ 1 −  − 1  __ 2 27 2000 1750 1500 1250 1000 750 500 250 10 11 12 Month x 31. 49 33. 254 35. S 7 = 39. S 7 = 5208.4 45. S = 9.2 51. ak = 30 − k 57. $400 per month 41. S 10 = − 47. $3,705.42 53. 9 terms 29. Sample answer: The graph of Sn seems to be approaching 1. This makes sense because ∑ 1   _ 2 defined infinite geometric k is a k = 1 ∞ series with S = 1 __ 2 _________ = 1. 1 −  1  __ 2 55 _ 2 4 _ 43. S = − 3 37. S 11 = 147 _ 2 1023 _ 256 49. $695,823.97 4 _ 55. r = 5 59. 420 feet 61. 12 feet Section 11.5 1. There are m + n ways for either event A or event B to occur. 3. The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem. C(n, r) = 5. A combination; 9. 5 + 4 + 7 = 16 7. 4 + 2 = 6 n! _ (n − r)!r! 13. 103 = 1,000 11. 2 × 6 = 12 17. P(3, 3) = 6 19. P(11, 5) = 55,440 23. C(7, 6) = 7 25. 210 = 1,024 15. P(5, 2) = 20 21. C(12, 4) = 495 29. 29 = 512 31. = 6,720 8! _ 3! 27. 212 = 4,096 12! _ 3!2!3!4! 33. 35. 9 37. Yes, for the trivial cases r = 0 and r = 1. If r = 0, then C(n, r) = P(n, r) = 1. If r = 1, then r = 1, C(n, r) = P(n, r) = n. 6! ___ 2! 39. × 4! = 8,640 41 43. 4 × 2 × 5 = 40 45. 4 × 12 × 3 = 144 49. C(10, 3) × C(6, 5) × C(5, 2) = 7,200 47. P(15, 9) = 1,816,214,400 51. 211 = 2,048 53. 20! ______ 6!6!8! = 116,396,280 Section 11.6 n k = 0 n r 7. 35 9. 10 11. 12,376 n! _________ . r !(n − r)!  x n − ky k and 1. A binomial coefficient is an alternative way of denoting the combination C(n, r). It is defined as   = C(n, r) =  n 3. The Binomial Theorem is defined as (x + y)n = ∑ k can be used to expand any binomial. 5. 15 13. 64a3 − 48a2b + 12ab2 − b3 17. 1024x5 + 2560x4y + 2560x3y2 + 1280x2y3 + 320xy4 + 32y5 19. 1024x5 − 3840x4y + 5760x3y2 − 4320x2y3 + 1620xy4 − 243y5 16 ___ y4 32 ___ xy3 25. a15 − 30a14b + 420a13b2 27. 3,486,784,401a20 + 23,245,229,340a19b + 73,609,892,910a18b2 29. x 24 − 8x 21 √ y + 28x 18y 33. 220,812,466,875,000y 7 23. a17 + 17a16b + 136a15b2 15. 27a3 + 54a2b + 36ab2 + 8b3 24 ____ x2y2 8 ___ x3y 1 ___ x4 21. + + + + — 37. 1,082,565a 3b 16 39. 41. f2(x) = x 4 + 12x 3 43. f4(x) = x 4 + 12x 3 + 54x 2 + 108x 31. −720x 2y 3 35. 35x 3y 4 1152y2 _ x7 7 6 5 4 3 2 1 −2−3−4−5 −1 −1 −2 −3 −4 −5 45. 590,625x 5y 2 47. k − 1 y y f2(x) f4(x) x 1 2 90 80 70 60 50 40 30 20 10 x 1 2 −2−3−4−5−6−7−8−9 −1 −10 −20 −30 −40 −50 −60 −70 −80 −90 49. The expression (x3 + 2y2 − z)5 cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial. Section 11.7 1. Probability; the probability of an event is restricted to values between 0 and 1, inclusive of 0 and 1. 3. An experiment is an activity with an observable result. 5. The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets A and B and a union of events A and B, the union includes either A or B or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between 0 and 1. 5 __ 9. 8 1 __ 21. 8 5 __ 11. 8 15 ___ 16 23. 3 __ 13. 8 5 __ 25. 8 1 __ 15. 4 27. 1 __ 7. 2 3 3 __ __ 17. 19. 4 8 1 ___ 26 29. 31. 1 ___ 13 12 ___ 13 ODD ANSWERS 33. 1 (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 (5, 1) 6 (6, 1) 7 2 (1, 2) 3 (2, 2) 4 (3, 2) 5 (4, 2) 6 (5, 2) 7 (6, 2) 8 3 (1, 3) 4 (2, 3) 5 (3, 3) 6 (4, 3) 7 (5, 3) 8 (6, 3) 9 4 (1, 4) 5 (2, 4) 6 (3, 4) 7 (4, 4) 8 (5, 4) 9 (6, 4) 10 5 (1, 5) 6 (2, 5) 7 (3, 5) 8 (4, 5) 9 (5, 5) 10 (6, 5) 11 6 (1, 6) 7 (2, 6) 8 (3, 6) 9 (4, 6) 10 (5, 6) 11 (6, 6) 12 1 2 3 4 5 6 35. 5 ___ 12 21 ___ 26 37. 0. 4 __ 39. 9 45. 47. C(12, 5) ________ = C(48, 5) 1 _____ 2162 3 __ 43. 4 1 __ 41. 4 C(12, 3)C(36, 2) ______________ = C(48, 5) 49. 175 _____ 2162 51. C(20, 3)C(60, 17) _______________ C(80, 20) ≈ 12.49% 53. C(20, 5)C(60, 15) _______________ C(80, 20) ≈ 23.33% 57. 55. 20.50 + 23.33 − 12.49 = 31.34% C(40000000, 1)C(277000000, 4) ___________________________ C(317000000, 5) C(40000000, 4)C(277000000, 1) ___________________________ C(317000000, 5) 59. = 36.78% = 0.11% Chapter 11 Review exercises 13 _ 24 13. r = 2 3. 13, 103, 1003, 10003 9. a 1 = −20, an = an − 1 + 10 1. 2, 4, 7, 11 5 _ 5. The sequence is arithmetic. The common difference is d = . 3 7. 18, 10, 2, −6, −14 1 _ 11. an = n + 3 17. 3, 12, 48, 192, 768 21. ∑ 1 __ m + 5   2 135 ____ 4 35. P(18, 4) = 73,440 1 1  ∙  _ _ 19. an = − 5 3 25. S9 ≈ 23.95 15. 4, 16, 64, 256, 1024 n − 1 37. C(15, 6) = 5,005 33. 104 = 10,000 23. S 11 = 110 29. $5,617.61 27. S = 31. 6 m = 0 5 39. 250 = 1.13 × 1015 41. = 3,360 43. 490,314 8! ____ 3!2! 45. 131,072a 17 + 1,114,112a 16b + 4,456,448a 15b 2 47, 1 2, 1 3, 1 4, 1 5, 1 6, 1 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 1, 3 2, 3 3, 3 4, 3 5, 3 6, 3 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4 5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 5 6 1, 6 2, 6 3, 6 4, 6 5, 6 6, 6 4 __ 53. 9 5 __ 51. 9 1 __ 49. 6 C(150, 3)C(350, 5) ________________ C(500, 8) 57. ≈ 25.6% 55. 1 − C(350, 8) ________ C(500, 8 ) ≈ 94.4% Chapter 11 practice test 1. −14, −6, −2, 0 common difference is d = 0.9. 3 _ ; a 22 = − 5. a 1 = −2, an = an − 1 − 2 67 _ 2 3. The sequence is arithmetic. The C-45 1 _ 7. The sequence is geometric. The common ratio is r = . 2 1 _ 9. a 1 = 1, an = − ∙ an −1 2 13. S 7 = −2,604.2 earned: $14,355.75 5 __ k   3k 2 − 6 15. Total in account: $140,355.75; Interest 17 = 180 11. ∑ k = −3 15 21. 29. = 151,200 10! _ 2!3!2! C(14, 3)C(26, 4) ______________ C(40, 7) ≈ 29.2% 23. 429x 14 _ 16 19. C(15, 3) = 455 4 _ 25. 7 5 _ 27. 7 ChapteR 12 9. Does not exist 7. 2 15. Answers will vary Section 12.1 1. The value of the function, the output, at x = a is f (a). When the lim f (x) is taken, the
values of x get infinitely close to a but x → a never equal a. As the values of x approach a from the left and right, the limit is the value that the function is approaching. 3. −4 5. −4 11. 4 13. Does not exist 17. Answers 19. Answers will vary 21. Answers will vary will vary 23. 7.38906 25. 54.59815 27. e 6 ≈ 403.428794, e 7 ≈ 1096.633158, en _________ _ ≈ 0.83 6 x 2 − 9   = −2.00 35. lim 10 − 10x 2  = 20.00 __ x 2 − 3x + 2  does not exist. Function values decrease  x 2 − 1 __ x 2 − 3x + 2 x  __ 37. lim 4x 2 + 4x + 1 1 _ x → − 2 29. lim x → −2 33. lim x → 1 31. lim f (x) = 1 x → 3 x → 1 without bound as x approaches −0.5 from either left or right. 39. lim x → 0 7tan x _ 3x 7 _ = 3 x f(x) −0.1 −0.01 7tan x _ 2.34114234 lim x → 0− 3x ↓ 2.33341114 7 _ 3 −0.001 2.33333411 0 Error 0.001 2.33333411 0.01 2.33341114 0.1 2.34114234 7 _ 3 ↑ 7tan x _ 3x x → 0+ lim 41. lim x → 0 2sin x _ 4tan x 1 _ = 2 x −0.1 f(x) x → 0− 0.49750208 lim −0.01 0.49997500 −0.001 0.49999975 0 Error 0.001 0.49999975 0.01 0.49997500 0.1 0.49750208 2sin x _ 4tan x ↓ 1 _ 2 1 _ 2 ↑ 2sin x _ 4tan x x → 0+ lim − 1 ___ x 2 e 43. lim e x → 0 = 1.0 45. lim x → −1− \|x + 1\| _______ x + 1 = −(x + 1) ________ (x + 1) = −1 and lim x → −1+ \|x + 1\| _______ = x + 1 (x + 1) _______ (x + 1) = 1 since the right-hand limit does not equal the left-hand limit, lim does not exist. \|x + 1\| _______ x + 1 x → −1 does not exist. The function increases without 47. lim x → −1 1 _______ (x + 1)2 bound as x approaches −1 from either side. 49. lim x → 0 5 _______ 1 − e 2 _ x does not exist. Function values approach 5 from the left and approach 0 from the right. 51. Through examination of the postulates and an understanding of relativistic physics, as v → c, m → ∞. Take this one step v → c− m = lim v → c − further to the solution, lim mo __ = ∞ √ _________ 1 −  v 2  _ c2 ODD ANSWERS C-46 Section 12.2 1. If f is a polynomial function, the limit of a polynomial function 3. It could mean either as x approaches a will always be f (a). (1) the values of the function increase or decrease without bound as x approaches c, or (2) the left and right-hand limits are not equal. 5. − 10 _ 3 13. Does not exist 7. 6 1 _ 9. 2 15. −12 17. − 19. −108 — 11. 6 5 √ _ 10 23. 6 21. 1 31. 6 + √ — 5 25. 1 3 _ 33. 5 27. 1 29. Does not exist 35. 0 37. −3 39. Does not exist; right-hand limit is not the same as the left-hand limit. 41. 2 43. Limit does not exist; limit approaches infinity. 45. 4x + 2h 51. −1 ________ x(x + h) cos (x + h) − cos(x) __ 47. 2x + h + 4 h x 2 + 5x + 6 −1 __________ __ 55. f (x √ 49. 53. — 57. Does not exist 59. 32 Section 12.3 1. Informally, if a function is continuous at x = c, then there is no break in the graph of the function at f (c), and f (c) is defined. 3. Discontinuous at a = −3; f (−3) does not exist 5. Removable discontinuity at a = −4; f (−4) is not defined 7. Discontinuous at a = 3; lim x → 3 not equal to the limit. f (x) = 3, but f (3) = 6, which is 9. lim x → 2 f (x) does not exist 11. lim x → 1− 13. lim x → 1− f (x) = 4; lim f (x) = 1, therefore, lim x → 1 x → 1+ f (x) does not exist. f (x) = 5 ≠ lim f (x) = −1, thus lim f (x) does not exist. x → 1+ x → 1 15. lim x → −3− f (x) = −6, lim x → −3+ 1 _ f (x) = − , therefore, lim 3 x → −3 f (x) does 23. Continuous on ( −∞, ∞) not exist. 17. f (2) is not defined 19. f (−3) is not defined. 21. f (0) is not defined. 25. Continuous on ( − ∞, ∞) and x = 2 on (0, ∞) ( −∞, ∞) 41. f (0) is undefined 45. At x = −1, the limit does not exist. At x = 1, f (1) does not exist. At x = 2, there appears to be a vertical asymptote, and the 29. Discontinuous at x = 0 33. Continuous on [4, ∞) 37. 1, but not 2 or 3 27. Discontinuous at x = 0 31. Continuous 35. Continuous on 43. ( −∞, 0) ∪ (0, ∞) 39. 1 and 2, but not 3 limit does not exist. 47. x 3 + 6x 2 − 7x _____________ (x + 7)(x − 1) 49. The function is discontinuous at x = 1 because the limit as x approaches 1 is 5 and f (1) = 2. Section 12.4 1. The slope of a linear function stays the same. The derivative of a general function varies according to x. Both the slope of a line and the derivative at a point measure the rate of change of 3. Average velocity is 55 miles per hour. The the function. instantaneous velocity at 2:30 p.m. is 62 miles per hour. The instantaneous velocity measures the velocity of the car at an instant of time whereas the average velocity gives the velocity of 5. The average rate of change of the the car over an interval. amount of water in the tank is 45 gallons per minute. If f (x) is the function giving the amount of water in the tank at any time t, then the average rate of change of f (x) between t = a and t = b is f (a) + 45(b − a). 9. f ′(x) = 4x + 1 7. f ′(x) = −2 11. f ′(x) = 17. f ′(x) = 0 1 _______ (x − 2)2 13. − 1 _ 19. − 3 16 ________ (3 + 2x)2 21. Undefined 15. f ′(x) = 9x 2 − 2x + 2 33. Discontinuous at 31. Discontinuous at x = −2 and 23. f ′(x) = −6x − 7 25. f ′(x) = 9x 2 + 4x + 1 27. y = 12x − 15 29. k = −10 or k = 2 x = 0. Not differentiable at −2, 0, 2. x = 5. Not differentiable at −4, −2, 0, 1, 3, 4, 5. 35. f (0) = −2 37. f (2) = −6 39. f ′(−1) = 9 41. f ′(1) = −3 43. f ′(3) = 9 45. Answers vary. The slope of the tangent line near x = 1 is 2. 47. At 12:30 p.m., the rate of change of the number of gallons in the tank is −20 gallons per minute. That is, the tank is losing 20 gallons per minute. noon, the volume of gallons in the tank is changing at the rate of 51. The height of the projectile after 30 gallons per minute. 53. The height of the projectile at t = 3 2 seconds is 96 feet. 55. The height of the projectile is zero at seconds is 96 feet. t = 0 and again at t = 5. In other words, the projectile starts on 57. 36π the ground and falls to earth again after 5 seconds. 59. $50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold. 63. $36 49. At 200 minutes after 65. f ′(x) = 10a − 1 61. $21 per unit 4 _ (3 − x)2 67. Chapter 12 Review exercises f (x) = 0 3. Does not exist 5. Discontinuous at x = −1 f (x) does not exist), x = 3 (jump discontinuity), and x = 7 7. lim a f (x) does not exist). x → 1 3 _ 11. − 17. 500 13. 1 5 21. At x = 4, the function has a vertical asymptote. 1. 2 ( lim x → a ( lim x → a 9. Does not exist 6 _ 19. − 7 23. At x = 3, the function has a vertical asymptote. 25. Removable discontinuity at a = 9 discontinuity at x = 5 discontinuity at x = 1 29. Removable discontinuity at x = 5, 31. Removable discontinuity at x = −2, 27. Removable 15. 6 discontinuity at x = 5 33. 3 37. e 2x + 2h − e 2x __ h 39. 10x − 3 35. 1 __ (x + 1)(x + h + 1) 41. The function would not be differentiable at however, 0 is not in its domain. So it is differentiable everywhere in its domain. Chapter 12 practice test 1. 3 3. 0 5. −1 7. lim 5 _ f (x) = − a and lim 2 x → 2+ f (x) = 9 x → 2− thus, the limit of the function as x approaches 2 does not exist. 1 _ 50 9. − 11. 1 15. f '(x) = − 3 ____ 3 _ 2 a 2 13. Removable discontinuity at x = 3 17. Discontinuous at −2, 0, not differentiable at −2, 0, 2 19. Not differentiable at x = 0 (no limit) 21. The height of the projectile at t = 2 seconds 23. The average velocity from t = 1 to t = 2 1 _ 27. 0 29. 2 31. x = 1 33. y = −14x − 18 25. 3 35. The graph is not differentiable at x = 1 (cusp). 37. f '(x) = 8x 39. f '(x) = − 41. f '(x) = −3x 2 43. f '(x) = 1 _______ (2 + x)2 1 _ — x − 1 2 √ ODD ANSWERS Index A AAS (angle-angle-side) 644 absolute maximum 47, 113 absolute minimum 47, 113 absolute value 31, 89 absolute value equation 92, 113 absolute value function 29, 89, 92 absolute value inequality 94, 113 addition method 762, 767, 854 Addition Principle 982, 1008 adjacent side 486, 498 altitude 644, 747 ambiguous case 646, 747 amplitude 509, 510, 552, 617, 629 angle 440, 458, 498 angle of depression 492, 498 angle of elevation 492, 498, 644 angle of rotation 913, 931 angular speed 453, 498 annual interest 977 annual percentage rate (APR) 336, 429 annuity 977, 1008 apoapsis 922 arc 443 arc length 444, 450, 458, 498 arccosine 552 arccosine function 542 Archimedes’ spiral 693, 747 arcsine 552 arcsine function 542 arctangent 552 arctangent function 542 area of a circle 224 area of a sector 451, 498 argument 700, 747 arithmetic sequence 951, 952, 954, 955, 970, 971, 1008 arithmetic series 971, 1008 arrow notation 278, 317 ASA (angle-side-angle) 644 asymptote 880 augmented matrix 816, 820, 821, 833, 854 average rate of change 38, 113, 1052, 1070 axis of symmetry 208, 211, 317, 880, 902, 903 B binomial 259, 1008 binomial coefficient 992 binomial expansion 993, 995, 1008 Binomial Theorem 993, 994, 1008 break-even point 769, 854 C cardioid 686, 747 carrying capacity 408, 429 Cartesian equation 676 Celsius 100 center of a hyperbola 880, 931 center of an ellipse 865, 931 central rectangle 880 change-of-base formula 387, 429 circle 787, 789 circular motion 517 circumference 443 coefficient 224, 268, 317 coefficient matrix 816, 818, 835, 854 cofunction 578 cofunction identities 491, 578 column 805, 854 column matrix 806 combination 987, 992, 1008 combining functions 52 common base 391 common difference 951, 970, 1008 common logarithm 358, 429 common ratio 961, 973, 1008 commutative 53 complement of an event 1003, correlation coefficient 180, 187 cosecant 473, 498, 528 cosecant function 529, 562 cosine 561, 596, 597 cosine function 458, 498, 507, divisor 258 domain 2, 10, 22, 23, 113 domain and range 22 domain and range of inverse functions 104 508, 510, 517, 528 domain of a composite function cost function 51, 768, 854 cotangent 473, 498, 534 cotangent function 534, 562 coterminal angles 448, 450, 498 co-vertex 865, 867, 880 Cramer’s Rule 843, 846, 850, 854 cube root 225 cube root function 30 cubic function 29, 302 curvilinear path 708 D damped harmonic motion 625, 634 decompose a composite function 59 decomposition 795 decreasing function 43, 113, 128 decreasing linear function 129, 187 degenerate conic sections 909 degree 229, 317, 441, 498 De Moivre 697, 702 De Moivre’s T
heorem 703, 704, 747 58 domain of a rational function 283 dot product 739, 747 double-angle formulas 584, 585, 634 doubling time 401, 405, 429 Dürer 688 E eccentricity 923, 931 electrostatic force 41 elimination 788 ellipse 721, 788, 865, 866, 867, 869, 872, 896, 923, 927, 931 ellipsis 938 end behavior 226, 287, 317 endpoint 40, 440 entry 805, 854 equation 8 Euler 697 even function 75, 113, 477, 561 even-odd identities 561, 563, 634 event 999, 1008 experiment 999, 1008 explicit formula 939, 955, 964, 1008 dependent system 759, 767, 1008 complex conjugate 202, 317 Complex Conjugate Theorem 272 complex number 198, 317, 697 complex plane 199, 317, 697 composite function 51, 52, 53, 113 compound interest 336, 429 compression 146, 348, 370 conic 864, 879, 928 conic section 714, 931 conjugate 1032 conjugate axis 880, 931 consistent system 758, 854 constant function 29 constant of variation 311, 317 constant rate of change 162 continuity 1040 continuous 239, 1040 continuous function 234, 317, 1037, 1070 convex limaçon 687, 747 coordinate plane 897 779, 854 dependent variable 2, 113 derivative 1053, 1054, 1055, 1056, 1060, 1070 Descartes 697 Descartes’ Rule of Signs 273, 317 determinant 843, 845, 846, 854 difference formula 571 difference quotient 1053 differentiable 1060, 1070 dimpled limaçon 687, 747 directrix 897, 900, 902, 923, 927, 928, 931 direct variation 311, 317 discontinuou 1040 discontinuous function 1038, 1070 displacement 452 distance formula 881, 897 diverge 974, 1008 dividend 258 Division Algorithm 258, 259, 266, 317 exponential 344 exponential decay 328, 334, 343, 401, 403, 406, 416 exponential equation 390 exponential function 328 exponential growth 328, 331, 401, 405, 407, 416, 429 exponential regression 417 extraneous solution 394, 429 extrapolation 177, 178, 187 F factorial 946 Factor Theorem 267, 317 Fahrenheit 100 feasible region 791, 854 finite arithmetic sequence 956 finite sequence 939, 1008 foci 865, 867, 880, 931 focus 865, 897, 900, 902, 922, 927, 928 focus (of an ellipse) 931 D-1 D-2 focus (of a parabola) 931 formula 8 function 2, 31, 113 function notation 4 Fundamental Counting Principle 984, 1008 Fundamental Theorem of Algebra 271, 272, 317 G Gauss 697, 774, 816 Gaussian elimination 774, 819, 854 general form 209 general form of a quadratic function 209, 317 Generalized Pythagorean Theorem 658, 747 geometric sequence 961, 963, 973, 1008 geometric series 973, 1008 global maximum 251, 252, 317 global minimum 251, 252, 317 Graphical Interpretation of a Linear Function 145 gravity 724 growth 344 H half-angle formulas 589, 634 half-life 397, 401, 403, 429 harmonic motion 624 Heaviside method 797 Heron of Alexandria 663 Heron’s formula 663 horizontal asymptote 280, 281, 286, 317 horizontal compression 79, 113, 611 horizontal line 150, 151, 187 horizontal line test 15, 113 horizontal reflection 71, 72, 113 horizontal shift 67, 113, 346, 367, 507 horizontal stretch 79, 113 hyperbola 879, 882, 883, 884, 887, 888, 891, 897, 924, 926, 931 hypotenuse 486, 498 I identities 468, 479, 498 identity function 29 identity matrix 829, 833, 854 imaginary number 198, 317 inconsistent system 759, 766, 778, 854 increasing function 43, 113, 128 increasing linear function 129, 187 independent variable 2, 113 index of summation 969, 970, 1008 inequality 790 infinite geometric series 974 infinite sequence 939, 1008 infinite series 974, 1008 initial point 729, 732, 747 initial side 441, 498 inner-loop limaçon 688, 747 input 2, 113 instantaneous rate of change 1053, 1070 instantaneous velocity 1065, 1070 Intermediate Value Theorem 249, 317 interpolation 177, 187 intersection 1001 interval notation 22, 26, 43, 113 inverse cosine function 542, 552 inverse function 101, 113, 299, 302 inversely proportional 312, 317 inverse matrix 833, 835 inverse of a radical function 305 inverse of a rational function 307 inverse sine function 542, 552 inverse tangent function 542, 552 inverse trigonometric functions 541, 542, 544, 548 inverse variation 312, 317 invertible function 301, 317 invertible matrix 829, 843 J K Kronecker 697 L latus rectum 897, 902, 931 Law of Cosines 659, 747 Law of Sines 645, 659, 747 leading coefficient 229, 317 leading term 229, 318 least common denominator (LCD) 1032 least squares regression 178, 187 left-hand limit 1021, 1040, 1070 lemniscate 689, 747 limaçon 687, 688 limit 1018, 1019, 1023, 1029, 1030, 1031, 1070 linear function 126, 143, 147, nth root of a complex number 162, 187 704 linear growth 328 linear model 163, 175 linear relationship 175 linear speed 452, 453, 498 local extrema 42, 113 local maximum 42, 113, 252 local minimum 42, 113, 252 logarithm 356, 429 logarithmic equation 395 logarithmic model 419 logistic growth model 408, 429 logistic regression 422 long division 257 lower limit 1008 lower limit of summation 969, 970 M magnitude 31, 66, 698, 729, 731, 747 main diagonal 818, 854 major and minor axes 867 major axis 865, 869, 931 marginal cost 1059 matrix 805, 806, 810, 816, 854 matrix multiplication 810, 830, 835 matrix operations 806 maximum value 208 measure of an angle 441, 498 midline 509, 510, 552, 617 minimum value 208 minor axis 865, 869, 931 model breakdown 177, 187 modeling 162 modulus 31, 700, 747 Multiplication Principle 983, 984, 1008 matrix 829, 854 multiplicity 243, 318 mutually exclusive events 1002, 1008 N natural logarithm 360, 393, 429 natural numbers 2 negative angle 441, 448, 467, 498 nth term of a sequence 939, 1008 O oblique triangle 644, 747 odd function 75, 113, 477, 561 one-loop limaçon 687, 747 one-to-one 299, 344, 356, 381, 387 one-to-one function 12, 103, 113, 541 opposite side 486, 498 ordered pair 2, 23 ordered triple 774 order of magnitude 402, 429 origin 90 outcomes 999, 1008 output 2, 113 P parabola 208, 214, 720, 791, 896, 901, 903, 922, 925, 931 parallel lines 151, 152, 187 parallelograms 733 parameter 708, 747 parametric equation 709, 719, 721 parametric form 722 parent function 367 partial fraction 795, 854 partial fraction decomposition 795, 854 Pascal 688 Pascal’s Triangle 994 periapsis 922 period 481, 498, 507, 523, 525, 603, 617 periodic function 481, 507, 552 periodic motion 617, 624 permutation 984, 1008 perpendicular lines 152, 153, 187 pH 380 phase shift 511, 552 piecewise function 31, 113, 942, 1041, 1044, 1046 point-slope form 131, 187 point-slope formula 885 polar axis 670, 747 polar coordinates 670, 672, 673, 674, 681, 747 Newton’s Law of Cooling 406, polar equation 676, 682, 683, 429 n factorial 946, 1008 nominal rate 336 nondegenerate conic section 909, 911, 931 nonlinear inequality 790, 854 non-right triangles 644 nth partial sum 969, 1008 747, 923, 931 polar form 698 polar form of a complex number 699, 747 polar form of a conic 928 polar grid 670 pole 670, 747 polynomial 268, 1044 joint variation 314, 317 jump discontinuity 1040, 1070 multiplicative inverse 831 multiplicative inverse of a independent system 758, 759, Linear Factorization Theorem 854 272, 318 INDEX polynomial function 228, 239, 246, 250, 318 position vector 729, 731 positive angle 441, 448, 498 power function 224, 318 power rule for logarithms 383, 387, 429 probability 999, 1008 probability model 999, 1008 product of two matrices 810 product rule for logarithms 381, 383, 429 product-to-sum formula 596, 598, 634 profit function 769, 854 properties of determinants 849 properties of limits 1029, 1070 Proxima Centauri 402 Pythagoras 697 Pythagorean identities 560, 563, 634 Pythagorean Identity 460, 461, 480, 498, 571 Pythagorean Theorem 585, 612, 658, 723 Q quadrantal angle 442, 498 quadratic 799, 801 quadratic equation 607 quadratic formula 219, 607 quadratic function 29, 211, 213 quotient 258 quotient identities 562, 563, 634 quotient rule for logarithms 382, 429 R radian 444, 445, 498 radian measure 445, 450, 498 radical functions 301 radiocarbon dating 404 range 2, 113, 542 rate of change 38, 114, 126 rational expression 795, 801 rational function 282, 289, 292, 318, 1028 Rational Zero Theorem 268, 318 ray 440, 498 reciprocal 101, 225 reciprocal function 30, 278 reciprocal identities 562, 563, 634 reciprocal identity 528, 534 reciprocal squared function 30 rectangular coordinates 670, 672, 673, 674 rectangular equation 676, 713 rectangular form 699, 722 recursive formula 944, 954, 963, 1009 D-3 upper limit of summation 969, 970, 1009 upper triangular form 774 V varies directly 311, 318 varies inversely 312, 318 vector 729, 748 vector addition 733, 748 velocity 1053, 1065 vertex 208, 318, 440, 498, 865, 897, 903 vertex form of a quadratic function 210, 318 vertical asymptote 280, 283, 287, 318, 542 vertical compression 76, 114 vertical line 151, 187 vertical line test 13, 114 vertical reflection 71, 72, 114 vertical shift 64, 65, 114, 146, 345, 368, 406, 511 vertical stretch 76, 114, 146, 370 vertical tangent 1062 vertices 865, 867 volume of a sphere 224 reduction formulas 588, 634 reference angle 448, 466, 467, SSA (side-side-angle) 644 SSS (side-side-side) triangle 468, 476, 498 reflection 349, 372 regression analysis 416, 419, 422 relation 2, 114 relative 42 remainder 258 Remainder Theorem 266, 318 removable discontinuity 285, 658 standard form of a quadratic function 210, 318 standard position 441, 498, 729, 748 stepwise function 1038 stretch 348 stretching/compressing factor 318, 1041, 1070 524, 525 Restricting the domain 108 resultant 733, 748 revenue function 768, 854 Richter Scale 355 right-hand limit 1021, 1040, 1070 right triangle 486, 541 roots 209 rose curve 691, 748 row 805, 854 row-echelon form 818, 820, 854 row-equivalent 818, 854 row matrix 806 row operations 818, 822, 832, 833, 834, 854 S sample space 999, 1009 SAS (side-angle-side) triangle 658 scalar 734, 748, 808 scalar multiple 808, 854 scalar multiplication 734, 748, 808 scatter plot 175 secant 473, 498, 528 secant function 528 secant line 1052, 1070 sector of a circle 451 sequence 938, 951, 1009, 1018 series 969, 1009 set-builder notation 25, 26, 114 sigma 969 simple harmonic motion 624, 634 subs
titution method 761, 854 sum and difference formulas for cosine 572 sum and difference formulas for sine 573 sum and difference formulas for tangent 575 summation notation 969, 970, 1009 sum-to-product formula 599, 634 surface area 299 symmetry test 682 synthetic division 261, 270, 318 system of equations 817, 818, 820, 821, 835 system of linear equations 168, X 758, 760, 761, 855 system of nonlinear equations 785, 855 system of nonlinear inequalities 791, 855 x-intercept 149, 187 Y y-intercept 127, 128, 145, 187 System of Three Equations in Three Variables 846 Z zeros 209, 240, 243, 268, 318, 684 T tangent 473, 498, 523 tangent function 524, 525, 526, 536, 562 tangent line 1051, 1070 term 938, 952, 1009 terminal point 729, 732, 748 terminal side 441, 498 term of a polynomial function 228, 318 sine 561, 597, 598 sine function 458, 477, 498, 506, 511, 516, 518 sinusoidal function 508, 552, transformation 64, 146 translation 901 transverse axis 880, 931 trigonometric equations 713, 617 714 slope 127, 128, 130, 187 slope-intercept form 127, 187 slope of the curve 1051 slope of the tangent 1051 smooth curve 234, 318 solution set 775, 854 solving systems of linear equations 760, 762 special angles 464, 489, 570 square matrix 806, 843 square root function 30 trigonometric functions 486 trigonometric identities 659 turning point 232, 247, 318 two-sided limit 1021, 1053, 1070 U union of two events 1001, 1009 unit circle 445, 458, 461, 465, 477, 486, 498, 604 unit vector 736, 748 INDEX
ponential functions to model and solve real-life problems (p. 223). Section 3.2 Recognize and evaluate logarithmic functions with base a (p. 229). Graph logarithmic functions (p. 231). Recognize, evaluate, and graph natural logarithmic functions (p. 233). Use logarithmic functions to model and solve real-life problems (p. 235). 3 In Exercises 1–6, evaluate the function at the indicated Review Exercises 3.1 value of Round your result to three decimal places. Section 3.3 Use the change-of-base formula to rewrite and evaluate logarithmic expressions (p. 239). Use properties of logarithms to evaluate or rewrite logarithmic expressions (p. 240). 19. Use properties of logarithms to expand or condense logarithmic expressions (p. 241). 21. Use logarithmic functions to model and solve real-life problems (p. 242). Function f x 6.1x f x 30x f x 20.5x f x 1278x5 f x 70.2x f x 145x Value Section 3.4 x 2.4 Solve simple exponential and logarithmic equations (p. 246). x 3 2. Solve more complicated exponential equations (p. 247). x 3. Solve more complicated logarithmic equations (p. 249). x 1 4. Use exponential and logarithmic equations to model and solve x 11 x 0.8 5. real-life problems (p. 251). 6. 25. 23. 1. x. f x 5 x2 4 79–94 f x 1 x 3 95, 96 2 x. 97–104 3x2 1 105–118 9 119–134 e5x7 e15 135, 136 In Exercises 23–26, use the One-to-One Property to solve the equation for 81 x2 1 3 e82x e3 24. 26. Section 3.5 Recognize the five most common types of models involving exponential In Exercises 7–10, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] and logarithmic functions (p. 257). y (a) real-life problems (p. 258). y Use exponential growth and decay functions to model and solve 5 4 3 2 Use Gaussian functions to model and solve real-life problems (p. 261). −1 Use logistic growth functions to model and solve real-life problems (p. 262). −2 −3 Use logarithmic functions to model and solve real-life problems (p. 263). −4 −5 (b) −3 −2 −1 −3 −2 1 3 2 1 2 3 x In Exercises 27–30, evaluate the function given by at the indicated value of decimal places. fx ex Round your result to three x. 137–142 27. 29. x 8 x 1.7 143–148 28. 30. x 5 8 x 0.278 In Exercises 31–34, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 149 150 x 31. 33. hx ex2 f x e x2 151, 152 32. 34. hx 2 ex2 st 4e2t, t > 0 (c) y (d3 −2 −1 1 2 3 x −3 −2 −1 21 3 x 7. 9. f x 4x f x 4x 8. 10. f x 4x f x 4x 1 In Exercises 11–14, use the graph of transformation that yields the graph of f g. to describe the 11. 12. 13. 14. f x 5x, f x 4x gx 5x1 gx 4x 3 gx 1 x2 x gx 8 2 2 3 In Exercises 15–22, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 15. 17. f x 4x 4 f x 2.65x1 16. 18. f x 4x 3 f x 2.65x1 Compound Interest table to determine the balance rate In Exercises 35 and 36, complete the for dollars invested at n P times per year. for years and compounded A r t 1 2 4 12 365 Continuous n A 35. 36. P $3500, r 6.5%, t 10 years P $2000, r 5%, t 30 years 37. Waiting Times The average time between incoming calls at a switchboard is 3 minutes. The probability of waiting t less than minutes until the next incoming call is approxiFt 1 et 3. mated by the model A call has just come in. Find the probability that the next call will be within F (a) minute. 1 2 (b) 2 minutes. (c) 5 minutes. 38. Depreciation After t years, the value originally cost $14,000 is given by V Vt 14,0003 of a car that t . 4 (a) Use a graphing utility to graph the function. (b) Find the value of the car 2 years after it was purchased. (c) According to the model, when does the car depreciate most rapidly? Is this realistic? Explain. 3 Chapter Test Chapter Test 275 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1– 4, evaluate the expression. Approximate your result to three decimal places. 1. 12.42.79 2. 432 3. e710 4. e3.1 • Chapter Tests and Cumulative Tests Chapter Tests, at the end of each chapter, and periodic Cumulative Tests offer students frequent opportunities for self-assessment and to develop strong study- and test-taking skills. 276 3 y 4 2 −2 2 4 12,000 10,000 8,000 6,000 4,000 2,000 −4 Exponential Growth y FIGURE FOR 6 (9, 11,277) (0, 2745) 2 4 6 8 10 t FIGURE FOR 27 In Exercises 5–7, construct a table of values. Then sketch the graph of the function. f x 6 x2 f x 1 e2x f x 10x 6. 7. 5. 8. Evaluate (a) log7 70.89 and (b) 4.6 ln e2. Chapter 3 Exponential and Logarithmic Functions In Exercises 9–11, construct a table of values. Then sketch the graph of the function. Identify any asymptotes. f x log x 6 9. 10. Cumulative Test for Chapters 1–3 In Exercises 12–14, evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. f x 1 lnx 6 f x lnx 4 11. 13. 12. log7 44 In Exercises 15–17, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. 14. log24 68 Take this test to review the material from earlier chapters. When you are finished, check your work against the answers given in the back of the book. 1, 1. segment joining the points and the distance between the points. Find the coordinates of the midpoint of the line 1. Plot the points log25 0.9 3, 4 and F E A T U R E S 15. log2 3a4 x 16. 17. In Exercises 2– 4, graph the equation without using a graphing utility. log ln 7x2 yz3 5x 6 In Exercises 18–20, condense the expression to the logarithm of a single quantity. 2. x 3y 12 0 3. y x 2 9 4. y 4 x 18. 20. log3 13 log3 y 2 ln x lnx 5 3 ln y 5. Find an equation of the line passing through 19. 4 ln x 4 ln y 1 2, 1 and 3, 8. 6. Explain why the graph at the left does not represent as a function of x. In Exercises 21– 26, solve the equation algebraically. Approximate your result to x 2 three decimal places. f s 2 7. Evaluate (if possible) the function given by f 2 (b) (c) for each value. y f x x 5x 1 25 5 1025 8 e4x 18 4 ln x 7 21. 23. 25. (a) 22. f 6 3e5x 132 necessary to sketch the graphs.) 24. (a) 26. ln x 1 r x 1 2 3x 2 log x log8 5x 2 (b) hx 3x 2 (c) gx 3x 2 8. Compare the graph of each function with the graph of y 3x. (Note: It is not (b) In Exercises 9 and 10, find (a) 27. Find an exponential growth model for the graph shown in the figure. is the domain of f x x 3, is 21.77 years. What percent of a present gx 4x 1 amount of radioactive actinium will remain after 19 years? 28. The half-life of radioactive actinium f/g? 227Ac f x x 1, 10. 9. f gx, f gx, (c) fgx, and (d) f/gx. What gx x2 1 29. A model that can be used for predicting the height In Exercises 11 and 12, find (a) H 70.228 5.104x 9.222 ln x, function. on his or her age is age of the child in years. x (Source: Snapshots of Applications in Mathematics) f x 2x2, (a) Construct a table of values. Then sketch the graph of the model. gx x 6 and (b) ≤ x ≤ 6, f g 1 4 (in centimeters) of a child based is the f x x 2, where g f. 11. 12. H gx x Find the domain of each composite (b) Use the graph from part (a) to estimate the height of a four-year-old child. Then hx 5x 2 has an inverse function. If so, find the inverse 13. Determine whether calculate the actual height using the model. function. 14. The power produced by a wind turbine is proportional to the cube of the wind speed A wind speed of 27 miles per hour produces a power output of 750 kilowatts. Find P S. the output for a wind speed of 40 miles per hour. 15. Find the quadratic function whose graph has a vertex at the point 4, 7. 8, 5 and passes through In Exercises 16–18, sketch the graph of the function without the aid of a graphing utility. 16. hx x 2 4x 17. f t 1 4tt 22 18. gs s2 4s 10 In Exercises 19–21, find all the zeros of the function and write the function as a product of linear factors. 19. 20. 21. f x x3 2x2 4x 8 f x x4 4x3 21x2 f x 2x4 11x3 30x2 62x 40 333200__SE_FM.qxd 12/7/05 10:20 AM Page xvi xvi Textbook Features and Highlights • Proofs in Mathematics At the end of every chapter, proofs of important mathematical properties and theorems are presented as well as discussions of various proof techniques. • P.S. Problem Solving Each chapter concludes with a collection of thought-provoking and challenging exercises that further explore and expand upon the chapter concepts. These exercises have unusual characteristics that set them apart from traditional text exercises. Proofs in Mathematics What does the word proof mean to you? In mathematics, the word proof is used to mean simply a valid argument. When you are proving a statement or theorem, you must use facts, definitions, and accepted properties in a logical order. You can also use previously proved theorems in your proof. For instance, the Distance Formula is used in the proof of the Midpoint Formula below. There are several different proof methods, which you will see in later chapters. The Midpoint Formula The midpoint of the line segment joining the points given by the Midpoint Formula (p. ) x1, y1 and x2, y2 is Midpoint x1 x2 2 y1 , y2 2 . Proof Using the figure, you must show that y (x1, y1) d1 d2 and d1 d2 d3. d1 d 3 ( x1 + x2 2 , y1 + y2 2 ) d 2 (x 2, y 2) x By the Distance Formula, you obtain x1 d1 x2 2 x12 y1 y2 2 y12 1 2 x2 x1 2 y2 y1 2 x2 d2 x1 x2 2 2 y2 y1 y2 2 2 x1 2 y2 y1 2 x2 1 2 x2 d3 So, it follows that x1 d1 2 y2 d2 y1 and d1 2 d2 d3. The Cartesian Plane The Cartesian plane was named after the French mathematician René Descartes (1596–1650). While Descartes was lying in bed, he noticed a fly buzzing around on the square ceiling tiles. He discovered that the position of the fly could be described by which ceiling tile the fly landed on. This led to the development of the Cartesian plane. Descartes felt that a coordinate plane could be used to facilitate description of the positions of objects. 124 P.S. Problem Solving This collection of thought-provoking and challenging exe
rcises further explores and expands upon concepts learned in this chapter. 1. As a salesperson, you receive a monthly salary of $2000, plus a commission of 7% of sales. You are offered a new job at $2300 per month, plus a commission of 5% of sales. (a) Write a linear equation for your current monthly wage W1 in terms of your monthly sales S. W2 (b) Write a linear equation for the monthly wage S. new job offer in terms of the monthly sales of your (c) Use a graphing utility to graph both equations in the same viewing window. Find the point of intersection. What does it signify? (d) You think you can sell $20,000 per month. Should you change jobs? Explain. 2. For the numbers 2 through 9 on a telephone keypad (see figure), create two relations: one mapping numbers onto letters, and the other mapping letters onto numbers. Are both relations functions? Explain. y (x, y) 12 ft FIGURE FOR 6 8 ft x 7. At 2:00 P.M. on April 11, 1912, the Titanic left Cobh, Ireland, on her voyage to New York City. At 11:40 P.M. on April 14, the Titanic struck an iceberg and sank, having covered only about 2100 miles of the approximately 3400-mile trip. (a) What was the total duration of the voyage in hours? (b) What was the average speed in miles per hour? (c) Write a function relating the distance of the Titantic from New York City and the number of hours traveled. Find the domain and range of the function. (d) Graph the function from part (c). 3. What can be said about the sum and difference of each of the following? (a) Two even functions (b) Two odd functions (c) An odd function and an even function 4. The two functions given by gx x f x x and are their own inverse functions. Graph each function and explain why this is true. Graph other linear functions that are their own inverse functions. Find a general formula for a family of linear functions that are their own inverse functions. 5. Prove that a function of the following form is even. y a2nx2n a2n2x2n2 . . . a2x2 a0 6. A miniature golf professional is trying to make a hole-inone on the miniature golf green shown. A coordinate plane is placed over the golf green. The golf ball is at the point 2.5, 2 The professional wants to bank the ball off the side wall of the green at the point Then Find the coordinates of the point write an equation for the path of the ball. and the hole is at the point 9.5, 2. x, y. x, y. 8. Consider the function given by the average rate of change of the function from fx x2 4x 3. x2. to Find x1 1.5 (b) x1 1, x2 (a) (c) (d) (e) x1 x1 x1 x1 1, x2 1, x2 1, x2 1, x2 2 1.25 1.125 1.0625 (f) Does the average rate of change seem to be approaching one value? If so, what value? (g) Find the equations of the secant lines through the points x1, fx1 and x2, fx2 for parts (a)–(e). (h) Find the equation of the line through the point 1, f1 using your answer from part (f ) as the slope of the line. gx x 6. f x 4x and (a) Find 9. Consider the functions given by f gx. f g1x. f 1x and g1 f 1x (b) Find (c) Find (d) Find g1x. and compare the result with that of part (b). (e) Repeat parts (a) through (d) for gx 2x. f x x3 1 and (f) Write two one-to-one functions and parts (a) through (d) for these functions. f g, and repeat (g) Make a conjecture about f g1x and g1 f 1x. 125 126 10. You are in a boat 2 miles from the nearest point on the coast. You are to travel to a point 3 miles down the coast and 1 mile inland (see figure). You can row at 2 miles per hour and you can walk at 4 miles per hour. Q, 13. Show that the Associative Property holds for compositions of functions—that is, f g hx f g hx. 2 mi x 3 − x 1 mi 3 mi Q Not drawn to scale. (a) Write the total time of the trip as a function of T x. (b) Determine the domain of the function. (c) Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. (d) Use the zoom and trace features to find the value of x that minimizes T. 11. The Heaviside function (e) Write a brief paragraph interpreting these values. Hx is widely used in engineering applications. (See figure.) To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Hx 1, 0, x ≥ 0 x < 0 Sketch the graph of each function by hand. Hx Hx 2 2 Hx 2 Hx (b) (d) (e) (a) (c) (f) Hx 2 2 Hx 1 y 3 2 1 −3 −2 −1 1 2 3 x −2 −3 12. Let f x 1 1 x . (a) What are the domain and range of f ? (b) Find (c) Find f f x. f f f x. What is the domain of this function? Is the graph a line? Why or why not? 14. Consider the graph of the function shown in the figure. Use this graph to sketch the graph of each function. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. f (a) (e) f x 1 f x (b) (f) f x 1 f x (c) (g) 2f x f x (d) f x y 4 2 −4 −2 2 4 x −2 −4 15. Use the graphs of function values. f and f1 to complete each table of y 4 2 −2 −2 −1 −2 −2 −4 (a) x 4 2 0 4 f f 1x (b) x 3 2 0 1 (c) (d) f f 1x x f f 1x x f 1x 3 2 0 1 4 3 0 4 333200__SE_FM.qxd 12/7/05 10:20 AM Page xvii Supplements Supplements for the Instructor Precalculus, Seventh Edition, has an extensive support package for the instructor that includes: Instructor’s Annotated Edition (IAE) Online Complete Solutions Guide Online Instructor Success Organizer Online Teaching Center: This free companion website contains an abundance of instructor resources. HM ClassPrep™ with HM Testing (powered by Diploma™): This CD-ROM is a combination of two course management tools. • HM Testing (powered by Diploma™) offers instructors a flexible and powerful tool for test generation and test management. Now supported by the Brownstone Research Group’s market-leading Diploma™ software, this new version of HM Testing significantly improves on functionality and ease of use by offering all the tools needed to create, author, deliver, and customize multiple types of tests—including authoring and editing algorithmic questions. Diploma™ is currently in use at thousands of college and university campuses throughout the United States and Canada. • HM ClassPrep™ also features supplements and text-specific resources for the instructor. Eduspace®: Eduspace®, powered by Blackboard®, is Houghton Mifflin’s customizable and interactive online learning tool. Eduspace® provides instructors with online courses and content. By pairing the widely recognized tools of Blackboard® with quality, text-specific content from Houghton Mifflin Company, Eduspace® makes it easy for instructors to create all or part of a course online. This online learning tool also contains ready-to-use homework exercises, quizzes, tests, tutorials, and supplemental study materials. Visit www.eduspace.com for more information. Eduspace ® with eSolutions: Eduspace® with eSolutions combines all the features of Eduspace® with an electronic version of the textbook exercises and the complete solutions to the odd-numbered exercises, providing students with a convenient and comprehensive way to do homework and view course materials xvii 333200__SE_FM.qxd 12/7/05 10:20 AM Page xviii xviii Supplements Supplements for the Student Precalculus, Seventh Edition, has an extensive support package for the student that includes: Study and Solutions Guide Online Student Notetaking Guide Instructional DVDs Online Study Center: This free companion website contains an abundance of student resources. HM mathSpace® CD-ROM: This tutorial CD-ROM provides opportunities for self-paced review and practice with algorithmically generated exercises and stepby-step solutions. Eduspace®: Eduspace®, powered by Blackboard®, is Houghton Mifflin’s customizable and interactive online learning tool for instructors and students. Eduspace® is a text-specific, web-based learning environment that your instructor can use to offer students a combination of practice exercises, multimedia tutorials, video explanations, online algorithmic homework and more. Specific content is available 24 hours a day to help you succeed in your course. Eduspace® with eSolutions: Eduspace® with eSolutions combines all the features of Eduspace® with an electronic version of the textbook exercises and the complete solutions to the odd-numbered exercises. The result is a convenient and comprehensive way to do homework and view your course materials. Smarthinking®: Houghton Mifflin has partnered with Smarthinking® to provide an easy-to-use, effective, online tutorial service. Through state-of-theart tools and whiteboard technology, students communicate in real-time with qualified e-instructors who can help the students understand difficult concepts and guide them through the problem-solving process while studying or completing homework. Three levels of service are offered to the students. Live Tutorial Help provides real-time, one-on-one instruction. Question Submission allows students to submit questions to the tutor outside the scheduled hours and receive a reply usually within 24 hours. Independent Study Resources connects students around-the-clock to additional educational resources, ranging from interactive websites to Frequently Asked Questions. Visit smarthinking.com for more information. *Limits apply; terms and hours of SMARTHINKING ® service are subject to change. 333202_0100.qxd 12/7/05 8:28 AM Page 1 Functions and Their Graphs 11 Rectangular Coordinates Graphs of Equations Linear Equations in Two Variables Functions Analyzing Graphs of Functions 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 A Library of Parent Functions 1.9 Inverse Functions Transformation of Functions 1.10 Mathematical Modeling and Variation Combinations of Functions: Composite Functions Functions play a primary role in modeling real-life situations. The estimated growth in the number of digital music sales in the United States can be modeled by a cubic function AT I O N S Functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Data Analysis: Mail, Exercise 69, page 12 • Population Statistics, Exercise 75, page 24 • College Enrollment, Exercise 109
, page 37 • Cost, Revenue, and Profit, • Fuel Use, Exercise 97, page 52 Exercise 67, page 82 • Digital Music Sales, Exercise 89, page 64 • Fluid Flow, Exercise 70, page 68 • Consumer Awareness, Exercise 68, page 92 • Diesel Mechanics, Exercise 83, page 102 1 333202_0101.qxd 12/7/05 8:29 AM Page 2 2 Chapter 1 Functions and Their Graphs 1.1 Rectangular Coordinates What you should learn • Plot points in the Cartesian plane. • Use the Distance Formula to find the distance between two points. • Use the Midpoint Formula to find the midpoint of a line segment. • Use a coordinate plane and geometric formulas to model and solve real-life problems. Why you should learn it The Cartesian plane can be used to represent relationships between two variables. For instance, in Exercise 60 on page 12, a graph represents the minimum wage in the United States from 1950 to 2004. The Cartesian Plane Just as you can represent real numbers by points on a real number line, you can represent ordered pairs of real numbers by points in a plane called the rectangular coordinate system, or the Cartesian plane, named after the French mathematician René Descartes (1596–1650). The Cartesian plane is formed by using two real number lines intersecting at right angles, as shown in Figure 1.1. The horizontal real number line is usually called the x-axis, and the vertical real number line is usually called the y-axis. The point of intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants. y-axis y-axis Quadrant II Origin −3 −2 −1 3 2 1 Quadrant I (Vertical number line) 11 2 3 (Horizontal number line) −1 −2 Directed distance x x-axis (x, y) y Directed distance x-axis Quadrant III −3 Quadrant IV FIGURE 1.1 FIGURE 1.2 y, and Each point in the plane corresponds to an ordered pair of real numbers x called coordinates of the point. The x-coordinate represents the directed distance from the -axis to the point, and the y-coordinate represents the directed distance from the -axis to the point, as shown in Figure 1.2. (x, y) y x Directed distance from y-axis x, y Directed distance from x-axis © Ariel Skelly/Corbis The notation x, y denotes both a point in the plane and an open interval on the real number line. The context will tell you which meaning is intended. (3, 4) Example 1 Plotting Points in the Cartesian Plane Plot the points (1, 2), (3, 4), (0, 0), (3, 0), and (2, 3). (0, 0) (3, 0) x 1 2 3 4 Solution on the -axis and a To plot the point horizontal line through 2 on the -axis. The intersection of these two lines is the point The other four points can be plotted in a similar way, as shown in Figure 1.3. imagine a vertical line through 1, 2. (1, 2), 1 y x Now try Exercise 3. y 4 3 1 −1 −2 − ( 1, 2) −4 −3 −1 − − ( 2, 3) −4 FIGURE 1.3 333202_0101.qxd 12/7/05 8:29 AM Page 3 Year, t Amount, A 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 475 577 521 569 609 562 707 723 718 648 495 476 527 464 Section 1.1 Rectangular Coordinates 3 The beauty of a rectangular coordinate system is that it allows you to see relationships between two variables. It would be difficult to overestimate the importance of Descartes’s introduction of coordinates in the plane. Today, his ideas are in common use in virtually every scientific and business-related field. Example 2 Sketching a Scatter Plot From 1990 through 2003, the amounts equipment in the United States are shown in the table, where year. Sketch a scatter plot of the data. Association) (in millions of dollars) spent on skiing represents the (Source: National Sporting Goods t A Solution To sketch a scatter plot of the data shown in the table, you simply represent each pair of values by an ordered pair and plot the resulting points, as shown in Figure 1.4. For instance, the first pair of values is represented by the ordered pair 1990, 475. t Note that the break in the -axis indicates that the numbers between 0 and 1990 have been omitted. (t, A) Amount Spent on Skiing Equipment 800 700 600 500 400 300 200 100 1991 1995 1999 2003 t Year FIGURE 1.4 Now try Exercise 21. In Example 2, you could have let t 1 represent the year 1990. In that case, the horizontal axis would not have been broken, and the tick marks would have been labeled 1 through 14 (instead of 1990 through 2003). Te c h n o l o g y The scatter plot in Example 2 is only one way to represent the data graphically. You could also represent the data using a bar graph and a line graph. If you have access to a graphing utility, try using it to represent graphically the data given in Example 2. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. 333202_0101.qxd 12/7/05 8:29 AM Page 4 4 a FIGURE 1.5  FIGURE 1.6 Chapter 1 Functions and Their Graphs a2 + b2 = c2 The Pythagorean Theorem and the Distance Formula The following famous theorem is used extensively throughout this course. c b (x , y ) 1 1 d Pythagorean Theorem For a right triangle with hypotenuse of length and sides of lengths and as shown in Figure 1.5. (The converse is also true. you have then the triangle is a right triangle.) That is, if a 2 b2 c 2, a 2 b2 c 2, a c b, d Suppose you want to determine the distance between two points x2, y2 y2 x1, y1 in the plane. With these two points, a right triangle can be formed, as y1, x1. By the Pythagorean Theorem, and shown in Figure 1.6. The length of the vertical side of the triangle is and the length of the horizontal side is you can write d 2 x2 d x2 x12 y2 x12 y2 2 y2 x2 y12 y12 This result is the Distance Formula. y1 x1 x2 2. (x , y ) 2 1 (x , y ) 2 2 x1 x x2 x − x  1 2 The Distance Formula x1, y1 d The distance between the points 2. d x2 y1 x1 2 y2 and x2, y2 in the plane is Example 3 Finding a Distance Find the distance between the points 2, 1 and 3, 4. Algebraic Solution 2, 1 x1, y1 Let Formula. and x2, y2 3, 4. Then apply the Distance 2 x1 y1 2 y2 d x2 3 22 4 12 52 32 34 5.83 Distance Formula Substitute for x1, y1, x2, and y2. Simplify. Simplify. Use a calculator. So, the distance between the points is about 5.83 units. You can use the Pythagorean Theorem to check that the distance is correct. d2 ? 34 2 ? 32 52 32 52 34 34 Pythagorean Theorem Substitute for d. Distance checks. ✓ Now try Exercises 31(a) and (b). Graphical Solution Use centimeter graph paper to plot the points A2, 1 B3, 4. Carefully sketch the line and A B. to Then use a centimeter segment from ruler to measure the length of the segment. cm 1 2 3 4 5 6 7 FIGURE 1.7 The line segment measures about 5.8 centimeters, as shown in Figure 1.7. So, the distance between the points is about 5.8 units. 333202_0101.qxd 12/7/05 8:29 AM Page 5, 7) d1 = 45 d3 = 50 (2, 1) d2 = 5 (4, 0) x Section 1.1 Rectangular Coordinates 5 Example 4 Verifying a Right Triangle Show that the points 2, 1, 4, 0, and 5, 7 are vertices of a right triangle. Solution The three points are plotted in Figure 1.8. Using the Distance Formula, you can find the lengths of the three sides as follows. 5 2 2 7 12 9 36 45 4 2 2 0 12 4 1 5 5 4 2 7 02 1 49 50 d1 d2 d3 1 2 3 4 5 6 7 FIGURE 1.8 Because d1 2 d2 2 45 5 50 d3 2 you can conclude by the Pythagorean Theorem that the triangle must be a right triangle. Now try Exercise 41. The Midpoint Formula To find the midpoint of the line segment that joins two points in a coordinate plane, you can simply find the average values of the respective coordinates of the two endpoints using the Midpoint Formula. The Midpoint Formula The midpoint of the line segment joining the points given by the Midpoint Formula x1, y1 and x2, y2 is Midpoint x1 x2 2 y1 , y2 2 . For a proof of the Midpoint Formula, see Proofs in Mathematics on page 124. Example 5 Finding a Line Segment’s Midpoint Find the midpoint of the line segment joining the points 5, 3 and 9, 3. y (9, 3) (2, 0) 3 6 9 x Midpoint 6 3 −3 −6 −6 −3 − − ( 5, 3) FIGURE 1.9 Solution x1, y1 Let Midpoint x1 5, 3 9, 3. x2, y2 and y2 y1 2 3 3 2 , x2 , 2 5 9 2 2, 0 Simplify. Midpoint Formula Substitute for x1, y1, x2, and y2. The midpoint of the line segment is 2, 0, as shown in Figure 1.9. Now try Exercise 31(c). 333202_0101.qxd 12/7/05 8:29 AM Page 6 6 Chapter 1 Functions and Their Graphs Applications Example 6 Finding the Length of a Pass Football Pass (40, 28) During the third quarter of the 2004 Sugar Bowl, the quarterback for Louisiana State University threw a pass from the 28-yard line, 40 yards from the sideline. The pass was caught by a wide receiver on the 5-yard line, 20 yards from the same sideline, as shown in Figure 1.10. How long was the pass? Solution You can find the length of the pass by finding the distance between the points 40, 28 35 30 25 20 15 10 20, 5) 5 10 15 20 25 30 35 40 Distance (in yards) FIGURE 1.10 e u n e v e R 26 25 24 23 22 21 20 ) ( FedEx Annual Revenue (2004, 24.7) (2003, 22.65) Midpoint (2002, 20.6) 2002 2003 2004 Year x1 2 y1 2 y2 20, 5. and d x2 40 20 2 28 52 400 529 929 30 Distance Formula Substitute for x1, y1, x2, and y2. Simplify. Simplify. Use a calculator. So, the pass was about 30 yards long. Now try Exercise 47. In Example 6, the scale along the goal line does not normally appear on a football field. However, when you use coordinate geometry to solve real-life problems, you are free to place the coordinate system in any way that is convenient for the solution of the problem. Example 7 Estimating Annual Revenue FedEx Corporation had annual revenues of $20.6 billion in 2002 and $24.7 billion in 2004. Without knowing any additional information, what would you estimate the 2003 revenue to have been? (Source: FedEx Corp.) Solution One solution to the problem is to assume that revenue followed a linear pattern. With this assumption, you can estimate the 2003 revenue by finding the midpoint of the line segment connecting the points 2004, 24.7. 2002, 20.6 and Midpoint x1 , y1 y2 x2 2 2 2002 2004 2 , Midpoint Formula 20.6 24.7 2 Substitute for x1, y1, x2, and y2. 2003
, 22.65 Simplify. So, you would estimate the 2003 revenue to have been about $22.65 billion, as shown in Figure 1.11. (The actual 2003 revenue was $22.5 billion.) FIGURE 1.11 Now try Exercise 49. 333202_0101.qxd 12/7/05 8:30 AM Page Much of computer graphics, including this computer-generated goldfish tessellation, consists of transformations of points in a coordinate plane. One type of transformation, a translation, is illustrated in Example 8. Other types include reflections, rotations, and stretches. Section 1.1 Rectangular Coordinates 7 Example 8 Translating Points in the Plane 2, 3. The triangle in Figure 1.12 has vertices at the points Shift the triangle three units to the right and two units upward and find the vertices of the shifted triangle, as shown in Figure 1.13. 1, 2, 1, 4, and y 5 4 − ( 1, 2) (2, 3) y 5 4 3 2 1 −2 −2 −1 1 2 3 5 6 7 x −2 −3 −4 (1, 4)− −2 −3 −4 FIGURE 1.12 FIGURE 1.13 Solution To shift the vertices three units to the right, add 3 to each of the -coordinates. To shift the vertices two units upward, add 2 to each of the -coordinates. x y Original Point 1, 2 1, 4 2, 3 Translated Point 1 3, 2 2 2, 4 1 3, 4 2 4, 2 2 3, 3 2 5, 5 Now try Exercise 51. The figures provided with Example 8 were not really essential to the solution. Nevertheless, it is strongly recommended that you develop the habit of including sketches with your solutions—even if they are not required. The following geometric formulas are used at various times throughout this course. For your convenience, these formulas along with several others are also provided on the inside back cover of this text. Common Formulas for Area A, Perimeter P, Circumference C, and Volume V Rectangle Rectangular Solid Circular Cylinder Circle A lw P 2l 2w A r2 C 2r Triangle A 1 2 P a b c bh V lwh V r 2h h l w r h Sphere V 4 3 r3 r w l r a h c b 333202_0101.qxd 12/7/05 8:30 AM Page 8 8 Chapter 1 Functions and Their Graphs Example 9 Using a Geometric Formula 4 cm h FIGURE 1.14 A cylindrical can has a volume of 200 cubic centimeters 4 centimeters (cm), as shown in Figure 1.14. Find the height of the can. and a radius of cm3 Solution The formula for the volume of a cylinder is can, solve for h V r 2 h. V r2h. To find the height of the V 200 Then, using h 200 42 200 16 and r 4, find the height. Substitute 200 for V and 4 for r. Simplify denominator. 3.98 Use a calculator. Because the value of was rounded in the solution, a check of the solution will not result in an equality. If the solution is valid, the expressions on each side of the equal sign will be approximately equal to each other. h V r2 h 200 ? 423.98 200 200.06 Write original equation. Substitute 200 for V, 4 for r, and 3.98 for h. Solution checks. ✓ You can also use unit analysis to check that your answer is reasonable. 200 cm3 16 cm2 3.98 cm Now try Exercise 63. W RITING ABOUT MATHEMATICS Extending the Example Example 8 shows how to translate points in a coordinate plane. Write a short paragraph describing how each of the following transformed points is related to the original point. Original Point x, y x, y x, y Transformed Point x, y x, y x, y 333202_0101.qxd 12/7/05 8:30 AM Page 9 1.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. Section 1.1 Rectangular Coordinates 9 VOCABULARY CHECK 1. Match each term with its definition. (a) x -axis (b) y -axis (c) origin (d) quadrants (e) x -coordinate (f) y -coordinate (i) point of intersection of vertical axis and horizontal axis (ii) directed distance from the x-axis (iii) directed distance from the y-axis (iv) four regions of the coordinate plane (v) horizontal real number line (vi) vertical real number line In Exercises 2–4, fill in the blanks. 2. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ________ plane. 3. The ________ ________ is a result derived from the Pythagorean Theorem. 4. Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the ________ ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1 and 2, approximate the coordinates of the points. In Exercises 11–20, determine the quadrant(s) in which (x, y) is located so that the condition(s) is (are) satisfied. 1. y 6 4 2 D 2. A y 4 2 C −6 −4 B −2 −2 −4 x 2 4 C D −6 −4 −2 x 2 B A −2 −4 In Exercises 3–6, plot the points in the Cartesian plane. 3. 4. 5. 6. 4, 2, 0, 0, 3, 8, 1, 1 3 3, 1, 0.5, 1, , 4, 3, 3 3, 6, 0, 5, 1, 4 2, 4, 1, 1 5, 6, 3, 4, 2, 2.5 3, 3 2 4 In Exercises 7–10, find the coordinates of the point. 7. The point is located three units to the left of the -axis and y four units above the -axis. x 8. The point is located eight units below the -axis and four x units to the right of the -axis. y 9. The point is located five units below the -axis and the x coordinates of the point are equal. 10. The point is on the -axis and 12 units to the left of the x y -axis. and y < 0 and y > 0 and y > 0 11. 13. 15. 17 19. xy > 0 12. x < 0 and 14. x > 2 and y < 0 y 3 16. 18. 20. x > 4 x > 0 xy < 0 and y < 0 In Exercises 21 and 22, sketch a scatter plot of the data shown in the table. 21. Number of Stores The table shows the number of from 1996 through 2003. y x Wal-Mart stores for each year (Source: Wal-Mart Stores, Inc.) Year, x Number of stores, y 1996 1997 1998 1999 2000 2001 2002 2003 3054 3406 3599 3985 4189 4414 4688 4906 333202_0101.qxd 12/7/05 8:30 AM Page 10 10 Chapter 1 Functions and Their Graphs 22. Meteorology The table shows the lowest temperature on (in degrees Fahrenheit) in Duluth, Minnesota for (Source: represents January. x 1 x, y record each month where NOAA) Month, x Temperature 10 11 12 39 39 29 5 17 27 35 32 22 8 23 34 In Exercises 31–40, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. 32. 34. 36. 1, 12, 6, 0 7, 4, 2, 8 2, 10, 10, 2 31. 33. 35. 37. 38. 39. 40. 1, 1, 9, 7 4, 10, 4, 5 1, 2, 5, 4 2, 1, 5 1 2, 4 , 1 1 6, 1 3, 1 6.2, 5.4, 3.7, 1.8 16.8, 12.3, 5.6, 4.9 3 3 2 In Exercises 41 and 42, show that the points form the vertices of the indicated polygon. 41. Right triangle: 4, 0, 2, 1, 1, 5 42. Isosceles triangle: 1, 3, 3, 2, 2, 4 43. A line segment has x1, y1 as one endpoint and its midpoint. Find the other endpoint segment in terms of and x1, y1, xm, ym. xm, ym as of the line x2, y2 23. In Exercises 23–26, find the distance between the points. (Note: In each case, the two points lie on the same horizontal or vertical line.) 6, 3, 6, 5 1, 4, 8, 4 3, 1, 2, 1 3, 4, 3, 6 24. 25. 26. In Exercises 27–30, (a) find the length of each side of the right triangle, and (b) show that these lengths satisfy the Pythagorean Theorem. 27. y 28. y 5 4 3 2 1 (4, 5) (0, 2) (4, 2) 1 2 3 4 5 29. y 6 4 2 (9, 4) (9, 1) (−1, 1) 6 8 x x 8 4 (1, 0) (13, 5) x 4 8 (13, 0) 30. y (1, 5) 4 2 −2 (5, −2) x 6 (1, −2) 44. Use the result of Exercise 43 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and midpoint are, respectively, (a) 1, 2, 4, 1 and (b) 5, 11, 2, 4. 45. Use the Midpoint Formula three times to find the three and points that divide the line segment joining x2, y2 into four parts. x1, y1 46. Use the result of Exercise 45 to find the points that divide the line segment joining the given points into four equal parts. (a) 1, 2, 4, 1 (b) 2, 3, 0, 0 47. Sports A soccer player passes the ball from a point that is 18 yards from the endline and 12 yards from the sideline. The pass is received by a teammate who is 42 yards from the same endline and 50 yards from the same sideline, as shown in the figure. How long is the pass? 50 40 30 20 10 ) 50, 42) (12, 18) 10 20 30 40 50 Distance (in yards) 60 48. Flying Distance An airplane flies from Naples, Italy in a straight line to Rome, Italy, which is 120 kilometers north and 150 kilometers west of Naples. How far does the plane fly? 333202_0101.qxd 12/7/05 8:30 AM Page 11 Sales In Exercises 49 and 50, use the Midpoint Formula to estimate the sales of Big Lots, Inc. and Dollar Tree Stores, Inc. in 2002, given the sales in 2001 and 2003. Assume that the sales followed a linear pattern. (Source: Big Lots, Inc.; Dollar Tree Stores, Inc.) 55. Approximate the highest price of a pound of butter shown in the graph. When did this occur? 56. Approximate the percent change in the price of butter from the price in 1995 to the highest price shown in the graph. Section 1.1 Rectangular Coordinates 11 49. Big Lots Year 2001 2003 Sales (in millions) $3433 $4174 50. Dollar Tree Year 2001 2003 Sales (in millions) $1987 $2800 In Exercises 51–54, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position. 51. y 4 − − ( 1, 1) −4 − units (2, 3)− − − ( 2, 4) 52. y − ( 3, 6, 3) 6 units − ( 3, 0) − ( 5, 3) 1 3 x 53. Original coordinates of vertices: 7, 4 2, 4, Shift: eight units upward, four units to the right 7, 2, 2, 2, 54. Original coordinates of vertices: 3, 6, Shift: 6 units downward, 10 units to the left 5, 8, 7, 6, 5, 2 Retail Price In Exercises 55 and 56, use the graph below, which shows the average retail price of 1 pound of butter from 1995 to 2003. (Source: U.S. Bureau of Labor Statistics.50 3.25 3.00 2.75 2.50 2.25 2.00 1.75 1.50 1995 1997 1999 Year 2001 2003 Advertising In Exercises 57 and 58, use the graph below, which shows the cost of a 30-second television spot (in thousands of dollars) during the Super Bowl from 1989 to 2003. (Source: USA Today Research and CNN ( 2400 2200 2000 1800 1600 1400 1200 1000 800 600 1989 1991 1993 1995 1997 1999 2001 2003 Year 57. Approximate the percent increase in the cost of a 30-second spot from Super Bowl XXIII in 1989 to Super Bowl XXXV in 2001. 58. Estimate the perc
ent increase in the cost of a 30-second spot (a) from Super Bowl XXIII in 1989 to Super Bowl XXVII in 1993 and (b) from Super Bowl XXVII in 1993 to Super Bowl XXXVII in 2003. 59. Music The graph shows the numbers of recording artists who were elected to the Rock and Roll Hall of Fame from 1986 to 2004. 16 14 12 10 1987 1989 1991 1993 1995 1997 1999 2001 2003 Year (a) Describe any trends in the data. From these trends, predict the number of artists elected in 2008. (b) Why do you think the numbers elected in 1986 and 1987 were greater in other years? 333202_0101.qxd 12/7/05 8:30 AM Page 12 12 Chapter 1 Functions and Their Graphs Model It 60. Labor Force Use the graph below, which shows the minimum wage in the United States (in dollars) from 1950 to 2004. (Source: U.S. Department of Labor 1950 1960 1970 1980 1990 2000 Year (a) Which decade shows the greatest increase in minimum wage? (b) Approximate the percent increases in the minimum wage from 1990 to 1995 and from 1995 to 2004. (c) Use the percent increase from 1995 to 2004 to pre- dict the minimum wage in 2008. (d) Do you believe that your prediction in part (c) is reasonable? Explain. 61. Sales The Coca-Cola Company had sales of $18,546 million in 1996 and $21,900 million in 2004. Use the Midpoint Formula to estimate the sales in 1998, 2000, and 2002. Assume that the sales followed a linear pattern. (Source: The Coca-Cola Company) 62. Data Analysis: Exam Scores The table shows the mathand the final examination in an algebra course for a sample of 10 students. ematics entrance test scores scores x y x y x y 22 53 48 90 29 74 53 76 35 57 58 93 40 66 65 83 44 79 76 99 64. Length of a Tank The diameter of a cylindrical propane gas tank is 4 feet. The total volume of the tank is 603.2 cubic feet. Find the length of the tank. 65. Geometry A “Slow Moving Vehicle” sign has the shape of an equilateral triangle. The sign has a perimeter of 129 centimeters. Find the length of each side of the sign. Find the area of the sign. 66. Geometry The radius of a traffic cone is 14 centimeters and the lateral surface of the cone is 1617 square centimeters. Find the height of the cone. 67. Dimensions of a Room A room is 1.5 times as long as it is wide, and its perimeter is 25 meters. (a) Draw a diagram that represents the problem. Identify the length as and the width as l w. (b) Write w in terms of w. perimeter in terms of l and write an equation for the (c) Find the dimensions of the room. 68. Dimensions of a Container The width of a rectangular storage container is 1.25 times its height. The length of the container is 16 inches and the volume of the container is 2000 cubic inches. (a) Draw a diagram that represents the problem. Label the height, width, and length accordingly. (b) Write w volume in terms of in terms of h. h and write an equation for the (c) Find the dimensions of the container. 69. Data Analysis: Mail The table shows the number of pieces of mail handled (in billions) by the U.S. Postal Service for each year from 1996 through 2003. (Source: U.S. Postal Service) y x Year, x Pieces of mail, y 1996 1997 1998 1999 2000 2001 2002 2003 183 191 197 202 208 207 203 202 (a) Sketch a scatter plot of the data. (b) Find the entrance exam score of any student with a final exam score in the 80s. (c) Does a higher entrance exam score imply a higher final exam score? Explain. 63. Volume of a Billiard Ball A billiard ball has a volume of 5.96 cubic inches. Find the radius of a billiard ball. (a) Sketch a scatter plot of the data. (b) Approximate the year in which there was the greatest decrease in the number of pieces of mail handled. (c) Why do you think the number of pieces of mail handled decreased? 333202_0101.qxd 12/7/05 8:30 AM Page 13 70. Data Analysis: Athletics The table shows the numbers of men’s M and women’s W college basketball teams for each year (Source: National Collegiate Athletic Association) from 1994 through 2003. x Year, x Men’s teams, M Women’s teams, W 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 858 868 866 865 895 926 932 937 936 967 859 864 874 879 911 940 956 958 975 1009 (a) Sketch scatter plots of these two sets of data on the same set of coordinate axes. (b) Find the year in which the numbers of men’s and women’s teams were nearly equal. (c) Find the year in which the difference between the numbers of men’s and women’s teams was the greatest. What was this difference? 2, 1, 3, 5, 71. Make a Conjecture Plot the points and 7, 3 on a rectangular coordinate system. Then change the sign of the -coordinate of each point and plot the three new points on the same rectangular coordinate system. Make a conjecture about the location of a point when each of the following occurs. x (a) The sign of the -coordinate is changed. x (b) The sign of the -coordinate is changed. y (c) The signs of both the - and -coordinates are changed. y x 72. Collinear Points Three or more points are collinear if they all lie on the same line. Use the steps below to deterA2, 3, mine if the set of points and the set of points C6, 3 are collinear. A8, 3, C2, 1 B5, 2, B2, 6, (a) For each set of points, use the Distance Formula to find B, to from What relationship exists among these distances for and from to to C, A A B the distances from C. each set of points? (b) Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line? (c) Compare your conclusions from part (a) with the conclusions you made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearity. Section 1.1 Rectangular Coordinates 13 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 73 and 74, determine whether 73. In order to divide a line segment into 16 equal parts, you 74. The points would have to use the Midpoint Formula 16 times. 2, 11, 8, 4, vertices of an isosceles triangle. 5, 1 and represent the 75. Think About It When plotting points on the rectangular coordinate system, is it true that the scales on the - and y -axes must be the same? Explain. x 76. Proof Prove that the diagonals of the parallelogram in the figure intersect at their midpoints. y y )x , y ) 0 0 x (0, 0) a ( , 0) x FIGURE FOR 76 FIGURE FOR 77–80 In Exercises 77–80, use the plot of the point in the figure. Match the transformation of the point with the correct plot. [The plots are labeled (a), (b), (c), and (d).] x0, y0 (a) (c) y y y y (b) (d) x x x x 77. 79. x0, y0 x0, 1 2 y0 Skills Review 78. 80. 2x0, y0 x0, y0 In Exercises 81– 88, solve the equation or inequality. 81. 83. 85. 87. 2x 1 7x 4 x2 4x 7 0 3x 1 < 22 x x 18 < 4 1 82. 84. 3x 2 5 1 6x 2x2 3x 8 0 3x 8 ≥ 1 86. 2 88. 2x 15 ≥ 11 10x 7 333202_0102.qxd 12/7/05 8:31 AM Page 14 14 Chapter 1 Function and Their Graphs 1.2 Graphs of Equations What you should learn • Sketch graphs of equations. • Find x- and y-intercepts of graphs of equations. • Use symmetry to sketch graphs of equations. • Find equations of and sketch graphs of circles. • Use graphs of equations in solving real-life problems. Why you should learn it The graph of an equation can help you see relationships between real-life quantities. For example, in Exercise 75 on page 24, a graph can be used to estimate the life expectancies of children who are born in the years 2005 and 2010. The Graph of an Equation In Section 1.1, you used a coordinate system to represent graphically the relationship between two quantities. There, the graphical picture consisted of a collection of points in a coordinate plane. Frequently, a relationship between two quantities is expressed as an equay 7 3x y. and An if the is substituted for For instance, is a solution or solution point of an equation in and b is substituted for and tion in two variables. For instance, ordered pair a equation is true when 1, 4 y 7 3x is a solution of x is an equation in x y. is a true statement. x because 4 7 31 a, b y In this section you will review some basic procedures for sketching the graph of an equation in two variables. The graph of an equation is the set of all points that are solutions of the equation. Example 1 Determining Solutions Determine whether (a) y 10x 7. 2, 13 and (b) 1, 3 are solutions of the equation Solution a. y 10x 7 13 ? 13 13 102 7 Write original equation. Substitute 2 for x and 13 for y. is a solution. ✓ 2, 13 Because the substitution does satisfy the original equation, you can conclude that the ordered pair y 10x 7 is a solution of the original equation. Write original equation. 2, 13 b. 101 7 3 ? 3 17 Substitute 1, 3 1 for x and 3 for y. is not a solution. Because the substitution does not satisfy the original equation, you can conis not a solution of the original equation. clude that the ordered pair 1, 3 Now try Exercise 1. © John Griffin/The Image Works point-plotting method. The basic technique used for sketching the graph of an equation is the Sketching the Graph of an Equation by Point Plotting 1. If possible, rewrite the equation so that one of the variables is isolated on one side of the equation. 2. Make a table of values showing several solution points. 3. Plot these points on a rectangular coordinate system. 4. Connect the points with a smooth curve or line. 333202_0102.qxd 12/7/05 8:31 AM Page 15 Section 1.2 Graphs of Equations 15 Example 2 Sketching the Graph of an Equation Sketch the graph of y 7 3x. Solution Because the equation is already solved for y, construct a table of values that x 1, consists of several solution points of the equation. For instance, when y 7 31 10 which implies that 1, 10 is a solution point of the graph 3x 10 7 4 1 2 5 x, y 1, 10 0, 7 1, 4 2, 1 3, 2 4, 5 From the table, it follows that 1, 10, 0, 7, 1, 4, 2, 1, 3, 2, and 4, 5 are solution points of the equation. After plotting these points, you can see that they appear to lie on a line, as shown in Figure 1.15. The graph of the equation is the line that passes through the six plotted points. y (− 1, 10) 8 6 4 2
(0, 7) (1, 4) (2, 1) x 2 6 4 (3, − 2) 8 10 (4, − 5) −4 −2 −2 −4 −6 FIGURE 1.15 Now try Exercise 5. 333202_0102.qxd 12/7/05 8:31 AM Page 16 16 Chapter 1 Function and Their Graphs One of your goals in this course is to learn to classify the basic shape of a graph from its equation. For instance, you will learn that the linear equation in Example 2 has the form y mx b and its graph is a line. Similarly, the quadratic equation in Example 3 has the form y ax2 bx c and its graph is a parabola. Example 3 Sketching the Graph of an Equation Sketch the graph of y x 2 2. Solution Because the equation is already solved for values. y, begin by constructing a table of x y x2 2 x, y 2 2 2, 2 1 1 1, 1 0 2 0, 2 1 1 1, 1 2 3 2 2, 2 7 3, 7 Next, plot the points given in the table, as shown in Figure 1.16. Finally, connect the points with a smooth curve, as shown in Figure 1.17. y 6 4 2 (−2, 2) −4 −2 (−1, −1) FIGURE 1.16 (3, 7) (2, 2) 2 (1, −1) 4 (0, −2) x (−2, 2) −4 −2 (−1, −1) FIGURE 1.17 y 6 4 2 (3, 7) y = x2 − 2 (2, 2) x 2 (1, −1) 4 (0, −2) Now try Exercise 7. The point-plotting method demonstrated in Examples 2 and 3 is easy to use, but it has some shortcomings. With too few solution points, you can misrepresent the graph of an equation. For instance, if only the four points 2, 2, 1, 1, 1, 1, and 2, 2 in Figure 1.16 were plotted, any one of the three graphs in Figure 1.18 would be reasonable2 2 x −2 2 x −2 2 x FIGURE 1.18 333202_0102.qxd 12/7/05 8:31 AM Page 17 Section 1.2 Graphs of Equations 17 Te c h n o l o g y To graph an equation involving x and y on a graphing utility, use the following procedure. 1. Rewrite the equation so that y is isolated on the left side. 2. Enter the equation into the graphing utility. 3. Determine a viewing window that shows all important features of the graph. 4. Graph the equation. For more extensive instructions on how to use a graphing utility to graph an equation, see the Graphing Technology Guide on the text website at college.hmco.com. Intercepts of a Graph y It is often easy to determine the solution points that have zero as either the x -coordinate or the -coordinate. These points are called intercepts because they are the points at which the graph intersects or touches the - or -axis. It is possible for a graph to have no intercepts, one intercept, or several intercepts, as shown in Figure 1.19. Note that an and a y -intercept can be written as the ordered pair Some texts denote the x [and the y-intercept as the -intercept as the -coordinate of the point y -coordinate of the point ] rather than the point itself. Unless it is necessary to make a distinction, we will use the term intercept to mean either the point or the coordinate. -intercept can be written as the ordered pair 0, y. 0, b a, 0 x, 0 x x y x Finding Intercepts x 1. To find -intercepts, let be zero and solve the equation for y x. 2. To find -intercepts, let be zero and solve the equation for y. x y Example 4 Finding x- and y-Intercepts Find the - and -intercepts of the graph of y x y x3 4x. y x No x-intercepts; one y-intercept y x Three x-intercepts; one y-intercept y x One x-intercept; two y-intercepts y x No intercepts FIGURE 1.19 y = x − 4x 3 y 4 Then Solution y 0. Let 0 x3 4x xx2 4 x 0 has solutions and x ±2. 0, 0, 2, 0, 2, 0 (−2, 0) − 4 (0, 0) (2, 0) 4 x Let -intercepts: x x 0. Then y 03 40 −2 − 4 has one solution, y -intercept: y 0. 0, 0 See Figure 1.20. FIGURE 1.20 Now try Exercise 11. 333202_0102.qxd 12/7/05 8:31 AM Page 18 18 Chapter 1 Function and Their Graphs Symmetry Graphs of equations can have symmetry with respect to one of the coordinate axes or with respect to the origin. Symmetry with respect to the -axis means that if the Cartesian plane were folded along the -axis, the portion of the graph above the -axis would coincide with the portion below the -axis. Symmetry with y respect to the -axis or the origin can be described in a similar manner, as shown in Figure 1.21. x x x x y y (−x, y) (x, y) (x, y) x (x, −y) y (x, y) x x (−x, −y) x-axis symmetry FIGURE 1.21 y-axis symmetry Origin symmetry Knowing the symmetry of a graph before attempting to sketch it is helpful, because then you need only half as many solution points to sketch the graph. There are three basic types of symmetry, described as follows. Graphical Tests for Symmetry 1. A graph is symmetric with respect to the x-axis if, whenever is also on the graph. the graph, x, y x, y is on 2. A graph is symmetric with respect to the y-axis if, whenever is also on the graph. the graph, x, y x, y is on 3. A graph is symmetric with respect to the origin if, whenever is also on the graph. x, y the graph, x, y is on Example 5 Testing for Symmetry The graph of x, y point below confirms that the graph is symmetric with respect to the -axis. is symmetric with respect to the -axis because the (See Figure 1.22.) The table y x2 2 is also on the graph of y x2 2. y y x y x, y 3 2 7 3, 7 2 2, 2 1 1 1, 1 1 1 1, 1 2 3 2 2, 2 7 3, 7 Now try Exercise 23. (−3, 7) (−2, 23, 7) (2, 2) x −4 −3 −2 (−1, −1) −3 3 4 5 2 (1, −1) 2 y = x − 2 FIGURE 1.22 y-axis symmetry 333202_0102.qxd 12/7/05 8:31 AM Page 19 Section 1.2 Graphs of Equations 19 Algebraic Tests for Symmetry 1. The graph of an equation is symmetric with respect to the -axis if x replacing with y y yields an equivalent equation. 2. The graph of an equation is symmetric with respect to the -axis if y replacing with x x yields an equivalent equation. 3. The graph of an equation is symmetric with respect to the origin if yields an equivalent equation. replacing with y and with x y x Example 6 Using Symmetry as a Sketching Aid (5, 2) Use symmetry to sketch the graph of x y 2 1. Solution Of the three tests for symmetry, the only one that is satisfied is the test for -axis symmetry because . So, the graph is symmetric with respect to the -axis. Using symmetry, you only need to find the solution points above the -axis and then reflect them to obtain the graph, as shown in Figure 1.23. x y2 1 is equivalent to x y2 1 x x x y 0 1 2 x y2 1 1 2 5 x, y 1, 0 2, 1 5, 2 Now try Exercise 37. Example 7 Sketching the Graph of an Equation Sketch the graph of y x 1. Solution This equation fails all three tests for symmetry and consequently its graph is not symmetric with respect to either axis or to the origin. The absolute value sign indiis always nonnegative. Create a table of values and plot the points as cates that shown in Figure 1.24. From the table, you can see that So, the y -intercept is y 1. So, the -intercept is x 0 x Similarly, x 1. 0, 1. 1, 0. y 0 when when y x y x 1 x, 3 2 1, 2 1 0, 1 0 1, 0 1 2, 1 2 3, 2 3 4, 3 Now try Exercise 41. y 2 1 (1, 0) x − y = 1 2 (2, 1) 2 3 4 5 x −1 −2 FIGURE 1.23 Notice that when creating the table in Example 6, it is easier to choose y-values and then find the corresponding x-values of the ordered pairs. y 6 5 4 3 2 (−2, 3) (−1, 2) y  −  x= 1 (4, 3) (3, 2) (0, 1) (2, 1) −3 −2 −1 (1, 0) 2 3 4 5 x −2 FIGURE 1.24 333202_0102.qxd 12/7/05 8:31 AM Page 20 20 y Chapter 1 Function and Their Graphs Throughout this course, you will learn to recognize several types of graphs from their equations. For instance, you will learn to recognize that the graph of a second-degree equation of the form y ax 2 bx c Center: (h, k) is a parabola (see Example 3). The graph of a circle is also easy to recognize. Radius: r Circles Point on circle: (x, y) x Consider the circle shown in Figure 1.25. A point if its distance from the center x, y is r. By the Distance Formula, h, k is on the circle if and only x h2 y k2 r. FIGURE 1.25 By squaring each side of this equation, you obtain the standard form of the equation of a circle. Standard Form of the Equation of a Circle lies on the circle of radius r and center The point x, y (h, k) if and only if x h2 y k2 r 2. To find the correct h and k, from the equation of the circle in Example 8, it may be helpful to rewrite the quantities and using subtraction. x 12 y 22, x 12 x 12, y 22 y 22 h 1 and k 2. So, From this result, you can see that the standard form of the equation of a h, k 0, 0, circle with its center at the origin, is simply x2 y 2 r 2. Circle with center at origin Example 8 Finding the Equation of a Circle 1, 2, The point 1.26. Write the standard form of the equation of this circle. lies on a circle whose center is at 3, 4 as shown in Figure y 6 4 (−1, 2) (3, 4) −6 −2 2 4 x −2 −4 Solution The radius of the circle is the distance between r x h2 y k2 3 12 4 22 42 22 16 4 20 h, k 1, 2 and 1, 2 and 3, 4. Distance Formula Substitute for x, y, h, and k. Simplify. Simplify. Radius Using r 20, the equation of the circle is x h2 y k2 r2 x 12 y 22 202 x 12 y 22 20. Equation of circle Substitute for h, k, and r. Standard form FIGURE 1.26 Now try Exercise 61. 333202_0102.qxd 12/7/05 8:31 AM Page 21 You should develop the habit of using at least two approaches to solve every problem. This helps build your intuition and helps you check that your answer is reasonable. Section 1.2 Graphs of Equations 21 Application In this course, you will learn that there are many ways to approach a problem. Three common approaches are illustrated in Example 9. A Numerical Approach: Construct and use a table. A Graphical Approach: Draw and use a graph. An Algebraic Approach: Use the rules of algebra. Example 9 Recommended Weight The median recommended weight y (in pounds) for men of medium frame who are 25 to 59 years old can be approximated by the mathematical model y 0.073x 2 6.99x 289.0, 62 ≤ x ≤ 76 is the man’s height (in inches). x where Company) (Source: Metropolitan Life Insurance Height, x Weight, y 62 64 66 68 70 72 74 76 136.2 140.6 145.6 151.2 157.4 164.2 171.5 179.4 a. Construct a table of values that shows the median recommended weights for men with heights of 62, 64, 66, 68, 70, 72, 74, and 76 inches. b. Use the table of values to sketch a graph of the model. Then use the graph to estimate graphically the median recommended weight for a man whose height is 71 inches. c. Use the model to c
onfirm algebraically the estimate you found in part (b). Solution a. You can use a calculator to complete the table, as shown at the left. b. The table of values can be used to sketch the graph of the equation, as shown in Figure 1.27. From the graph, you can estimate that a height of 71 inches corresponds to a weight of about 161 pounds. Recommended Weight 180 170 160 150 140 130 FIGURE 1.27 62 64 66 68 70 72 74 76 Height (in inches) x c. To confirm algebraically the estimate found in part (b), you can substitute 71 for x in the model. y 0.073(71)2 6.99(71) 289.0 160.70 So, the graphical estimate of 161 pounds is fairly good. Now try Exercise 75. 333202_0102.qxd 12/7/05 8:31 AM Page 22 22 Chapter 1 Function and Their Graphs 1.2 Exercises VOCABULARY CHECK: Fill in the blanks. a, b 1. An ordered pair b is substituted for and y. x is a ________ of an equation in and x y if the equation is true when a is substituted for 2. The set of all solution points of an equation is the ________ of the equation. 3. The points at which a graph intersects or touches an axis are called the ________ of the graph. x, y 4. A graph is symmetric with respect to the ________ if, whenever is on the graph, x, y is also on the graph. 5. The equation x h2 y k2 r2 is the standard form of the equation of a ________ with center ________ and radius ________. 6. When you construct and use a table to solve a problem, you are using a ________ approach. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, determine whether each point lies on the graph of the equation. 8. y 5 x2 Equation y x 4 y x 2 3x 2 y 4 x 2 y 1 3x3 2x2 1. 2. 3. 4. Points (a) (a) (a) (a) 0, 2 2, 0 1, 5 2, 16 3 (b) (b) (b) (b) 5, 3 2, 8 6, 0 3, 9 In Exercises 5–8, complete the table. Use the resulting solution points to sketch the graph of the equation. 5. y 2x , y 6. y 3 4x , y 7. y x2 3x 1 0 1 2 3 x y x, y In Exercises 9–20, find the x- and y-intercepts of the graph of the equation. 9. y 16 4x2 10. y x 32 y 20 8 4 y 10 8 6 −1 1 3 x −6 −4 −2 2 4 x 12. 11. 13. 14. 15. y 5x 6 y 8 3x y x 4 y 2x 1 y 3x 7 y x 10 y 2x3 4x2 y x4 25 y2 6 x 19. 20. y2 x 1 17. 16. 18. 333202_0102.qxd 12/7/05 8:31 AM Page 23 In Exercises 21–24, assume that the graph has the indicated type of symmetry. Sketch the complete graph of the equation. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 21. 23. y 4 2 −4 x 2 4 22. y 4 2 −2 −4 2 4 6 8 y -axis symmetry x -axis symmetry y 4 2 24. y 4 2 −4 −2 2 4 x −4 −2 2 4 −2 −4 −2 −4 Origin symmetry y -axis symmetry x x In Exercises 25–32, use the algebraic tests to check for symmetry with respect to both axes and the origin. 25. 27. 29. 31. x2 y 0 y x3 y x x2 1 xy 2 10 0 26. 28. 30. x y 2 0 y x4 x2 3 y 1 x2 1 32. xy 4 In Exercises 33– 44, use symmetry to sketch the graph of the equation. 33. 35. 37. 39. 41. 43. y 3x 1 y x 2 2x y x3 34. 36. 38. 40. 42. 44. y 2x 3 y x 2 2x y x3 In Exercises 45– 56, use a graphing utility to graph the equation. Use a standard setting. Approximate any intercepts. 45. 47. 49. y 3 1 2x y x2 4x 3 y 2x x 1 51. y 3x 46. 48. 50. 52. y 2 3x 1 y x2 x 2 y 4 x2 1 y 3x 1 Section 1.2 Graphs of Equations 23 53. 55. y xx 6 y x 3 54. 56. y 6 xx y 2 x In Exercises 57–64, write the standard form of the equation of the circle with the given characteristics. radius: 4 58. Center: 0, 0; radius: 5 57. Center: 59. Center: 60. Center: 61. Center: 62. Center: 0, 0; 2, 1; 7, 4; 1, 2; 3, 2; radius: 4 radius: 7 solution point: solution point: 0, 0 1, 1 63. Endpoints of a diameter: 64. Endpoints of a diameter: 0, 0, 6, 8 4, 1, 4, 1 67. 65. In Exercises 65–70, find the center and radius of the circle, and sketch its graph. x2 y 2 25 x 12 y 32 9 x2 y 12 22 y 32 16 9 x2 y 2 16 70. 69. 66. 68. 2 2 71. Depreciation A manufacturing plant purchases a new y by Sketch the graph of the molding machine for $225,000. The depreciated value (reduced y 225,000 20,000t, equation. t 0 ≤ t ≤ 8. value) given years after is 72. Consumerism You purchase a jet ski for $8100. The is given by Sketch the graph of the 0 ≤ t ≤ 6. years after y t depreciated value y 8100 929t, equation. 73. Geometry A regulation NFL playing field (including the has a perimeter of and width x y end zones) of length 1040 3 yards. 346 2 3 or (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is A x520 3 and its area is x. y 520 3 x (c) Use a graphing utility to graph the area equation. Be sure to adjust your window settings. (d) From the graph in part (c), estimate the dimensions of the rectangle that yield a maximum area. (e) Use your school’s library, the Internet, or some other reference source to find the actual dimensions and area of a regulation NFL playing field and compare your findings with the results of part (d). The symbol indicates an exercise or a part of an exercise in which you are instructed to use a graphing utility. 333202_0102.qxd 12/7/05 8:31 AM Page 24 24 Chapter 1 Function and Their Graphs 74. Geometry A soccer playing field of length and width x y has a perimeter of 360 meters. (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is and its area is A x180 x. w 180 x (c) Use a graphing utility to graph the area equation. Be sure to adjust your window settings. (d) From the graph in part (c), estimate the dimensions of the rectangle that yield a maximum area. (e) Use your school’s library, the Internet, or some other reference source to find the actual dimensions and area of a regulation Major League Soccer field and compare your findings with the results of part(d). Model It 75. Population Statistics The table shows the life expectancies of a child (at birth) in the United States for selected years from 1920 to 2000. (Source: U.S. National Center for Health Statistics) 76. Electronics The resistance y (in ohms) of 1000 feet of solid copper wire at 68 degrees Fahrenheit can be approxiy 10,770 5 ≤ x ≤ 100 is the diameter of the wire in mils (0.001 inch). mated by the model 0.37, x2 x where (Source: American Wire Gage) (a) Complete the table. 5 10 20 30 40 50 60 70 80 90 100 x y x y (b) Use the table of values in part (a) to sketch a graph of the model. Then use your graph to estimate the resistance when x 85.5. (c) Use the model to confirm algebraically the estimate you found in part (b). (d) What can you conclude in general about the relationship between the diameter of the copper wire and the resistance? Year Life expectancy, y Synthesis 1920 1930 1940 1950 1960 1970 1980 1990 2000 54.1 59.7 62.9 68.2 69.7 70.8 73.7 75.4 77.0 A model for the life expectancy during this period is y 0.0025t2 0.574t 44.25, 20 ≤ t ≤ 100 y where in years, with represents the life expectancy and t corresponding to 1920. t 20 (a) Sketch a scatter plot of the data. is the time (b) Graph the model for the data and compare the scatter plot and the graph. (c) Determine the life expectancy in 1948 both graph- ically and algebraically. (d) Use the graph of the model to estimate the life expectancies of a child for the years 2005 and 2010. (e) Do you think this model can be used to predict the life expectancy of a child 50 years from now? Explain. True or False? the statement is true or false. Justify your answer. In Exercises 77 and 78, determine whether 77. A graph is symmetric with respect to the -axis if, when- x ever x, y is on the graph, x, y is also on the graph. 78. A graph of an equation can have more than one -intercept. y y 79. Think About It Suppose you correctly enter an expression for the variable on a graphing utility. However, no graph appears on the display when you graph the equation. Give a possible explanation and the steps you could take to remedy the problem. Illustrate your explanation with an example. 80. Think About It Find b if the graph of y ax 2 bx3 is symmetric with respect to (a) the y -axis and (b) the origin. (There are many correct answers.) and a Skills Review 81. Identify the terms: 9x5 4x3 7. 82. Rewrite the expression using exponential notation. (7 7 7 7) In Exercises 83–88, simplify the expression. 83. 85. 87. 18x 2x 70 7x 6t 2 84. 4x5 55 20 3 3y 86. 88. 333202_0103.qxd 12/7/05 8:33 AM Page 25 1.3 Linear Equations in Two Variables Section 1.3 Linear Equations in Two Variables 25 What you should learn • Use slope to graph linear equations in two variables. • Find slopes of lines. • Write linear equations in two variables. • Use slope to identify parallel and perpendicular lines. • Use slope and linear equations in two variables to model and solve real-life problems. Why you should learn it Linear equations in two variables can be used to model and solve real-life problems. For instance, in Exercise 109 on page 37, you will use a linear equation to model student enrollment at the Pennsylvania State University. Courtesy of Pennsylvania State University Using Slope y mx b. The simplest mathematical model for relating two variables is the linear equation The equation is called linear because its graph is a in two variables line. (In mathematics, the term line means straight line.) By letting you can see that the line crosses the -axis at as shown in Figure 1.28. In other y words, the -intercept is y mx b The steepness or slope of the line is y 0, b. y b, x 0, m. Slope y-Intercept The slope of a nonvertical line is the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right, as shown in Figure 1.28 and Figure 1.29. y y-intercept y = mx + b (0, b) 1 unit m units, m > 0 x y (0, b) 1 unit y-intercept m units, m < 0 y = mx + b x Positive slope, line rises. FIGURE 1.28 Negative slope, line falls. FIGURE 1.29 A linear equation that is written in the form y mx b is said to be written in slope-in
tercept form. The Slope-Intercept Form of the Equation of a Line The graph of the equation y mx b is a line whose slope is m and whose y- intercept is 0, b. Exploration Use a graphing utility to compare the slopes of the lines m 0.5, 1, 2, 1, 2, and obtain a true geometric perspective. What can you conclude about the slope and the “rate” at which the line rises or falls? where m 0.5, Which line falls most quickly? Use a square setting to and 4. Which line rises most quickly? Now, let 4. y mx, 333202_0103.qxd 12/7/05 8:33 AM Page 26 Chapter 1 Functions and Their Graphs 26 y 5 4 3 2 1 (3, 5) x = 3 (3, 1) 1 2 4 5 x FIGURE 1.30 Slope is undefined. Once you have determined the slope and the -intercept of a line, it is a relatively simple matter to sketch its graph. In the next example, note that none of the lines is vertical. A vertical line has an equation of the form y x a. Vertical line The equation of a vertical line cannot be written in the form the slope of a vertical line is undefined, as indicated in Figure 1.30. y mx b because Example 1 Graphing a Linear Equation Sketch the graph of each linear equation. a. b. c. y 2x 1 y 2 x y 2 Solution a. Because m 2, shown in Figure 1.31. the b 1, Moreover, because the slope is the line rises two units for each unit the line moves to the right, as -intercept is 0, 1. y b. By writing this equation in the form y 0x 2, you can see that the and the slope is zero. A zero slope implies that the line is y -intercept is horizontal—that is, it doesn’t rise or fall, as shown in Figure 1.32. 0, 2 c. By writing this equation in slope-intercept form x y 2 y x 2 y 1x 2 Write original equation. Subtract x from each side. Write in slope-intercept form. Moreover, because the slope is the line falls one unit for each unit the line moves to the right, as -intercept is y 0, 2. you can see that the m 1, shown in Figure 1.33. y 5 4 3 2 y = 2x + 1 m = 2 (0, 10, 2− (0, 2 When m is positive, the line rises. FIGURE 1.31 When m is 0, the line is horizontal. FIGURE 1.32 When m is negative, the line falls. FIGURE 1.33 Now try Exercise 9. 333202_0103.qxd 12/7/05 8:33 AM Page 27 Section 1.3 Linear Equations in Two Variables 27 Finding the Slope of a Line y y 2 y 1 (x 1, y 1) (x 2, y 2) y2 − y1 x 2 − x1 x1 x2 x FIGURE 1.34 Given an equation of a line, you can find its slope by writing the equation in slope-intercept form. If you are not given an equation, you can still find the slope of a line. For instance, suppose you want to find the slope of the line passing x1, y1 , as shown in Figure 1.34. As you move through the points from left to right along this line, a change of units in the vertical direction corresponds to a change of units in the horizontal direction. x2, y2 y1 y2 and x2 x1 y2 y1 the change in y rise and the change in x run y2 x2 and x1 x2, y2 x2 x1 The ratio of through the points y1 to x1, y1 Slope change in y change in x represents the slope of the line that passes . rise run y2 x2 y1 x1 The Slope of a Line Passing Through Two Points x2, y2 The slope x1, y1 and m is of the nonvertical line through y1 x1 m where x1 y2 x2 x2. . x2, y2 When this formula is used for slope, the order of subtraction is important. and However, once you have done this, you must form the numer- Given two points on a line, you are free to label either one of them as the other as ator and denominator using the same order of subtraction. y1 x2 y1 x1 y2 x2 x1, y1 m m m y1 x1 y2 x2 y2 x1 Correct Correct Incorrect For instance, the slope of the line passing through the points be calculated as m 7 4 5 3 3 2 3, 4 and 5, 7 can or, reversing the subtraction order in both the numerator and denominator, as m 4 7 3 5 3 2 3 2 . 333202_0103.qxd 12/7/05 8:33 AM Page 28 28 Chapter 1 Functions and Their Graphs Example 2 Finding the Slope of a Line Through Two Points Find the slope of the line passing through each pair of points. a. c. 2, 0 0, 4 and 3, 1 and 1, 1 b. d. 1, 2 3, 4 and and 3, 1 2, 2 Solution a. Letting x1, y1 2, 0 and x2, y2 3, 1 , you obtain a slope of m y2 x2 y1 x1 1 0 3 2 1 5 . See Figure 1.35. b. The slope of the line passing through 1, 2 and 2, 2 is m 2 2 2 1 0 3 0. See Figure 1.36. c. The slope of the line passing through 0, 4 and 1, 1 is m 1 4 1 0 5 1 5. See Figure 1.37. d. The slope of the line passing through 3, 4 and 3, 1 is m 1 4 3 3 3 0 . See Figure 1.38. Because division by 0 is undefined, the slope is undefined and the line is vertical. In Figures 1.35 to 1.38, note the relationships between slope and the orientation of the line. a. Positive slope: line rises from left to right b. Zero slope: line is horizontal c. Negative slope: line falls from left to right d. Undefined slope: line is vertical y 4 3 2 1 −1 (−2, 0) −2 −1 FIGURE 1.35 m = 1 5 (3, 1) (−1, 22, 2) 1 2 3 −2 −1 −1 FIGURE 1.36 x x y (0, 4) 4 3 2 1 m = −5 −1 −1 3 2 (1, −1) 4 x x y 4 3 2 1 Slope is undefined. (3, 4) (3, 1) −1 −1 1 2 4 FIGURE 1.37 FIGURE 1.38 Now try Exercise 21. 333202_0103.qxd 12/7/05 8:33 AM Page 29 Section 1.3 Linear Equations in Two Variables 29 Writing Linear Equations in Two Variables x1, y1 If then is a point on a line of slope m and x, y is any other point on the line, y y1 x x1 m. This equation, involving the variables and x y, can be rewritten in the form y y1 mx x1 which is the point-slope form of the equation of a line. Point-Slope Form of the Equation of a Line The equation of the line with slope mx x1 y y1 . m passing through the point x1, y1 is The point-slope form is most useful for finding the equation of a line. You should remember this form. −2 −1 y 1 −1 −2 −3 −4 −5 FIGURE 1.39 y = 3x − 5 Example 3 Using the Point-Slope Form Find the slope-intercept form of the equation of the line that has a slope of 3 and passes through the point 1, 2. 1 3 4 3 x Solution Use the point-slope form with m 3 and x1, y1 1, 2. 1 (1, −2) y y1 mx x1 y 2 3x 1 y 2 3x 3 y 3x 5 Point-slope form Substitute for m, x1, and y1. Simplify. Write in slope-intercept form. The slope-intercept form of the equation of the line is this line is shown in Figure 1.39. y 3x 5. The graph of Now try Exercise 39. When you find an equation of the line that passes through two given points, you only need to substitute the coordinates of one of the points into the point-slope form. It does not matter which point you choose because both points will yield the same result. The point-slope form can be used to find an equation of the line passing To do this, first find the slope of the line through two points and . x2, y2 x1, y1 m y2 x2 y1 x1 , x1 x2 and then use the point-slope form to obtain the equation y y1 y2 x2 y1 x1 x x1 . Two-point form This is sometimes called the two-point form of the equation of a line. 333202_0103.qxd 12/7/05 8:33 AM Page 30 30 Chapter 1 Functions and Their Graphs in terms of Exploration m1 d1 d2 Find and m2, and respectively (see figure). Then use the Pythagorean Theorem to find a relationship and between m2. m1 y d1 (0, 0) (1, m1) x Parallel and Perpendicular Lines Slope can be used to decide whether two nonvertical lines in a plane are parallel, perpendicular, or neither. Parallel and Perpendicular Lines 1. Two distinct nonvertical lines are parallel if and only if their slopes are equal. That is, m1 m2. 2. Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is, m1 1m2. d2 (1, m2) Example 4 Finding Parallel and Perpendicular Lines Find the slope-intercept forms of the equations of the lines that pass through the 2x 3y 5. and are (a) parallel to and (b) perpendicular to the line point 2, 1 Solution By writing the equation of the given line in slope-intercept form 3 y = − x + 2 2 2x − 3y = 5 2x 3y 5 3y 2x 5 y 2 3x 5 3 Write original equation. Subtract 2x from each side. Write in slope-intercept form. x you can see that it has a slope of m 2 3, as shown in Figure 1.40. a. Any line parallel to the given line must also have a slope of So, the line that is parallel to the given line has the following equation. through 2 3. y 3 2 1 −1 1 4 5 (2, −1) 2 y = x − 3 7 3 FIGURE 1.40 Te c h n o l o g y On a graphing utility, lines will not appear to have the correct slope unless you use a viewing window that has a square setting. For instance, try graphing the lines in Example 4 using the standard 10 ≤ x ≤ 10 setting 10 ≤ y ≤ 10. viewing window with the square setting 6 ≤ y ≤ 6. the lines y 3 perpendicular? and On which setting do 3 x 5 and 3 appear to be Then reset the and 2, 1 y 1 2 x 2 3 3y 1 2x 2 3y 3 2x 4 y 2 3x 7 3 Write in point-slope form. Multiply each side by 3. Distributive Property Write in slope-intercept form. 3 3 because 2 2 that is perpendi- b. Any line perpendicular to the given line must have a slope of . 2, 1 2 3 So, the line through is the negative reciprocal of cular to the given line has the following equation. x 2 y 1 3 2 2y 1 3x 2 2y 2 3x 6 y 3 2x 2 Now try Exercise 69. Write in point-slope form. Multiply each side by 2. Distributive Property Write in slope-intercept form. Notice in Example 4 how the slope-intercept form is used to obtain information about the graph of a line, whereas the point-slope form is used to write the equation of a line. 333202_0103.qxd 12/7/05 8:33 AM Page 31 Section 1.3 Linear Equations in Two Variables 31 Applications In real-life problems, the slope of a line can be interpreted as either a ratio or a rate. If the -axis and -axis have the same unit of measure, then the slope has no units and is a ratio. If the -axis and -axis have different units of measure, then the slope is a rate or rate of change. x y x y Example 5 Using Slope as a Ratio The maximum recommended slope of a wheelchair ramp is A business is installing a wheelchair ramp that rises 22 inches over a horizontal length of 24 feet. (Source: Americans with Disabilities Is the ramp steeper than recommended? Act Handbook) 1 12. Solution The horizontal length of the ramp is 24 feet or Figure 1.41. So, the slope of the ramp is 1224 288 inches, as shown in Slope vertical change
horizontal change 22 in. 288 in. 0.076. Because 1 12 0.083, the slope of the ramp is not steeper than recommended. y 22 in. 24 ft x FIGURE 1.41 Now try Exercise 97. Example 6 Using Slope as a Rate of Change A kitchen appliance manufacturing company determines that the total cost in x dollars of producing units of a blender is C 25x 3500. Cost equation Describe the practical significance of the -intercept and slope of this line. y 0, 3500 Solution y The -intercept tells you that the cost of producing zero units is $3500. This is the fixed cost of production—it includes costs that must be paid regardless of the number of units produced. The slope of tells you that the cost of producing each unit is $25, as shown in Figure 1.42. Economists call the cost per unit the marginal cost. If the production increases by one unit, then the “margin,” or extra amount of cost, is $25. So, the cost increases at a rate of $25 per unit. m 25 Manufacturing C C = 25 + 3500 x Marginal cost: m = $25 Fixed cost: $3500 ) 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 x 50 100 150 Number of units FIGURE 1.42 Production cost Now try Exercise 101. 333202_0103.qxd 12/7/05 8:33 AM Page 32 32 Chapter 1 Functions and Their Graphs Most business expenses can be deducted in the same year they occur. One exception is the cost of property that has a useful life of more than 1 year. Such costs must be depreciated (decreased in value) over the useful life of the property. If the same amount is depreciated each year, the procedure is called linear or straight-line depreciation. The book value is the difference between the original value and the total amount of depreciation accumulated to date. Example 7 Straight-Line Depreciation A college purchased exercise equipment worth $12,000 for the new campus fitness center. The equipment has a useful life of 8 years. The salvage value at the end of 8 years is $2000. Write a linear equation that describes the book value of the equipment each year. Solution V Let the initial value of the equipment by the data point value of the equipment by the data point represent the value of the equipment at the end of year You can represent and the salvage 0, 12,000 The slope of the line is 8, 2000. t. m 2000 12,000 8 0 $1250 which represents the annual depreciation in dollars per year. Using the pointslope form, you can write the equation of the line as follows. V 12,000 1250t 0 Write in point-slope form. V 1250t 12,000 Write in slope-intercept form. Useful Life of Equipment V The table shows the book value at the end of each year, and the graph of the equation is shown in Figure 1.43. 12,000 (0, 12,000) V − t = 1250 +12,000 Year, t Value 10,000 8,000 6,000 4,000 2,000 (8, 2000) 4 6 2 Number of years 8 t 10 FIGURE 1.43 Straight-line depreciation 0 1 2 3 4 5 6 7 8 12,000 10,750 9,500 8,250 7,000 5,750 4,500 3,250 2,000 Now try Exercise 107. In many real-life applications, the two data points that determine the line are often given in a disguised form. Note how the data points are described in Example 7. 333202_0103.qxd 12/7/05 8:33 AM Page 33 Section 1.3 Linear Equations in Two Variables 33 Example 8 Predicting Sales per Share The sales per share for Starbucks Corporation were $6.97 in 2001 and $8.47 in 2002. Using only this information, write a linear equation that gives the sales per (Source: share in terms of the year. Then predict the sales per share for 2003. Starbucks Corporation) Starbucks Corporation (3, 9.97) (2, 8.47) (1, 6.97) y = 1.5t + 5.47 1 2 3 4 Year (1 ↔ 2001) t y 10 FIGURE 1.44 y Given points Estimated point Linear extrapolation FIGURE 1.45 y Given points Estimated point Linear interpolation FIGURE 1.46 x x represent 2001. Then the two given values are represented by the data 2, 8.47. The slope of the line through these points is Solution t 1 Let points 1, 6.97 and m 8.47 6.97 2 1 1.5. Using the point-slope form, you can find the equation that relates the sales per share and the year t y to be y 6.97 1.5t 1 Write in point-slope form. y 1.5t 5.47. Write in slope-intercept form. to According y 1.53 5.47 $9.97, tion is quite good—the actual sales per share in 2003 was $10.35.) in 2003 was as shown in Figure 1.44. (In this case, the predic- the sales per share this equation, Now try Exercise 109. The prediction method illustrated in Example 8 is called linear extrapolation. Note in Figure 1.45 that an extrapolated point does not lie between the given points. When the estimated point lies between two given points, as shown in Figure 1.46, the procedure is called linear interpolation. Because the slope of a vertical line is not defined, its equation cannot be written in slope-intercept form. However, every line has an equation that can be written in the general form Ax By C 0 General form A where can be represented by the general form x a 0. are not both zero. For instance, the vertical line given by and B x a Summary of Equations of Lines 1. General form: 2. Vertical line: 3. Horizontal line: 4. Slope-intercept form: 5. Point-slope form: 6. Two-point form: y y1 Ax By C 0 x a y b y mx b y y1 mx x1 y1 x1 y2 x2 x x1 333202_0103.qxd 12/7/05 8:33 AM Page 34 34 Chapter 1 Functions and Their Graphs 1.3 Exercises VOCABULARY CHECK: In Exercises 1–6, fill in the blanks. 1. The simplest mathematical model for relating two variables is the ________ equation in two variables y mx b. 2. For a line, the ratio of the change in y to the change in x is called the ________ of the line. 3. Two lines are ________ if and only if their slopes are equal. 4. Two lines are ________ if and only if their slopes are negative reciprocals of each other. 5. When the -axis and -axis have different units of measure, the slope can be interpreted as a ________. y x 6. The prediction method ________ ________ is the method used to estimate a point on a line that does not lie between the given points. 7. Match each equation of a line with its form. (a) (b) (c) (d) (e) Ax By C 0 x a y b y mx b y y1 mx x1 (i) Vertical line (ii) Slope-intercept form (iii) General form (iv) Point-slope form (v) Horizontal line PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1 and 2, identify the line that has each slope. 7. y 8. y 1. (a) (b) (c) is undefined. m 2 3 m m 2 y L1 L3 2. (a) (b) (c) m 0 m 3 4 m 1 y L1 L2 x L3 x L2 In Exercises 3 and 4, sketch the lines through the point with the indicated slopes on the same set of coordinate axes. Point 3. 4. 2, 3 4, 1 (a) 0 (b) 1 (a) 3 (b) 3 Slopes (c) 2 (d) 1 2 3 (d) Undefined (c) In Exercises 5–8, estimate the slope of the line. 5. y 6 In Exercises 9–20, find the slope and ble) of the equation of the line. Sketch the line. y -intercept (if possi- 9. 11. 13. 15. 17. 19. 2x 4 y 5x 3 y 1 5x 2 0 7x 6y 30 y 3 0 x 5 0 10. 12. 14. 16. 18. 20. 2x 6 y x 10 y 3 3y 5 0 2x 3y 9 y 4 0 x 2 0 In Exercises 21–28, plot the points and find the slope of the line passing through the pair of points. 21. 23. 25. 27. 28. 3, 2, 1, 6 6, 1, 6, 4 11 , 3 2, 1 2 , 4 4.8, 3.1, 5.2, 1.6 1.75, 8.3, 3 3 2.25, 2.6 22. 24. 26. 2, 4, 4, 4 0, 10, 4, 0 7 4,1 8, 3 , 5 4 4 333202_0103.qxd 12/7/05 8:33 AM Page 35 In Exercises 29–38, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) Section 1.3 Linear Equations in Two Variables 35 and passing through the pairs of points are parallel, perpen- In Exercises 65–68, determine whether the lines L2 dicular, or neither. L1 Point 2, 1 4, 1 5, 6 10, 6 8, 1 3, 1 5, 4 0, 9 7, 2 1, 6 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. is undefined. is undefined. Slope In Exercises 39–50, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope. Sketch the line. Point 0, 2 0, 10 3, 6 0, 0 4, 0 2, 5 6, 1 10, 4 4, 5 2 1 2, 3 2 5.1, 1.8 2.3, 8.5 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. Slope is undefined. is undefined In Exercises 51– 64, find the slope-intercept form of the equation of the line passing through the points. Sketch the line. 52. 54. 56. 58. (4, 3), (4, 4) 1, 4, 6, 4 1, 1, 6, 2 3 , 4 3, 7 4, 3 3 2 4 51. 53. 55. 57. 59. 60. 61. 62. 63. 64. 5 2 4 5, 1, 5, 5 8, 1, 8, 7 2, 1 , 1 2, 5 , 9 1 10, 3 10, 9 5 1, 0.6, 2, 0.6 8, 0.6, 2, 2.4 2, 1, 1 5, 2, 6, 2 1 7 3, 1 3, 8, 7 1.5, 2, 1.5, 0.2 3, 1 65. 67. L1: 0, 1, 5, 9 L2: 0, 3, 4, 1 L1: 3, 6, 6, 0 L2: 0, 1, 5, 7 3 66. 68. L1: 2, 1, 1, 5 L2: 1, 3, 5, 5 L1: (4, 8), (4, 2) L2: 3, 5, 1, 1 3 In Exercises 69–78, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. Point 8 2, 1 3, 2 2 3, 7 7 8, 3 4 1, 0 4, 2 2, 5 5, 1 2.5, 6.8 3.9, 1.4 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. Line 4x 2y 3 x y 7 3x 4y 7 5x 3y 6x 2y 9 In Exercises 79–84, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts and 0, b a, 0 is x a y b 1, a 0, b 0. 79. x -intercept: y -intercept: 81. x -intercept: y -intercept: 83. Point on line: 2, 0 0, 3 1 6, 0 0, 2 3 1, 2 80. x -intercept: y -intercept: 82. x -intercept: y -intercept: 3, 0 0, 4 2 3, 0 0, 2 84. Point on line: 3, 4 x -intercept: y -intercept: c, 0 0, c, c 0 x -intercept: y -intercept: d, 0 0, d, d 0 Graphical Interpretation In Exercises 85–88, identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that the slope appears visually correct—that is, so that parallel lines appear parallel and perpendicular lines appear to intersect at right angles. 85. (a) 86. (a) y 2x y 2 3x (b) (b) y 2x y 3 2x y 1 2x (c) (c) y 2 3x 2 333202_0103.qxd 12/7/05 8:33 AM Page 36 36 Chapter 1 Functions and Their Graphs 87. (a) 88. (a) y 1 2x y x 8 (b) (b) 2x 3 y 1 y x 1 (c) (c) y 2x 4 y x 3 96. Net Profit The graph shows the net p
rofits (in millions) for Applebee’s International, Inc. for the years 1994 through 2003. (Source: Applebee’s International, Inc.) In Exercises 89–92, find a relationship between such that two points. y is equidistant (the same distance) from the x, y and x 89. 90. 91. 92. 4, 1, 2, 3 6, 5, 1, 8 , 7, 1 3, 5 2 2, 4, 7 1 2, 5 4 93. Sales The following are the slopes of lines representing in years. Use the slopes to annual sales interpret any change in annual sales for a one-year increase in time. in terms of time x y (a) The line has a slope of (b) The line has a slope of (c) The line has a slope of m 135. m 0. m 40. 94. Revenue The following are lines y representing daily revenues in days. Use the slopes to interpret any change in daily revenues for a one-day increase in time. in terms of time the slopes of x (a) The line has a slope of (b) The line has a slope of (c) The line has a slope of m 400. m 100. m 0. 95. Average Salary The graph shows the average salaries for senior high school principals from 1990 through 2002. (Source: Educational Research Service) 85,000 80,000 75,000 70,000 65,000 60,000 55,000 ) 12, 83,944) (10, 79,839) (8, 74,380) (6, 69,277) (4, 64,993) (2, 61,768) (0, 55,722) 2 6 4 Year (0 ↔ 1990) 8 10 12 (a) Use the slopes to determine the time periods in which the average salary increased the greatest and the least. (b) Find the slope of the line segment connecting the years 1990 and 2002. (c) Interpret the meaning of the slope in part (b) in the con- text of the problem ( 100 90 80 70 60 50 40 30 20 10 (13, 99.2) (12, 83.0) (10, 63.2) (11, 68.6) (8, 50.7) (9, 57.2) (6, 38.0) (7, 45.1) (5, 29.2) (4, 16.6) 4 5 6 8 7 Year (4 9 10 11 12 13 14 ↔ 1994) (a) Use the slopes to determine the years in which the net profit showed the greatest increase and the least increase. (b) Find the slope of the line segment connecting the years 1994 and 2003. (c) Interpret the meaning of the slope in part (b) in the con- text of the problem. 97. Road Grade You are driving on a road that has a 6% uphill grade (see figure). This means that the slope of the road is Approximate the amount of vertical change in your position if you drive 200 feet. 6 100. 98. Road Grade From the top of a mountain road, a and x as shown in the table ( surveyor takes several horizontal measurements several vertical measurements and are measured in feet). y, y x x y 300 600 900 1200 1500 1800 2100 25 50 75 100 125 150 175 (a) Sketch a scatter plot of the data. (b) Use a straightedge to sketch the line that you think best fits the data. (c) Find an equation for the line you sketched in part (b). (d) Interpret the meaning of the slope of the line in part (c) in the context of the problem. (e) The surveyor needs to put up a road sign that indicates the steepness of the road. For instance, a surveyor would put up a sign that states “8% grade” on a road 100. What with a downhill grade that has a slope of should the sign state for the road in this problem? 8 333202_0103.qxd 12/7/05 8:33 AM Page 37 Rate of Change In Exercises 99 and 100, you are given the dollar value of a product in 2005 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value of the product in terms of the year t. represent 2005.) (Let V t 5 2005 Value Rate 99. $2540 100. $156 $125 decrease per year $4.50 increase per year Graphical Interpretation In Exercises 101–104, match the description of the situation with its graph. Also determine the slope and y-intercept of each graph and interpret the slope and y-intercept in the context of the situation. [The graphs are labeled (a), (b), (c), and (d).] (a) y (b) y 2 4 6 8 40 30 20 10 (c) y 24 18 12 6 200 150 100 50 −2 (d) y 800 600 400 200 x x 2 4 6 8 10 101. A person is paying $20 per week to a friend to repay a $200 loan. 102. An employee is paid $8.50 per hour plus $2 for each unit produced per hour. 103. A sales representative receives $30 per day for food plus $0.32 for each mile traveled. 104. A computer that was purchased for $750 depreciates $100 per year. 105. Cash Flow per Share The cash flow per share for the Timberland Co. was $0.18 in 1995 and $4.04 in 2003. Write a linear equation that gives the cash flow per share t 5 in terms of the year. Let represent 1995. Then predict the cash flows for the years 2008 and 2010. (Source: The Timberland Co.) 106. Number of Stores In 1999 there were 4076 J.C. Penney stores and in 2003 there were 1078 stores. Write a linear equation that gives the number of stores in terms of the year. Let represent 1999. Then predict the numbers of stores for the years 2008 and 2010. Are your answers reasonable? Explain. (Source: J.C. Penney Co.) t 9 Section 1.3 Linear Equations in Two Variables 37 107. Depreciation A sub shop purchases a used pizza oven for $875. After 5 years, the oven will have to be replaced. Write a linear equation giving the value of the equipment during the 5 years it will be in use. V 108. Depreciation A school district purchases a high-volume printer, copier, and scanner for $25,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value of the equipment during the 10 years it will be in use. V 109. College Enrollment The Pennsylvania State University had enrollments of 40,571 students in 2000 and 41,289 students in 2004 at its main campus in University Park, Pennsylvania. (Source: Penn State Fact Book) (a) Assuming the enrollment growth is linear, find a linear model that gives the enrollment in terms of the t, year where corresponds to 2000. t 0 (b) Use your model from part (a) to predict the enroll- ments in 2008 and 2010. (c) What is the slope of your model? Explain its meaning in the context of the situation. 110. College Enrollment The University of Florida had enrollments of 36,531 students in 1990 and 48,673 students in 2003. (Source: University of Florida) (a) What was the average annual change in enrollment from 1990 to 2003? (b) Use the average annual change in enrollment to estimate the enrollments in 1994, 1998, and 2002. (c) Write the equation of a line that represents the given data. What is its slope? Interpret the slope in the context of the problem. (d) Using the results of parts (a)–(c), write a short paragraph discussing the concepts of slope and average rate of change. 111. Sales A discount outlet is offering a 15% discount on S all items. Write a linear equation giving the sale price L. for an item with a list price 112. Hourly Wage A microchip manufacturer pays its assembly line workers $11.50 per hour. In addition, workers receive a piecework rate of $0.75 per unit W produced. Write a linear equation for the hourly wage in terms of the number of units produced per hour. x 113. Cost, Revenue, and Profit A roofing contractor purchases a shingle delivery truck with a shingle elevator for $36,500. The vehicle requires an average expenditure of $5.25 per hour for fuel and maintenance, and the operator is paid $11.50 per hour. (a) Write a linear equation giving the total cost of opert ating this equipment for hours. (Include the purchase cost of the equipment.) C 333202_0103.qxd 12/7/05 8:33 AM Page 38 38 Chapter 1 Functions and Their Graphs (b) Assuming that customers are charged $27 per hour of R machine use, write an equation for the revenue t derived from hours of use. (c) Use the formula for profit P R C to write an equation for the profit derived from hours of use. t (d) Use the result of part (c) to find the break-even point—that is, the number of hours this equipment must be used to yield a profit of 0 dollars. 114. Rental Demand A real estate office handles an apartment complex with 50 units. When the rent per unit is $580 per month, all 50 units are occupied. However, when the rent is $625 per month, the average number of occupied units drops to 47. Assume that the relationship between the monthly rent and the demand is linear. p x (a) Write the equation of the line giving the demand x in terms of the rent p. (b) Use this equation to predict the number of units occu- pied when the rent is $655. (c) Predict the number of units occupied when the rent is $595. 115. Geometry The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width surrounds the garden. x (a) Draw a diagram that gives a visual representation of the problem. (b) Write the equation for the perimeter of the walkway y in terms of x. (c) Use a graphing utility to graph the equation for the perimeter. (d) Determine the slope of the graph in part (c). For each additional one-meter increase in the width of the walkway, determine the increase in its perimeter. 116. Monthly Salary A pharmaceutical salesperson receives a monthly salary of $2500 plus a commission of 7% of sales. Write a linear equation for the salesperson’s monthly wage W S. in terms of monthly sales 117. Business Costs A sales representative of a company using a personal car receives $120 per day for lodging and meals plus $0.38 per mile driven. Write a linear equation giving the daily cost the number of miles driven. to the company in terms of x, C 118. Sports The median salaries (in thousands of dollars) for players on the Los Angeles Dodgers from 1996 to 2003 are shown in the scatter plot. Find the equation of the line represent the that you think best fits these data. (Let t 6 t median salary and let corresponding to 1996.) represent the year, with (Source: USA TODAY) y y 2500 2000 1500 1000 500 FIGURE FOR 118 9 10 11 12 13 8 7 Year (6 ↔ 1996) t Model It 119. Data Analysis: Cell Phone Suscribers The num(in millions) in the bers of cellular phone suscribers is the United States from 1990 through 2002, where year, are shown as data points (Source: Cellular Telecommunications & Internet Association) x, y. x y (1990, (1991, (1992, (1993, (1994, (1995, (1996, (1997, (1998, (1999, (20
00, (2001, (2002, 5.3) 7.6) 11.0) 16.0) 24.1) 33.8) 44.0) 55.3) 69.2) 86.0) 109.5) 128.4) 140.8) (a) Sketch a scatter plot of the data. Let x 0 corre- spond to 1990. (b) Use a straightedge to sketch the line that you think best fits the data. (c) Find the equation of the line from part (b). Explain the procedure you used. (d) Write a short paragraph explaining the meanings of the slope and -intercept of the line in terms of the data. y (e) Compare the values obtained using your model with the actual values. (f) Use your model to estimate the number of cellular phone suscribers in 2008. 333202_0103.qxd 12/7/05 8:33 AM Page 39 120. Data Analysis: Average Scores An instructor gives regular 20-point quizzes and 100-point exams in an algebra course. Average scores for six students, given as data y is the points 19, 96, average test score, are 16, 79, 13, 76, [Note: There are many correct answers for parts (b)–(d).] is the average quiz score and 15, 82. 18, 87, 10, 55, where x, y and x (a) Sketch a scatter plot of the data. (b) Use a straightedge to sketch the line that you think best fits the data. (c) Find an equation for the line you sketched in part (b). (d) Use the equation in part (c) to estimate the average test score for a person with an average quiz score of 17. (e) The instructor adds 4 points to the average test score of each student in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line. Synthesis True or False? In Exercises 121 and 122, determine whether the statement is true or false. Justify your answer. 121. A line with a slope of 5 7 is steeper than a line with a slope of 6 7. 122. The line through 0, 4 and 7, 7 8, 2 and are parallel. 1, 4 and the line through 123. Explain how you could show that the points B 2, 9 , and C 4, 3 are the vertices of a right triangle. A 2, 3, 124. Explain why the slope of a vertical line is said to be undefined. 125. With the information shown in the graphs, is it possible to determine the slope of each line? Is it possible that the lines could have the same slope? Explain. (a) y (b) y x 2 4 x 2 4 126. The slopes of two lines are 4 and Which is steeper? 5 2. Explain. 127. The value V purchased is of a molding machine years after it is t V 4000t 58,500, 0 ≤ t ≤ 5. Explain what the V -intercept and slope measure. Section 1.3 Linear Equations in Two Variables 39 128. Think About It Is it possible for two lines with positive slopes to be perpendicular? Explain. Skills Review In Exercises 129–132, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) y (b) y 6 4 2 −2 −6 −4 6 4 2 −2 x 2 x 2 −6 −4 (c) y (d) y 12 8 4 −4 −4 x 4 8 12 8 4 −4 −4 x 4 8 12 129. 130. 131. 132. y 8 3x y 8 x y 1 2 x 2 2x 1 y x 2 1 In Exercises 133–138, find all the solutions of the equation. Check your solution(s) in the original equation. 133. 134. 135. 136. 137. 138. 9 4x 73 x 14x 1 4 8 2x 7 2x2 21x 49 0 x2 8x 3 0 x 9 15 0 3x 16x 5 0 139. Make a Decision To work an extended application analyzing the numbers of bachelor’s degrees earned by women in the United States from 1985 to 2002, visit this text’s website at college.hmco.com. (Data Source: U.S. Census Bureau) 333202_0104.qxd 12/7/05 8:35 AM Page 40 40 Chapter 1 Functions and Their Graphs 1.4 Functions What you should learn • Determine whether relations between two variables are functions. • Use function notation and evaluate functions. • Find the domains of functions. • Use functions to model and solve real-life problems. • Evaluate difference quotients. Why you should learn it Functions can be used to model and solve real-life problems. For instance, in Exercise 100 on page 52, you will use a function to model the force of water against the face of a dam. Introduction to Functions Many everyday phenomena involve two quantities that are related to each other by some rule of correspondence. The mathematical term for such a rule of correspondence is a relation. In mathematics, relations are often represented by I mathematical equations and formulas. For instance, the simple interest earned on I 1000r. $1000 for 1 year is related to the annual interest rate by the formula represents a special kind of relation that matches each item from one set with exactly one item from a different set. Such a relation is called a function. The formula I 1000r r Definition of Function f A function from a set element domain (or set of inputs) of the function and the set (or set of outputs). exactly one element f, in the set to a set A A B x y is a relation that assigns to each B. in the set The set B A contains the range is the To help understand this definition, look at the function that relates the time of day to the temperature in Figure 1.47. Time of day (P.M.) Temperature (in degrees C) 1 2 3 6 4 5 A is the domain. Set Inputs: 1, 2, 3, 4, 5, 6 FIGURE 1.47 13 6 15 9 12 1 4 14 3 7 10 16 2 5 8 11 B contains the range. Set Outputs: 9, 10, 12, 13, 15 © Lester Lefkowitz/Corbis This function can be represented by the following ordered pairs, in which the first coordinate ( -value) is the input and the second coordinate ( -value) is the output. x y 1, 9, 2, 13, 3, 15, 4, 15, 5, 12, 6, 10 Characteristics of a Function from Set A to Set B 1. Each element in must be matched with an element in A B. 2. Some elements in may not be matched with any element in B A. 3. Two or more elements in may be matched with the same element in A B. 4. An element in elements in B. A (the domain) cannot be matched with two different 333202_0104.qxd 12/7/05 8:35 AM Page 41 Functions are commonly represented in four ways. Section 1.4 Functions 41 Four Ways to Represent a Function 1. Verbally by a sentence that describes how the input variable is related to the output variable 2. Numerically by a table or a list of ordered pairs that matches input values with output values 3. Graphically by points on a graph in a coordinate plane in which the input values are represented by the horizontal axis and the output values are represented by the vertical axis 4. Algebraically by an equation in two variables To determine whether or not a relation is a function, you must decide whether each input value is matched with exactly one output value. If any input value is matched with two or more output values, the relation is not a function. Example 1 Testing for Functions Determine whether the relation represents as a function of y x. a. The input value y value is the number of senators. x is the number of representatives from a state, and the output b. Input, x Output, y c. 2 2 3 4 5 11 10 8 5 1 y 3 2 1 −3 −2 −1 1 2 3 x −2 −3 FIGURE 1.48 Solution a. This verbal description does describe as a function of Regardless of the is always 2. Such functions are called constant the value of x. x, y y value of functions. b. This table does not describe as a function of The input value 2 is matched x. y with two different -values. y c. The graph in Figure 1.48 does describe as a function of Each input value x. y is matched with exactly one output value. Now try Exercise 5. Representing functions by sets of ordered pairs is common in discrete mathematics. In algebra, however, it is more common to represent functions by equations or formulas involving two variables. For instance, the equation y x 2 y is a function of x. represents the variable y as a function of the variable x. In this equation, isx 333202_0104.qxd 12/7/05 8:35 AM Page 42 42 Chapter 1 Functions and Their Graphs the independent variable and function is the set of all values taken on by the independent variable range of the function is the set of all values taken on by the dependent variable is the dependent variable. The domain of the and the y. x © Historical Note Leonhard Euler (1707–1783), a Swiss mathematician, is considered to have been the most prolific and productive mathematician in history. One of his greatest influences on mathematics was his use of symbols, or notation. The y fx function notation was introduced by Euler. Example 2 Testing for Functions Represented Algebraically Which of the equations represent(s) as a function of y x? a. x2 y 1 b. x y2 1 Solution To determine whether y is a function of x, try to solve for y in terms of x. a. Solving for yields y x2 y 1 Write original equation. y 1 x2. Solve for y. x To each value of of x. there corresponds exactly one value of So, y. y is a function b. Solving for yields y x y2 1 y2 1 x y ± 1 x. Write original equation. Add x to each side. Solve for y. indicates that to a given value of x there correspond two values of y. ± The y So, is not a function of x. Now try Exercise 15. Function Notation When an equation is used to represent a function, it is convenient to name the function so that it can be referenced easily. For example, you know that the Suppose you give this equation function the name “ ” Then you can use the following function notation. y 1 x2 f. as a function of describes x. y Input x Output f x Equation f x 1 x2 f x is read as the value of The symbol y corresponds to the -value for a given mind that at For instance, the function given by x. f is the name of the function, whereas f x. f x f x at or simply of x. The symbol y f x. So, you can write Keep in f x is the value of the function f x 3 2x f 2, f1, f0, For has function values denoted by substitute the specified input values into the given equation. f 1 3 21 3 2 5. f 0 3 20 3 0 3. f 2 3 22 3 4 1. x 1, x 0, x 2, For For and so on. To find these values, 333202_0104.qxd 12/7/05 8:35 AM Page 43 Section 1.4 Functions 43 Although x as the independent variable, you can use other letters. For instance, is often used as a convenient function name and f is often used f x x2 4x 7, f t t2 4t 7, and gs s2 4s 7 all define the same function. In fact, the role of the independent variable is that of a “placeholder.” Consequently, the function could be described by f 2 4 7. Example 3 Evaluating a Function gx 2 In Example 3, note that In is not equal
to general, gu v gu gv. gx g2. Let a. gx x2 4x 1. g2 gt b. c. Find each function value. gx 2 Solution a. Replacing with 2 in x gx x2 4x 1 yields the following. g2 22 42 1 4 8 1 5 b. Replacing with yields the following. t x gt t2 4t 1 t 2 4t 1 c. Replacing with x yields the following. gx 2 x 22 4x 2 1 x 2 x 2 4x 4 4x 8 1 x 2 4x 4 4x 8 1 x 2 5 Now try Exercise 29. A function defined by two or more equations over a specified domain is called a piecewise-defined function. Example 4 A Piecewise-Defined Function Evaluate the function when x 1, 0, and 1. f x x2 1, x 1, x < 0 x ≥ 0 Solution Because x 1 is less than 0, use f x x2 1 to obtain For For use f1 12 1 2. fx x 1 x 0, f0 0 1 1. fx x 1 x 1, use f1 1 1 0. to obtain to obtain Now try Exercise 35. 333202_0104.qxd 12/7/05 8:35 AM Page 44 44 Chapter 1 Functions and Their Graphs Te x2 What is the Use a graphing utility to graph the functions given by y x2 4. and domain of each function? Do the domains of these two functions overlap? If so, for what values do the domains overlap? The Domain of a Function The domain of a function can be described explicitly or it can be implied by the expression used to define the function. The implied domain is the set of all real numbers for which the expression is defined. For instance, the function given by fx 1 x2 4 Domain excludes x-values that result in division by zero. These two has an implied domain that consists of all real other than values are excluded from the domain because division by zero is undefined. Another common type of implied domain is that used to avoid even roots of negative numbers. For example, the function given by x x ±2. fx x Domain excludes x-values that result in even roots of negative numbers. is defined only for In general, the domain of a function excludes values that would cause division by zero or that would result in the even root of a negative number. So, its implied domain is the interval x ≥ 0. 0, . Example 5 Finding the Domain of a Function Find the domain of each function. a. f : 3, 0, 1, 4, 0, 2, 2, 2, 4, 1 c. Volume of a sphere: V 4 3 r3 b. gx 1 x 5 d. hx 4 x2 Solution a. The domain of consists of all first coordinates in the set of ordered pairs. f Domain 3, 1, 0, 2, 4 b. Excluding -values that yield zero in the denominator, the domain of x g is the x set of all real numbers except x 5. c. Because this function represents the volume of a sphere, the values of the radius must be positive. So, the domain is the set of all real numbers such that r r > 0. r d. This function is defined only for -values for which x 4 x 2 ≥ 0. By solving this inequality (see Section 2.7), you can conclude that 2 ≤ x ≤ 2. So, the domain is the interval 2, 2. Now try Exercise 59. In Example 5(c), note that the domain of a function may be implied by the physical context. For instance, from the equation V 4 3 r3 you would have no reason to restrict implies that a sphere cannot have a negative or zero radius. r to positive values, but the physical context 333202_0104.qxd 12/7/05 8:35 AM Page 45 h r = 4 r Applications Section 1.4 Functions 45 Example 6 The Dimensions of a Container You work in the marketing department of a soft-drink company and are experimenting with a new can for iced tea that is slightly narrower and taller than a standard can. For your experimental can, the ratio of the height to the radius is 4, as shown in Figure 1.49. h a. Write the volume of the can as a function of the radius b. Write the volume of the can as a function of the height r. h. Solution a. Vr r 2h r 24r 4r3 2 Vh h 4 h3 16 h b. Write V as a function of r. Write V as a function of h. FIGURE 1.49 Now try Exercise 87. Example 7 The Path of a Baseball A baseball is hit at a point 3 feet above ground at a velocity of 100 feet per second and an angle of 45º. The path of the baseball is given by the function fx 0.0032x2 x 3 y where and are measured in feet, as shown in Figure 1.50. Will the baseball clear a 10-foot fence located 300 feet from home plate? x f(x) 80 60 40 20 ) 30 60 90 Baseball Path f(x) = − 0.0032x 2 + x + 3 150 120 Distance (in feet) 180 210 240 270 300 x FIGURE 1.50 Solution When x 300, the height of the baseball is f 300 0.00323002 300 3 15 feet. So, the baseball will clear the fence. Now try Exercise 93. In the equation in Example 7, the height of the baseball is a function of the distance from home plate. 333202_0104.qxd 12/7/05 8:35 AM Page 46 46 Chapter 1 Functions and Their Graphs Number of Alternative-Fueled Vehicles in the U.S. Example 8 Alternative-Fueled Vehicles V 500 450 400 350 300 250 200 ( FIGURE 1.51 5 6 7 8 9 10 11 Year (5 ↔ 1995) 12 t V The number (in thousands) of alternative-fueled vehicles in the United States increased in a linear pattern from 1995 to 1999, as shown in Figure 1.51. Then, in 2000, the number of vehicles took a jump and, until 2002, increased in a different linear pattern. These two patterns can be approximated by the function Vt 18.08t 155.3 38.20t 10.2, 5 ≤ t ≤ 9 10 ≤ t ≤ 12 t 5 t where represents the year, with corresponding to 1995. Use this function to approximate the number of alternative-fueled vehicles for each year from 1995 to 2002. (Source: Science Applications International Corporation; Energy Information Administration) Solution From 1995 to 1999, use 245.7 263.8 Vt 18.08t 155.3. 281.9 299.9 318.0 1995 1996 1997 1998 1999 From 2000 to 2002, use 392.2 430.4 Vt 38.20t 10.2. 468.6 2000 2001 2002 Now try Exercise 95. Difference Quotients One of the basic definitions in calculus employs the ratio f x h f x h , h 0. This ratio is called a difference quotient, as illustrated in Example 9. Example 9 Evaluating a Difference Quotient For f x x2 4x 7, find f x h f x h . Solution f x h f x h x h2 4x h 7 x 2 4x 7 h x 2 2xh h2 4x 4h 7 x 2 4x 7 h h2x h 4 h 2xh h2 4h h 2x h 4, h 0 Now try Exercise 79. The symbol indicates an example or exercise that highlights algebraic techniques specifically used in calculus. 333202_0104.qxd 12/7/05 2:47 PM Page 47 Section 1.4 Functions 47 You may find it easier to calculate the difference quotient in Example 9 by and then substituting the resulting expression into the f x h, first finding difference quotient, as follows. f x h x h2 4x h 7 x2 2xh h2 4x 4h 7 f x h f x h x2 2xh h2 4x 4h 7 x2 4x 7 h h2x h 4 h 2xh h2 4h h 2x h 4, h 0 Summary of Function Terminology Function: A function is a relationship between two variables such that to each value of the independent variable there corresponds exactly one value of the dependent variable. Function Notation: y f x f y is the name of the function. is the dependent variable. is the independent variable. x f x is the value of the function at x. Domain: The domain of a function is the set of all values (inputs) of the is in the domain independent variable for which the function is defined. If ff, is said to be of undefined at x is not in the domain of is said to be defined at ff, If x. x. x Range: The range of a function is the set of all values (outputs) assumed by the dependent variable (that is, the set of all function values). Implied Domain: is not specified, the implied domain consists of all real numbers for which the expression is defined. is defined by an algebraic expression and the domain If f W RITING ABOUT MATHEMATICS Everyday Functions In groups of two or three, identify common real-life functions. Consider everyday activities, events, and expenses, such as long distance telephone calls and car insurance. Here are two examples. a. The statement,“Your happiness is a function of the grade you receive in this course” is not a correct mathematical use of the word “function.“ The word ”happiness” is ambiguous. b. The statement,“Your federal income tax is a function of your adjusted gross income” is a correct mathematical use of the word “function.” Once you have determined your adjusted gross income, your income tax can be determined. Describe your functions in words. Avoid using ambiguous words. Can you find an example of a piecewise-defined function? 333202_0104.qxd 12/7/05 8:35 AM Page 48 48 Chapter 1 Functions and Their Graphs 1.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A relation that assigns to each element x from a set of inputs, or ________, exactly one element y in a set of outputs, or ________, is called a ________. 2. Functions are commonly represented in four different ways, ________, ________, ________, and ________. 3. For an equation that represents as a function of y x, the domain, and the set of all values taken on by the ________ variable the set of all values taken on by the ________ variable is the range. y x is 4. The function given by f x 2x 1, x2 4, x < 0 x ≥ 0 is an example of a ________ function. 5. If the domain of the function f is not given, then the set of values of the independent variable for which the expression is defined is called the ________ ________. 6. In calculus, one of the basic definitions is that of a ________ ________, given by f x h f x h , h 0. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, is the relationship a function? 1. Domain −2 −1 0 1 2 Range 2. 5 6 7 8 3. Domain Range 4. National League American League Cubs Pirates Dodgers Orioles Yankees Twins Range 3 4 5 Range (Number of North Atlantic tropical storms and hurricanes) 7 8 12 13 14 15 19 Domain − − 2 1 0 1 2 Domain (Year) 1994 1995 1996 1997 1998 1999 2000 2001 2002 In Exercises 5–8, does the table describe a function? Explain your reasoning. 5. Input value Output value . 7. 8. Input value 0 1 Output value 4 2 Input value 10 Output value Input value Output value 12 12 3 0 4 10 15 15 3 In Exercises 9 and 10, which sets of ordered pairs represent functions from B? A 9. 10. to A 0, 1, 2, 3 (a) Explain. B 2, 1, 0, 1, 2 and 0, 1, 1, 2, 2, 0, 3, 2 0, 1, 2, 2, 1, 2, 3, 0, 1, 1 0, 0, 1, 0, 2, 0, 3, 0 0, 2, 3, 0, 1, 1 (b) (c) B 0, 1, 2, 3 (d) A a, b, c (a) and a, 1, c, 2, c, 3, b, 3 a, 1, b, 2, c, 3 1, a, 0, a, 2, c, 3,
b (c) (d) c, 0, b, 0, a, 3 (b) 333202_0104.qxd 12/7/05 8:35 AM Page 49 Circulation of Newspapers In Exercises 11 and 12, use the graph, which shows the circulation (in millions) of daily newspapers in the United States. (Source: Editor & Publisher Company) 50 40 30 20 10 Morning Evening ) 1992 1994 1996 1998 2000 2002 Year 11. Is the circulation of morning newspapers a function of the year? Is the circulation of evening newspapers a function of the year? Explain. 12. Let f x x. year Find f 1998. represent the circulation of evening newspapers in In Exercises 13–24, determine whether the equation represents as a function of x. y 13. 15. 17. 19. 21. 23. x2 y 2 4 x2 y 4 2x 3y 14 14. 16. 18. 20. 22. 24. x y 2 x y 2 4 x 22 75 In Exercises 25–38, evaluate the function at each specified value of the independent variable and simplify. 32. 33. 34. 35. 36. 37. 38. 31. qx 1 x2 9 q0 (a) qt 2t 2 3 t 2 x (a) q2 f x x f 2 (a) f x x 4 f 2 (a) Section 1.4 Functions 49 (b) q3 (c) qy 3 (b) q0 (c) qx (b) f 2 (c) f x 1 (b) f 2 (c) f x2 (a) (b) f 1 f x 2x 1, x < 0 2x 2, x ≥ 0 f 0 f x x2 2, x ≤ 1 2x2 2, x > 1 f 1 (b 3x 1, f 2 (a) 4, x2, f 2 (a) f x 4 5x, 0, x2 1, 2 f 1 (bc) f 2 (c) f 2 (c) f 3 (a) f 3 (b) f 4 (c) f 1 In Exercises 39 –44, complete the table. 39. f x x2 3 2 1 0 1 2 x f x 40. gx x 3 25. 26. 27. 28. 29. 30. r3 f x 2x 3 f 1 (a) g y 7 3y g0 (a) Vr 4 3 V3 (a) ht t2 2t h2 (a) f y 3 y (a) f x x 8 2 (a) f 8 f 4 (b) (b) (b) (b) (b) (b) f 3 (cc) gs 2 (c) V2r 41. h1.5 (c) hx 2 f 0.25 (c) f 4x2 f 1 (c gx ht 1 2t 3 5 t ht 4 3 2 1 42 333202_0104.qxd 12/7/05 8:35 AM Page 50 50 43. Chapter 1 Functions and Their Graphs f x 1 2x 4, x ≤ 0 x 22 44. f x 9 x 2, x < 3 x 3 In Exercises 45–52, find all real values of f x 0. x such that 45. 47. 49. 51. f x 15 3x f x 3x 4 f x x 2 9 f x x3 x 5 46. 48. 50. 52. f x 5x 1 f x 12 x2 5 f x x2 8x 15 f x x3 x2 4x 4 In Exercises 53–56, find the value(s) of f x gx. x for which 53. 54. 55. 56. gx 3x 3 f x x2 2x 1, f x x4 2x 2, f x 3x 1, f x x 4, gx 2x2 gx x 1 gx 2 x In Exercises 57–70, find the domain of the function. 57. 59. 61. 63. 65. 67. 69. f x 5x2 2x 1 ht 4 t g y y 10 f x 41 x2 gx 58. 60. 62. 64. 66. 68. 70. gx 1 2x2 s y 3y y 5 f t 3t 4 f x 4x2 3x hx 10 f x x 2 2x x 6 6 x f x x 5 x2 9 In Exercises 71–74, assume that the domain of A {2, 1, 0, 1, 2}. the set pairs that represents the function is Determine the set of ordered f 71. f x x2 72. f. f x x2 3 73. f x x 2 74. f x x 1 Exploration In Exercises 75–78, match the data with one of the following functions f x cx, gx cx2, hx cx, and rx c x and determine the value of the constant the function fit the data in the table. c that will make 75. 76. 77. 78 32 1 2 0 0 1 4 2 32 32 1 1 4 0 4 1 1 Undef. 32 In Exercises 79–86, find the difference quotient and simplify your answer. 79. f x x 2 x 1 80. f x 5x x 2, 81. f x x3 3x, 82. f x 4x2 2x 83. 84. gx 1 x2, f t 1 t 2 , 85. f x 5x, , x 3 gx g3 86. f x x23 1, x 8 , 87. Geometry Write the area A of a square as a function of its perimeter P. A 88. Geometry Write the area of a circle as a function of its circumference C. The symbol indicates an example or exercise that highlights algebraic techniques specifically used in calculus. 333202_0104.qxd 12/7/05 8:35 AM Page 51 89. Maximum Volume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). x 24 2− x x 24 2− x x V (a) The table shows the volume x the box for various heights table to estimate the maximum volume. (in cubic centimeters) of (in centimeters). Use the Height, x 1 2 3 4 5 6 Volume, V 484 800 972 1024 980 864 (b) Plot the points x, V the relation defined by the ordered pairs represent a function of x? from the table in part (a). Does as V (c) If V is a function of write the function and determine x, its domain. 90. Maximum Profit The cost per unit in the production of a portable CD player is $60. The manufacturer charges $90 per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by $0.15 per CD player for each unit ordered in excess of 100 (for example, there would be a charge of $87 per CD player for an order size of 120). (a) The table shows the profit (in dollars) for various x. numbers of units ordered, Use the table to estimate the maximum profit. P Units, x 110 120 130 140 Profit, P 3135 3240 3315 3360 Units, x 150 160 170 Profit, P 3375 3360 3315 x, P (b) Plot the points from the table in part (a). Does the relation defined by the ordered pairs represent as a function of x? P (c) If P is a function of write the function and determine x, its domain. Section 1.4 Functions 51 91. Geometry A right triangle is formed in the first quadrant (see x, by the - and -axes and a line through the point A figure). Write the area of the triangle as a function of and determine the domain of the function. 20, )b (2, 1) a ( , 0 = 36 − x 2 (x, y) x −6 −4 −2 2 4 6 x FIGURE FOR 91 FIGURE FOR 92 92. Geometry A rectangle is bounded by the -axis and the (see figure). Write the area of and determine the domain semicircle the rectangle as a function of of the function. y 36 x2 x, A x 93. Path of a Ball The height y (in feet) of a baseball thrown by a child is y 1 10 x2 3x 6 x where is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.) 94. Prescription Drugs The amounts (in billions of dollars) spent on prescription drugs in the United States from 1991 to 2002 (see figure) can be approximated by the model d dt 5.0t 37, 18.7t 64, 1 ≤ t ≤ 7 8 ≤ t ≤ 12 t represents the year, with where corresponding to 1991. Use this model to find the amount spent on prescription drugs in each year from 1991 to 2002. (Source: U.S. Centers for Medicare & Medicaid Services ( 180 150 120 90 60 30 1 2 3 4 5 6 7 8 9 10 11 12 Year (1 ↔ 1991) t 333202_0104.qxd 12/7/05 8:35 AM Page 52 52 Chapter 1 Functions and Their Graphs 95. Average Price The average prices (in thousands of dollars) of a new mobile home in the United States from 1990 to 2002 (see figure) can be approximated by the model p pt 0.182t2 0.57t 27.3, 2.50t 21.3, 0 ≤ t ≤ 7 8 ≤ t ≤ 12 t represents the year, with where corresponding to 1990. Use this model to find the average price of a mobile (Source: U.S. home in each year from 1990 to 2002. Census Bureau) t 0 p 55 50 45 40 35 30 25 20 15 10 Year (0 ↔ 1990) t 10 11 12 96. Postal Regulations A rectangular package to be sent by the U.S. Postal Service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure). (b) Write the revenue R as a function of the number of units sold. (c) Write the profit as a function of the number of units ) P R C sold. (Note: P 98. Average Cost The inventor of a new game believes that the variable cost for producing the game is $0.95 per unit and the fixed costs are $6000. The inventor sells each game for $1.69. Let be the number of games sold. x (a) The total cost for a business is the sum of the variable as a cost and the fixed costs. Write the total cost function of the number of games sold. C (b) Write the average cost per unit C Cx as a function of x. 99. Transportation For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate 8 0.05n 80, where the rate is given in dollars and people. is the number of n ≥ 80 n (a) Write the revenue R for the bus company as a function of n. (b) Use the function in part (a) to complete the table. What can you conclude? 90 100 110 120 130 140 150 n Rn F 100. Physics The force is a dam of Fy 149.7610 y52, (in feet). (in tons) of water against the face estimated function y is the depth of the water where the by x x y (a) Write the volume V What is the domain of the function? of the package as a function of (b) Use a graphing utility to graph your function. Be sure to use an appropriate window setting. (c) What dimensions will maximize the volume of the package? Explain your answer. 97. Cost, Revenue, and Profit A company produces a product for which the variable cost is $12.30 per unit and the fixed costs are $98,000. The product sells for $17.98. Let be the number of units produced and sold. x (a) The total cost for a business is the sum of the variable as a cost and the fixed costs. Write the total cost function of the number of units produced. C (a) Complete the table. What can you conclude from the table? y F y 5 10 20 30 40 x. (b) Use the table to approximate the depth at which the force against the dam is 1,000,000 tons. (c) Find the depth at which the force against the dam is 1,000,000 tons algebraically. 101. Height of a Balloon A balloon carrying a transmitter ascends vertically from a point 3000 feet from the receiving station. (a) Draw a diagram that gives a visual representation of represent the height of the balloon represent the distance between the balloon the problem. Let d and let and the receiving station. h (b) Write the height of the balloon as a function of d. What is the domain of the function? 333202_0104.qxd 12/7/05 8:35 AM Page 53 Model It Synthesis Section 1.4 Functions 53 102. Wildlife The graph shows the numbers of threatened and endangered fish species in the world from 1996 represent the number of threatthrough 2003. Let t. ened and endangered fish species in the year (Source: U.S. Fish and Wildlife Service) f t f t 126 125 124 123 122 121 120 119 118 117 116 1996 1998 2000 2002 Year t (a) Find f 2003 f 1996 2003 1996 in the context of the problem. and interpret the result N (b) Find a linear model for the data algebraically. represent the number of threatened and correspond Let endangered fish species and let to 1996. x 6 (c) Use the model found in part (b) to complete the table. 6 7 8 9 10 11 12 13 x N (d) Compare your results from
part (c) with the actual data. (e) Use a graphing utility to find a linear model for the data. Let correspond to 1996. How does the model you found in part (b) compare with the model given by the graphing utility? x 6 True or False? In Exercises 103 and 104, determine whether the statement is true or false. Justify your answer. 103. The domain of the function given by f x x 4 1 is , , and the range of 104. The set of ordered pairs 0, 4, 2, 2 2, 2, 0, . f x is 8, 2, represents a function. 6, 0, 4, 0, 105. Writing In your own words, explain the meanings of domain and range. 106. Think About gx 3x 2. It Consider and Why are the domains of and different? f x x 2 g f In Exercises 107 and 108, determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. 107. (a) The sales tax on a purchased item is a function of the selling price. (b) Your score on the next algebra exam is a function of the number of hours you study the night before the exam. 108. (a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped. Skills Review In Exercises 109–112, solve the equation. 109. 110. 111. 112 xx 1 12 In Exercises 113–116, find the equation of the line passing through the pair of points. 113. 115. 2, 5, 6, 5, 3, 5 4, 1 114. 116. 10, 0, 1 2, 3, 1, 9 11 2 , 1 3 333202_0105.qxd 12/7/05 8:36 AM Page 54 54 Chapter 1 Functions and Their Graphs 1.5 Analyzing Graphs of Functions What you should learn • Use the Vertical Line Test for functions. • Find the zeros of functions. • Determine intervals on which functions are increasing or decreasing and determine relative maximum and relative minimum values of functions. • Determine the average rate of change of a function. • Identify even and odd functions. Why you should learn it Graphs of functions can help you visualize relationships between variables in real life. For instance, in Exercise 86 on page 64, you will use the graph of a function to represent visually the temperature for a city over a 24-hour period. y 5 4 1 (−1, 1) Range −3 −2 (0, 3) y f x= ( ) (5, 2) 2 3 4 6 x (2, −3) Domain −5 FIGURE 1.53 The Graph of a Function In Section 1.4, you studied functions from an algebraic point of view. In this section, you will study functions from a graphical perspective. f x, f x such is in the domain of As you study this section, remember that The graph of a function is the collection of ordered pairs x x y f x the directed distance from the -axis the directed distance from the -axis f. x y that as shown in Figure 1.52. y 2 1 y = f(x) f(x) −1 1 2 x x −1 FIGURE 1.52 Example 1 Finding the Domain and Range of a Function Use the graph of the function f, (b) the function values f, f1 shown in Figure 1.53, to find (a) the domain of and , and (c) the range of f 2 f. Solution a. The closed dot at the open dot at is all of f b. Because 1, 1 indicates that x 5 5, 2 in the interval indicates that 1, 5. is a point on the graph of x 1, 1 x 1 Similarly, because f 2 3. 2, 3 is a point on the graph of f, f, it follows that f 1 1. it follows that is in the domain of whereas is not in the domain. So, the domain f, c. Because the graph does not extend below range of f is the interval 3, 3. f 2 3 or above f 0 3, the Now try Exercise 1. The use of dots (open or closed) at the extreme left and right points of a graph indicates that the graph does not extend beyond these points. If no such dots are shown, assume that the graph extends beyond these points. 333202_0105.qxd 12/7/05 8:36 AM Page 55 Section 1.5 Analyzing Graphs of Functions 55 By the definition of a function, at most one -value corresponds to a given x -value. This means that the graph of a function cannot have two or more different points with the same coordinate, and no two points on the graph of a function can be vertically above or below each other. It follows, then, that a vertical line can intersect the graph of a function at most once. This observation provides a convenient visual test called the Vertical Line Test for functions. x- y Vertical Line Test for Functions x A set of points in a coordinate plane is the graph of as a function of and only if no vertical line intersects the graph at more than one point. y if Example 2 Vertical Line Test for Functions Use the Vertical Line Test to decide whether the graphs in Figure 1.54 represent y as a function of x. y 4 2 x 1 −3 −2 −1 (a) FIGURE 1.54 y 4 3 2 1 (b) y 4 3 1 −1 −1 (c Solution a. This is not a graph of as a function of because you can find a vertical line there is more that intersects the graph twice. That is, for a particular input than one output x, x, y. y b. This is a graph of as a function of y x, the graph at most once. That is, for a particular input output y. because every vertical line intersects there is at most one x, y c. This is a graph of as a function of x. (Note that if a vertical line does not intersect the graph, it simply means that the function is undefined for that particular value of there is at most one ) That is, for a particular input output x, x. y. Now try Exercise 9. 333202_0105.qxd 12/7/05 8:36 AM Page 56 56 Chapter 1 Functions and Their Graphs Zeros of a Function If the graph of a function of has an -intercept at function. x x a, 0, then a is a zero of the Zeros of a Function The zeros of a function of are the -values for which f x 0. f x x − x 10 3 −1 1 2 − ( 2, 0) ( ) 5 3 , 0 −2 −4 −6 −8 x Example 3 Finding the Zeros of a Function Find the zeros of each function. a. f x 3x2 x 10 b. gx 10 x2 c. ht 2t 3 t 5 Solution To find the zeros of a function, set the function equal to zero and solve for the independent variable. Zeros of f: FIGURE 1.55 x 22 −4 ) ( − 10, 0 −6 −4 −2 ( ) = 10 − 2 g x x ( 10, 0 ) 2 4 6 x ± 10 Zeros of g: FIGURE 1.56 y 2 2( , 03 ) 2 4 6 h t( ) = 2t − 3 t + 5 −4 −2 −2 −4 −6 −8 Zero of h: FIGURE 1.57 t 3 2 x t 3x2 x 10 0 3x 5x 2 0 3x 5 0 x 2 0 Set f x equal to 0. Factor. x 5 3 x 2 Set 1st factor equal to 0. Set 2nd factor equal to 0. x 5 3 as its -intercepts. x 2. and x 2, 0 In Figure 1.55, note that the graph of f a. b. f and The zeros of are 3, 0 5 has 10 x2 0 10 x2 0 10 x2 ± 10 x Set gx equal to 0. Square each side. Add x2 to each side. Extract square roots. g The zeros of graph of has g c. 0 2t 3 t 5 2t 3 0 2t 3 t 3 2 are 10, 0 x 10 and and 10, 0 x 10. In Figure 1.56, note that the x as its -intercepts. Set ht equal to 0. Multiply each side by t 5. Add 3 to each side. Divide each side by 2. h The zero of its -intercept. t is t 3 2. In Figure 1.57, note that the graph of has h 2, 0 3 as Now try Exercise 15. 333202_0105.qxd 12/7/05 8:36 AM Page 57 Section 1.5 Analyzing Graphs of Functions 57 Increasing and Decreasing Functions 1 −2 −1 FIGURE 1.58 Increasing Constant 1 2 3 4 x y 1 f(x) = x3 The more you know about the graph of a function, the more you know about the function itself. Consider the graph shown in Figure 1.58. As you move from left x 2, to right, this graph falls from to and rises from is constant from x 2 x 0, x 4. x 2 x 0 to to Increasing, Decreasing, and Constant Functions x2 is increasing on an interval if, for any A function x1 < x2 f implies < f x2 f x1 and x1 . in the interval, A function x1 < x2 f implies f x1 > f x2 . is decreasing on an interval if, for any x1 and x2 in the interval, f A function f x 2 f x1 is constant on an interval if, for any . x1 and x2 in the interval, Example 4 Increasing and Decreasing Functions Use the graphs in Figure 1.59 to describe the increasing or decreasing behavior of each function. Solution a. This function is increasing over the entire real line. b. This function is increasing on the interval interval 1, 1, and increasing on the interval , 1, 1, . decreasing on the c. This function is increasing on the interval , 0, constant on the interval 0, 2, 2, . and decreasing on the interval y f(x) = x − 3x 3 (−1, 2) 2 y 2 1 (0, 1) (2, 1) −1 x 1 −2 −1 1 2 x 1 2 3 t −1 (a) FIGURE 1.59 −1 −2 (1, −2) −1 −2 f(t) = t + 1, t < 0 1, 0 ≤ t ≤ 2 −t + 3, t > 2 (b) (c) Now try Exercise 33. To help you decide whether a function is increasing, decreasing, or constant on an interval, you can evaluate the function for several values of However, calculus is needed to determine, for certain, all intervals on which a function is increasing, decreasing, or constant. x. 333202_0105.qxd 12/7/05 8:36 AM Page 58 58 Chapter 1 Functions and Their Graphs A relative minimum or relative maximum is also referred to as a local minimum or local maximum. y Relative maxima Relative minima FIGURE 1.60 4 5 −4 FIGURE 1.61 The points at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative minimum or relative maximum values of the function. Definitions of Relative Minimum and Relative Maximum A function value if there exists an interval is called a relative minimum of that contains fa f x1, x2 x1 < x < x2 a such that fa ≤ f x. implies fa is called a relative maximum of f if there exists an A function value interval x1, x2 x1 < x < x2 that contains implies a such that f a ≥ f x. Figure 1.60 shows several different examples of relative minima and relative maxima. In Section 2.1, you will study a technique for finding the exact point at which a second-degree polynomial function has a relative minimum or relative maximum. For the time being, however, you can use a graphing utility to find reasonable approximations of these points. x Example 5 Approximating a Relative Minimum Use a graphing utility to approximate the relative minimum of the function given by f x 3x2 4x 2. Solution is shown in Figure 1.61. By using the zoom and trace features or The graph of the minimum feature of a graphing utility, you can estimate that the function has a relative minimum at the point f 0.67, 3.33. Relative minimum Later, in Section 2.1, you will be able to determine that the exact point at which . the relativ
e minimum occurs is 2 3, 10 Now try Exercise 49. 3 You can also use the table feature of a graphing utility to approximate numerically the relative minimum of the function in Example 5. Using a table that begins at 0.6 and increments the value of by 0.01, you can approximate that the minimum of occurs at the point 0.67, 3.33. f x 3x2 4x 2 x Te c h n o l o g y If you use a graphing utility to estimate the x- and y-values of a relative minimum or relative maximum, the zoom feature will often produce graphs that are nearly flat. To overcome this problem, you can manually change the vertical setting of the viewing window. The graph will stretch vertically if the values of Ymin and Ymax are closer together. 333202_0105.qxd 12/7/05 8:36 AM Page 59 Section 1.5 Analyzing Graphs of Functions 59 Average Rate of Change y (x1, f(x1)) (x2, f(x2)) Secant line f x2 − x1 f(x2) − f(x1) x1 x2 FIGURE 1.62 y f(x) = x 3 − 3x 2 (0, 0) −3 −2 −1 1 2 3 −1 (−2, −2) (1, −2) −3 FIGURE 1.63 x x Exploration Use the information in Example 7 to find the average speed of 0 9 t1 the car from seconds. Explain why the result is less than the value obtained in part (b). to t2 In Section 1.3, you learned that the slope of a line can be interpreted as a rate of change. For a nonlinear graph whose slope changes at each point, the average rate of change between any two points is the slope of the line through the two points (see Figure 1.62). The line through the two points is called the secant line, and the slope of this line is denoted as x2, fx2 x1, fx1 and msec. Average rate of change of f from x1 to x2 fx2 fx1 x1 x2 change in y change in x msec Example 6 Average Rate of Change of a Function Find the average rates of change of 0 and (b) from 1 to x1 x2 (see Figure 1.63). fx x3 3x (a) from x1 2 to x2 0 from 0 is x2 f Solution a. The average rate of change of fx1 fx2 x1 x2 f b. The average rate of change of f1 f0 fx1 fx2 1 0 x1 x2 f0 f2 0 2 1. x1 2 to 0 2 2 0 x1 2 0 1 to x2 2. from 1 is Secant line has positive slope. Secant line has negative slope. Now try Exercise 63. Example 7 Finding Average Speed s The distance (in feet) a moving car is from a stoplight is given by the function st 20t32, t where is the time (in seconds). Find the average speed of the car 4 4 t2 (a) from to seconds and (b) from seconds. 9 0 to t2 t1 t1 Solution a. The average speed of the car from st1 st2 t1 t2 s4 s0 4 0 b. The average speed of the car from st1 st2 t1 t2 s9 s4 9 4 Now try Exercise 89. 0 to t1 t2 160 0 4 4 t1 t2 540 160 5 to 4 seconds is 40 9 feet per second. seconds is 76 feet per second. 333202_0105.qxd 12/7/05 8:36 AM Page 60 60 Chapter 1 Functions and Their Graphs Even and Odd Functions In Section 1.2, you studied different types of symmetry of a graph. In the terminology of functions, a function is said to be even if its graph is symmetric with respect to the -axis and to be odd if its graph is symmetric with respect to the origin. The symmetry tests in Section 1.2 yield the following tests for even and odd functions. y Exploration Graph each of the functions with a graphing utility. Determine whether the function is even, odd, or neither. f x x2 x 4 gx 2x 3 1 hx x 5 2x3 x jx 2 x6 x8 kx x 5 2x 4 x 2 px x9 3x5 x3 x What do you notice about the equations of functions that are odd? What do you notice about the equations of functions that are even? Can you describe a way to identify a function as odd or even by inspecting the equation? Can you describe a way to identify a function as neither odd nor even by inspecting the equation? Tests for Even and Odd Functions A function is even if, for each x in the domain of f, A function is odd if, for each x in the domain of f, y f x f x f x. y f x f x f x. Example 8 Even and Odd Functions a. The function gx x 3 x gx x3 x x3 x x 3 x gx is odd because gx gx, as follows. Substitute x for x. Simplify. Distributive Property Test for odd function b. The function hx x 2 1 is even because hx hx, as follows. hx x2 1 x 2 1 hx Substitute x for x. Simplify. Test for even function The graphs and symmetry of these two functions are shown in Figure 1.64. y 3 1 g(x) = x − x 3 −3 −2 (−x, −y) −1 −2 −3 (x, y) 1 2 3 x (− x, y) y 6 5 4 3 2 (x, y) h(x) = x + 1 2 −3 −2 −1 1 2 3 x (b) Symmetric to y-axis: Even Function (a) Symmetric to origin: Odd Function FIGURE 1.64 Now try Exercise 71. 333202_0105.qxd 12/7/05 8:36 AM Page 61 1.5 Exercises Section 1.5 Analyzing Graphs of Functions 61 VOCABULARY CHECK: Fill in the blanks. 1. The graph of a function f is the collection of ________ ________ or x, f x such that x is in the domain of f. 2. The ________ ________ ________ is used to determine whether the graph of an equation is a function of y in terms of x. 3. The ________ of a function are the values of f x 4. A function f is ________ on an interval if, for any for which x2 and x1 f x 0. in the interval, 5. A function value is a relative ________ of f if there exists an interval containing f a f a ≥ f x. implies implies x1 < x2 x1, x2 f x1 a . > f x2 x1 < x < x2 such that 6. The ________ ________ ________ ________ between any two points through the two points, and this line is called the ________ line. x1, f x1 and x2, f x2 is the slope of the line 7. A function 8. A function f f is ________ if for the each x in the domain of f, is ________ if its graph is symmetric with respect to the -axis. f x f x. y PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. y = f(x) −4 −2 In Exercises 1– 4, use the graph of the function to find the domain and range of f. y = f(x) x 2 4 2. y 6 4 2 −2 −2 4. y = f(x) 1. 3. −4 −2 y 6 4 2 −2 y 6 2 −4 −2 −2 −4 y = f(x) x 2 4 In Exercises 5–8, use the graph of the function to find the indicated function values. (a) 6. (a) (b) (b) (d) (d) (c) (c) 2 y y = f(x) 4 3 2 −3 −4 y = f(x) y x 43 −4 x 2 4 2 −2 −4 7. (a) (c) f 2 f 0 (b) (d) f 1 f 2 y = f(x) y 8. (a) (c) f 2 f 3 y (b) (d) f 1 f 1 y = f(x) x 2 4 −2 −4 −6 4 2 −2 −2 x 2 4 In Exercises 9–14, use the Vertical Line Test to determine whether y is a function of x.To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 9. y 1 2x 2 10. y 1 4x3 y 6 4 2 −4 −2 2 4 11. x y 2 1 y 4 2 −2 4 6 y 4 2 −2 −4 −4 12. x2 y 2 25 x 2 4 y 6 4 2 − 333202_0105.qxd 12/7/05 8:36 AM Page 62 62 Chapter 1 Functions and Their Graphs 13. x2 2xy 1 y 4 2 −2 −4 −4 x 2 4 142 −4 −6 In Exercises 15–24, find the zeros of the function algebraically. 15. 17. 19. 20. 21. 22. 23. 24. 16. 18. f x 3x2 22x 16 f x x2 9x 14 4x f x f x 2x2 7x 30 x 9x2 4 2 x3 x f x 1 f x x3 4x2 9x 36 f x 4x3 24x2 x 6 f x 9x4 25x2 f x 2x 1 f x 3x 2 In Exercises 25–30, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. 25. 26. 27. 28. 29. 30. f x 3 5 x f x xx 7 f x 2x 11 f x 3x 14 8 f x 3x 1 x 6 f x 2x2 9 3 x In Exercises 31–38, determine the intervals over which the function is increasing, decreasing, or constant. 31. f x 3 2 x y 4 2 32. f x x2 4x y −4 −2 2 4 x −4 −2 2 x 6 −2 −4 (2, −4) 33. f x x3 3x2 2 34. f x x2 1 y 4 2 (0, 2) −2 2 4 x (2, −2) (−1, 0) −4 −2 35. f x x 3, 3, 2x 12 (1, 0) 2 4 x y 6 4 −2 2 4 x 36. f x 2x 1, x2 22 2 4 x −4 37. f x x 1 x 1 38. y 6 4 (−1, 2) (1, 2) −2 2 4 x f x x2 x 1 x 1 y (0, 1) −4 −2 (−2, −3) −2 x 2 333202_0105.qxd 12/7/05 8:36 AM Page 63 In Exercises 39– 48, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a). 39. 41. 43. 45. 47. f x 3 gs s2 32 40. gx x 42. hx x2 4 44. 46. 48. f x 3x4 6x2 f x xx 3 f x x23 In Exercises 49–54, use a graphing utility to graph the function and approximate (to two decimal places) any relative minimum or relative maximum values. 49. 51. 53. 54. f x x 4x 2 f x x2 3x 2 f x xx 2x 3 f x x3 3x2 x 1 50. 52. f x 3x2 2x 5 f x 2x2 9x 55. f x ≥ 0. In Exercises 55– 62, graph the function and determine the interval(s) for which 4x 2 f x x 2 4x 62. 58. 59. 56. 61. 57. 60. In Exercises 63–70, find the average rate of change of the function from to x1 x2. Function 63. 64. 65. 66. 67. 68. 69. 70. f x 2x 15 f (x 3x 8 f x x2 12x 4 f x x2 2x 8 f x x3 3x2 x f x x3 6x2 -Values 0, x2 0, x2 1, x2 1, x2 1, x2 1, x2 3, x2 3, x2 3 3 5 5 3 6 11 8 x1 x1 x1 x1 x1 x1 x1 x1 In Exercises 71–76, determine whether the function is even, odd, or neither. Then describe the symmetry. 71. 73. 75. f x x6 2x2 3 gx x3 5x f t t 2 2t 3 72. 74. 76. hx x3 5 f x x1 x 2 gs 4s 23 Section 1.5 Analyzing Graphs of Functions 63 In Exercises 77–80, write the height of the rectangle as a function of x. h 77. y 781, 2) (3, 2) 4 3 2 1 792, 4) h y x= 2 x1 1, 3 80. y 4 h −2 (8, 2) 2 x 8 4 y 6 x= 3 x x In Exercises 81– 84, write the length of the rectangle as a function of y. L 81. y 82. y x 6 4 y −2 L (8, 4) x = 21 = 23 y (2, 4) L 1 2 3 4 83. y 84. y 4 3 2 y y= 2 x (4, 21, 2 85. Electronics The number of lumens (time rate of flow of from a fluorescent lamp can be approximated by L light) the model L 0.294x2 97.744x 664.875, 20 ≤ x ≤ 90 where x is the wattage of the lamp. (a) Use a graphing utility to graph the function. (b) Use the graph from part (a) to estimate the wattage necessary to obtain 2000 lumens. 333202_0105.qxd 12/7/05 8:36 AM Page 64 64 Chapter 1 Functions and Their Graphs Model It 88. Geometry Corners of equal size are cut from a square with sides of length 8 meters (see figure). a) Write the area of the resulting figure as a function of A x. Determine the domain of the function. (b) Use a graphing utility to graph the area function over its domain. Use the graph to find the range of the function. (c) Identify the figure that would result if were chosen to be the maximum value in the domain of the function. What would be the length of each side of the figure? x 89. Digital Music Sales The estimated revenues (in billio
ns of dollars) from sales of digital music from 2002 to 2007 can be approximated by the model r r 15.639t3 104.75t2 303.5t 301, 2 ≤ t ≤ 7 t where 2002. represents the year, with (Source: Fortune) t 2 corresponding to (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 2002 to 2007. Interpret your answer in the context of the problem. 90. Foreign College Students The numbers of foreign students (in thousands) enrolled in colleges in the United States from 1992 to 2002 can be approximated by the model. F F 0.004t 4 0.46t2 431.6, 2 ≤ t ≤ 12 t where 1992. represents the year, with (Source: Institute of International Education) corresponding to t 2 (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 1992 to 2002. Interpret your answer in the context of the problem. (c) Find the five-year time periods when the rate of change was the greatest and the least. 86. Data Analysis: Temperature The table shows the (in degrees Fahrenheit) of a certain city represent the time of day, y temperature over a 24-hour period. Let where corresponds to 6 A.M. x 0 x Time, x Temperature, y 0 2 4 6 8 10 12 14 16 18 20 22 24 34 50 60 64 63 59 53 46 40 36 34 37 45 A model that represents these data is given by y 0.026x3 1.03x2 10.2x 34, 0 ≤ x ≤ 24. (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use the graph to approximate the times when the temperature was increasing and decreasing. (d) Use the graph to approximate the maximum and minimum temperatures during this 24-hour period. (e) Could this model be used to predict the temperature for the city during the next 24-hour period? Why or why not? 87. Coordinate Axis Scale Each function models the specit 5 fied data for the years 1995 through 2005, with corresponding to 1995. Estimate a reasonable scale for the vertical axis (e.g., hundreds, thousands, millions, etc.) of the graph and justify your answer. (There are many correct answers.) f t f t f t that is unemployed. represents the average salary of college professors. represents the percent of the civilian work force represents the U.S. population. (b) (c) (a) 333202_0105.qxd 12/7/05 8:36 AM Page 65 Physics In Exercises 91– 96, (a) use the position equation s 16t2 v0t s0 to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function (d) interpret your answer to part (c) in the from context of the problem, (e) find the equation of the secant line through and (f) graph the secant line in the same viewing window as your position function. and to t2, t2, t1 t1 91. An object is thrown upward from a height of 6 feet at a velocity of 64 feet per second. t1 0, t2 3 92. An object is thrown upward from a height of 6.5 feet at a velocity of 72 feet per second. t1 0, t2 4 93. An object is thrown upward from ground level at a veloc- ity of 120 feet per second. t1 3, t2 5 94. An object is thrown upward from ground level at a veloc- ity of 96 feet per second. t1 2, t2 5 Section 1.5 Analyzing Graphs of Functions 65 Think About It In Exercises 101–104, find the coordinates of a second point on the graph of a function if the given point is on the graph and the function is (a) even and (b) odd. f 101. 102. 103. 104. 2, 4 3, 7 3 5 4, 9 5, 1 105. Writing Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) (c) (e) y x y x3 y x5 (b) (d) (f 106. Conjecture Use the results of Exercise 105 to make a Use a conjecture about the graphs of graphing utility to graph the functions and compare the results with your conjecture. y x8. y x 7 and Skills Review 95. An object is dropped from a height of 120 feet. In Exercises 107–110, solve the equation. t1 0, t2 2 96. An object is dropped from a height of 80 feet. t1 1, t2 2 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 97 and 98, determine whether 97. A function with a square root cannot have a domain that is the set of real numbers. 98. It is possible for an odd function to have the interval 0, as its domain. 99. If f g is an even function, determine whether is even, odd, (a) (b) or neither. Explain. gx f x gx f x gx f x 2 gx f x 2 (d) (c) 100. Think About It Does the graph in Exercise 11 represent Explain. as a function of y? x 107. 108. 109. 110. x2 10x 0 100 x 52 0 x3 x 0 16x2 40x 25 0 In Exercises 111–114, evaluate the function at each specified value of the independent variable and simplify. 111. 112. 113. 114. f 4 f x 5x 8 f 9 (a) (b) f x x2 10x (a) (b) f x x 12 9 (a) (b) f x x4 x 5 (a) (b) f 1 f 12 f 4 f 8 f 40 (c) f x 7 (c) f x 4 f 36 (c) f 1 2 f 23 (c) In Exercises 115 and 116, find the difference quotient and simplify your answer. 115. f x x2 2x 9, 116. f x 5 6x x2 333202_0106.qxd 12/7/05 8:40 AM Page 66 66 Chapter 1 Functions and Their Graphs 1.6 A Library of Parent Functions What you should learn • Identify and graph linear and squaring functions. • Identify and graph cubic, square root, and reciprocal functions. • Identify and graph step and other piecewise-defined functions. • Recognize graphs of parent functions. Why you should learn it Step functions can be used to model real-life situations. For instance, in Exercise 63 on page 72, you will use a step function to model the cost of sending an overnight package from Los Angeles to Miami. Linear and Squaring Functions One of the goals of this text is to enable you to recognize the basic shapes of the graphs of different types of functions. For instance, you know that the graph of the linear function intercept at 0, b. The graph of the linear function has the following characteristics. is a line with slope f x ax b m a and y- • The domain of the function is the set of all real numbers. • The range of the function is the set of all real numbers. • The graph has an -intercept of bm, 0 y x and a -intercept of 0, b. • The graph is increasing if if m 0. m > 0, decreasing if m < 0, and constant Example 1 Writing a Linear Function Write the linear function f for which f 1 3 and f 4 0. Solution To find the equation of the line that passes through x2, y2 m y2 x2 4, 0, y1 x1 first find the slope of the line. 0 3 3 4 1 3 1 x1, y1 1, 3 and Next, use the point-slope form of the equation of a line. mx x1 y y1 y 3 1x 1 y x 4 f x x 4 Point-slope form Substitute for x1, y1, and m. Simplify. Function notation The graph of this function is shown in Figure 1.65. © Getty Images y 5 4 3 2 1 f(x) = −1 −1 FIGURE 1.65 Now try Exercise 1. 333202_0106.qxd 12/7/05 8:40 AM Page 67 Section 1.6 A Library of Parent Functions 67 There are two special types of linear functions, the constant function and the identity function. A constant function has the form f x c and has the domain of all real numbers with a range consisting of a single real number The graph of a constant function is a horizontal line, as shown in Figure 1.66. The identity function has the form c. f x x. m 1 Its domain and range are the set of all real numbers. The identity function has a slope of The graph of the identity function is a line for which each -coordinate equals the corresponding -coordinate. The graph is always increasing, as shown in Figure 1.67 and a -intercept 0, 0. y x y y 3 2 1 f(x) = c 1 2 3 x y f(x) = x 2 1 −2 −1 1 2 x −1 −2 FIGURE 1.66 FIGURE 1.67 The graph of the squaring function f x x2 is a U-shaped curve with the following characteristics. • The domain of the function is the set of all real numbers. • The range of the function is the set of all nonnegative real numbers. • The function is even. • The graph has an intercept at • The graph is decreasing on the interval and increasing on , 0 0, 0. the interval 0, . • The graph is symmetric with respect to the -axis. • The graph has a relative minimum at 0, 0. y The graph of the squaring function is shown in Figure 1.68. y f(x3 −2 −1 −1 2 1 (0, 0) 3 x FIGURE 1.68 333202_0106.qxd 12/7/05 8:40 AM Page 68 68 Chapter 1 Functions and Their Graphs Cubic, Square Root, and Reciprocal Functions The basic characteristics of the graphs of the cubic, square root, and reciprocal functions are summarized below. 1. The graph of the cubic function f x x3 has the following characteristics. • The domain of the function is the set of all real numbers. • The range of the function is the set of all real numbers. • The function is odd. • The graph has an intercept at 0, 0. • The graph is increasing on the interval , . • The graph is symmetric with respect to the origin. The graph of the cubic function is shown in Figure 1.69. f x x 2. The graph of the square root function has the following characteristics. • The domain of the function is the set of all nonnegative real numbers. • The range of the function is the set of all nonnegative real numbers. 0, 0. • The graph has an intercept at • The graph is increasing on the interval 0, . The graph of the square root function is shown in Figure 1.70. 3. The graph of the reciprocal function characteristics. f x 1 x has the following • The domain of the function is , 0 0, . • The range of the function is , 0 0, . • The function is odd. • The graph does not have any intercepts. • The graph is decreasing on the intervals , 0 • The graph is symmetric with respect to the origin. and 0, . The graph of the reciprocal function is shown in Figure 1.71. y 3 2 1 (0, 0) −1 −2 −3 −3 −2 f(x0, 0) −1 −1 −2 f(x(x) = 1 x −1 1 2 3 x Cubic function FIGURE 1.69 Square root function FIGURE 1.70 Reciprocal function FIGURE 1.71 333202_0106.qxd 12/7/05 8:40 AM Page 69 Section 1.6 A Library of Parent Functions 69 Step and Piecewise-Defined Functions Functions whose graphs resemble sets of stairsteps are known as step functions. The most famous of the step functions is the greatest integer function, which is denoted by and defined
as x f x x the greatest integer less than or equal to x. Some values of the greatest integer function are as follows. 1 greatest integer ≤ 1 1 1 2 greatest integer ≤ 1 1 10 greatest integer ≤ 1 1 1.5 greatest integer ≤ 1.5 1 0 10 2 The graph of the greatest integer function f x x y 3 2 1 −4 −3 −2 −1 1 2 3 4 x ( ) = [[ ]] f x x −3 −4 FIGURE 1.72 has the following characteristics, as shown in Figure 1.72. Te c h n o l o g y When graphing a step function, you should set your graphing utility to dot mode. • The domain of the function is the set of all real numbers. • The range of the function is the set of all integers. • The graph has a -intercept at y 0, 0 and -intercepts in the interval x 0, 1. • The graph is constant between each pair of consecutive integers. • The graph jumps vertically one unit at each integer value. Example 2 Evaluating a Step Function Evaluate the function when x 1, 2, and 3 22 FIGURE 1.73 For f x ( ) = [[ ]] x + 1 −3 −2 −1 1 2 3 4 5 x For Solution For 1, so ≤ 1 x 1, the greatest integer is f 1 1 1 1 1 0. x 2, ≤ 2 the greatest integer f 2 2 1 2 1 3. x 3 ≤ 3 2, the greatest integer 2 3 2 1 1 1 2. f 3 is 2, so is so 1, 2 You can verify your answers by examining the graph of Figure 1.73. f x x 1 shown in Now try Exercise 29. Recall from Section 1.4 that a piecewise-defined function is defined by two or more equations over a specified domain. To graph a piecewise-defined function, graph each equation separately over the specified domain, as shown in Example 3. 333202_0106.qxd 12/7/05 8:40 AM Page 70 70 Chapter 1 Functions and Their Graphs y − x= + 4 21 = + 3 −5 −4 −3 −1 −2 −3 − 4 −5 −6 FIGURE 1.74 Example 3 Graphing a Piecewise-Defined Function Sketch the graph of f x 2x 3, x 4, x ≤ 1 x > 1 . Solution This piecewise-defined function is composed of two linear functions. At and to the left of and to the right of the graph is the line 1, 5 f1 21 3 5. x 1 x 1 as shown in Figure 1.74. Notice that the point is an open dot. This is because is a solid dot and the point the graph is the line y x 4, y 2x 3, x 1 1, 3 Now try Exercise 43. Parent Functions The eight graphs shown in Figure 1.75 represent the most commonly used functions in algebra. Familiarity with the basic characteristics of these simple graphs will help you analyze the shapes of more complicated graphs—in particular, graphs obtained from these graphs by the rigid and nonrigid transformations studied in the next section. y 3 2 1 f(x1 −2 −2 −1 y f(x) = x y f(x) = x  x 1 2 −2 −1 2 1 −1 −2 x 1 2 y 3 2 1 f(x) = x 1 2 3 (a) Constant Function (b) Identity Function (c) Absolute Value Function (d) Square Root Function y 4 3 2 1 −2 −1 y 2 1 −1 −2 x 1 2 f(x) = x3 y 3 2 1 f(x3 −2 −1 1 2 3 ( ) = [[ ]] f x x −3 −2 −1 f(x) = x2 x 1 2 x x (e) Quadratic Function FIGURE 1.75 (f) Cubic Function (g) Reciprocal Function (h) Greatest Integer Function 333202_0106.qxd 12/7/05 8:40 AM Page 71 1.6 Exercises VOCABULARY CHECK: Match each function with its name. 1. 4. 7. f x x f x x2 f x x 2. 5. 8. f x x f x x f x x3 (a) squaring function (b) square root function Section 1.6 A Library of Parent Functions 71 3. 6 ax b (c) cubic function 9. (d) linear function (e) constant function (f) absolute value function (e) greatest integer function (h) reciprocal function (i) identity function PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. f 1. 3. f 0 6 In Exercises 1–8, (a) write the linear function such that it has the indicated function values and (b) sketch the graph of the function. f 1 4, f 5 4, f 5 1, f 10 12, f 1 6, 15 f 2 2 , f 2 17 f 5 1 f 16 1 f 4 3 f 4 11 f 1 2 f 1 11 f 3 8, f 3 9, 4. 2. 8. 6. 7. 5. 2 3 In Exercises 9–28, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27 x2 2x hx x2 4x 12 f x x3 1 f x x 13 2 f x 4x gx 2 x 4 f x 1 x hx 1 x 2 10. 12. 14. 16. 18. 20. 22. 24. 26. 28. f x 3x 5 2 f x 5 2 3x 6 f x x2 8x gx x2 6x 16 f x 8 x3 gx 2x 33 1 f x 4 2x hx x 2 3 f x 4 1 x kx 1 x 3 In Exercises 29–36, evaluate the function for the indicated values. 29. 30. f x x f 2.1 (a) gx 2x g3 (a) (b) f 2.9 (c) f 3.1 (d) f 7 2 (b) g0.25 (c) g9.5 (d) g 11 3 31. 32. 33. 34. 35. 36. 2 h1 f 0 hx x 3 h2 (a) (b) f x 4x 7 (a) (b) hx 3x 1 h2.5 (a) (b) kx 1 2x 6 k5 (a) (b) gx 3x 2 5 (a) gx 7x 4 6 g9 (a) g 2.7 g1 (b) (b) 8 f 1.5 h3.2 k6.1 g 1 (c) h4.2 (d) h21.6 (c) f 6 (d) f 5 3 (c) h7 3 (d) h21 3 (c) k0.1 (d) k15 (c) g 0.8 (d) g 14.5 (c) g4 (d) g3 2 In Exercises 37–42, sketch the graph of the function. 37. 39. 41. gx x gx x 2 gx x 1 38. 40. 42. gx 4 x gx x 1 gx x 3 In Exercises 43–50, graph the function. 43. 44. 45. 46. 47 2x 3, 3 x, gx x 6, 2x 4, f x 4 x, 4 x, f x 1 x 12, x 2, f x x 2 5, x 2 4x 3 333202_0106.qxd 12/7/05 8:40 AM Page 72 72 48. 49. 50. Chapter 1 Functions and Their Graphs hx 3 x2, x2 2, hx 4 x2, kx 2x 1, 3 x, x2 1, 2x2 1, 1 x2 In Exercises 51 and 52, (a) use a graphing utility to graph the function, (b) state the domain and range of the function, and (c) describe the pattern of the graph. gx 21 sx 21 51. 52. 4x 1 4x 4x 1 4x2 In Exercises 53–60, (a) identify the parent function and the transformed parent function shown in the graph, (b) write an equation for the function shown in the graph, and (c) use a graphing utility to verify your answers in parts (a) and (b). 53. y 54. y 4 2 −2 −4 2 1 543 −6 −4 55. y 2 1 −1 −2 −3 −4 57. 592 −1 321 −2 −1 32 −2 −2 −1 321 56. y 1 −2 −1 32 −2 58. 60. y 2 1 −4 −2 −1 1 −3 −4 y 2 1 −2 −1 32 −4 61. Communications The cost of a telephone call between Denver and Boise is $0.60 for the first minute and $0.42 for each additional minute or portion of a minute. A model for the total cost (in dollars) of the phone call is C 0.60 0.421 t, where is the length of the phone call in minutes. t > 0 C t (a) Sketch the graph of the model. (b) Determine the cost of a call lasting 12 minutes and 30 seconds. 62. Communications The cost of using a telephone calling card is $1.05 for the first minute and $0.38 for each additional minute or portion of a minute. (a) A customer needs a model for the cost of using a calling card for a call lasting minutes. Which of the following is the appropriate model? Explain. C1 C2 t 1.05 0.38t 1 t 1.05 0.38t 1 C t (b) Graph the appropriate model. Determine the cost of a call lasting 18 minutes and 45 seconds. 63. Delivery Charges The cost of sending an overnight package from Los Angeles to Miami is $10.75 for a package weighing up to but not including 1 pound and $3.95 for each additional pound or portion of a pound. A model for the total cost (in dollars) of sending the package is C 10.75 3.95x, x is the weight in pounds. where x > 0 C (a) Sketch a graph of the model. (b) Determine the cost of sending a package that weighs 10.33 pounds. 64. Delivery Charges The cost of sending an overnight package from New York to Atlanta is $9.80 for a package weighing up to but not including 1 pound and $2.50 for each additional pound or portion of a pound. (a) Use the greatest integer function to create a model for the cost of overnight delivery of a package weighing x C pounds, x > 0. (b) Sketch the graph of the function. 65. Wages A mechanic is paid $12.00 per hour for regular time and time-and-a-half for overtime. The weekly wage function is given by Wh 12h, 18h 40 480, 0 < h ≤ 40 h > 40 where h is the number of hours worked in a week. (a) Evaluate W30, W40, W45, and W50. (b) The company increased the regular work week to 45 hours. What is the new weekly wage function? 333202_0106.qxd 12/7/05 8:40 AM Page 73 Section 1.6 A Library of Parent Functions 73 66. Snowstorm During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate of 2 inches per hour for the next 6 hours, and at a rate of 0.5 inch per hour for the final hour. Write and graph a piecewise-defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm? Model It V 100 75 50 25 ) 60, 100) (10, 75) (20, 75) (45, 50) (5, 50) (50, 50) (30, 25) (40, 25) 20 30 50 Time (in minutes) 40 t 60 (0, 0) 10 FIGURE FOR 68 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 69 and 70, determine whether 69. A piecewise-defined function will always have at least one y x -intercept or at least one -intercept, 4, 6, 70. can be rewritten as f x 2x, 1 ≤ x < 4. Exploration In Exercises 71 and 72, write equations for the piecewise-defined function shown in the graph. 71. y 72. y 8 6 4 2 (0, 6) (3, 2) (8, 0) x 2 4 6 8 10 8 6 4 (3, 4) (7, 0) (1, 1) (− 1, 1) 4 (0, 0) 6 x Skills Review 67. Revenue The table shows the monthly revenue (in thousands of dollars) of a landscaping business for each month of the year 2005, with representing January. x 1 y Month, x Revenue 10 11 12 5.2 5.6 6.6 8.3 11.5 15.8 12.8 10.1 8.6 6.9 4.5 2.7 A mathematical model that represents these data is f x 1.97x 26.3 . 0.505x2 1.47x 6.3 (a) What is the domain of each part of the piecewisedefined function? How can you tell? Explain your reasoning. (b) Sketch a graph of the model. (c) Find and the context of the problem. f 11, f 5 and interpret your results in (d) How do the values obtained from the model in part (b) compare with the actual data values? In Exercises 73 and 74, solve the inequality and sketch the solution on the real number line. 73. 3x 4 ≤ 12 5x 74. 2x 1 > 6x 9 68. Fluid Flow The intake pipe of a 100-gallon tank has a flow rate of 10 gallons per minute, and two drainpipes have flow rates of 5 gallons per minute each. The figure shows t. the volume Determine the combination of the input pipe and drain pipes in which the fluid is flowing in specific subintervals of the 1 hour of time shown on the graph. (There are many correct answers.) of fluid in the tank as a function of time V and passing through the pairs of points are parallel, perpen- In Exercises 75 and 76, determine whether the lines L2 dicular, or neither. L1 75. L1: 2, 2, 2, 10 L2
: 1, 3, 3, 9 76. L1: 1, 7, 4, 3 L2: 1, 5, 2, 7 333202_0107.qxd 12/7/05 8:41 AM Page 74 74 Chapter 1 Functions and Their Graphs 1.7 Transformations of Functions What you should learn • Use vertical and horizontal shifts to sketch graphs of functions. • Use reflections to sketch graphs of functions. • Use nonrigid transformations to sketch graphs of functions. Why you should learn it Knowing the graphs of common functions and knowing how to shift, reflect, and stretch graphs of functions can help you sketch a wide variety of simple functions by hand. This skill is useful in sketching graphs of functions that model real-life data, such as in Exercise 68 on page 83, where you are asked to sketch the graph of a function that models the amounts of mortgage debt outstanding from 1990 through 2002. Shifting Graphs Many functions have graphs that are simple transformations of the parent graphs summarized in Section 1.6. For example, you can obtain the graph of hx x2 2 by shifting the graph of h function notation, hx x2 2 f x x2 f and are related as follows. f x 2 Upward shift of two units upward two units, as shown in Figure 1.76. In Similarly, you can obtain the graph of gx x 22 f x x 2 by shifting the graph of f g In this case, the functions and have the following relationship. f x 2 gx x 22 Right shift of two units to the right two units, as shown in Figure 1.77. h(x) = x2 + 2 y 4 3 1 f(x) = x2 f(x) = x2 y g(x) = (x − 2)2 4 3 2 1 −2 −1 1 2 x −1 1 2 3 x FIGURE 1.76 FIGURE 1.77 © Ken Fisher/Getty Images The following list summarizes this discussion about horizontal and vertical shifts. Vertical and Horizontal Shifts Let be a positive real number. Vertical and horizontal shifts in the graph of are represented as follows. c y f x In items 3 and 4, be sure you hx f x c see that corresponds to a right shift and hx f x c corresponds to a left shift for c > 0. 1. Vertical shift units upward: c 2. Vertical shift units downward: c 3. Horizontal shift units to the right: c 4. Horizontal shift units to the left: c hx f x c hx f x c hx f x c hx f x c 333202_0107.qxd 12/7/05 8:41 AM Page 75 Section 1.7 Transformations of Functions 75 Some graphs can be obtained from combinations of vertical and horizontal shifts, as demonstrated in Example 1(b). Vertical and horizontal shifts generate a family of functions, each with the same shape but at different locations in the plane. Example 1 Shifts in the Graphs of a Function f x x3 Use the graph of gx x3 1 hx x 23 1 b. a. to sketch the graph of each function. Solution a. Relative to the graph of f x x3, the graph of gx x3 1 is a downward shift of one unit, as shown in Figure 1.78. b. Relative to the graph of involves a left shift of two units and an upward shift of one unit, as shown in Figure 1.79. the graph of hx x 23 1 f x x3(x) = (x + 2) + 1 3 y f(x2 −1 1 2 x −4 −2 −−3 x −2 FIGURE 1.78 FIGURE 1.79 Now try Exercise 1. −1 −2 −3 In Figure 1.79, notice that the same result is obtained if the vertical shift precedes the horizontal shift or if the horizontal shift precedes the vertical shift. Exploration Graphing utilities are ideal tools for exploring translations of functions. Graph try to predict how the graphs of and in same viewing window. Before looking at the graphs, relate to the graph of and f, g, f. h h g a. f x x 2, gx x 42, hx x 42 3 f x x 2, gx x 12, hx x 12 2 b. c. f x x 2, gx x 42, hx x 42 2 333202_0107.qxd 12/7/05 8:41 AM Page 76 76 Chapter 1 Functions and Their Graphs y 2 1 −1 −2 −2 −1 FIGURE 1.80 Reflecting Graphs The second common type of transformation is a reflection. For instance, if you consider the -axis to be a mirror, the graph of x f (x) = x2 1 2 h(x) = −x 2 hx x2 is the mirror image (or reflection) of the graph of x f x x2, as shown in Figure 1.80. Reflections in the Coordinate Axes Reflections in the coordinate axes of the graph of as follows. 1. Reflection in the -axis: x 2. Reflection in the -axis: y hx f x hx f x y f x are represented ( ) = 4 f x x 3 Example 2 Finding Equations from Graphs −3 3 The graph of the function given by f x x 4 −1 FIGURE 1.81 is shown in Figure 1.81. Each of the graphs in Figure 1.82 is a transformation of the graph of Find an equation for each of these functions. f. 1 −3 −1 (b) 5 y h x= ( ) 3 −3 3 y g x= ( ) −1 (a) FIGURE 1.82 Solution a. The graph of g units of the graph of gx x 4 2. Exploration Reverse the order of transformations in Example 2(a). Do you obtain the same graph? Do the same for Example 2(b). Do you obtain the same graph? Explain. is a reflection in the -axis followed by an upward shift of two x f x x 4. So, the equation for g is b. The graph of h reflection in the -axis of the graph of is a horizontal shift of three units to the right followed by a x So, the equation for f x x 4. is h hx x 34. Now try Exercise 9. 333202_0107.qxd 12/7/05 8:41 AM Page 77 Section 1.7 Transformations of Functions 77 Example 3 Reflections and Shifts Compare the graph of each function with the graph of f x x . a. gx x b. hx x c. kx x 2 Algebraic Solution a. The graph of g is a reflection of the graph of f x in the -axis because gx x f x. b. The graph of h is a reflection of the graph of f y in the -axis because hx x f x. c. The graph of is a left shift of two units x followed by a reflection in the -axis because k kx x 2 f x 2. Graphical Solution f a. Graph and on the same set of coordinate axes. From the graph is a reflection of g g b. Graph and on the same set of coordinate axes. From the graph is a reflection of h in Figure 1.83, you can see that the graph of x in the -axis. the graph of f h f in Figure 1.84, you can see that the graph of y in the -axis. the graph of f f k c. Graph and on the same set of coordinate axes. From the graph is a left shift of two x k in Figure 1.85, you can see that the graph of f, units of the graph of followed by a reflection in the -axis. y 2 1 f(x) = x −1 1 2 3 −1 −2 g(x) = x− FIGURE 1.83 f(x) = x x 1 2 h(x) = − x −2 −1 FIGURE 1.84 Now try Exercise 19. FIGURE 1.85 When sketching the graphs of functions involving square roots, remember that the domain must be restricted to exclude negative numbers inside the radical. For instance, here are the domains of the functions in Example 3. Domain of Domain of Domain of gx x: hx x: kx x 2 333202_0107.qxd 12/7/05 8:41 AM Page 78 78 Chapter 1 Functions and Their Graphs Nonrigid Transformations y h(x) = 3 x  4 3 2 y 4 −2 −1 FIGURE 1.86 f(x) = x  x 1 2 g(x) = 1 3 x  f(x) = x  2 1 y 6 − 2 −1 FIGURE 1.87 x 1 2 Horizontal shifts, vertical shifts, and reflections are rigid transformations because the basic shape of the graph is unchanged. These transformations change only the position of the graph in the coordinate plane. Nonrigid transformations are those that cause a distortion—a change in the shape of the original graph. For is represented instance, a nonrigid transformation of the graph of gx cf x, by and a vertical shrink if Another nonrigid transformation of the graph of y f x where the transformation is a horizontal shrink if where the transformation is a vertical stretch if is represented by c > 1 and a horizontal stretch if hx f cx, 0 < c < 1. 0 < c < 1. y f x c > 1 Example 4 Nonrigid Transformations Compare the graph of each function with the graph of gx 1 hx 3x b. a. 3x f x x. Solution a. Relative to the graph of hx 3x 3f x f x x, the graph of is a vertical stretch (each -value is multiplied by 3) of the graph of Figure 1.86.) y f. (See b. Similarly, the graph of gx 1 3x 3 f x 1 y is a vertical shrink each -value is multiplied by Figure 1.87.) 1 3 of the graph of f. (See g(x) = 2 − 8x 3 Now try Exercise 23. Example 5 Nonrigid Transformations f(x) = 2 − x 3 − 4 −3 −2 −1−1 −2 FIGURE 1.88 3 −1 −2 − 4 f(x) = 2 − x 3 FIGURE 1.89 h(x) = 2 − x 31 8 1 2 3 4 x Compare the graph of each function with the graph of 2x hx f 1 gx f 2x b. a. f x 2 x3. Solution a. Relative to the graph of f x 2 x3, the graph of gx f 2x 2 2x3 2 8x3 is a horizontal shrink c > 1 of the graph of f. (See Figure 1.88.) b. Similarly, the graph of hx f 1 2x 2 1 2x3 2 1 8x3 is a horizontal stretch 0 < c < 1 of the graph of f. (See Figure 1.89.) Now try Exercise 27. 333202_0107.qxd 12/7/05 8:41 AM Page 79 Section 1.7 Transformations of Functions 79 1.7 Exercises VOCABULARY CHECK: In Exercises 1–5, fill in the blanks. 1. Horizontal shifts, vertical shifts, and reflections are called ________ transformations. 2. A reflection in the -axis of x is represented by y f x hx 3. Transformations that cause a distortion in the shape of the graph of is represented by ________. y f x hx of y f x are ________, while a reflection in the -axis y called ________ transformations. 4. A nonrigid transformation of a ________ ________ if 5. A nonrigid transformation of a ________ ________ if y f x 0 < c < 1. y f x 0 < c < 1. represented by hx f cx is a ________ ________ if c > 1 and represented by gx cf x is a ________ ________ if c > 1 and 6. Match the rigid transformation of y f x with the correct representation of the graph of where h, c > 0. (a) (b) (c) (d) hx f x c hx f x c hx f x c hx f x c (i) A horizontal shift of cf, units to the right (ii) A vertical shift of cf, units downward (iii) A horizontal shift of cf, units to the left (iv) A vertical shift of cf, units upward PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1. For each function, sketch (on the same set of coordinate axes) a graph of each function for c 1, 1, and 3. (a) (b) (c. For each function, sketch (on the same set of coordinate axes) a graph of each function for c 3, 1, 1, and 3. (a) (b) (c. For each function, sketch (on the same set of coordinate axes) a graph of each function for c 2, 0, and 2. (a) (b) (c. (a) (b) (c) (d) (e) (f) (g 2x y 6 4 2 (1, 0) −4 −2 −4 (3, 1) f (4, 2) 6 2 4 (0, 1)− x 6. (a) (b) (c) (d) (e) (f) (g 2x y 8 − ( 4, 2) −4 − − ( 2, 2) −6 (6, 2) f x 4 (0, 2)− 8 4. For each function, sketch (on the same set of coordinate axes) a graph of each function for c 3, 1, 1,
and 3. FIGURE FOR 5 FIGURE FOR 6 (a) (b) f x x2 c, x2 c, f x x c2, x c2 In Exercises 5–8, use the graph of to sketch each graph.To print an enlarged copy of the graph go to the website www.mathgraphs.com. f 7. (a) (b) (c) (d) (e) (f) (g 2x 8. (a) (b) (c 10 (f) 3x (g) y f 1 (d) (e) 333202_0107.qxd 12/7/05 8:41 AM Page 80 80 Chapter 1 Functions and Their Graphs y (3, 0) x 11. Use the graph of f x x to write an equation for each function whose graph is shown. (a) y (b) y y 6 2 − ( 2, 4) f −4 −2 −2 −4 (0, 3) (1, 0) 6 4 (3, 1)− x (0, 5) −3 , 0) ( 2 −10 −6 − − ( 6, 4) −2 −6 −10 −14 2 6 f (6, 4)− 4 2 FIGURE FOR 7 FIGURE FOR 8 9. Use the graph of f x x2 to write an equation for each function whose graph is shown. −2 y (c) 2 4 −6 x x (d) y −4 −6 x x (a) y 2 1 −2 −1 1 2 −2 (c) y 6 4 2 −2 2 4 6 x x (b) y x 1 −3 −1 −1 −2 −3 (d) y 4 2 2 4 6 8 x 10. Use the graph of f x x3 to write an equation for each function whose graph is shown. (a) y 3 2 (b) y 3 2 1 −2 −1 −1 x 2 −1 1 2 3 (c) y (d) 4 2 −2 x 2 −6 −4 y 4 −4 −4 −8 −12 4 8 x x 2 4 6 4 8 12 −4 −2 −4 −6 12. Use the graph of f x x to write an equation for each function whose graph is shown. (a) y 4 2 −2 −4 −6 −8 x 6 8 10 (b) y 2 −2 −4 −8 −10 2 4 6 8 10 (c) y (d) 8 6 4 2 −2 −4 2 4 6 8 10 x y 2 −4 −2 2 4 6 −4 −8 −10 x x In Exercises 13–18, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. 13. y 14. 2 −2 x 2 4 y 2 −2 x 2 333202_0107.qxd 12/7/05 8:41 AM Page 81 15. y 16. −2 x 2 y 6 4 −2 −4 17. y 18. 2 −2 x 4 x 2 4 −2 −2 y 4 −4 −2 x In Exercises 19–42, g is related to one of the parent functions described in this chapter. (a) Identify the parent function f. (b) Describe the sequence of tranformations from f to g. (c) Sketch the graph of g. (d) Use function notation to write g in terms of f. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 39. 41. g x 12 x2 g x x3 7 gx 2 3x2 4 g x 2 x 52 gx 3x g x x 13 2x 4 20. 22. 24. 26. 28. 30. 32. 34. 36. 38. 40. 42. g x x 82 g x x 3 1 gx 2x 72 gx x 102 5 gx 1 4 x g x x 33 10 2x 3x 1 In Exercises 43–50, write an equation for the function that is described by the given characteristics. 43. The shape of f x x2 , but moved two units to the right and eight units downward 44. The shape of f x x2 seven units upward, and reflected in the -axis , but moved three units to the left, x 45. The shape of 46. The shape of f x x3 f x x3 , but moved 13 units to the right , but moved six units to the left, six 47. The shape of units downward, and reflected in the -axis f x x, x reflected in the -axis y but moved 10 units upward and 48. The shape of f x x, seven units downward but moved one unit to the left and Section 1.7 Transformations of Functions 81 49. The shape of f x x, but moved six units to the left and 50. The shape of x reflected in both the -axis and the -axis f x x, but moved nine units downward x and reflected in both the -axis and the -axis y y 51. Use the graph of f x x2 to write an equation for each function whose graph is shown. (a) y 1 −3 −2 −1 1 2 3 x (b) y (1, 7) (1, 3)− 2 −5 −2 2 4 x 52. Use the graph of f x x3 to write an equation for each function whose graph is shown. (a) (b) y 6 4 2 (2, 2) y 3 2 −6 −4 2 4 6 x −3 −2 −1 −4 −6 −2 −3 x 1 3 2 (1, 2)− 53. Use the graph of f x x to write an equation for each function whose graph is shown. (a) −4 y 4 2 −4 −6 −8 (4, 2)− (b) x 6 − ( 2, 3) y 8 6 4 −4 −2 2 4 6 x −4 54. Use the graph of f x x to write an equation for each function whose graph is shown. (a) y (b) y 20 16 12 8 4 −4 (4, 16) 4 8 12 16 20 x 1 −1 −2 −3 1 x )− ( 4, 1 2 333202_0107.qxd 12/7/05 8:41 AM Page 82 82 Chapter 1 Functions and Their Graphs In Exercises 55–60, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer. 56. 58. 55. 57. y 2 1 −2 −1 1 2 x −2 y 4 2 −4 −6 −8 −4 x 4 6 59. y 60. y 5 4 −3 −2 −1 1 2 3 y 3 2 1 −3 −1 1 2 3 −2 −3 y 4 2 x x x x −6 −4 −2 42 6 −4 −3 −2 −1 2 1 −1 −2 Graphical Analysis In Exercises 61–64, use the viewing window shown to write a possible equation for the transformation of the parent function. 61. −4 63. −4 6 −2 1 −7 62. −10 64. −4 8 8 7 −1 5 −3 2 8 Graphical Reasoning In Exercises 65 and 66, use the graph of f to sketch the graph of g. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 65. 66. y 4 3 2 f 1 432 5 x −4 −3 −2 −1 −2 −3 (a) (c) (e) gx f x 2 gx f x gx f 4x (b) (d) (f) gx f x 1 gx 2f x gx f 1 2x y 6 4 f −4 −2 −4 −6 2 4 86 10 12 x (a) (c) (e) gx f x 5 gx f x gx f 2x 1 (b) gx f x 1 2 gx 4 f x 4x 2 (d) (f) gx f 1 Model It 67. Fuel Use The amounts of fuel (in billions of gallons) used by trucks from 1980 through 2002 can be approximated by the function F ft 20.6 0.035t2, 0 ≤ t ≤ 22 F t where represents the year, with 1980. corresponding to (Source: U.S. Federal Highway Administration) t 0 (a) Describe the transformation of the parent function Then sketch the graph over the specified f x x2. domain. (b) Find the average rate of change of the function from 1980 to 2002. Interpret your answer in the context of the problem. (c) Rewrite the function so that t 0 Explain how you got your answer. represents 1990. (d) Use the model from part (c) to predict the amount of fuel used by trucks in 2010. Does your answer seem reasonable? Explain. 333202_0107.qxd 12/7/05 2:48 PM Page 83 M 68. Finance The amounts (in trillions of dollars) of mortgage debt outstanding in the United States from 1990 through 2002 can be approximated by the function M f t 0.0054t 20.3962, 0 ≤ t ≤ 12 represents the year, with corresponding to (Source: Board of Governors of the Federal t 0 t where 1990. Reserve System) (a) Describe the transformation of the parent function Then sketch the graph over the specified f x x2. domain. (b) Rewrite the function so that t 0 represents 2000. Explain how you got your answer. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 69 and 70, determine whether 69. The graphs of f x x 6 are identical. and f x x 6 70. If the graph of the parent function is moved six units to the right, three units upward, and reflected in the x -axis, then the point will lie on the graph of the transformation. 2, 19 f x x2 71. Describing Profits Management originally predicted that the profits from the sales of a new product would be f approximated by the graph of the function shown. The g along with a actual profits are shown by the function verbal description. Use the concepts of transformations of f. graphs to write in terms of g y 40,000 20,000 f (a) The profits were only three-fourths as large as expected. (b) The profits were consistently $10,000 greater than predicted. t 2 4 40,000 20,000 y y g 2 4 60,000 30,000 g 2 4 t t Section 1.7 Transformations of Functions 83 (c) There was a two-year y delay in the introduction of the product. After sales began, profits grew as expected. 40,000 20,000 g t 2 4 6 72. Explain why the graph of y f x is a reflection of the graph of y f x about the -axis. x 73. The graph of y f x 0, 1, passes through the points Find the corresponding points on the 2, 3. 1, 2, graph of and y f x 2 1. 74. Think About It You can use either of two methods to graph a function: plotting points or translating a parent function as shown in this section. Which method of graphing do you prefer to use for each function? Explain. (a) f x 3x2 4x 1 (b) f x 2x 12 6 Skills Review In Exercises 75–82, perform the operation and simplify. 75. 77. 78. 79. 80. 81. 82. 4 x 4 1 x 76 xx x2 4 x2 x 2 x x2 4 x2 x2 9 x 3 5 x x2 3x 28 x2 3x x2 5x 4 In Exercises 83 and 84, evaluate the function at the specified values of the independent variable and simplify. 83. 84. f 3 f x x2 6x 11 (a) f x x 10 3 (a) f 10 (b) (b) f 1 2 f 26 (c) f x 3 (c) f x 10 In Exercises 85–88, find the domain of the function. 85. f x 2 11 x 87. f x 81 x2 86. f x x 3 x 8 88. f x 34 x2 333202_0108.qxd 12/7/05 8:43 AM Page 84 84 Chapter 1 Functions and Their Graphs 1.8 Combinations of Functions: Composite Functions What you should learn • Add, subtract, multiply, and divide functions. • Find the composition of one function with another function. • Use combinations and compositions of functions to model and solve real-life problems. Why you should learn it Compositions of functions can be used to model and solve real-life problems. For instance, in Exercise 68 on page 92, compositions of functions are used to determine the price of a new hybrid car. © Jim West/The Image Works Arithmetic Combinations of Functions Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be the functions given by combined to create new functions. For example, f x 2x 3 gx x 2 1 can be combined to form the sum, difference, g. product, and quotient of and and f f x gx 2x 3 x 2 1 x 2 2x 4 f x gx 2x 3 x 2 1 Sum x 2 2x 2 Difference f xgx 2x 3x 2 1 2x 3 3x 2 2x 3 2x 3 x2 1 , x ±1 f x gx Product Quotient The domain of an arithmetic combination of functions and real numbers that are common to the domains of and tient there is the further restriction that f g. gx 0. fxgx, g f consists of all In the case of the quo- f Sum, Difference, Product, and Quotient of Functions x Let and be two functions with overlapping domains. Then, for all common to both domains, the sum, difference, product, and quotient of and are defined as follows. g g f 1. Sum: 2. Difference: 3. Product: 4. Quotient: f gx f x gx f gx f x gx fgx f x gx x f x f gx , g gx 0 Example 1 Finding the Sum of Two Functions Given f x 2x 1 and gx x 2 2x 1, find f gx. Solution f gx f x gx 2x 1 x 2 2x 1 x 2 4x Now try Exercise 5(a). 333202_0108.qxd 12/7/05 8:43 AM Page 85 Section 1.8 Combinations of Functions: Composite Functions 85 Example 2 Finding the Difference of Two Functions f x 2x 1 Given the difference when and x 2. gx x 2 2x 1, find f gx. Then evaluate Solution The difference
of and f f gx f x gx is g 2x 1 x 2 2x 1 x 2 2. When x 2, the value of this difference is f g2 22 2 2. Now try Exercise 5(b). In Examples 1 and 2, both numbers. So, the domains of numbers. Remember that any restrictions on the domains of considered when forming the sum, difference, product, or quotient of and have domains that consist of all real are also the set of all real and must be f and f g f g g and g f g. f Example 3 Finding the Domains of Quotients of Functions Find x f g and x g f for the functions given by f x x and gx 4 x 2. Then find the domains of fg and gf. Solution The quotient of and f g is and the quotient of and is g f g x f x gx g f x gx . The domain of f these domains is is 0, 2. 0, Domain of f g : 0, 2 So, the domains of g and the domain of f g Domain of g f x 0, includes is and 2, 2. g f : 0, 2 fg Note that the domain of but not yields a zero in the denominator, whereas the domain of but not x 2, gf yields a zero in the denominator. because x 0, x 0 because includes x 2 x 2, The intersection of are as follows. Now try Exercise 5(d). 333202_0108.qxd 12/7/05 8:43 AM Page 86 86 Chapter 1 Functions and Their Graphs Composition of Functions f ˚ g x g(x) g f f(g(x)) Domain of g FIGURE 1.90 Domain of f The following tables of values help illustrate the composition f gx given in Example 4. x gx gx f gx x f gx Note that the first two tables can be combined (or “composed”) to produce the values given in the third table. Another way of combining two functions is to form the composition of one with the other. For instance, if the composition of with g gx x 1, f x x2 and is f f gx f x 1 x 12. This composition is denoted as f g and reads as “f composed with g.” Definition of Composition of Two Functions The composition of the function with the function is g f f gx f gx. The domain of in the domain of f g f. is the set of all (See Figure 1.90.) x in the domain of g such that gx is Example 4 Composition of Functions Given f x x 2 a. f gx b. and g f x c. g f 2 gx 4 x2, find the following. Solution a. The composition of with f g is as follows. f gx f gx . The composition of with is as follows. g f g f x g f x gx 2 4 x 22 4 x2 4x 4 x2 4x Definition of f g Definition of Definition of gx f x Simplify. Definition of g f Definition of Definition of f x gx Expand. Simplify. Note that, in this case, f gx g f x. c. Using the result of part (b), you can write the following. g f 2 22 42 4 8 4 Now try Exercise 31. Substitute. Simplify. Simplify. 333202_0108.qxd 12/7/05 8:43 AM Page 87 Te c h n o l o g y You can use a graphing utility to determine the domain of a composition of functions. For the composition in Example 5, enter the function composition as y 9 x22 9. You should obtain the graph shown below. Use the trace feature to determine that the x-coordinates of points on the graph extend from f gx domain of 3 is to 3. So, the 3 ≤ x ≤ 3. 1 −5 5 Section 1.8 Combinations of Functions: Composite Functions 87 Example 5 Finding the Domain of a Composite Function f x x2 9 Given find the domain of and f g. gx 9 x2, find the composition f gx. Then Solution f gx f gx f 9 x2 9 x22 9 9 x2 9 x2 From this, it might appear that the domain of the composition is the set of all real is the set of all real numbers. This, however is not true. Because the domain of numbers and the domain of is 3 ≤ x ≤ 3. 3 ≤ x ≤ 3, the domain of f g is g f Now try Exercise 35. −10 In Examples 4 and 5, you formed the composition of two given functions. In calculus, it is also important to be able to identify two functions that make up a given composite function. For instance, the function given by h hx 3x 53 is the composition of with where f hx 3x 53 gx3 fgx. g, f x x3 and gx 3x 5. That is, Basically, to “decompose” a composite function, look for an “inner” function and is the inner function an “outer” function. In the function above, and is the outer function. gx 3x 5 f x x3 h Example 6 Decomposing a Composite Function Write the function given by hx 1 x 22 as a composition of two functions. Solution One way to write as a composition of two functions is to take the inner function to be and the outer function to be gx x 2 h f x 1 x2 x2. Then you can write hx 1 x 22 x 22 f x 2 f gx. Now try Exercise 47. 333202_0108.qxd 12/7/05 8:43 AM Page 88 88 Chapter 1 Functions and Their Graphs Application Example 7 Bacteria Count The number N of bacteria in a refrigerated food is given by NT 20T 2 80T 500, 2 ≤ T ≤ 14 T where removed from refrigeration, the temperature of the food is given by is the temperature of the food in degrees Celsius. When the food is Tt 4t 2, 0 ≤ t ≤ 3 t is the time in hours. (a) Find the composition where meaning in context. (b) Find the time when the bacterial count reaches 2000. and interpret its NTt Solution a. NTt 204t 22 804t 2 500 2016t 2 16t 4 320t 160 500 320t 2 320t 80 320t 160 500 320t 2 420 The composite function as a function of the amount of time the food has been out of refrigeration. represents the number of bacteria in the food NTt b. The bacterial count will reach 2000 when Solve this hours. When you equation to find that the count will reach 2000 when solve this equation, note that the negative value is rejected because it is not in the domain of the composite function. 320t 2 420 2000. t 2.2 Now try Exercise 65. W RITING ABOUT MATHEMATICS Analyzing Arithmetic Combinations of Functions a. Use the graphs of and when x 1, 2, 3, 4, 5 gx of f b. Use the graphs of and when x 1, 2, 3, 4, 5 hx of f f g in Figure 1.91 to make a table showing the values , and 6. Explain your reasoning. f h in Figure 1.91 to make a table showing the values , and 6. Explain your reasoning FIGURE 1.91 333202_0108.qxd 12/7/05 8:43 AM Page 89 Section 1.8 Combinations of Functions: Composite Functions 89 1.8 Exercises VOCABULARY CHECK: Fill in the blanks. 1. Two functions and can be combined by the arithmetic operations of ________, ________, ________, g f and _________ to create new functions. 2. The ________ of the function with f g 3. The domain of is all x f g is f gx fgx. in the domain of such that _______ is in the domain of g f. 4. To decompose a composite function, look for an ________ function and an ________ function. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, use the graphs of hx f gx. go to the website www.mathgraphs.com. to graph To print an enlarged copy of the graph, and g f 1. y 2 3. 4. y 2 g −2 2 f −2 −2 2 x 6 −2 2 g −2 In Exercises 5–12, (c) find (a) f gx, What is the domain of (b) f /gx. f gx, f /g? fgx, and (d) f x x 2, f x 2x 5, f x x 2, f x 2x 5, f x x 2 6, 5. 6. 7. 8. 9. 10. f x x2 4, 11. 12 , gx x 2 gx 2 x gx 4x 5 gx 4 gx 1 x gx x2 x2 1 gx 1 x2 gx x3 16. 17. 15. 13. 14. and gx x 4. In Exercises 13 –24, evaluate the indicated function for fx x 2 1 f g2 f g0 f g3t fg6 5 f g f g f g1 f g1 f gt 2 fg6 0 f g 1 g3 fg 5 f 4 20. 19. 24. 22. 18. 23. 21. In Exercises 25 –28, graph the functions the same set of coordinate axes. f, g, and f g on 25. 26. 27. 28. f x 1 2 x, f x 1 3 x, f x x 2, f x 4 x 2, gx x 1 gx x 4 gx 2x gx x Graphical Reasoning In Exercises 29 and 30, use a graphf g ing utility to graph in the same viewing winand dow. Which function contributes most to the magnitude of the sum when Which function contributes most to the magnitude of the sum when 0 ≤ x ≤ 2? f, g, 29. f x 3x, 30. f x x , 2 x > 6? gx x3 10 gx x In Exercises 31–34, find (a) f g, (b) g f, and (c) f f. 31. 32. 33. f x x2, f x 3x 5, f x 3x 1, 34. f x x3, gx x 1 gx 5 x gx x3 1 gx 1 x 333202_0108.qxd 12/7/05 8:43 AM Page 90 90 Chapter 1 Functions and Their Graphs In Exercises 35–42, find (a) domain of each function and each composite function. and (b) g f. f g Find the 35. 36. 37. 38. 39. 40. 41. 42. f x x 4, f x 3x 5, f x x2 1, f x x23, f x x, f x x 4, f x 1 x , f x 3 , x2 1 gx x 2 gx x3 1 gx x gx x6 gx x 6 gx 3 x gx x 3 gx x 1 In Exercises 43–46, use the graphs of and to evaluate the functions. g f y y = f(x) 4 3 2 1 y = g(x 43. (a) 44. (a) 45. (a) 46. (a) f g3 f g1 f g2 f g1 (b) (b) (b) (b) fg2 fg4 g f 2 g f 3 In Exercises 47–54, find two functions and f gx hx. (There are many correct answers.) f g such that 47. 49. 51. hx 2x 12 hx 3x 2 4 hx 1 53. hx x 2 x2 3 4 x2 48. 50. 52. 54. hx 1 x3 hx 9 x hx 4 5x 22 hx 27x 3 6x 10 27x 3 55. Stopping Distance The research and development department of an automobile manufacturer has determined that when a driver is required to stop quickly to avoid an accident, the distance (in feet) the car travels during the driver’s reaction time is given by is the speed of the car in miles per hour. The distance (in feet) 15x 2. traveled while the driver is braking is given by Find the function that represents the total stopping distance T. and on the same set of coordinate axes for Graph the functions 0 ≤ x ≤ 60. Bx 1 Rx 3 where B,R, 4x, T x 56. Sales From 2000 to 2005, the sales (in thousands of dollars) for one of two restaurants owned by the same parent company can be modeled by R1 R1 480 8t 0.8t 2, t 0, 1, 2, 3, 4, 5 t 0 where period, the sales restaurant can be modeled by R2 represents 2000. During the same six-year (in thousands of dollars) for the second R2 254 0.78t, t 0, 1, 2, 3, 4, 5. (a) Write a function R3 that represents the total sales of the two restaurants owned by the same parent company. (b) Use a graphing utility to graph R1, R2, and R3 in the same viewing window. bt t, 57. Vital Statistics Let United States in year of deaths in the United States in year where corresponds to 2000. be the number of births in the represent the number t 0 and let dt t, (a) If pt is the population of the United States in year t, that represents the percent change ct find the function in the population of the United States. c5. (b) Interpret the value of dt and let t, 58. Pets Let in year States in year where ct be the number of dogs in the United States be the number of cats in the United t 0 corresponds to 2000. pt of dogs and cats in the United States. t
hat represents the total number (a) Find the function t, (b) Interpret the value of p5. (c) Let represent the population of the United States in corresponds to 2000. Find and t 0 nt t, year where interpret ht pt nt. 59. Military Personnel The total numbers of Army personnel from (in thousands) and Navy personnel (in thousands) 1990 to 2002 can be approximated by the models At 3.36t2 59.8t 735 N A and Nt 1.95t2 42.2t 603 t where 1990. t 0 represents the year, with (Source: Department of Defense) corresponding to (a) Find and interpret A Nt. Evaluate this function for t 4, 8, and 12. (b) Find and interpret t 4, 8, and 12. for A Nt. Evaluate this function 333202_0108.qxd 12/7/05 8:43 AM Page 91 Section 1.8 Combinations of Functions: Composite Functions 91 60. Sales The sales of exercise equipment (in millions of dollars) in the United States from 1997 to 2003 can be approximated by the function Et 25.95t2 231.2t 3356 E 62. Graphical Reasoning An electronically controlled thermostat in a home is programmed to lower the temperature automatically during the night. The temperature in the house the time in hours on a 24-hour clock (see figure). (in degrees Fahrenheit) is given in terms of T t, and the U.S. population can be approximated by the function Pt 3.02t 252.0 P (in millions) from 1997 to 2003 corresponding to represents the year, with (Source: National Sporting Goods Association, t 7 t where 1997. U.S. Census Bureau) (a) Find and interpret ht Et Pt. (b) Evaluate the function in part (a) for t 7, 10, and 12. Model It 61. Health Care Costs The table shows the total amounts (in billions of dollars) spent on health services and supplies in the United States (including Puerto Rico) y2, for the years 1995 through 2001. The variables represent out-of-pocket payments, insurance and premiums, and other types of payments, respectively. (Source: Centers for Medicare and Medicaid Services) y1, y3 Year 1995 1996 1997 1998 1999 2000 2001 y1 146.2 152.0 162.2 175.2 184.4 194.7 205.5 y2 329.1 344.1 359.9 382.0 412.1 449.0 496.1 y3 44.8 48.1 52.1 55.6 57.8 57.4 57.8 (a) Use the regression feature of a graphing utility to y1 and quadratic models for find a linear model for y3. y2 Let y2 (b) Find t 5 y3. (c) Use a graphing utility to graph and y1 represent 1995. What does this sum represent? y2, in the same viewing window. y1, y3, and y1 y2 y3 (d) Use the model from part (b) to estimate the total amounts spent on health services and supplies in the years 2008 and 2010. T 80 70 60 50 ) 15 6 12 Time (in hours) 18 21 t 24 T T4 (a) Explain why is a function of t. (b) Approximate T15. (c) The thermostat is reprogrammed to produce a temperHt How does this Tt 1. for which and H ature change the temperature? (d) The thermostat is reprogrammed to produce a temperHow does this Ht Tt 1. H ature change the temperature? for which (e) Write a piecewise-defined function that represents the graph. 63. Geometry A square concrete foundation is prepared as a base for a cylindrical tank (see figure). r x (a) Write the radius of the tank as a function of the length r x of the sides of the square. (b) Write the area A of the circular base of the tank as a function of the radius (c) Find and interpret A rx. r. 333202_0108.qxd 12/7/05 8:43 AM Page 92 92 Chapter 1 Functions and Their Graphs 64. Physics A pebble is dropped into a calm pond, causing ripples in the form of concentric circles (see figure). The t radius where is the time in seconds after the pebble strikes the water. The Ar r 2. area Find and interpret of the circle is given by the function (in feet) of the outer ripple is rt 0.6t, A rt. A r 65. Bacteria Count The number N of bacteria in a refriger- ated food is given by NT 10T 2 20T 600, 1 ≤ T ≤ 20 T where is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by Tt 3t 2, 0 ≤ t ≤ 6 where t is the time in hours. (a) Find the composition NTt and interpret its meaning in context. (b) Find the time when the bacterial count reaches 1500. 66. Cost The weekly cost C of producing units in a manu- x facturing process is given by Cx 60x 750. The number of units produced in hours is given by xt 50t. x t (a) Find and interpret C xt. (b) Find the time that must elapse in order for the cost to increase to $15,000. 67. Salary You are a sales representative for a clothing manufacturer. You are paid an annual salary, plus a bonus of 3% of your sales over $500,000. Consider the two functions given by f x x 500,000 g(x) 0.03x. and x is greater than $500,000, which of the following repre- If sents your bonus? Explain your reasoning. (a) f gx (b) g f x 68. Consumer Awareness The suggested retail price of a new hybrid car is dollars. The dealership advertises a factory rebate of $2000 and a 10% discount. p (a) Write a function R in terms of giving the cost of the p hybrid car after receiving the rebate from the factory. (b) Write a function S in terms of giving the cost of the p hybrid car after receiving the dealership discount. (c) Form the composite functions R Sp and S Rp and interpret each. R S20,500 (d) Find and S R20,500. Which yields the lower cost for the hybrid car? Explain. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 69 and 70, determine whether 69. If f x x 1 and f g)x g f )x. gx 6x, then 70. If you are given two functions f gx calculate f. of the domain of if and only if the range of f x and gx g , you can is a subset 71. Proof Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function. 72. Conjecture Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis. Skills Review Average Rate of Change difference quotient f x h f x h and simplify your answer. 73. 75. f x 3x 4 f x 4 x In Exercises 73–76, find the 74. f x 1 x2 76. f x 2x 1 In Exercises 77–80, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line. 2, 4, 8, 1, 6, 3, 7, 0 77. 80. 79. 78. 333202_0109.qxd 12/7/05 8:45 AM Page 93 1.9 Inverse Functions Section 1.9 Inverse Functions 93 What you should learn • Find inverse functions informally and verify that two functions are inverse functions of each other. • Use graphs of functions to determine whether functions have inverse functions. • Use the Horizontal Line Test to determine if functions are one-to-one. • Find inverse functions alge- braically. Why you should learn it Inverse functions can be used to model and solve real-life problems. For instance, in Exercise 80 on page 101, an inverse function can be used to determine the year in which there was a given dollar amount of sales of digital cameras in the United States. Inverse Functions Recall from Section 1.4, that a function can be represented by a set of ordered pairs. For instance, the function to the set A 1, 2, 3, 4 from the set B 5, 6, 7, 8 fx x 4: f x x 4 can be written as follows. 1, 5, 2, 6, 3, 7, 4, 8 In this case, by interchanging the first and second coordinates of each of these f 1. ordered pairs, you can form the inverse function of which is denoted by It is a function from the set f 1x x 4: to the set 5, 1, 6, 2, 7, 3, 8, 4 and can be written as follows. A, B f, is equal to the range of f Note that the domain of Figure 1.92. Also note that the functions and each other. In other words, when you form the composition of with composition of and vice versa, as shown in have the effect of “undoing” or the you obtain the identity function. f 1 f 1 f f f 1, f 1 with f, f f 1x 1x 4 x 4 4 x Domain of f Range of f f (x) = x + 4 x Range of f −1 FIGURE 1.92 −1(x) = x − 4 f f(x) Domain of f −1 © Tim Boyle/Getty Images Example 1 Finding Inverse Functions Informally Find the inverse function of f 1 f x are equal to the identity function. f(x) 4x. Then verify that both f f 1x and Solution The function multiplies each input by 4. To “undo” this function, you need to divide each input by 4. So, the inverse function of f x 4x is f f 1x x 4 . You can verify that both f f 1x fx 4 4x 4 f f 1x x x Now try Exercise 1. as follows. f 1 f x x and f 1 f x f 14x 4x 4 x 333202_0109.qxd 12/7/05 8:45 AM Page 94 94 Chapter 1 Functions and Their Graphs Exploration Consider the functions given by x 2 f x and f 1x x 2. f f 1x Evaluate f 1 f x for the indicated x. values of What can you conclude about the functions? and 10 0 7 45 x f f1x f1 f x Definition of Inverse Function Let and be two functions such that f g f gx x for every x in the domain of g and g f x x for every x in the domain of f. Under these conditions, the function f. tion The function is denoted by f f 1x x g f 1 f 1 f x x. and g is the inverse function of the func- (read “ -inverse”). So, f f The domain of must be equal to the range of be equal to the domain of f 1. f 1, and the range of must f In this is written, it always refers to the inverse function of the func- to denote the inverse function f 1. 1 Don’t be confused by the use of f 1 text, whenever tion and not to the reciprocal of f If the function f x. f, is the inverse function of the function it must also be true g. that the function is the inverse function of the function For this reason, you can say that the functions and are inverse functions of each other. g g f f Example 2 Verifying Inverse Functions Which of the functions is the inverse function of gx x 2 5 hx 5 x 2 fx 5 x 2 ? Solution By forming the composition of with f g, you have f gx f x 2 5 5 x 2 5 25 x 12 Substitute x 2 5 for x. 2 x. g Because this composition is not equal to the identity function is not the inverse function of By forming the composition of with you have it follows that x, f h, f. f hx f5 . So, it appears that that the composition of with is the inverse function of You can confirm this by showing is also equal to the identity function. f. h h f Now try Exercise 5. 333202_0109.qxd 12/7/05 8:4
5 AM Page 95 y y = x The Graph of an Inverse Function Section 1.9 Inverse Functions 95 y = f (x) (a, b) y = f −1(x) (b, a) x x FIGURE 1.93 f −1( ) = ( + 3− y 6 (1, 2) (3, 3) (2, 1) (1, 1)− 6 (0, 3)− − ( 1, 1) − ( 3, 0) −6 − − ( 5, 1) y x= − − ( 1, 5) FIGURE 1.94 y (3, 9) f (x) = x2 (2, 4) (4, 2) y = x (9, 3) (1, 1) −1 (x The graphs of a function and its inverse function in the following way. If the point must lie on the graph of a reflection of the graph of are related to each other b, a then the point f 1 is and vice versa. This means that the graph of in the line as shown in Figure 1.93. lies on the graph of f 1, f y x, a, b f, f 1 Example 3 Finding Inverse Functions Graphically x 3 Sketch the graphs of the inverse functions on the same rectangular coordinate system and show that the graphs are reflections of each other in the line f x 2x 3 f 1x 1 2 y x. and f Solution are shown in Figure 1.94. It appears that the graphs are The graphs of and y x. reflections of each other in the line You can further verify this reflective property by testing a few points on each graph. Note in the following list that if the point the point a, b f 1. f 1 f, is on the graph of fx 2x 3 Graph of Graph of is on the graph of x 3 1, 5 0, 3 1, 1 2, 1 3, 3 b, a f 1x 1 2 5, 1 3, 0 1, 1 1, 2 3, 3 Now try Exercise 15. Example 4 Finding Inverse Functions Graphically Sketch the graphs of the inverse functions on the same rectangular coordinate system and show that the graphs are reflections of each other in the line y x. and f x x 2 x ≥ 0 f 1x x f Solution are shown in Figure 1.95. It appears that the graphs are The graphs of and y x. reflections of each other in the line You can further verify this reflective property by testing a few points on each graph. Note in the following list that if the point the point is on the graph of b, a a, b f 1. f 1 f, Graph of x ≥ 0 is on the graph of fx x 2, 0, 0 1, 1 2, 4 3, 9 Graph of f 1x x 0, 0 1, 1 4, 2 9, 3 (0, 0) 3 4 5 6 7 8 9 x Try showing that f f 1x x and f 1 f x x. FIGURE 1.95 Now try Exercise 17. 333202_0109.qxd 12/7/05 8:46 AM Page 96 96 Chapter 1 Functions and Their Graphs One-to-One Functions The reflective property of the graphs of inverse functions gives you a nice geometric test for determining whether a function has an inverse function. This test is called the Horizontal Line Test for inverse functions. Horizontal Line Test for Inverse Functions A function has an inverse function if and only if no horizontal line f intersects the graph of at more than one point. f If no horizontal line intersects the graph of at more than one point, then no y x -value is matched with more than one -value. This is the essential characteristic of what are called one-to-one functions. f One-to-One Functions A function sponds to exactly one value of the independent variable. A function has an is one-to-one. inverse function if and only if is one-to-one if each value of the dependent variable corre- f f f f x x2. f x x2. Consider the function given by The table on the left is a table of values for The table of values on the right is made up by interchanging the columns of the first table. The table on the right does not represent a funcand tion because the input y 2. x 4 y 2 is not one-to-one and does not have an inverse function. is matched with two different outputs: f x x2 So x2 Example 5 Applying the Horizontal Line Test a. The graph of the function given by f x x3 1 Because no horizontal line intersects the graph of you can conclude that function. is shown in Figure 1.96. f at more than one point, is a one-to-one function and does have an inverse f b. The graph of the function given by is shown in Figure 1.97. Because it is possible to find a horizontal line that intersects the graph of at is not a one-to-one function and more than one point, you can conclude that does not have an inverse function. f f fx x 2 1 Now try Exercise 29. y 3 1 −2 −3 y 3 2 −2 −3 −3 −2 −1 FIGURE 1.96 −3 −2 FIGURE 1.97 x 2 3 f (xx) = x 2 − 1 333202_0109.qxd 12/7/05 8:46 AM Page 97 Section 1.9 Inverse Functions 97 Finding Inverse Functions Algebraically Note what happens when you try to find the inverse function of a function that is not one-to-one. f x x2 1 y x2 1 x y2 1 x 1 y2 Original function Replace f(x) by y. Interchange x and y. Isolate y-term. For simple functions (such as the one in Example 1), you can find inverse functions by inspection. For more complicated functions, however, it is best to use the following guidelines. The key step in these guidelines is Step 3—interchanging the roles of and This step corresponds to the fact that inverse functions have ordered pairs with the coordinates reversed. y. x Finding an Inverse Function 1. Use the Horizontal Line Test to decide whether has an inverse function. f x 2. In the equation for replace f x, by y. f 3. Interchange the roles of and x y, and solve for y. y ± x 1 Solve for y. 4. Replace by y f 1x in the new equation. You obtain two -values for each x. y f x( ) = 5 − 3x 2 x 4 6 y 6 4 −2 − 4 −6 −6 −4 −2 FIGURE 1.98 Exploration Restrict the domain of f x x2 1 Use to a graphing utility to graph the function. Does the restricted function have an inverse function? Explain. x ≥ 0. f 5. Verify that and f the domain of f 1, domain of are inverse functions of each other by showing that is equal to the f 1 is equal to the range of and f f 1x x and f 1 f x x. the range of f 1, f Example 6 Finding an Inverse Function Algebraically Find the inverse function of f x 5 3x . 2 is a line, as shown in Figure 1.98. This graph passes the Horizontal f is one-to-one and has an inverse function. Solution f The graph of Line Test. So, you know that f x 5 3x 2 y 5 3x 2 x 5 3y 2 2x 5 3y 3y 5 2x y 5 2x 3 f 1x 5 2x 3 Write original function. Replace f x by y. Interchange and x y. Multiply each side by 2. Isolate the -term. y Solve for y. Replace by y f 1x. f 1 Note that both and real numbers. Check that f have domains and ranges that consist of the entire set of f f 1x x f 1 f x x. and Now try Exercise 55. 333202_0109.qxd 12/7/05 8:46 AM Page 98 98 Chapter 1 Functions and Their Graphs Example 7 Finding an Inverse Function Find the inverse function of f x 3x 13 −2 1 2 3 x −1 −2 −3 FIGURE 1.99 Solution The graph of Horizontal Line Test, you know that f is a curve, as shown in Figure 1.99. Because this graph passes the is one-to-one and has an inverse function. f f x 3x 1 y 3x 1 x 3y 1 x3 y 1 x3 1 y x 3 1 f 1x f 1 f Write original function. Replace f x by y. Interchange and x y. Cube each side. Solve for y. Replace by y f 1x. and Both numbers. You can verify this result numerically as shown in the tables below. have domains and ranges that consist of the entire set of real x 28 9 2 1 0 7 26 f x 3 2 1 0 1 2 3 Now try Exercise 61 1x 28 9 2 1 0 7 26 W RITING ABOUT MATHEMATICS The Existence of an Inverse Function Write a short paragraph describing why the following functions do or do not have inverse functions. x f x represent the retail price of an item (in dollars), and represent the sales tax on the item. Assume that a. Let let the sales tax is 6% of the retail price and that the sales tax is rounded to the nearest cent. Does this function have an inverse function? (Hint: Can you undo this function? For instance, if you know that the sales tax is $0.12, can you determine exactly what the retail price is?) x f x represent the temperature in degrees Celsius, and represent the temperature in degrees Fahrenheit. b. Let let Does this function have an inverse function? (Hint: The formula for converting from degrees Celsius to degrees ) Fahrenheit is F 9 5 C 32. 333202_0109.qxd 12/7/05 8:46 AM Page 99 1.9 Exercises Section 1.9 Inverse Functions 99 g is the ________ function of f. VOCABULARY CHECK: Fill in the blanks. 1. If the composite functions and 2. The domain of f fgx x is the ________ of f 1, g fx x and the ________ of then the function f 1 is the range of f. 3. The graphs of and f f 1 are reflections of each other in the line ________. 4. A function f is ________ if each value of the dependent variable corresponds to exactly one value of the independent variable. 5. A graphical test for the existence of an inverse function of f is called the _______ Line Test. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 8, find the inverse function of Verify that f 1f x x. f f 1x x and f informally. 11. y 12. 1. 3. 5. 7. f x 6x f x x 9 f x 3x 1 f x 3x 2. 4. 6. 8 x5 In Exercises 9–12, match the graph of the function with the graph of its inverse function. [The graphs of the inverse functions are labeled (a), (b), (c), and (d).] (a) y (b) y 4 3 2 1 (c) 91 1 2 3 −2 y 4 3 2 1 −2 −d) y 3 2 1 −2 −3 1 2 3 −3 −2 103 −3 f In Exercises 13–24, show that and are inverse functions (a) algebraically and (b) graphically. g 13. f x 2x, 14. f x x 5, 15. f x 7x 1, 16. f x 3 4x, 17. 18. 19. 20. 21. 22. 23. 24. , f x x3 , f x 1 x3, f x 9 x 2, x ≥ 0, gx x 2 gx x 5 gx x 1 7 gx 3 x 4 gx 38x gx 1 x gx x2 4, gx 31 x gx 9 x, gx 1 x x gx 5x 1 x 1 gx 2x 3 x 1 , 333202_0109.qxd 12/7/05 8:46 AM Page 100 100 Chapter 1 Functions and Their Graphs In Exercises 25 and 26, does the function have an inverse function? 25. 26 10 10 In Exercises 27 and 28, use the table of values for to complete a table for y f 1x. y f x 27. x f x 28. x 2 2 3 f x 10 In Exercises 29–32, does the function have an inverse function? 29. y 30. y 6 4 2 −2 2 4 6 31. y 2 −2 2 −2 6 2 x x −4 −2 −2 2 4 32. y 4 2 −2 2 4 6 x x In Exercises 33–38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. 34. f x 10 gx 4 x 6 hx x 4 x 4 gx x 53 f x 2x16 x2 f x 1 x 22 1 8 33. 35. 36. 37. 38. f, In Exercises 39–54, (a) find the inverse function of on the same set of coordinate axes, (b) graph both and (c) describe the relationship between the graphs of and 42. 40. , and (d) state the domain and range of and f f x
2x 3 f x 3x 1 f x x5 2 f x x3 1 f x x f x x 2, f x 4 x2, f x x2 2 44 50. 48. f x 3x 1 f x 6x 4 4x 5 52. 54. f x x35 f x 8x 4 2x 6 39. 41. 43. 45. 46. 47. 49. 51. 53. In Exercises 55–68, determine whether the function has an inverse function. If it does, find the inverse function. 55. f x x4 57. gx x 8 59. px 4 56. f x 1 x 2 58. f x 3x 5 60. f x 3x 4 5 x ≥ 3 62. x < 0 x ≥ 0 64. qx x 52 f x x, x2 3x, x ≤ 0 x > 0 61. 63. 65. f x x 32, f x x 3, 6 x, hx 4 x2 y 1 1 −1 −2 67. f x 2x 3 y 4 3 2 1 −4 −3 −2 −1 1 2 66. f x x 2, x ≤ 2 y 4 2 1 −1 −2 −3 −4 1 2 3 4 5 6 682 −2 −2 333202_0109.qxd 12/7/05 8:46 AM Page 101 Section 1.9 Inverse Functions 101 In Exercises 69–74, use the f x 1 and function. gx x 3 8x 3 functions given by to find the indicated value or 69. 71. 73. f 1 g11 f 1 f 16 ( f g)1 70. 72. 74. g1 f 13 g1 g14 g1 f 1 80. Digital Camera Sales The factory sales (in millions of dollars) of digital cameras in the United States from 1998 through 2003 are shown in the table. The time (in years) is given by with (Source: Consumer Electronincs Association) corresponding to 1998. t 8 t, f f x x 4 Year, t Sales, f t In Exercises 75–78, use the functions given by and to find the specified function. gx 2x 5 75. 77. f 1 g1 f g1 76. 78. f 1 g1 g f 1 Model It 79. U.S. Households The numbers of households (in thousands) in the United States from 1995 to 2003 are t, shown in the table. The time (in years) is given by t 5 (Source: U.S. with corresponding to 1995. Census Bureau) f 8 9 10 11 12 13 519 1209 1825 1972 2794 3421 exist? f 1 exists, what does it represent in the context of the (b) If (a) Does f 1 problem? f 1 (c) If exists, find f 11825. Year, t Households, f t (d) If the table was extended to 2004 and if the factory sales of digital cameras for that year was $2794 exist? Explain. million, would f1 5 6 7 8 9 10 11 12 13 98,990 99,627 101,018 102,528 103,874 104,705 108,209 109,297 111,278 (a) Find f 1108,209. f 1 (b) What does mean in the context of the problem? (c) Use the regression feature of a graphing utility to y mx b. find a linear model for the data, (Round to two decimal places.) and m b (d) Algebraically find the inverse function of the linear model in part (c). (e) Use the inverse function of the linear model you found in part (d) to approximate f 1117, 022. (f) Use the inverse function of the linear model you found in part (d) to approximate How does this value compare with the original data shown in the table? f1108,209. 81. Miles Traveled The total numbers (in billions) of miles traveled by motor vehicles in the United States from 1995 through 2002 are shown in the table. The time t, (in years) is given by with corresponding to 1995. (Source: U.S. Federal Highway Administration) t 5 f Year, t Miles traveled, f t 5 6 7 8 9 10 11 12 2423 2486 2562 2632 2691 2747 2797 2856 exist? f 1 exists, what does it mean in the context of the (b) If (a) Does f 1 problem? f 1 (c) If exists, find f 12632. (d) If the table was extended to 2003 and if the total number of miles traveled by motor vehicles for that year was 2747 billion, would exist? Explain. f 1 333202_0109.qxd 12/7/05 8:46 AM Page 102 102 Chapter 1 Functions and Their Graphs 82. Hourly Wage Your wage is $8.00 per hour plus $0.75 for each unit produced per hour. So, your hourly wage in terms of the number of units produced is y y 8 0.75x. (a) Find the inverse function. (b) What does each variable represent in the inverse function? (c) Determine the number of units produced when your hourly wage is $22.25. 83. Diesel Mechanics The function given by y 0.03x 2 245.50, 0 < x < 100 approximates the exhaust temperature Fahrenheit, where in degrees is the percent load for a diesel engine. x y (a) Find the inverse function. What does each variable represent in the inverse function? (b) Use a graphing utility to graph the inverse function. (c) The exhaust temperature of the engine must not exceed 500 degrees Fahrenheit. What is the percent load interval? 84. Cost You need a total of 50 pounds of two types of ground beef costing $1.25 and $1.60 per pound, respectively. A model for the total cost of the two types of beef is y 1.25x 1.6050 x y x where ground beef. is the number of pounds of the less expensive (a) Find the inverse function of the cost function. What does each variable represent in the inverse function? (b) Use the context of the problem to determine the domain of the inverse function. (c) Determine the number of pounds of the less expensive ground beef purchased when the total cost is $73. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 85 and 86, determine whether 85. If f is an even function, exists. 86. If the inverse function of exists and the graph of f f 1. is an -intercept of y y -intercept, the -intercept of x f has a f 1 f In Exercises 89– 92, use the graph of the function to create a table of values for the given points. Then create a second table that can be used to find , and sketch the graph of f 1 if possible. f 1 f 89. y 90 91. 4 f −2 −4 −2 2 4 x 92. y f x 4 −4 −2 −2 −4 93. Think About It The function given by f x k2 x x 3 has an inverse function, and f 1(3) 2. Find k. 94. Think About It The function given by f x kx3 3x 4 has an inverse function, and f 1(5) 2. Find k. Skills Review In Exercises 95–102, solve the equation using any convenient method. 95. 96. 97. 98. 99. 100. 101. 102. x 2 64 x 52 8 4x 2 12x 9 0 9x 2 12x 3 0 x 2 6x 4 0 2x 2 4x 6 0 50 5x 3x 2 2x 2 4x 9 2x 12 87. Proof Prove that if and are one-to-one functions, then g f g1x g1 f 1x. f 88. Proof Prove that if f is an odd function. f 1 is a one-to-one odd function, then 103. Find two consecutive positive even integers whose product is 288. 104. Geometry A triangular sign has a height that is twice its base. The area of the sign is 10 square feet. Find the base and height of the sign. 333202_0110.qxd_pg 103 1/9/06 8:52 AM Page 103 Section 1.10 Mathematical Modeling and Variation 103 1.10 Mathematical Modeling and Variation What you should learn • Use mathematical models to approximate sets of data points. • Use the regression feature of a graphing utility to find the equation of a least squares regression line. • Write mathematical models for direct variation. • Write mathematical models for direct variation as an nth power. • Write mathematical models for inverse variation. • Write mathematical models for joint variation. Why you should learn it You can use functions as models to represent a wide variety of real-life data sets. For instance, in Exercise 71 on page 113, a variation model can be used to model the water temperature of the ocean at various depths. U.S. Banks y y = −0.283t + 11.14 11 10 ( Introduction You have already studied some techniques for fitting models to data. For instance, in Section 1.3, you learned how to find the equation of a line that passes through two points. In this section, you will study other techniques for fitting models to data: least squares regression and direct and inverse variation. The resulting models are either polynomial functions or rational functions. (Rational functions will be studied in Chapter 2.) Example 1 A Mathematical Model The numbers of insured commercial banks for the years 1996 to 2001 are shown in the table. Insurance Corporation) y (in thousands) in the United States (Source: Federal Deposit Year 1996 1997 1998 1999 2000 2001 Insured commercial banks, y 9.53 9.14 8.77 8.58 8.32 8.08 t y 0.283t 11.14 where is the year, with for A linear model that approximates the data is t 6 6 ≤ t ≤ 11, corresponding to 1996. Plot the actual data and the model on the same graph. How closely does the model represent the data? Solution The actual data are plotted in Figure 1.100, along with the graph of the linear model. From the graph, it appears that the model is a “good fit” for the actual data. You can see how well the model fits by comparing the actual values of with y* the values of given by the model. The values given by the model are labeled in the table below. y y t y 6 7 8 9 10 11 9.53 9.14 8.77 8.58 8.32 8.08 y* 9.44 9.16 8.88 8.59 8.31 8.03 6 FIGURE 1.100 7 10 8 Year (6 ↔ 1996) 9 t 11 Now try Exercise 1. Note in Example 1 that you could have chosen any two points to find a line that fits the data. However, the given linear model was found using the regression feature of a graphing utility and is the line that best fits the data. This concept of a “best-fitting” line is discussed on the next page. 333202_0110.qxd 12/7/05 2:49 PM Page 104 104 Chapter 1 Functions and Their Graphs Least Squares Regression and Graphing Utilities So far in this text, you have worked with many different types of mathematical models that approximate real-life data. In some instances the model was given (as in Example 1), whereas in other instances you were asked to find the model using simple algebraic techniques or a graphing utility. To find a model that approximates the data most accurately, statisticians use a measure called the sum of square differences, which is the sum of the squares of the differences between actual data values and model values. The “bestfitting” linear model, called the least squares regression line, is the one with the least sum of square differences. Recall that you can approximate this line visually by plotting the data points and drawing the line that appears to fit best—or you can enter the data points into a calculator or computer and use the linear regression feature of the calculator or computer. When you use the regression feauture of a graphing calculator or computer program, you will notice that the program may also output an “ -value.” This -value is the correlation coefficient of the data and gives a measure of how well the model fits the data. The closer the value of is to 1, the better the fit. r r r Example 2 Finding a Least Squares Regression Line p The amounts (in millions of dollars) of total annual prize money awarded at the Indianapolis 500 race from 1995 to 2004
are shown in the table. Construct a scatter plot that represents the data and find the least squares regression line for the data. (Source: indy500.com) Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Prize money, p 8.06 8.11 8.61 8.72 9.05 9.48 9.61 10.03 10.15 10.25 Solution t 5 represent 1995. The scatter plot for the points is shown in Figure 1.101. Let Using the regression feature of a graphing utility, you can determine that the equation of the least squares regression line is p 0.268t 6.66. To check this model, compare the actual -values with the -values given by the model, which are labeled in the table at the left. The correlation coefficient for this model is which implies that the model is a good fit. p* r 0.991, p p Now try Exercise 7. Indianapolis 500 p 11 10 FIGURE 1.101 5 6 7 8 9 10 11 12 13 14 Year (5 ↔ 1995) t t 5 6 7 8 9 10 11 12 13 14 p p* 8.06 8.11 8.61 8.72 9.05 9.48 9.61 10.03 10.15 10.25 8.00 8.27 8.54 8.80 9.07 9.34 9.61 9.88 10.14 10.41 333202_0110.qxd 12/7/05 8:47 AM Page 105 Section 1.10 Mathematical Modeling and Variation 105 Direct Variation There are two basic types of linear models. The more general model has a y -intercept that is nonzero. b 0 y mx b, The simpler model y kx y has a -intercept that is zero. In the simpler model, x, or to be directly proportional to x. y is said to vary directly as Direct Variation The following statements are equivalent. 1. y varies directly as x. 2. 3. y y kx is directly proportional to x. for some nonzero constant k. k is the constant of variation or the constant of proportionality. Example 3 Direct Variation In Pennsylvania, the state income tax is directly proportional to gross income. You are working in Pennsylvania and your state income tax deduction is $46.05 for a gross monthly income of $1500. Find a mathematical model that gives the Pennsylvania state income tax in terms of gross income. Solution Verbal Model: Labels: Equation: State income tax k Gross income State income tax y Gross income x Income tax rate k y kx (dollars) (dollars) (percent in decimal form) substitute the given information into the equation y kx, and then k, To solve for k. solve for y kx 46.05 k1500 0.0307 k Write direct variation model. Substitute y 46.05 and x 1500. Simplify. So, the equation (or model) for state income tax in Pennsylvania is x y 0.0307x. In other words, Pennsylvania has a state income tax rate of 3.07% of gross income. The graph of this equation is shown in Figure 1.102. Now try Exercise 33. Pennsylvania Taxes y y = 0.0307x (1500, 46.05) 100 80 60 40 20 ) 1000 2000 3000 4000 Gross income (in dollars) FIGURE 1.102 333202_0110.qxd 12/7/05 8:47 AM Page 106 106 Chapter 1 Functions and Their Graphs Direct Variation as an nth Power Another type of direct variation relates one variable to a power of another variable. For example, in the formula for the area of a circle A r2 the area formula, A is directly proportional to the square of the radius Note that for this is the constant of proportionality. r. Note that the direct variation model of with n 1. is a special case y kx n y kx Direct Variation as an nth Power The following statements are equivalent. 1. y varies directly as the nth power of x. is directly proportional to the nth power of x. 2. 3. y y kx n for some constant k. t = 0 sec t = 1 sec 10 20 30 FIGURE 1.103 Example 4 Direct Variation as nth Power The distance a ball rolls down an inclined plane is directly proportional to the square of the time it rolls. During the first second, the ball rolls 8 feet. (See Figure 1.103.) 40 50 60 t = 3 sec 70 a. Write an equation relating the distance traveled to the time. b. How far will the ball roll during the first 3 seconds? Solution a. Letting be the distance (in feet) the ball rolls and letting be the time (in t d seconds), you have d kt 2. when t 1, you can see that k 8, as follows. d 8 Now, because d kt 2 8 k12 8 k So, the equation relating distance to time is d 8t 2. t 3, b. When the distance traveled is d 832 89 72 feet. Now try Exercise 63. d 1 In Examples 3 and 4, the direct variations are such that an increase in one variable corresponds to an increase in the other variable. This is also true in the model results in an increase in You should not, however, assume that this always occurs with direct variation. For example, in the model and an increase in yet y 3x, is said to vary directly as x. results in a decrease in where an increase in 5F, F > 0, d. y, F x y 333202_0110.qxd 12/7/05 8:47 AM Page 107 Section 1.10 Mathematical Modeling and Variation 107 Inverse Variation Inverse Variation The following statements are equivalent. 1. 3. x. 2. y is inversely proportional to x. y varies inversely as y k x for some constant k. x y If and are related by an equation of the form as the th power of (or n y x is inversely proportional to the th power of ). y then varies inversely n x y kx n, Some applications of variation involve problems with both direct and inverse variation in the same model. These types of models are said to have combined variation. Example 5 Direct and Inverse Variation P1 V1 A gas law states that the volume of an enclosed gas varies directly as the temperature and inversely as the pressure, as shown in Figure 1.104. The pres294 sure of a gas is 0.75 kilogram per square centimeter when the temperature is K and the volume is 8000 cubic centimeters. (a) Write an equation relating pressure, temperature, and volume. (b) Find the pressure when the temperature is 300 K and the volume is 7000 cubic centimeters. P2 V2 P2 > P1 then < V2 V1 If the temperature is held FIGURE 1.104 constant and pressure increases, volume decreases. Solution V a. Let be volume (in cubic centimeters), let P be pressure (in kilograms per varies V be temperature (in Kelvin). Because P, you have T and inversely as square centimeter), and let T directly as V kT P . Now, because when T 294 and V 8000, you have P 0.75 8000 k294 0.75 k 6000 294 1000 49 . So, the equation relating pressure, temperature, and volume is b. When the pressure is . T P V 1000 49 T 300 P 1000 49 and 300 7000 V 7000, 300 343 Now try Exercise 65. 0.87 kilogram per square centimeter. 333202_0110.qxd 12/7/05 8:47 AM Page 108 108 Chapter 1 Functions and Their Graphs Joint Variation In Example 5, note that when a direct variation and an inverse variation occur in the same statement, they are coupled with the word “and.” To describe two different direct variations in the same statement, the word jointly is used. Joint Variation The following statements are equivalent. 1. z varies jointly as and x y. is jointly proportional to and x y. for some constant k. z z kxy 2. 3. If x, and are related by an equation of the form z y, z kx ny m then varies jointly as the th power of and the n x z m th power of y. Example 6 Joint Variation The simple interest for a certain savings account is jointly proportional to the time and the principal. After one quarter (3 months), the interest on a principal of $5000 is $43.75. a. Write an equation relating the interest, principal, and time. b. Find the interest after three quarters. Solution a. Let I interest (in dollars), P is jointly proportional to principal (in dollars), and t, and you have P t time (in I years). Because I kPt. I 43.75, P 5000, 43.75 k50001 4 For and t 1 4, you have which implies that interest, principal, and time is k 443.755000 0.035. So, the equation relating I 0.035Pt which is the familiar equation for simple interest where the constant of proportionality, 0.035, represents an annual interest rate of 3.5%. b. When P $5000 and I 0.03550003 4 t 3 4, the interest is $131.25. Now try Exercise 67. 333202_0110.qxd 12/7/05 8:47 AM Page 109 Section 1.10 Mathematical Modeling and Variation 109 1.10 Exercises VOCABULARY CHECK: Fill in the blanks. 1. Two techniques for fitting models to data are called direct ________ and least squares ________. 2. Statisticians use a measure called ________ of________ ________ to find a model that approximates a set of data most accurately. 3. An -value of a set of data, also called a ________ ________, gives a measure of how well a model fits a set of data. r 4. Direct variation models can be described as varies directly as x, or y is ________ ________ to x. 5. In direct variation models of the form k is called the ________ of ________. y y kx, 6. The direct variation model y kxn or y is ________ ________ to the th power of y k x 7. The mathematical model is an example of ________ variation. can be described as varies directly as the th power of n x. n y x, 8. Mathematical models that involve both direct and inverse variation are said to have ________ variation. 9. The joint variation model z kxy x y. is ________ ________ to and can be described as varies jointly as and x z y, or z PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1998, 137,673 1999, 139,368 2000, 142,583 2001, 143,734 2002, 144,683 1. Employment The total numbers of employees (in thousands) in the United States from 1992 to 2002 are given by the following ordered pairs. 1992, 128,105 1993, 129,200 1994, 131,056 1995, 132,304 1996, 133,943 1997, 136,297 A is linear model y 1767.0t 123,916, represents the number of where employees (in thousands) and represents 1992. Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? (Source: U.S. Bureau of Labor Statistics) that approximates y t 2 the data 2. Sports The winning times (in minutes) in the women’s 400-meter freestyle swimming event in the Olympics from 1948 to 2004 are given by the following ordered pairs. 1948, 5.30 1952, 5.20 1956, 4.91 1960, 4.84 1964, 4.72 1968, 4.53 1972, 4.32 1976, 4.16 1980, 4.15 1984, 4.12 1988, 4.06 1992, 4.12 1996, 4.12 2000, 4.10 2004, 4.09 y the data A is that approximates linear model y 0.022t 5.03, where represents the winning time t 0 (in minutes) and represents 1950. Plot the actual data and the mo
del on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: The World Almanac and Book of Facts) In Exercises 3– 6, sketch the line that you think best approximates the data in the scatter plot.Then find an equation of the line. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 3. y 4 333202_0110.qxd 12/7/05 8:47 AM Page 110 110 Chapter 1 Functions and Their Graphs 7. Sports The lengths (in feet) of the winning men’s discus throws in the Olympics from 1912 to 2004 are listed below. (Source: The World Almanac and Book of Facts) 1912 148.3 1920 146.6 1924 151.3 1928 155.3 1932 162.3 1936 165.6 1948 173.2 1952 180.5 1956 184.9 1960 194.2 1964 200.1 1968 212.5 1972 211.3 1976 221.5 1980 218.7 1984 218.5 1988 225.8 1992 213.7 1996 227.7 2000 227.3 2004 229.3 (a) Sketch a scatter plot of the data. Let represent the length of the winning discus throw (in feet) and let t 12 represent 1912. y (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the winning men’s discus throw in the year 2008. (f) Use your school’s library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e). 8. Revenue The total revenues (in millions of dollars) for Outback Steakhouse from 1995 to 2003 are listed below. (Source: Outback Steakhouse, Inc.) 1995 664.0 1996 937.4 1997 1151.6 1998 1358.9 1999 1646.0 2000 1906.0 2001 2127.0 2002 2362.1 2003 2744.4 (a) Sketch a scatter plot of the data. Let y revenue (in millions of dollars) and let 1995. represent the total represent t 5 (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the revenues of Outback Steakhouse in 2005. (f) Use your school’s library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e). 9. Data Analysis: Broadway Shows The table shows the S annual gross ticket sales (in millions of dollars) for Broadway shows in New York City from 1995 through 2004. (Source: The League of American Theatres and Producers, Inc.) Year Sales, S 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 406 436 499 558 588 603 666 643 721 771 (a) Use a graphing utility to create a scatter plot of the data. Let t 5 represent 1995. (b) Use the regression feature of a graphing utility to find the equation of the least squares regression line that fits the data. (c) Use the graphing utility to graph the scatter plot you found in part (a) and the model you found in part (b) in the same viewing window. How closely does the model represent the data? (d) Use the model to estimate the annual gross ticket sales in 2005 and 2007. (e) Interpret the meaning of the slope of the linear model in the context of the problem. x 10. Data Analysis: Television Households The table shows (in millions) of households with cable televithe numbers sion and the numbers (in millions) of households with color television sets in the United States from 1995 through 2002. (Source: Nielson Media Research; Television Bureau of Advertising, Inc.) y Households with cable, x Households with color TV, y 63 65 66 67 75 77 80 86 94 95 97 98 99 101 102 105 333202_0110.qxd 12/7/05 8:47 AM Page 111 Section 1.10 Mathematical Modeling and Variation 111 (a) Use the regression feature of a graphing utility to find the equation of the least squares regression line that fits the data. (b) Use the graphing utility to create a scatter plot of the data. Then graph the model you found in part (a) and the scatter plot in the same viewing window. How closely does the model represent the data? (c) Use the model to estimate the number of households with color television sets if the number of households with cable television is 90 million. (d) Interpret the meaning of the slope of the linear model in the context of the problem. Think About It In Exercises 11 and 12, use the graph to determine whether varies directly as some power of or inversely as some power of Explain. x. y x 11. y 12 In Exercises 13–16, use the given value of k y kx2. table for the direct variation model on a rectangular coordinate system. to complete the Plot the points x 2 4 6 8 10 y kx2 13. 15. k 1 k 1 2 14. 16. k 2 k 1 4 In Exercises 17–20, use the given value of k table for the inverse variation model to complete the y k x 2. Plot the points on a rectangular coordinate system. 2 4 6 8 10 x y k x2 17. 19. k 2 k 10 18. 20. k 5 k 20 In Exercises 21–24, determine whether the variation model y k/x is of the form , and find y kx or k. 21. 22. 23. 24 10 15 20 25 1 2 1 3 1 4 1 5 10 15 20 4 6 8 25 10 5 3.5 10 7 15 20 25 10.5 14 17.5 5 10 15 20 25 24 12 8 6 24 5 Direct Variation In Exercises 25–28, assume that x. directly proportional to y x -value to find a linear model that relates and is -value and x. Use the given y y 25. 26. 27. 28. x 5, x 2, x 10, x 6, y 12 y 14 y 2050 y 580 29. Simple Interest The simple interest on an investment is directly proportional to the amount of the investment. By investing $2500 in a certain bond issue, you obtained an interest payment of $87.50 after 1 year. Find a mathematical model that gives the interest for this bond issue after 1 year in terms of the amount invested P. I 30. Simple Interest The simple interest on an investment is directly proportional to the amount of the investment. By investing $5000 in a municipal bond, you obtained an interest payment of $187.50 after 1 year. Find a mathematfor this municipal bond ical model that gives the interest P. after 1 year in terms of the amount invested I 31. Measurement On a yardstick with scales in inches and centimeters, you notice that 13 inches is approximately the same length as 33 centimeters. Use this information to find a mathematical model that relates centimeters to inches. Then use the model to find the numbers of centimeters in 10 inches and 20 inches. 32. Measurement When buying gasoline, you notice that 14 gallons of gasoline is approximately the same amount of gasoline as 53 liters. Then use this information to find a linear model that relates gallons to liters. Then use the model to find the numbers of liters in 5 gallons and 25 gallons. 333202_0110.qxd 12/7/05 8:47 AM Page 112 112 Chapter 1 Functions and Their Graphs 33. Taxes Property tax is based on the assessed value of a property. A house that has an assessed value of $150,000 has a property tax of $5520. Find a mathematical model that gives the amount of property tax in terms of the assessed value of the property. Use the model to find the property tax on a house that has an assessed value of $200,000. y x 8 ft 34. Taxes State sales tax is based on retail price. An item that sells for $145.99 has a sales tax of $10.22. Find a in mathematical model that gives the amount of sales tax x. terms of the retail price Use the model to find the sales tax on a $540.50 purchase. y Hooke’s Law In Exercises 35–38, use Hooke’s Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. 35. A force of 265 newtons stretches a spring 0.15 meter (see figure). FIGURE FOR 38 In Exercises 39–48, find a mathematical model for the verbal statement. 39. A varies directly as the square of r. 40. V varies directly as the cube of e. 41. y varies inversely as the square of x. Equilibrium 0.15 meter 265 newtons (a) How far will a force of 90 newtons stretch the spring? (b) What force is required to stretch the spring 0.1 meter? 36. A force of 220 newtons stretches a spring 0.12 meter. What force is required to stretch the spring 0.16 meter? 37. The coiled spring of a toy supports the weight of a child. The spring is compressed a distance of 1.9 inches by the weight of a 25-pound child. The toy will not work properly if its spring is compressed more than 3 inches. What is the weight of the heaviest child who should be allowed to use the toy? 38. An overhead garage door has two springs, one on each side of the door (see figure). A force of 15 pounds is required to stretch each spring 1 foot. Because of a pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the combined lifting force applied to the door by the springs when the door is closed. 42. h varies inversely as the square root of 43. F g varies directly as and inversely as s. r 2. 44. z is jointly proportional to the square of and the cube of x y. 45. Boyle’s Law: For a constant temperature, the pressure of a gas is inversely proportional to the volume of the gas. V P 46. Newton’s Law of Cooling: The rate of change R of the temperature of an object is proportional to the T difference between the temperature of the object and the of the environment in which the object is temperature placed. Te F 47. Newton’s Law of Universal Gravitation: The gravitationis al attraction between two objects of masses proportional to the product of the masses and inversely proportional to the square of the distance between the objects. and m2 m1 r 48. Logistic Growth: The rate of growth of a population is jointly proportional to the size of the population and the L difference between and the maximum population size that the environment can support. R S
S In Exercises 49–54, write a sentence using the variation terminology of this section to describe the formula. 49. Area of a triangle: A 1 50. Surface area of a sphere: 2bh 51. Volume of a sphere: S 4r 2 r3 V 4 3 52. Volume of a right circular cylinder: V r 2h 53. Average speed: r d t 54. Free vibrations: kg W 333202_0110.qxd 12/7/05 8:47 AM Page 113 Section 1.10 Mathematical Modeling and Variation 113 In Exercises 55–62, find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) when r 3. x 25. when x 4 and the third power of when y 7 when x 4. and y 8. s. 55. A 56. 57. 58. 59. 60. 61. 62. y y z x x. r 2. varies directly as varies jointly as and varies inversely as A 9 y 3 x. is inversely proportional to z 64 r s 3. is jointly proportional to and r 11 x varies directly as y 9. x 42 and when varies directly as the square of when y. 3 F F 4158 P P 28 z z 6 v v 1.5 x and inversely as y. and when y 4. q varies jointly as and and inversely as the square of q 6.3, x 6 p p 4.1, s 1.2. when and s. and inversely as the square of y. Ecology In Exercises 63 and 64, use the fact that the diameter of the largest particle that can be moved by a stream varies approximately directly as the square of the velocity of the stream. 63. A stream with a velocity of mile per hour can move coarse sand particles about 0.02 inch in diameter. Approximate the velocity required to carry particles 0.12 inch in diameter. 1 4 64. A stream of velocity can move particles of diameter or d increase when the velocity is d v less. By what factor does doubled? Resistance In Exercises 65 and 66, use the fact that the resistance of a wire carrying an electrical current is directly proportional to its length and inversely proportional to its cross-sectional area. 65. If #28 copper wire (which has a diameter of 0.0126 inch) has a resistance of 66.17 ohms per thousand feet, what length of #28 copper wire will produce a resistance of 33.5 ohms? 66. A 14-foot piece of copper wire produces a resistance of 0.05 ohm. Use the constant of proportionality from Exercise 65 to find the diameter of the wire. 67. Work The work W (in joules) done when lifting an object (in kilograms) of the object varies jointly with the mass and the height (in meters) that the object is lifted. The work done when a 120-kilogram object is lifted 1.8 meters is 2116.8 joules. How much work is done when lifting a 100-kilogram object 1.5 meters? m h 68. Spending The prices of three sizes of pizza at a pizza shop are as follows. 9-inch: $8.78, 12-inch: $11.78, 15-inch: $14.18 You would expect that the price of a certain size of pizza would be directly proportional to its surface area. Is that the case for this pizza shop? If not, which size of pizza is the best buy? 69. Fluid Flow The velocity of a fluid flowing in a conduit v is inversely proportional to the cross-sectional area of the conduit. (Assume that the volume of the flow per unit of time is held constant.) Determine the change in the velocity of water flowing from a hose when a person places a finger over the end of the hose to decrease its cross-sectional area by 25%. 70. Beam Load The maximum load that can be safely supported by a horizontal beam varies jointly as the width of the beam and the square of its depth, and inversely as the length of the beam. Determine the changes in the maximum safe load under the following conditions. (a) The width and length of the beam are doubled. (b) The width and depth of the beam are doubled. (c) All three of the dimensions are doubled. (d) The depth of the beam is halved. Model It 71. Data Analysis: Ocean Temperatures An oceanographer took readings of the water temperatures (in d (in meters). The degrees Celsius) at several depths data collected are shown in the table. C Depth, d Temperature, C 1000 2000 3000 4000 5000 4.2 1.9 1.4 1.2 0.9 (a) Sketch a scatter plot of the data. (b) Does it appear that the data can be modeled by the for C kd? If so, find k inverse variation model each pair of coordinates. k (c) Determine the mean value of C kd. the inverse variation model from part (b) to find (d) Use a graphing utility to plot the data points and the inverse model in part (c). (e) Use the model to approximate the depth at which the water temperature is C. 3 333202_0110.qxd 12/7/05 8:47 AM Page 114 114 Chapter 1 Functions and Their Graphs 72. Data Analysis: Physics Experiment An experiment in a physics lab requires a student to measure the compressed lengths (in centimeters) of a spring when various forces F of pounds are applied. The data are shown in the table. y Force, F Length, y 0 2 4 6 8 10 12 0 1.15 2.3 3.45 4.6 5.75 6.9 (a) Sketch a scatter plot of the data. (b) Does it appear that the data can be modeled by Hooke’s Law? If so, estimate k. (See Exercises 35–38.) (c) Use the model in part (b) to approximate the force required to compress the spring 9 centimeters. 73. Data Analysis: Light Intensity A light probe is located x (in centimeters from a light source, and the intensity microwatts per square centimeter) of the light is measured. The results are shown as ordered pairs 30, 0.1881 42, 0.0998 A model for the data is x, y. 38, 0.1172 50, 0.0645 34, 0.1543 46, 0.0775 y 262.76x 2.12. y (a) Use a graphing utility to plot the data points and the model in the same viewing window. (b) Use the model to approximate the light intensity 25 centimeters from the light source. 74. Illumination The illumination from a light source varies inversely as the square of the distance from the light source. When the distance from a light source is doubled, how does the illumination change? Discuss this model in terms of the data given in Exercise 73. Give a possible explanation of the difference. Synthesis 78. Discuss how well the data shown in each scatter plot can be approximated by a linear model. (a) y (bcd 79. Writing A linear mathematical model for predicting prize winnings at a race is based on data for 3 years. Write a paragraph discussing the potential accuracy or inaccuracy of such a model. 80. Research Project Use your school’s library, the Internet, or some other reference source to find data that you think describe a linear relationship. Create a scatter plot of the data and find the least squares regression line that represents the data points. Interpret the slope and -intercept in the context of the data. Write a summary of your findings. y Skills Review In Exercises 81– 84, solve the inequality and graph the solution on the real number line. 81. 82. 83. 3x 2 > 17 7x 10 ≤ 1 x 2x 1 < 9 84. 4 3x 7 ≥ 12 In Exercises 85 and 86, evaluate the function at each value of the independent variable and simplify. 85. f x x2 5 x 3 True or False? statement is true or false. Justify your answer. In Exercises 75–77, decide whether the 75. If varies directly as y as well. x, then if x increases, will increase y 86. (a) f 0 (b) f 3 (c) f 4 f x x2 10, 6x2 1, f 1 (b) f 2 (a) x ≥ 2 x < 2 (c) f 8 76. In the equation for kinetic energy, E 1 is directly proportional to the mass the amount m 2m v2, of kinetic energy of an object and the square of its velocity E v. 77. If the correlation coefficient for a least squares regression the regression line cannot be used to 1, line is close to describe the data. 87. Make a Decision To work an extended application analyzing registered voters in United States, visit this text’s website at college.hmco.com. (Data Source: U.S. Census Bureau) 333202_010R.qxd 12/7/05 8:49 AM Page 115 1 2 Chapter Summary What did you learn? Section 1.1 Plot points on the Cartesian plane (p. 2). Use the Distance Formula to find the distance between two points (p. 4). Use the Midpoint Formula to find the midpoint of a line segment (p. 5). Use a coordinate plane and geometric formulas to model and solve real-life problems (p. 6). Section 1.2 Sketch graphs of equations (p. 14). Find x- and y-intercepts of graphs of equations (p. 17). Use symmetry to sketch graphs of equations (p. 18). Find equations of and sketch graphs of circles (p. 20). Use graphs of equations in solving real-life problems (p. 21). Section 1.3 Use slope to graph linear equations in two variables (p. 25). Find slopes of lines (p. 27). Write linear equations in two variables (p. 29). Use slope to identify parallel and perpendicular lines (p. 30). Use slope and linear equations in two variables to model and solve real-life problems (p. 31). Section 1.4 Determine whether relations between two variables are functions (p. 40). Use function notation and evaluate functions (p. 42). Find the domains of functions (p. 44). Use functions to model and solve real-life problems (p. 45). Evaluate difference quotients (p. 46). Section 1.5 Use the Vertical Line Test for functions (p. 54). Find the zeros of functions (p. 56). Determine intervals on which functions are increasing or decreasing and determine relative maximum and relative minimum values of functions (p. 57). Determine the average rate of change of a function (p. 59). Identify even and odd functions (p. 60). Chapter Summary 115 Review Exercises 1–4 5–8 5–8 9–14 15–24 25–28 29–36 37– 44 45, 46 47–50 51–54 55–62 63, 64 65, 66 67–70 71, 72 73–76 77, 78 79, 80 81–84 85–88 89–94 95–98 99–102 333202_010R.qxd 12/7/05 8:49 AM Page 116 116 Chapter 1 Functions and Their Graphs Section 1.6 Identify and graph linear, squaring (p. 66), cubic, square root, reciprocal (p. 68), step, and other piecewise-defined functions (p. 69). Recognize graphs of parent functions (p. 70). Section 1.7 Use vertical and horizontal shifts to sketch graphs of functions (p. 74). Use reflections to sketch graphs of functions (p. 76). Use nonrigid transformations to sketch graphs of functions (p. 78). Section 1.8 Add, subtract, multiply, and divide functions (p. 84). Find the composition of one function with another function (p. 86). Use combinations and compositions of functions to model and solve real-life problems (p. 88). Section 1.9 Find inverse functions infor
mally and verify that two functions are inverse functions of each other (p. 93). Use graphs of functions to determine whether functions have inverse functions (p. 95). Use the Horizontal Line Test to determine if functions are one-to-one (p. 96). Find inverse functions algebraically (p. 97). Section 1.10 Use mathematical models to approximate sets of data points (p. 103). Use the regression feature of a graphing utility to find the equation of a least squares regression line (p. 104). Write mathematical models for direct variation (p. 105). Write mathematical models for direct variation as an nth power (p. 106). Write mathematical models for inverse variation (p. 107). Write mathematical models for joint variation (p. 108). 103–114 115, 116 117–120 121–126 127–130 131, 132 133–136 137, 138 139, 140 141, 142 143–146 147–152 153 154 155 156, 157 158, 159 160 333202_010R.qxd 12/7/05 8:49 AM Page 117 1 Review Exercises In Exercises 1 and 2, plot the points in the Cartesian 1.1 plane. 1. 2. 2, 2, 0, 4, 3, 6, 1, 7 5, 0, 8, 1, 4, 2, 3, 3 In Exercises 3 and 4, determine the quadrant(s) in which x, y is located so that the condition(s) is (are) satisfied. 3. x > 0 and y 2 4. y > 0 In Exercises 5–8, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. (a) plot the points, 5. 6. 7. 8. 1, 5 4, 3 0, 8.2 3, 8, 2, 6, 5.6, 0, 0, 1.2, Review Exercises 117 13. Geometry The volume of a globe is about 47,712.94 cubic centimeters. Find the radius of the globe. 14. Geometry The volume of a rectangular package is 2304 cubic inches. The length of the package is 3 times its width, and the height is 1.5 times its width. (a) Draw a diagram that represents the problem. Label the height, width, and length accordingly. (b) Find the dimensions of the package. In Exercises 15–18, complete a table of values. Use 1.2 the solution points to sketch the graph of the equation. 15. 16. 17. 18. y 3x 5 y 1 2x 2 y x2 3x y 2x2 x 9 3.6, 0 In Exercises 19–24, sketch the graph by hand. In Exercises 9 and 10, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position. 9. Original coordinates of vertices: 4, 8, 6, 8, 4, 3, 6, 3 Shift: three units downward, two units to the left 10. Original coordinates of vertices: 0, 1, 3, 3, 0, 5, 3, 3 Shift: five units upward, four units to the left 11. Sales The Cheesecake Factory had annual sales of $539.1 million in 2001 and $773.8 million in 2003. Use the Midpoint Formula to estimate the sales in 2002. (Source: The Cheesecake Factory, Inc.) 12. Meteorology The apparent temperature is a measure of relative discomfort to a person from heat and high x (in humidity. The table shows the actual temperatures y degrees Fahrenheit) versus the apparent temperatures (in degrees Fahrenheit) for a relative humidity of 75%. x y 70 70 75 77 80 85 85 95 90 95 100 109 130 150 (a) Sketch a scatter plot of the data shown in the table. (b) Find the change in the apparent temperature when the actual temperature changes from 70F to 100F. 19. 20. 21. 22. 23. 24. y 2x 3 0 3x 2y 6 0 y 5 x y x 2 y 2x2 0 y x2 4x In Exercises 25–28, find the - and -intercepts of the graph of the equation. x y 25. 26. 27. 28. y 2x 7 y x 1 3 y x 32 4 y x4 x2 In Exercises 29–36, use the algebraic tests to check for symmetry with respect to both axes and the origin. Then sketch the graph of the equation. 32. 30. 31. 29. y 4x 1 y 5x 6 y 5 x2 y x2 10 y x3 3 y 6 x3 y x 5 35. 36. y x 9 33. 34. 333202_010R.qxd 12/7/05 8:49 AM Page 118 118 Chapter 1 Functions and Their Graphs In Exercises 37–42, find the center and radius of the circle and sketch its graph. 37. 38. 39. 40. 41. 42. x2 y2 9 x2 y2 4 x 22 y2 16 x2 y 82 81 x 1 x 42 y 3 2 y 12 36 2 100 2 2 which the endpoints of a diameter are 43. Find the standard form of the equation of the circle for 4, 6. 44. Find the standard form of the equation of the circle for and which the endpoints of a diameter are 4, 10. 2, 3 0, 0 and 45. Physics The force x spring inches from its natural length (see figure) is F (in pounds) required to stretch a F 5 4 x, 0 ≤ x ≤ 20. Natural length x in. F (a) Use the model to complete the table. x 0 4 8 12 16 20 Force, F (b) Sketch a graph of the model. (c) Use the graph to estimate the force necessary to stretch the spring 10 inches. 46. Number of Stores The numbers of Target stores for the years 1994 to 2003 can be approximated by the model N N 3.69t2 939, 4 ≤ t ≤ 13 t where 1994. is the time (in years), with (Source: Target Corp.) (a) Sketch a graph of the model. t 4 corresponding to (b) Use the graph to estimate the year in which the number of stores was 1300. In Exercises 47–50, find the slope and -intercept (if 1.3 possible) of the equation of the line. Sketch the line. y 47. 48. 49. 50. y 6 x 3 y 3x 13 y 10x 9 In Exercises 51–54, plot the points and find the slope of the line passing through the pair of points. 51. 52. 53. 54. 7, 1 6, 5 3, 4, 1, 8, 4.5, 6, 2.1, 3 3, 2, 8, 2 In Exercises 55–58, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope. Sketch the line. Point 0, 5 2, 6 10, 3 8, 5 55. 56. 57. 58. Slope m 3 2 m 0 m 1 2 m is undefined. In Exercises 59–62, find the slope-intercept form of the equation of the line passing through the points. 59. 60. 61. 62. 2, 1 0, 0, 0, 10 2, 5, 1, 4, 2, 0 11, 2, 6, 1 In Exercises 63 and 64, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. Point 3, 2 8, 3 63. 64. Line 5x 4y 8 2x 3y 5 Rate of Change In Exercises 65 and 66, you are given the dollar value of a product in 2006 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value of the product in terms of the year represent 2006.) t 6 (Let V t. 2006 Value Rate 65. $12,500 66. $72.95 $850 increase per year $5.15 increase per year 333202_010R.qxd 12/7/05 8:49 AM Page 119 Review Exercises 119 In Exercises 67–70, determine whether the equation 1.4 represents as a function of y x. 67. 68. 69. 70. 16x y 4 0 2x y 3 0 y 1 x y x 2 In Exercises 71 and 72, evaluate the function at each specified value of the independent variable and simplify. 81. y x 32 y 5 4 3 2 1 71. 72. (a) f x x 2 1 f 2 hx 2x 1, x2 2, h2 (a) (b) (b) f 4 x ≤ 1 x > 1 h1 (c) f t 2 (d) f t 1 −1 1 2 3 4 5 83. x 4 y2 (c) h0 (d) h2 1.5 In Exercises 81–84, use the Vertical Line Test to determine whether is a function of To print an enlarged copy of the graph, go to the website www.mathgraphs.com. x. y 82. y 3 5x 3 2x 1 y 1 −3 −2 −1 1 2 3 −2 −3 84. x 4 y y 10 8 4 2 −8 −4 −2 −4 2 4 8 73. In Exercises 73–76, find the domain of the function. Verify your result with a graph. f x 25 x 2 f x 3x 4 x x2 x 6 75. 74. h(x) h(t) t 1 76. 77. Physics The velocity of a ball projected upward from vt is the 32t 48, is the velocity in feet per second. where ground level is given by time in seconds and v t (a) Find the velocity when t 1. (b) Find the time when the ball reaches its maximum height. [Hint: Find the time when (c) Find the velocity when t 2. vt 0. ] 78. Mixture Problem From a full 50-liter container of a 40% liters is removed and replaced with x concentration of acid, 100% acid. (a) Write the amount of acid in the final mixture as a function of x. (b) Determine the domain and range of the function. (c) Determine x if the final mixture is 50% acid. In Exercises 79 and 80, find the difference quotient and simplify your answer. 79. f x 2x2 3x 1, 80. f x x3 5x2 x In Exercises 85– 88, find the zeros of the function algebraically. 85. 86. 87. fx 3x2 16x 21 fx 5x2 4x 1 f x 8x 3 11 x 88. fx x3 x 2 25x 25 In Exercises 89 and 90, determine the intervals over which the function is increasing, decreasing, or constant. f x x2 42 89. 90 20 8 4 −2 −1 21 3 x −2 −1 21 3 x 333202_010R.qxd 12/7/05 8:49 AM Page 120 120 Chapter 1 Functions and Their Graphs In Exercises 91–94, use a graphing utility to graph the function and approximate (to two decimal places) any relative minimum or relative maximum values. 91. 92. 93. 94. fx x2 2x 1 fx x4 4x2 2 fx x3 6x4 fx x3 4x2 x 1 In Exercises 95–98, find the average rate of change of the function from to x1 x2. Function fx x2 8x 4 fx x3 12x 2 fx 2 x 1 fx 1 x 3 95. 96. 97. 98. x -Values 0, x2 0, x2 3, x2 1, x2 4 4 7 6 x1 x1 x1 x1 In Exercises 99–102, determine whether the function is even, odd, or neither. 99. 100. 101. 102. f x x 5 4x 7 f x x 4 20x2 f x 2xx2 3 f x 56x2 1.6 In Exercises 103–104, write the linear function such that it has the indicated function values. Then sketch the graph of the function. f 2 6, f 0 5, f 1 3 f 4 8 104. 103. f In Exercises 105–114, graph the function. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. f x 3 x2 hx x3 2 f x x f x x 1 gx 3 x gx 1 x 5 f x x 2 gx x 4 f x 5x 3, 4x 5, f x x2 2, 5, 8x 5 In Exercises 115 and 116, the figure shows the graph of a transformed parent function. Identify the parent function. 115. y 116. y 10 8 6 4 2 8 6 4 2 −8 −4 −2 2 x −2 −2 2 4 6 8 x f f. h. h. 117. 118. 121. 120. 119. In Exercises 117–130, (c) Sketch the graph of in terms of 1.7 h is related to one of the parent functions described in this chapter. (a) Identify the parent f. (b) Describe the sequence of transformations function to from (d) Use function h notation to write hx x2 9 hx x 23 2 hx x 7 hx x 3 5 hx x 32 1 hx x 53 5 hx x 6 hx x 1 9 hx x 4 6 hx x 12 3 hx 5x 9 hx 1 hx 2x 4 2x 1 hx 1 126. 125. 128. 123. 127. 129. 130. 122. 124. 3 x 3 In Exercises 131 and 132, find (a) and (d) (b) What is the domain of fgx, (c) f gx, f/gx. 1.8 f gx, f/g? 131. 132. f x x2 3, f x x2 4, gx 2x 1 gx 3 x In Exercises 133 and 134, find (a) Find the domain of each function and each composite function. gx 3x 1 gx 3x 7 f x 1 3 x 3, f x x3 4, and (b) g f. 133. 134. f g In Exercises 135 and 136, find two functions and that (There are
many correct answers.) f g such f gx hx. hx 6x 53 135. 136. hx 3x 2 333202_010R.qxd 12/7/05 8:49 AM Page 121 137. Electronics Sales The factory sales (in millions of from 1997 to vt dt dollars) for VCRs 2003 can be approximated by the functions vt 31.86t2 233.6t 2594 and DVD players and dt 4.18t2 571.0t 3706 t where 1997. represents the year, with (Source: Consumer Electronics Association) corresponding to t 7 (a) Find and interpret v dt. (b) Use a graphing utility to graph vt, dt, and the function from part (a) in the same viewing window. (c) Find v d10. Use the graph in part (b) to verify your result. 138. Bacteria Count The number N of bacteria in a refriger- ated food is given by NT 25T 2 50T 300, 2 ≤ T ≤ 20 T where is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by Tt 2t 1, 0 ≤ t ≤ 9 is the time in hours (a) Find the composition and interpret its meaning in context, and (b) find t where NTt, the time when the bacterial count reaches 750. In Exercises 139 and 140, find the inverse function of f f1x x and f 1fx x. 1.9 f informally. Verify that f x x 7 f x x 5 139. 140. In Exercises 141 and 142, determine whether the function has an inverse function. 141. y 4 2 −2 2 4 x −4 142. y x 2 4 −2 −2 −4 −6 In Exercises 143–146, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. 143. 144. f x 4 1 3x f x x 12 Review Exercises 121 145. ht 2 t 3 146. gx x 6 f, In Exercises 147–150, (a) find the inverse function of on the same set of coordinate axes, (b) graph both and (c) describe the relationship between the graphs of and f 1, and (d) state the domains and ranges of and f f 1. f 1 f f 147. 148. 149. 150. f x 1 2x 3 f x 5x 7 f x x 1 f x x3 2 In Exercises 151 and 152, restrict the domain of the function to an interval over which the function is increasing and determine over that interval. f 1 f 151. 152. f x 2x 42 f x x 2 1.10 I 153. Median Income The median incomes (in thousands of dollars) for married-couple families in the United States from 1995 through 2002 are shown in the table. A linear model that approximates these data is I 2.09t 37.2 t where 1995. t 5 represents the year, with (Source: U.S. Census Bureau) corresponding to Year 1995 1996 1997 1998 1999 2000 2001 2002 Median income, I 47.1 49.7 51.6 54.2 56.5 59.1 60.3 61.1 (a) Plot the actual data and the model on the same set of coordinate axes. (b) How closely does the model represent the data? 333202_010R.qxd 12/7/05 8:49 AM Page 122 122 Chapter 1 Functions and Their Graphs 154. Data Analysis: Electronic Games The table shows the factory sales (in millions of dollars) of electronic gaming software in the United States from 1995 through 2003. (Source: Consumer Electronics Association) S Year Sales, S 1995 1996 1997 1998 1999 2000 2001 2002 2003 3000 3500 3900 4480 5100 5850 6725 7375 7744 (a) Use a graphing utility to create a scatter plot of the data. corresponding represent the year, with t 5 t Let to 1995. (b) Use the regression feature of the graphing utility to find the equation of the least squares regression line that fits the data. Then graph the model and the scatter plot you found in part (a) in the same viewing window. How closely does the model represent the data? (c) Use the model to estimate the factory sales of electronic gaming software in the year 2008. (d) Interpret the meaning of the slope of the linear model in the context of the problem. 155. Measurement You notice a billboard indicating that it is 2.5 miles or 4 kilometers to the next restaurant of a national fast-food chain. Use this information to find a mathematical model that relates miles to kilometers. Then use the model to find the numbers of kilometers in 2 miles and 10 miles. 157. Frictional Force The frictional force between the tires and the road required to keep a car on a curved section of a highway is directly proportional to the square of the speed of the car. If the speed of the car is doubled, the force will change by what factor? F s 158. Demand A company has found that the daily demand x for its boxes of chocolates is inversely proportional to the price When the price is $5, the demand is 800 boxes. Approximate the demand when the price is increased to $6. p. 159. Travel Time The travel time between two cities is inversely proportional to the average speed. A train travels between the cities in 3 hours at an average speed of 65 miles per hour. How long would it take to travel between the cities at an average speed of 80 miles per hour? 160. Cost The cost of constructing a wooden box with a square base varies jointly as the height of the box and the square of the width of the box. A box of height 16 inches and width 6 inches costs $28.80. How much would a box of height 14 inches and width 8 inches cost? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 161–163, determine whether 161. Relative to the graph of hx x 9 13 13 units downward, then reflected in the -axis. f x x, the function given by is shifted 9 units to the left and x f 162. If and are two inverse functions, then the domain of f. is equal to the range of g g 163. If y is directly proportional to x, then x is directly proportional to y. 164. Writing Explain the difference between the Vertical Line Test and the Horizontal Line Test. 165. Writing Explain how to tell whether a relation between two variables is a function. 156. Energy The power P produced by a wind turbine is proportional to the cube of the wind speed A wind speed of 27 miles per hour produces a power output of 750 kilowatts. Find the output for a wind speed of 40 miles per hour. S. 333202_010R.qxd 12/7/05 8:49 AM Page 123 1 Chapter Test Chapter Test 123 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Plot the points 6, 0. ment joining the points and the distance between the points. 2, 5 and Find the coordinates of the midpoint of the line seg- 2. A cylindrical can has a volume of 600 cubic centimeters and a radius of 4 centimeters. Find the height of the can. (−3, 3) y 8 6 4 2 −2 −2 FIGURE FOR 6 In Exercises 3–5, use intercepts and symmetry to sketch the graph of the equation. y 4 x y 3 5x y x2 1 4. 3. 5. (5, 3) x 4 6 6. Write the standard form of the equation of the circle shown at the left. In Exercises 7 and 8, find an equation of the line passing through the points. 7. 2, 3, 4, 9 8. 3, 0.8, 7, 6 9. Find equations of the lines that pass through the point 3, 8 and are (a) parallel to and (b) perpendicular to the line 4x 7y 5. 10. Evaluate x 9 x2 81 11. Determine the domain of f x at each value: (a) f 7 (b) f 5 (c) f x 9. f x 100 x2. In Exercises 12–14, (a) find the zeros of the function, (b) use a graphing utility to graph the function, (c) approximate the intervals over which the function is increasing, decreasing, or constant, and (d) determine whether the function is even, odd, or neither. 12. f x 2x 6 5x 4 x2 13. f x 4x3 x 14. 15. Sketch the graph of f x 3x 7, 4x2 1 In Exercises 16 and 17, identify the parent function in the transformation. Then sketch a graph of the function. 16. hx x 17. hx x 5 8 In Exercises 18 and 19, find (a) and (f) (e) g f x. f gx, f gx, (b) f gx, (c) fgx, (d) f/gx, 18. f x 3x2 7, gx x2 4x 5 19. f x 1 x , gx 2x In Exercises 20–22, determine whether or not the function has an inverse function, and if so, find the inverse function. 20. f x x 3 8 21. f x x2 3 6 22. f x 3xx In Exercises 23–25, find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) v 24 when 23. s. v varies directly as the square root of A 500 varies jointly as and x 24. A s 16. y 8. y. b 32 when when a 1.5. x 15 and 25. b varies inversely as a. 333202_010R.qxd_pg 124 1/9/06 8:53 AM Page 124 Proofs in Mathematics What does the word proof mean to you? In mathematics, the word proof is used to mean simply a valid argument. When you are proving a statement or theorem, you must use facts, definitions, and accepted properties in a logical order. You can also use previously proved theorems in your proof. For instance, the Distance Formula is used in the proof of the Midpoint Formula below. There are several different proof methods, which you will see in later chapters. The Midpoint Formula The midpoint of the line segment joining the points given by the Midpoint Formula (p. 5) x1, y1 and x2, y2 is Midpoint x1 x2 2 y1 , y2 2 . Proof Using the figure, you must show that y (x1, y1) d1 d2 and d1 d2 d3. d1 d3 ( x1 + x 2 2 , y1 + y2 2 ) d2 (x 2, y2) x By the Distance Formula, you obtain x1 d1 x2 2 x12 y1 y2 2 y12 1 2 x2 x1 2 y2 y1 2 x2 d2 x1 x2 2 2 y2 y1 y2 2 2 x1 2 y2 y1 2 x2 1 2 x2 d3 So, it follows that x1 d1 2 y2 d2 y1 and d1 2 d2 d3. The Cartesian Plane The Cartesian plane was named after the French mathematician René Descartes (1596–1650). While Descartes was lying in bed, he noticed a fly buzzing around on the square ceiling tiles. He discovered that the position of the fly could be described by which ceiling tile the fly landed on. This led to the development of the Cartesian plane. Descartes felt that a coordinate plane could be used to facilitate description of the positions of objects. 124 333202_010R.qxd 12/7/05 2:49 PM Page 125 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. As a salesperson, you receive a monthly salary of $2000, plus a commission of 7% of sales. You are offered a new job at $2300 per month, plus a commission of 5% of sales. (a) Write a linear equation for your current monthly wage W1 in terms of your monthly sales S. W2 (b) Write a linear equation for the monthly wage S. new job offer in terms of the monthly sales of your (c) Use a graphing utility t
o graph both equations in the same viewing window. Find the point of intersection. What does it signify? (d) You think you can sell $20,000 per month. Should you change jobs? Explain. 2. For the numbers 2 through 9 on a telephone keypad (see figure), create two relations: one mapping numbers onto letters, and the other mapping letters onto numbers. Are both relations functions? Explain. y (x, y) 12 ft FIGURE FOR 6 8 ft x 7. At 2:00 P.M. on April 11, 1912, the Titanic left Cobh, Ireland, on her voyage to New York City. At 11:40 P.M. on April 14, the Titanic struck an iceberg and sank, having covered only about 2100 miles of the approximately 3400-mile trip. (a) What was the total duration of the voyage in hours? (b) What was the average speed in miles per hour? (c) Write a function relating the distance of the Titantic from New York City and the number of hours traveled. Find the domain and range of the function. (d) Graph the function from part (c). 3. What can be said about the sum and difference of each of the following? (a) Two even functions (b) Two odd functions (c) An odd function and an even function 4. The two functions given by gx x f x x and are their own inverse functions. Graph each function and explain why this is true. Graph other linear functions that are their own inverse functions. Find a general formula for a family of linear functions that are their own inverse functions. 5. Prove that a function of the following form is even. y a2nx2n a2n2x2n2 . . . a2x2 a0 6. A miniature golf professional is trying to make a hole-inone on the miniature golf green shown. A coordinate plane is placed over the golf green. The golf ball is at the point 2.5, 2 The professional wants to bank the ball off the side wall of the green at the point Then Find the coordinates of the point write an equation for the path of the ball. and the hole is at the point 9.5, 2. x, y. x, y. 8. Consider the function given by the average rate of change of the function from fx x2 4x 3. x2. to Find x1 1.5 (b) x1 1, x2 (a) (c) (d) (e) x1 x1 x1 x1 1, x2 1, x2 1, x2 1, x2 2 1.25 1.125 1.0625 (f) Does the average rate of change seem to be approaching one value? If so, what value? (g) Find the equations of the secant lines through the points x1, fx1 and x2, fx2 for parts (a)–(e). (h) Find the equation of the line through the point 1, f1 using your answer from part (f ) as the slope of the line. gx x 6. f x 4x and (a) Find 9. Consider the functions given by f gx. f g1x. f 1x and g1 f 1x (b) Find (c) Find (d) Find g1x. and compare the result with that of part (b). (e) Repeat parts (a) through (d) for gx 2x. f x x3 1 and (f) Write two one-to-one functions and (a) through (d) for these functions. f g, and repeat parts (g) Make a conjecture about f g1x and g1 f 1x. 125 333202_010R.qxd 12/7/05 8:49 AM Page 126 10. You are in a boat 2 miles from the nearest point on the coast. You are to travel to a point 3 miles down the coast and 1 mile inland (see figure). You can row at 2 miles per hour and you can walk at 4 miles per hour. Q, 13. Show that the Associative Property holds for compositions of functions—that is, f g hx f g hx. 14. Consider the graph of the function shown in the figure. Use this graph to sketch the graph of each function. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. f (a) (e) f x 1 f x (b) (f) f x 1 f x (c) (g) 2f x f x (d) f x y 4 2 −4 −2 2 4 x −2 −4 15. Use the graphs of function values. f and f1 to complete each table of y 4 2 −2 −2 −1 −2 −2 −4 (a) x 4 2 0 4 f f 1x (b) x 3 2 0 1 (c) (d) f f 1x x f f 1x x f 1x 3 2 0 1 4 3 0 4 2 mi x 3 − x 1 mi 3 mi Q Not drawn to scale. (a) Write the total time of the trip as a function of T x. (b) Determine the domain of the function. (c) Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. (d) Use the zoom and trace features to find the value of x that minimizes T. 11. The Heaviside function (e) Write a brief paragraph interpreting these values. Hx is widely used in engineering applications. (See figure.) To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Hx 1, 0, x ≥ 0 x < 0 Sketch the graph of each function by hand. Hx Hx 2 2 Hx 2 Hx (b) (d) (e) (c) (a) (f) Hx 2 2 Hx 1 y 3 2 1 −3 −2 −1 1 2 3 x −2 −3 12. Let f x 1 1 x . (a) What are the domain and range of f ? f f x. f f f x. What is the domain of this function? Is the graph a line? Why or why not? (b) Find (c) Find 126 22 333202_0200.qxd 12/7/05 9:08 AM Page 127 Polynomial and Rational Functions 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Quadratic Functions and Models Polynomial Functions of Higher Degree Polynomial and Synthetic Division Complex Numbers Zeros of Polynomial Functions Rational Functions Nonlinear Inequalities Quadratic functions are often used to model real-life phenomena, such as the path of a diver AT I O N S Polynomial and rational functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Path of a Diver, Exercise 77, page 136 • Data Analysis: Home Prices, Exercises 93–96, page 151 • Advertising Cost, Exercise 105, page 181 • Athletics, Exercise 109, page 182 • Data Analysis: Cable Television, • Recycling, Exercise 74, page 161 Exercise 112, page 195 • Average Speed, Exercise 79, page 196 • Height of a Projectile, Exercise 67, page 205 127 333202_0201.qxd 12/7/05 9:10 AM Page 128 128 Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions and Models What you should learn • Analyze graphs of quadratic functions. • Write quadratic functions in standard form and use the results to sketch graphs of functions. • Use quadratic functions to model and solve real-life problems. Why you should learn it Quadratic functions can be used to model data to analyze consumer behavior. For instance, in Exercise 83 on page 137, you will use a quadratic function to model the revenue earned from manufacturing handheld video games. © John Henley/Corbis The Graph of a Quadratic Function In this and the next section, you will study the graphs of polynomial functions. In Section 1.6, you were introduced to the following basic functions. f x ax b f x c f x x2 Linear function Constant function Squaring function These functions are examples of polynomial functions. Definition of Polynomial Function Let be a nonnegative integer and let 0. numbers with an, n an The function given by f x anx n an1x n1 . . . a 2x 2 a1x a 0 an1, . . . , a2, a1, a0 be real is called a polynomial function of x with degree n. Polynomial functions are classified by degree. For instance, a constant function has degree 0 and a linear function has degree 1. In this section, you will study second-degree polynomial functions, which are called quadratic functions. For instance, each of the following functions is a quadratic function. f x x2 6x 2 gx 2x 12 3 hx 9 1 kx 3x2 4 mx x 2x 1 4 x2 Note that the squaring function is a simple quadratic function that has degree 2. Definition of Quadratic Function a 0. and be real numbers with Let c a, b, f x ax 2 bx c Quadratic function The function given by is called a quadratic function. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. The graph of a quadratic function is a special type of “U”-shaped curve called a parabola. Parabolas occur in many real-life applications—especially those involving reflective properties of satellite dishes and flashlight reflectors. You will study these properties in Section 10.2. 333202_0201.qxd 12/7/05 9:10 AM Page 129 Section 2.1 Quadratic Functions and Models 129 All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola, as shown in Figure 2.1. If the leading coefficient is positive, the graph of f x ax 2 bx c is a parabola that opens upward. If the leading coefficient is negative, the graph of f x ax 2 bx c is a parabola that opens downward. y y Opens upward Axis f x ( ) = ax 2 + bx + c, a < 0 Vertex is highest point Axis Vertex is lowest point f x ( ) = ax 2 + bx + c, a 0 > x Leading coefficient is positive. FIGURE 2.1 x Opens downward Leading coefficient is negative. The simplest type of quadratic function is f x ax 2. Its graph is a parabola whose vertex is (0, 0). If the minimum -value on the graph, and if maximum -value on the graph, as shown in Figure 2.2. a < 0, a > 0, y y the vertex is the point with the vertex is the point with the y 3 2 1 −1 −2 −3 −3 −2 −1 f x ( ) = 2 ax a , > 0 x 1 2 3 Minimum: (0, 0) y 3 2 1 Maximum: (0, 0) −3 −2 − ax a , < 0 −1 −2 −3 Leading coefficient is positive. FIGURE 2.2 Leading coefficient is negative. When sketching the graph of f x ax 2, it is helpful to use the graph of y x 2 as a reference, as discussed in Section 1.7. Exploration y ax2 Graph for 0.5, 0.5, 1, and 2. How does changing the value of affect the graph? a 2, 1, a for y x h2 h 4, 2, and 4. How does chang- Graph 2, ing the value of affect the graph? h for y x2 k k 4, 2, and 4. How does chang- Graph 2, ing the value of affect the graph? k 333202_0201.qxd 12/7/05 9:10 AM Page 130 130 Chapter 2 Polynomial and Rational Functions Example 1 Sketching Graphs of Quadratic Functions a. Compare the graphs of b. Compare the graphs of y x2 y x2 and and f x 1 3x2. gx 2x2. Solution a. Compared with 3x 2 creating the broader parabola shown in Figure 2.3. each output of y x 2, f x 1 “shrinks” by a factor of 1 3, b. Compared with y x 2, each output of gx 2x 2 “stretches” by a factor of 2, creating the narrower parabola shown in Figure 2.4. y x2 −1 FIGURE 2.3 x 1 2 −2 −1 FIGURE 2.4 Now try Exercise 9. y x= 2 1 2 x a In Example 1, note that the coefficient determines how widely the parabola is small, the parabola opens more widely than f x ax 2 opens. If a given by a if is large. Recall from Section 1.7 that the graphs of and y f x
y f x, For instance, in Figure 2.5, notice how the graph of to produce the graphs of f x x 2 1 and y f x ± c, y f x ± c, y f x. can be transformed y x 2 gx x 22 3. are rigid transformations of the graph of y 2 (0, 1) y x= 2 −2 f(x) = − x2 + 1 x 2 −1 −2 Reflection in x-axis followed by an upward shift of one unit FIGURE 2.5 g(x) = (x + 2)4 −3 −1 1 2 x −2 −3 (−2, −3) Left shift of two units followed by a downward shift of three units 333202_0201.qxd 12/7/05 9:10 AM Page 131 Section 2.1 Quadratic Functions and Models 131 The Standard Form of a Quadratic Function The standard form of a quadratic function identifies four basic transformations of the graph of y x2. a. The factor a produces a vertical stretch or shrink. b. If a < 0, x in the -axis. the graph is reflected c. The factor x h2 a horizontal shift of units. represents h d. The term represents a k vertical shift of units. k f x ( ) = 2( + 2= 2 2 x 1 −1 x = 2− −3 − − ( 2, 1) FIGURE 2.6 The standard form of a quadratic function is This form is especially convenient for sketching a parabola because it identifies the vertex of the parabola as f x ax h 2 k. h, k. Standard Form of a Quadratic Function The quadratic function given by f x ax h 2 k, a 0 is in standard form. The graph of x h line upward, and if and whose vertex is the point a < 0, f the parabola opens downward. is a parabola whose axis is the vertical the parabola opens h, k. a > 0, If To graph a parabola, it is helpful to begin by writing the quadratic function in standard form using the process of completing the square, as illustrated in Example 2. In this example, notice that when completing the square, you add and subtract the square of half the coefficient of within the parentheses instead of adding the value to each side of the equation as is done in Appendix A.5. x Example 2 Graphing a Parabola in Standard Form Sketch the graph of the parabola. f x 2x 2 8x 7 and identify the vertex and the axis of Solution Begin by writing the quadratic function in standard form. Notice that the first step in completing the square is to factor out any coefficient of that is not 1. x2 f x 2x 2 8x 7 2x 2 4x 7 2x 2 4x 4 4 7 Write original function. Factor 2 out of -terms. x Add and subtract 4 within parentheses. 422 After adding and subtracting 4 within the parentheses, you must now regroup the can be removed from inside the terms to form a perfect square trinomial. The parentheses; however, because of the 2 outside of the parentheses, you must multiply by 2, as shown below. 4 4 f x 2x 2 4x 4 24 7 2x 2 4x 4 8 7 2x 22 1 Regroup terms. Simplify. Write in standard form. is a parabola that opens From this form, you can see that the graph of This corresponds to a left shift of two upward and has its vertex at units and a downward shift of one unit relative to the graph of as shown in Figure 2.6. In the figure, you can see that the axis of the parabola is the vertical line through the vertex, 2, 1. x 2. y 2x 2, f Now try Exercise 13. 333202_0201.qxd 12/7/05 9:10 AM Page 132 132 Chapter 2 Polynomial and Rational Functions To find the -intercepts of the graph of x you must solve the equation If does not factor, you can x use the Quadratic Formula to find the -intercepts. Remember, however, that a parabola may not have -intercepts. ax2 bx c 0. ax2 bx c f x ax2 bx c, x Example 3 Finding the Vertex and x-Intercepts of a Parabola Sketch the graph of f x x 2 6x 8 and identify the vertex and -intercepts. x Solution f x x 2 6x 8 x 2 6x 8 x 2 6x 9 9 8 622 x 2 6x 9 9 8 x 32 1 Write original function. Factor 1 x out of -terms. Add and subtract 9 within parentheses. Regroup terms. Write in standard form. f is a parabola that opens downward with vertex From this form, you can see that 3, 1. x The -intercepts of the graph are determined as follows. x2 6x 8 0 x 2x Factor out Factor. 1. Set 1st factor equal to 0. Set 2nd factor equal to 0. So, the -intercepts are x 2, 0 and 4, 0, as shown in Figure 2.7. Now try Exercise 19. Example 4 Writing the Equation of a Parabola Write the standard form of the equation of the parabola whose vertex is that passes through the point as shown in Figure 2.8. 0, 0, 1, 2 and Solution Because the vertex of the parabola is at h, k 1, 2, the equation has the form f x ax 12 2. k Substitute for and h Because the parabola passes through the point 0, 0, it follows that in standard form. f 0 0. So, f(x) = −(x − 3)2 + 1 (3, 1) (2, 0) (4, 0) 1 3 5 x y = −x2 (1, 2) y = f(x) −1 y 2 1 −1 −2 −3 −4 FIGURE 2.7 y 2 1 (0, 0) 1 x 0 a0 12 2 a 2 Substitute 0 for x; solve for a. which implies that the equation in standard form is f x 2x 12 2. FIGURE 2.8 Now try Exercise 43. 333202_0201.qxd 12/7/05 9:10 AM Page 133 Section 2.1 Quadratic Functions and Models 133 Applications Many applications involve finding the maximum or minimum value of a quadratic function. You can find the maximum or minimum value of a quadratic function by locating the vertex of the graph of the function. Vertex of a Parabola The vertex of the graph of f x ax2 bx c is b 2a , f b 2a . 1. If a > 0, 2. If a < 0, has a minimum at x b 2a has a maximum at x b 2a . . Example 5 The Maximum Height of a Baseball A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per with respect to the ground. The path of the baseball second and at an angle of is the height of is given by the function x the baseball (in feet) and is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball? 45 f x 0.0032x 2 x 3, where f x Solution From the given function, you can see that function has a maximum when reaches its maximum height when it is x b2a, and a 0.0032 b 1. Because the you can conclude that the baseball is feet from home plate, where x x x b 2a x b 2a 1 20.0032 156.25 feet. At this distance, 156.25 3 81.125 the maximum height is f 156.25 0.0032156.25 2 feet. The path of the baseball is shown in Figure 2.9. Now try Exercise 77. Example 6 Minimizing Cost A small local soft-drink manufacturer has daily production costs of C 70,000 120x 0.075x2, C is the number of units produced. How many units should be produced each day to yield a minimum cost? is the total cost (in dollars) and where x Baseball f(x) = −0.0032x 2 + x + 3 (156.25, 81.125) x 200 100 Distance (in feet) 300 y 100 ) 80 60 40 20 FIGURE 2.9 Solution Use the fact that the function has a minimum when function you can see that 120 2(0.075 x b 2a a 0.075 800 units and b 120. x b2a. From the given So, producing each day will yield a minimum cost. Now try Exercise 83. 333202_0201.qxd 12/7/05 9:10 AM Page 134 134 Chapter 2 Polynomial and Rational Functions 2.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. A polynomial function of degree and leading coefficient n f x anxn an1xn1 . . . a1x a0 an a1 2. A ________ function is a second-degree polynomial function, and its graph is called a ________. is a ________ ________ and 0 an where is a function of the form n are ________ numbers. 3. The graph of a quadratic function is symmetric about its ________. 4. If the graph of a quadratic function opens upward, then its leading coefficient is ________ and the vertex of the graph is a ________. 5. If the graph of a quadratic function opens downward, then its leading coefficient is ________ and the vertex of the graph is a ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 8, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] (a) (c) y 6 4 2 (b) y 6 4 2 −4 (−1, −2) (− 4, 0) − 6 − 4 −2 2 y 6 4 2 −2 (e) y 2 −2 −4 −6 (g) y 2 4 6 (3, 2)− 6 4 2 (2, 0) −4 −2 (0, −2) 2 4 (4, 0) 2 4 6 8 (d) y −2 −4 −6 (2, 4) x 6 (f) y 4 2 −2 (h) 2 y 4 (0, 3) x 4 −4 −2 −4 1. 3. 5. 7. f x x 22 f x x 2 2 f x 4 (x 2)2 f x x 32 2 2. 4. 6. 8. f x x 42 f x 3 x 2 f x x 12 2 f x x 42 In Exercises 9–12, graph each function. Compare the graph of each function with the graph of y x2. (b) (d) (b) (d) (b) (d) gx 1 8 x2 kx 3x 2 gx x 2 1 kx x 2 3 gx 3x2 1 kx x 32 9. (a) (c) 10. (a) (c) 11. (a) (c) 12. (a) (b) (c) (d) f x 1 2 x2 hx 3 2 x2 f x x 2 1 hx x 2 3 f x x 12 hx 1 3 x2 3 f x 1 x 22 1 2 gx 1 x 12 3 2 x 22 1 hx 1 2 kx 2x 12 4 In Exercises 13–28, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and -intercept(s). x 14. 16. 18. 20. 22. 24. hx 25 x 2 f x 16 1 4 x 2 f x x 62 3 gx x 2 2x 1 f x x 2 3x 1 4 f x x 2 4x 1 13. 15. 17. 19. 21. 23. 25. 26. 27. 28. f x x 2 5 f x 1 2x 2 4 f x x 52 6 hx x 2 8x 16 2x 5 hx 4x 2 4x 21 f x 2x 2 x 1 f x 1 f x 1 4x2 2x 12 3x2 3x 6 333202_0201.qxd 12/7/05 9:10 AM Page 135 Section 2.1 Quadratic Functions and Models 135 In Exercises 29–36, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and -intercepts. Then check your results algebraically by writing the quadratic function in standard form. x 50. Vertex: 51. Vertex: 52. Vertex: 4 ; 5 2, 3 2, 0; 5 6, 6; point: 2, 4 7 2, 16 point: 61 10, 3 2 3 point: 29. 31. 33. 35. f x x 2 2x 3 gx x 2 8x 11 f x 2x 2 16x 31 gx 1 x 2 4x 2 2 30. 32. 34. 36. f x x2 x 30 f x x2 10x 14 f x 4x2 24x 41 f x 3 5 x 2 6x 5 In Exercises 37– 42, find the standard form of the quadratic function. 37. 39. 41. y 8 6 (1, 0) 38. y 2 (−1, 0) (0, 1) (1, 0) −2 2 4 x (0, 1) −2 (−1, 4) (−3, 0) −4 −2 x x 4 2 y (1, 0) 2 2 −2 −4 y 2 (−2, 2) (−3, 0) −6 −4 x 2 (−1, 0) −6 40. −4 −6 y 6 (0, 3) 2 −6 −4 (−2, −1) x 2 42. y 8 6 4 2 (2, 0) (3, 2) −2 2 4 6 x In Exercises 43–52, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. 43. Vertex: 44. Vertex: 45. Vertex: 46. Vertex: 47. Vertex: 48. Vertex: 49. Vertex: point: point: 0, 9 2, 3 point: point: point: 1, 2 0, 2 7, 15
2, 5; 4, 1; 3, 4; 2, 3; 5, 12; 2, 2; 1 ; 4, 3 2 point: 2, 0 point: 1, 0 In Exercises 53–56, determine -intercept(s) of the graph visually. Then find the Graphical Reasoning the x x -intercepts algebraically to confirm your results. 53. y x 2 16 y −8 −4 x 8 55. y x 2 4x 5 y 54. y x 2 6x 56. y 2x 2 5x 3 x −6 −4 8 −4 −4 −8 y 2 −2 −4 x 2 58. 57. In Exercises 57–64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and x compare them with the solutions of the corresponding f x 0. quadratic equation when f x x 2 4x f x 2 x2 10x f x x 2 9x 18 f x x2 8x 20 f x 2x 2 7x 30 f x 4x2 25x 21 f x 1 x 2 6x 7 2 x2 12x 45 f x 7 10 60. 59. 62. 64. 63. 61. x In Exercises 65–70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given -intercepts. (There are many correct answers.) 1, 0, 3, 0 0, 0, 10, 0 3, 0, 1 5, 0, 5, 0 4, 0, 8, 0 2, 0, 2, 0 68. 70. 5 2, 0 65. 69. 66. 67. 333202_0201.qxd 12/7/05 9:10 AM Page 136 136 Chapter 2 Polynomial and Rational Functions In Exercises 71–74, find two positive real numbers whose product is a maximum. 71. The sum is 110. 72. The sum is S. (c) Use the result of part (b) to write the area of the rectangular region as a function of What dimensions will produce a maximum area of the rectangle? x. A 77. Path of a Diver The path of a diver is given by 73. The sum of the first and twice the second is 24. 74. The sum of the first and three times the second is 42. y 4 9 x 2 24 9 x 12 75. Numerical, Graphical, and Analytical Analysis A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure). x x y (a) Write the area of the corral as a function of A x. (b) Create a table showing possible values of and the corresponding areas of the corral. Use the table to estimate the dimensions that will produce the maximum enclosed area. x (c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area. (d) Write the area function in standard form to find analytically the dimensions that will produce the maximum area. (e) Compare your results from parts (b), (c), and (d). 76. Geometry An indoor physical fitness room consists of a rectangular region with a semicircle on each end (see figure). The perimeter of the room is to be a 200-meter single-lane running track. x y (a) Determine the radius of the semicircular ends of the room. Determine the distance, in terms of around the inside edge of the two semicircular parts of the track. y, (b) Use the result of part (a) to write an equation, in terms for the distance traveled in one lap around x y, of and the track. Solve for y. y is the height (in feet) and where is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver? x 78. Height of a Ball The height y (in feet) of a punted foot- ball is given by y 16 2025 x2 9 5 x 1.5 x is the horizontal distance (in feet) from the point at where which the ball is punted (see figure). y x Not drawn to scale (a) How high is the ball when it is punted? (b) What is the maximum height of the punt? (c) How long is the punt? 79. Minimum Cost A manufacturer of lighting fixtures has daily production costs of C 800 10x 0.25x 2 C is the total cost (in dollars) and where is the number of units produced. How many fixtures should be produced each day to yield a minimum cost? x 80. Minimum Cost A textile manufacturer has daily produc- tion costs of C 100,000 110x 0.045x 2 C is the total cost (in dollars) and where is the number of units produced. How many units should be produced each day to yield a minimum cost? x 81. Maximum Profit The profit P (in dollars) for a company that produces antivirus and system utilities software is P 0.0002x 2 140x 250,000 is the number of units sold. What sales level will x where yield a maximum profit? 333202_0201.qxd 12/7/05 9:10 AM Page 137 82. Maximum Profit The profit (in hundreds of dollars) that a company makes depends on the amount (in hundreds of dollars) the company spends on advertising according to the model P x P 230 20x 0.5x 2. What expenditure for advertising will yield a maximum profit? 83. Maximum Revenue The total revenue earned (in thousands of dollars) from manufacturing handheld video games is given by R p 25p2 1200p R where p is the price per unit (in dollars). (a) Find the revenue earned for each price per unit given below. $20 $25 $30 (b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results. 84. Maximum Revenue The total revenue R earned per day (in dollars) from a pet-sitting service is given by R p 12p2 150p where p is the price charged per pet (in dollars). (a) Find the revenue earned for each price per pet given below. $4 $6 $8 (b) Find the price that will yield a maximum revenue. What is the maximum revenue? Explain your results. 85. Graphical Analysis From 1960 to 2003, the per capita of cigarettes by Americans (age 18 and C consumption older) can be modeled by C 4299 1.8t 1.36t2, 0 ≤ t ≤ 43 t t 0 where is the year, with (Source: Tobacco Outlook Report) corresponding to 1960. (a) Use a graphing utility to graph the model. (b) Use the graph of the model to approximate the maximum average annual consumption. Beginning in 1966, all cigarette packages were required by law to carry a health warning. Do you think the warning had any effect? Explain. (c) In 2000, the U.S. population (age 18 and over) was 209,128,094. Of those, about 48,308,590 were smokers. What was the average annual cigarette consumption per smoker in 2000? What was the average daily cigarette consumption per smoker? Section 2.1 Quadratic Functions and Models 137 Model It 86. Data Analysis The numbers (in thousands) of hairdressers and cosmetologists in the United States for the years 1994 through 2002 are shown in the table. (Source: U.S. Bureau of Labor Statistics) y Year 1994 1995 1996 1997 1998 1999 2000 2001 2002 Number of hairdressers and cosmetologists, y 753 750 737 748 763 784 820 854 908 (a) Use a graphing utility to create a scatter plot of the corre- represent the year, with x 4 x data. Let sponding to 1994. (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model in the same viewing window as the scatter plot. How well does the model fit the data? (d) Use the trace feature of the graphing utility to approximate the year in which the number of hairdressers and cosmetologists was the least. (e) Verify your answer to part (d) algebraically. (f) Use the model to predict the number of hairdressers and cosmetologists in 2008. 87. Wind Drag The number of horsepower required to overcome wind drag on an automobile is approximated by y y 0.002s 2 0.005s 0.029, 0 ≤ s ≤ 100 where s is the speed of the car (in miles per hour). (a) Use a graphing utility to graph the function. (b) Graphically estimate the maximum speed of the car if the power required to overcome wind drag is not to exceed 10 horsepower. Verify your estimate algebraically. 333202_0201.qxd 12/7/05 9:10 AM Page 138 138 Chapter 2 Polynomial and Rational Functions 88. Maximum Fuel Economy A study was done to compare the speed (in miles (in miles per hour) with the mileage per gallon) of an automobile. The results are shown in the (Source: Federal Highway Administration) table. y x Speed, x Mileage, y 15 20 25 30 35 40 45 50 55 60 65 70 75 22.3 25.5 27.5 29.0 28.8 30.0 29.9 30.2 30.4 28.8 27.4 25.3 23.3 (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model in the same viewing window as the scatter plot. (d) Estimate the speed for which the miles per gallon is greatest. Synthesis In Exercises 89 and 90, determine whether True or False? the statement is true or false. Justify your answer. f x 12x2 1 89. The function given by has no x -intercepts. 90. The graphs of f x 4x2 10x 7 and gx 12x2 30x 1 have the same axis of symmetry. 91. Write the quadratic function f x ax2 bx c in standard form to verify that the vertex occurs at b 2a , f b 2a . 92. Profit The profit (in millions of dollars) for a recreational vehicle retailer is modeled by a quadratic function of the form P P at 2 bt c t where represents the year. If you were president of the company, which of the models below would you prefer? Explain your reasoning. a a (a) (b) is positive and is positive and b2a ≤ t. t ≤ b2a. b2a ≤ t. t ≤ b2a. 93. Is it possible for a quadratic equation to have only one is negative and is negative and (d) (c) a a x -intercept? Explain. 94. Assume that the function given by f x ax 2 bx c, a 0 has two real zeros. Show that the vertex of the graph is the average of the zeros of Use the Quadratic Formula.) x -coordinate of the (Hint: f. Skills Review In Exercises 95–98, find the equation of the line in slope-intercept form that has the given characteristics. 4, 3 95. Passes through the points 7 2, 2 0, 3 96. Passes through the point 97. Passes through the point 4x 5y 10 98. Passes through the point 8, 4 line and 2, 1 and has a slope of 3 2 and is perpendicular to the and is parallel to the line and let gx 8x2. y 3x 2 fx 14x 3 99. 100. In Exercises 99–104, let Find the indicated value. f g3 g f 2 fg4 1.5 f g f g1 g f 0 103. 104. 101. 102. 7 105. Make a Decision To work an extended application analyzing the height of a basketball after it has been dropped, visit this text’s website at college.hmco.com. 333202_0202.qxd 12/7/05 9:11 AM Page 139 Section 2.2 Polynomial Functions of Higher Degree 139 2.2 Polynomial Functions of Higher Degree What you should learn • Use transformations to sketch graphs of polynomial functions. • Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. • Find and use zeros of polynomial funct
ions as sketching aids. • Use the Intermediate Value Theorem to help locate zeros of polynomial functions. Why you should learn it You can use polynomial functions to analyze business situations such as how revenue is related to advertising expenses, as discussed in Exercise 98 on page 151. Graphs of Polynomial Functions In this section, you will study basic features of the graphs of polynomial functions. The first feature is that the graph of a polynomial function is continuous. Essentially, this means that the graph of a polynomial function has no breaks, holes, or gaps, as shown in Figure 2.10(a). The graph shown in Figure 2.10(b) is an example of a piecewise-defined function that is not continuous. y y x x (a) Polynomial functions have continuous graphs. FIGURE 2.10 (b) Functions with graphs that are not continuous are not polynomial functions. The second feature is that the graph of a polynomial function has only smooth, rounded turns, as shown in Figure 2.11. A polynomial function cannot have a sharp turn. For instance, the function given by which has a sharp turn at the point as shown in Figure 2.12, is not a polynomial function. f x x, 0, 0, y Bill Aron /PhotoEdit, Inc. f(x) = 4 −3 −2 −1 −2 3 4 1 2 (0, 0) x Polynomial functions have graphs with smooth rounded turns. FIGURE 2.11 Graphs of polynomial functions cannot have sharp turns. FIGURE 2.12 The graphs of polynomial functions of degree greater than 2 are more difficult to analyze than the graphs of polynomials of degree 0, 1, or 2. However, using the features presented in this section, coupled with your knowledge of point plotting, intercepts, and symmetry, you should be able to make reasonably accurate sketches by hand. 333202_0202.qxd 12/7/05 9:11 AM Page 140 140 Chapter 2 Polynomial and Rational Functions if For power functions given by f x xn, n is even, then the graph of the function is symmetric with respect to the y n -axis, and if is odd, then the graph of the function is symmetric with respect to the origin. (−1, 1) −1 f x x n, The polynomial functions that have the simplest graphs are monomials of is an integer greater than zero. From Figure 2.13, f x x 2, Moreover, the the flatter the graph near the origin. Polynomial functions are often referred to as power functions. the form you can see that when n and when greater the value of of the form is even, the graph is similar to the graph of is odd, the graph is similar to the graph of n, f x xn where n f x x 31, 11, 1) −1 x 1 x 1 y xn −1 (−1, −1) y xn (b) If n is odd, the graph of x crosses the axis at the -intercept. (a) If n is even, the graph of x touches the axis at the -intercept. FIGURE 2.13 Example 1 Sketching Transformations of Monomial Functions Sketch the graph of each function. a. f x x5 b. hx x 14 Solution a. Because the degree of is odd, its graph is similar to the graph of In Figure 2.14, note that the negative coefficient has the effect of f x x 5 y x 3. reflecting the graph in the -axis. x b. The graph of hx x 14, as shown in Figure 2.15, is a left shift by one y x 4. unit of the graph of y (−1, 1) 1 f(x) = −x 5 x 1 −1 −1 (1, −1) FIGURE 2.14 Now try Exercise 9. h(x) = (x + 1) 4 y 3 2 1 (0, 1) x 1 (−2, 1) (−1, 0) −2 −1 FIGURE 2.15 333202_0202.qxd 12/7/05 9:11 AM Page 141 Exploration For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each function. Describe the relationship between the degree and the sign of the leading coefficient of the function and the right-hand and left-hand behavior of the graph of the function. a. b. c. d. e. f. g. f x x3 2x2 x 1 f x 2x5 2x2 5x 1 f x 2x5 x2 5x 3 f x x3 5x 2 f x 2x2 3x 4 f x x 4 3x2 2x 1 f x x2 3x 2 as f x → ” indicates that the The notation “ x → graph falls to the left. The f x → x → ” notation “ indicates that the graph rises to the right. as Section 2.2 Polynomial Functions of Higher Degree 141 The Leading Coefficient Test x In Example 1, note that both graphs eventually rise or fall without bound as moves to the right. Whether the graph of a polynomial function eventually rises or falls can be determined by the function’s degree (even or odd) and by its leading coefficient, as indicated in the Leading Coefficient Test. x Leading Coefficient Test As moves without bound to the left or to the right, the graph of the polynomial function falls in the following manner. f x anxn . . . a1x a0 eventually rises or 1. When n is odd: y y f(x) → ∞ as x → ∞ f(x) → ∞ as x → −∞ f(x) → −∞ as x → −∞ x f(x) → − ∞ as x → ∞ x If the leading coefficient is an > 0, positive the graph falls to the left and rises to the right. If the leading coefficient is an < 0, negative to the left and falls to the right. the graph rises 2. When n is even: y y f(x) → ∞ as x → −∞ f(x) → ∞ as x → ∞ f(x) → −∞ as x → −∞ f(x) → −∞ as x → ∞ x x If the leading coefficient is an > 0, positive the graph rises to the left and right. If the leading coefficient is an < 0, negative falls to the left and right. the graph The dashed portions of the graphs indicate that the test determines only the right-hand and left-hand behavior of the graph. 333202_0202.qxd 12/7/05 9:11 AM Page 142 142 Chapter 2 Polynomial and Rational Functions Example 2 Applying the Leading Coefficient Test A polynomial function is written in standard form if its terms are written in descending order of exponents from left to right. Before applying the Leading Coefficient Test to a polynomial function, it is a good idea to check that the polynomial function is written in standard form. Exploration For each of the graphs in Example 2, count the number of zeros of the polynomial function and the number of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe? Remember that the zeros of a function of are the -values for which the function is zero. x x Describe the right-hand and left-hand behavior of the graph of each function. f x x 4 5x 2 4 f x x3 4x f x x 5 x b. a. c. Solution a. Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right, as shown in Figure 2.16. b. Because the degree is even and the leading coefficient is positive, the graph rises to the left and right, as shown in Figure 2.17. c. Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right, as shown in Figure 2.18. f(x) = − x3 + 4x y 3 2 1 −3 −1 1 3 x f(x) = x4 − 5x2 + 4 y 6 4 −4 −2 x 4 f(x) = x5 − x y x 2 2 1 −1 −2 FIGURE 2.16 FIGURE 2.17 FIGURE 2.18 Now try Exercise 15. In Example 2, note that the Leading Coefficient Test tells you only whether the graph eventually rises or falls to the right or left. Other characteristics of the graph, such as intercepts and minimum and maximum points, must be determined by other tests. Zeros of Polynomial Functions It can be shown that for a polynomial function statements are true. f of degree n, the following 1. The function has, at most, f n real zeros. (You will study this result in detail in the discussion of the Fundamental Theorem of Algebra in Section 2.5.) f 2. The graph of has, at most, turning points. (Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa.) n 1 Finding the zeros of polynomial functions is one of the most important problems in algebra. There is a strong interplay between graphical and algebraic approaches to this problem. Sometimes you can use information about the graph of a function to help find its zeros, and in other cases you can use information about the zeros of a function to help sketch its graph. Finding zeros of polynomial functions is closely related to factoring and finding -intercepts. x 333202_0202.qxd 12/7/05 9:11 AM Page 143 Section 2.2 Polynomial Functions of Higher Degree 143 Real Zeros of Polynomial Functions If ments are equivalent. is a polynomial function and a f is a real number, the following state- 1. 2. 3. 4. x a x a x a a, 0 is a zero of the function f. is a solution of the polynomial equation f x 0. is a factor of the polynomial f x. is an -intercept of the graph of f. x Example 3 Finding the Zeros of a Polynomial Function Find all real zeros of f (x) 2x4 2x 2. Then determine the number of turning points of the graph of the function. Algebraic Solution To find the real zeros of the function, set x. zero and solve for f x equal to 2x4 2x2 0 2x2x2 1 0 2x2x 1x 1 0 f x equal to 0. Set Remove common monomial factor. Factor completely. x 0, x 1. So, the real zeros are Because the function is a fourth-degree polynomial, the turning points. graph of can have at most 4 1 3 x 1, and f y 2x4 2x2. 0, 0, Graphical Solution In Figure Use a graphing utility to graph 1, 0, 2.19, the graph appears to have zeros at and 1, 0. Use the zero or root feature, or the zoom and trace features, of the graphing utility to verify these zeros. So, the real zeros are From the figure, you can see that the graph has three turning points. This is consistent with the fact that a fourth-degree polynomial can have at most three turning points. x 1. x 1, x 0, and 2 y = − 2x 4 + 2x 2 −3 3 Now try Exercise 27. −2 FIGURE 2.19 k In Example 3, note that because x 0. x The graph touches the -axis at is even, the factor x 0, zero 2x2 yields the repeated as shown in Figure 2.19. Repeated Zeros A factor x ak, k > 1, yields a repeated zero x a of multiplicity k. 1. If k is odd, the graph crosses the -axis at x x a. is even, the graph touches the -axis (but does not cross the -axis) x x 2. If k at x a. 333202_0202.qxd 12/7/05 9:12 AM Page 144 144 Chapter 2 Polynomial and Rational Functions Te c h n o l o g y Example 4 uses an algebraic approach to describe the graph of the function. A graphing utility is a complement to this approach. Remember that an import
ant aspect of using a graphing utility is to find a viewing window that shows all significant features of the graph. For instance, the viewing window in part (a) illustrates all of the significant features of the function in Example 4. a. 3 − 4 b. −2 − 3 0.5 −0.5 5 2 If you are unsure of the shape of a portion of the graph of a polynomial function, plot some additional points, such as the 0.5, 0.3125 point in Figure 2.21. as shown To graph polynomial functions, you can use the fact that a polynomial function can change signs only at its zeros. Between two consecutive zeros, a polynomial must be entirely positive or entirely negative. This means that when the real zeros of a polynomial function are put in order, they divide the real number line into intervals in which the function has no sign changes. These resulting intervals are test intervals in which a representative -value in the interval is chosen to determine if the value of the polynomial function is positive (the graph lies above the -axis) or negative (the graph lies below the -axis). x x x Example 4 Sketching the Graph of a Polynomial Function f x 3x 4 4x 3. Sketch the graph of Solution 1. Apply the Leading Coefficient Test. Because the leading coefficient is positive and the degree is even, you know that the graph eventually rises to the left and to the right (see Figure 2.20). 2. Find the Zeros of the Polynomial. By factoring you can see that the zeros of are 0, 0 f x x 33x 4, of odd multiplicity). So, the -intercepts occur at points to your graph, as shown in Figure 2.20. x f f x 3x 4 4x 3 x 0 and 3, 0. 4 and x 4 3 as (both Add these 3. Plot a Few Additional Points. Use the zeros of the polynomial to find the -value and test intervals. In each test interval, choose a representative evaluate the polynomial function, as shown in the table. x Test interval Representative Value of f Sign x-value , 0 1 f 1 7 Positive Point on graph 1, 7 0, 4 3 3, 4 1 1.5 f 1 1 Negative 1, 1 f 1.5 1.6875 Positive 1.5, 1.6875 4. Draw the Graph. Draw a continuous curve through the points, as shown in Figure 2.21. Because both zeros are of odd multiplicity, you know that the graph should cross the -axis at x 0 and x x 4 3. y 7 6 5 4 3 2 Up to left Up to right f(x) = 3x4 − 4x3 y 7 6 5 4 3 ) ) , 04 3 (0, 0) −4 −3 −2 −1 −1 1 2 3 4 FIGURE 2.20 x −4 −3 −2 −1 −1 FIGURE 2.21 Now try Exercise 67. x 2 3 4 333202_0202.qxd 12/7/05 9:12 AM Page 145 Section 2.2 Polynomial Functions of Higher Degree 145 Example 5 Sketching the Graph of a Polynomial Function Sketch the graph of f x 2x 3 6x2 9 2x. Solution 1. Apply the Leading Coefficient Test. Because the leading coefficient is negative and the degree is odd, you know that the graph eventually rises to the left and falls to the right (see Figure 2.22). 2. Find the Zeros of the Polynomial. By factoring f x 2x3 6x2 9 2 x 1 1 2x4x2 12x 9 2x2x 32 x 0 you can see that the zeros of x multiplicity). So, the -intercepts occur at to your graph, as shown in Figure 2.22. are f (odd multiplicity) and 2, 0. 3 0, 0 x 3 (even 2 Add these points and 3. Plot a Few Additional Points. Use the zeros of the polynomial to find the test intervals. In each test interval, choose a representative -value and evaluate the polynomial function, as shown in the table. x f x is positive to the Observe in Example 5 that the sign of left of and negative to the right x 0. of the zero Similarly, the f x sign of is negative to the left and to the right of the zero x 3 2. This suggests that if the zero of a polynomial function is of odd multiplicity, then the sign changes from one side of of the zero to the other side. If the zero is of even multiplicity, then the sign of change from one side of the zero to the other side. does not f x f x Test interval Representative Value of f Sign , 0 0, 3 2 2, 3 x-value 0.5 0.5 2 Point on graph 0.5, 4 f 0.5 4 Positive f 0.5 1 Negative 0.5, 1 f 2 1 Negative 2, 1 4. Draw the Graph. Draw a continuous curve through the points, as shown in Figure 2.23. As indicated by the multiplicities of the zeros, the graph crosses the -axis at but does not cross the -axis at 0, 0 x x 2, 0. 3 y y 6 5 4 3 2 Up to left Down to right − 0, 0) ) 2( , 03 1 2 3 4 −4 −3 −2 −1 −1 −2 FIGURE 2.22 x −4 −3 1 −2 −1 −1 −2 FIGURE 2.23 Now try Exercise 69. x 3 4 333202_0202.qxd 12/7/05 9:12 AM Page 146 146 Chapter 2 Polynomial and Rational Functions The Intermediate Value Theorem The next theorem, called the Intermediate Value Theorem, illustrates the existence of real zeros of polynomial functions. This theorem implies that if a, f a are two points on the graph of a polynomial function such f a that there must be a number between and then for any number between a f b and (See Figure 2.24.) and f a f b, f c d. b, f b such that b d c y f b( ) FIGURE 2.24 a c b x Intermediate Value Theorem a Let and be real numbers such that such that between b f a f b, and f b. f a then, in the interval a < b. f If a, b, is a polynomial function f takes on every value The Intermediate Value Theorem helps you locate the real zeros of a at which polynomial function in the following way. If you can find a value at which it is negaa polynomial function is positive, and another value tive, you can conclude that the function has at least one real zero between these is negative two values. For example, the function given by when it follows from the Intermediate Value Theorem that must have a real zero somewhere between 2 f x x 3 x 2 1 Therefore, and positive when f as shown in Figure 2.25. x 1. x 2 x a x b 1, and ( 1) = 1 x 1 2 f has a zero between − − 2 and 1. − − f( 2) = 3 − ( 1, 1) −2 −1 −2 −3 − − ( 2, 3) FIGURE 2.25 By continuing this line of reasoning, you can approximate any real zeros of a polynomial function to any desired accuracy. This concept is further demonstrated in Example 6. 333202_0202.qxd 12/7/05 9:12 AM Page 147 Section 2.2 Polynomial Functions of Higher Degree 147 Example 6 Approximating a Zero of a Polynomial Function Use the Intermediate Value Theorem to approximate the real zero of f x x 3 x 2 1. Solution Begin by computing a few function values, as follows. x 2 1 0 1 f x 11 1 1 1 f 1 is negative and f 0 Because Value Theorem to conclude that the function has a zero between pinpoint this zero more closely, divide the interval evaluate the function at each point. When you do this, you will find that is positive, you can apply the Intermediate and 0. To into tenths and 1, 0 1 f 0.8 0.152 and f 0.7 0.167. 0.7, and f So, must have a zero between more accurate approximation, compute function values between f 0.7 process, you can approximate this zero to any desired accuracy. as shown in Figure 2.26. For a and and apply the Intermediate Value Theorem again. By continuing this f 0.8 0.8 f x ( ) = 3 x 2− x + 1 y 2 (0, 1) (1, 1) x 2 1 f has a zero between 0.8− − and 0.7. −1 −1 − − ( 1, 1) FIGURE 2.26 Now try Exercise 85. Te c h n o l o g y You can use the table feature of a graphing utility to approximate the zeros of a polynomial function. For instance, for the function given by fx 2x3 3x2 3 , as 20 ≤ x ≤ 20 and f1 f0 create a table that shows the function values for shown in the first table at the right. Scroll through the table looking for consecutive function values that differ in sign. From the table, you can see that differ in sign. So, you can conclude from the Intermediate Value Theorem that the function has a zero between 0 and 1. You can adjust your table to show function values for using increments of 0.1, as shown in the second table at the right. By scrolling through the table you can see that and and 0.9. If you repeat this process several times, you should obtain x 0.806 as the zero of the function. Use the zero or root feature of a graphing utility to confirm this result. f0.8 differ in sign. So, the function has a zero between 0.8 0 ≤ x ≤ 1 f0.9 333202_0202.qxd 12/7/05 2:50 PM Page 148 148 Chapter 2 Polynomial and Rational Functions 2.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The graphs of all polynomial functions are ________, which means that the graphs have no breaks, holes, or gaps. 2. The ________ ________ ________ is used to determine the left-hand and right-hand behavior of the graph of a polynomial function. 3. A polynomial function of degree has at most ________ real zeros and at most ________ turning points. n 4. If x a is a zero of a polynomial function f, then the following three statements are true. (a) x a is a ________ of the polynomial equation f x 0. (b) ________ is a factor of the polynomial f x. (c) a, 0 is an ________ of the graph f. 5. If a real zero of a polynomial function is of even multiplicity, then the graph of ________ the -axis at x f x a, and if it is of odd multiplicity then the graph of ________ the -axis at x f x a. 6. A polynomial function is written in ________ form if its terms are written in descending order of exponents from left to right. 7. The ________ ________ Theorem states that if a, b, interval f b. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. takes on every value between f f f a is a polynomial function such that and f a f b, then in the In Exercises 1– 8, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] (a) y (b) y 8 −8 x 8 −8 −4 4 8 x (c) (e) −4 −8 y 8 4 x x −8 −4 4 8 −4 −8 y 8 −8 −4 4 8 −4 −8 (d) −4 (f) y 6 4 2 −2 y 4 (g) y (h) y 4 −2 2 x 6 −4 x 2 −2 −4 −2 −4 1. 3. 5. 7. f x 2x 3 f x 2x 2 5x f x 1 4x 4 3x 2 f x x 4 2x 3 2. 4. 6. 8. f x x 2 4x f x 2x3 3x 1 f x 1 3x3 x 2 4 3 f x 1 5x 5 2x 3 9 5x In Exercises 9–12, sketch the graph of transformation. y x n and each x 2 4 9. 10. 11. f x x 23 f x 1 2x 3 f x x 15 f x 1 1 2x 5 y x 3 (a) (c) y x 5 (a) (c) y x 4 (a) f x x 34 f x 4 x 4 fx 2x4 1 (c) (e) (b) (d) f x x 3 2 f x x 23 2 (b) (d 15 (bd) 2 (f) fx 1 x 14 2 x4 2 −4 −2 2 4 x −4 333202_0202.qxd 12/7/05 9:12 AM Page 149 Section 2.2 Polynomial Functions of Higher Degree 149 12. y x 6 (a)
8x 6 f x 1 f x x 6 4 fx 1 4 x6 2 (c) (e) (b) (d) (f) f x x 26 4 f x 1 4x 6 1 fx 2x6 1 In Exercises 13–22, describe the right-hand and left-hand behavior of the graph of the polynomial function. 13. 15. 17. 18. 19. 20. 21. 22. 14. 16. f x 2x 2 3x 1 hx 1 x 6 2x 3x 2 3x 3 5x f x 1 gx 5 7 f x 2.1x 5 4x 3 2 f x 2x 5 5x 7.5 f x 6 2x 4x 2 5x 3 f x 3x 4 2x 5 4 ht 2 3 fs 7 8 t 2 5t 3 s3 5s 2 7s 1 Graphical Analysis In Exercises 23–26, use a graphing f utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of appear identical. and g g f gx 3x 3 23. 24. 25. 26. x3 3x 2, f x 3x 3 9x 1, fx 1 3 f x x 4 4x 3 16x, fx 3x 4 6x 2, gx 3x 4 gx 1 3x 3 gx x 4 In Exercises 27– 42, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers. Graphical Analysis In Exercises 43–46, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation, and (d) compare the results of part (c) with any -intercepts of the graph. y 0 x x 43. 44. 45. 46. y 4x3 20x 2 25x y 4x 3 4x 2 8x 8 y x 5 5x 3 4x 4x 3x 2 9 y 1 In Exercises 47–56, find a polynomial function that has the given zeros. (There are many correct answers.) 47. 49. 51. 53. 55. 0, 10 2, 6 0, 2, 3 4, 3, 3, 0 1 3, 1 3 48. 50. 52. 54. 56. 0, 3 4, 5 0, 2, 5 2, 1, 0, 1, 2 2, 4 5, 4 5 In Exercises 57–66, find a polynomial of degree the given zero(s). (There are many correct answers.) n that has Zero(s) x 2 x 8, 4 x 3, 0, 1 x 2, 4, 7 x 0, 3, 3 x 9 x 5, 1, 2 x 4, 1, 3, 6 x 0, 4 x 3, 1, 5, 6 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. Degree 27. 29. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 28. 30. 3 2 3 x 2 2x 3 fx x 2 25 ht t 2 6t 2x 2 5 fx 3x3 12x2 3x gx 5xx 2 2x 1 f t t 3 4t 2 4t fx x 4 x 3 20x 2 gt t 5 6t 3 9t fx x 5 x 3 6x fx 5x 4 15x 2 10 fx 2x 4 2x 2 40 gx x3 3x 2 4x 12 fx x 3 4x 2 25x 100 fx 49 x 2 fx x 2 10x 25 In Exercises 67– 80, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. 73. 69. 71. 67. 70. f x x 3 9x f t 1 t 2 2t 15 4 gx x 2 10x 16 f x x 3 3x 2 f x 3x3 15x2 18x f x 4x 3 4x2 15x f x 5x2 x3 f x x 2x 4 gt 1 t 22t 22 79. 4 x 12x 33 80. gx 1 10 77. 75. 74. 68. gx x 4 4x2 72. f x 1 x 3 76. 78. f x 48x2 3x4 hx 1 3x 3x 42 333202_0202.qxd 12/7/05 9:12 AM Page 150 150 Chapter 2 Polynomial and Rational Functions In Exercises 81–84, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero. 81. 82. 83. 84. f x x 3 4x f x 1 gx 1 5 hx 1 5 4x 4 2x 2 x 12x 32x 9 x 223x 52 In Exercises 85– 88, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. Adjust the table to approximate the zeros of the function. Use the zero or root feature of a graphing utility to verify your results. 85. 86. 87. 88. f x x 3 3x 2 3 f x 0.11x 3 2.07x 2 9.81x 6.88 gx 3x 4 4x 3 3 hx x 4 10x 2 3 90. Maximum Volume An open box with locking tabs is to be made from a square piece of material 24 inches on a side. This is to be done by cutting equal squares from the corners and folding along the dashed lines shown in the figure. x x x 24 ina) Verify that the volume of the box is given by the function Vx 8x6 x12 x. (b) Determine the domain of the function V. (c) Sketch a graph of the function and estimate the value of x for which Vx is maximum. 89. Numerical and Graphical Analysis An open box is to be made from a square piece of material, 36 inches on a side, by cutting equal squares with sides of length from the corners and turning up the sides (see figure). x 91. Construction A roofing contractor is fabricating gutters from 12-inch aluminum sheeting. The contractor plans to use an aluminum siding folding press to create the gutter by creasing equal lengths for the sidewalls (see figure). x x 12 − 2x x x 36 2− x x (a) Let x (a) Verify that the volume of the box is given by the function Vx x36 2x2. (b) Determine the domain of the function. (c) Use a graphing utility to create a table that shows the box height and the corresponding volumes Use the table to estimate the dimensions that will produce a maximum volume. V. x (d) Use a graphing utility to graph V Vx estimate the value of Compare your result with that of part (c). for which x and use the graph to is maximum. represent the height of the sidewall of the gutter. that represents the cross-sectional A Write a function area of the gutter. (b) The length of the aluminum sheeting is 16 feet. Write that represents the volume of one run of V a function gutter in terms of x. (c) Determine the domain of the function in part (b). (d) Use a graphing utility to create a table that shows the V. sidewall height Use the table to estimate the dimensions that will produce a maximum volume. and the corresponding volumes x (e) Use a graphing utility to graph Use the graph to is a maximum. estimate the value of Compare your result with that of part (d). for which V. Vx x (f) Would the value of change if the aluminum sheeting x were of different lengths? Explain. 333202_0202.qxd 12/7/05 9:12 AM Page 151 Section 2.2 Polynomial Functions of Higher Degree 151 92. Construction An industrial propane tank is formed by adjoining two hemispheres to the ends of a right circular cylinder. The length of the cylindrical portion of the tank is four times the radius of the hemispherical components (see figure). 4r r 96. Use the graphs of the models in Exercises 93 and 94 to write a short paragraph about the relationship between the median prices of homes in the two regions. Model It 97. Tree Growth The growth of a red oak tree is approx- imated by the function G 0.003t 3 0.137t 2 0.458t 0.839 (a) Write a function that represents the total volume V of the tank in terms of r. (b) Find the domain of the function. (c) Use a graphing utility to graph the function. (d) The total volume of the tank is to be 120 cubic feet. Use the graph from part (c) to estimate the radius and length of the cylindrical portion of the tank. Data Analysis: Home Prices In Exercise 93–96, use the table, which shows the median prices (in thousands of dollars) of new privately owned U.S. homes in the Midwest for the years 1997 through 2003.The y1 data can be approximated by the following models. and in the South y2 0.139t3 4.42t2 51.1t 39 0.056t3 1.73t2 23.8t 29 y1 y2 In the models, t ding to 1997. Department of Housing and Urban Development) represents the year, with correspon(Source: U.S. Census Bureau; U.S. t 7 Year, t 7 8 9 10 11 12 13 y1 150 158 164 170 173 178 184 y2 130 136 146 148 155 163 168 93. Use a graphing utility to plot the data and graph the model in the same viewing window. How closely does the y1 for model represent the data? 94. Use a graphing utility to plot the data and graph the model in the same viewing window. How closely does the y2 for model represent the data? 95. Use the models to predict the median prices of a new privately owned home in both regions in 2008. Do your answers seem reasonable? Explain. G is the height of the tree (in feet) and where 2 ≤ t ≤ 34 (a) Use a graphing utility to graph the function. (Hint: 10 ≤ x ≤ 45 is its age (in years). Use a viewing window in which and 5 ≤ y ≤ 60.) t (b) Estimate the age of the tree when it is growing most rapidly. This point is called the point of diminishing returns because the increase in size will be less with each additional year. (c) Using calculus, the point of diminishing returns can also be found by finding the vertex of the parabola given by y 0.009t2 0.274t 0.458. Find the vertex of this parabola. (d) Compare your results from parts (b) and (c). 98. Revenue The total revenue R (in millions of dollars) for a company is related to its advertising expense by the function R 1 100,000 x3 600x 2, 0 ≤ x ≤ 400 x is the amount spent on advertising (in tens of thouwhere sands of dollars). Use the graph of this function, shown in the figure, to estimate the point on the graph at which the function is increasing most rapidly. This point is called the point of diminishing returns because any expense above this amount will yield less return per dollar invested in advertising ( 350 300 250 200 150 100 50 100 200 300 400 Advertising expense (in tens of thousands of dollars) x 333202_0202.qxd 12/7/05 9:12 AM Page 152 152 Chapter 2 Polynomial and Rational Functions Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 99–101, determine whether 99. A fifth-degree polynomial can have five turning points in its graph. 100. It is possible for a sixth-degree polynomial to have only one solution. 101. The graph of the function given by fx 2 x x2 x3 x 4 x5 x6 x7 rises to the left and falls to the right. 102. Graphical Analysis For each graph, describe a polynomial function that could represent the graph. (Indicate the degree of the function and the sign of its leading coefficient.) (a) y (b) y (c) y (d) y x x x x 103. Graphical Reasoning Sketch a graph of the function f x x 4. Explain how the graph of each differs (if it does) from the graph of each is odd, even, or neither. given by g function f. function Determine whether g (a) (b) (c) (d) (e) (f ) (g) (h) gx f x 2 gx f x 2 gx f x gx fx gx f 1 2x 2 f x gx 1 gx f x34 gx f f x 104. Exploration Explore the transformations of the form gx ax h5 k. (a) Use a graphing utility to graph the functions given by y1 1 x 25 1 3 and y2 3 x 25 3. 5 Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of always be increasin
g or decreask? ing? If so, is this behavior determined by Explain. a, h, or g (c) Use a graphing utility to graph the function given by Hx x 5 3x 3 2x 1. Use the graph and the result of part (b) to determine whether form can be written Hx ax h5 k. Explain. the in H Skills Review In Exercises 105–108, factor the expression completely. 105. 107. 5x2 7x 24 4x 4 7x3 15x2 106. 108. 6x3 61x2 10x y3 216 In Exercises 109 –112, solve the equation by factoring. 109. 110. 111. 112. 2x2 x 28 0 3x2 22x 16 0 12x2 11x 5 0 x2 24x 144 0 In Exercises 113–116, solve the equation by completing the square. 113. 115. x2 2x 21 0 2x2 5x 20 0 114. 116. x2 8x 2 0 3x2 4x 9 0 In Exercises 117–122, describe the transformation from a Then sketch its graph. common function that occurs in f x. 117. 118. 119. 120. 121. 122. f x x 42 f x 3 x2 2x 9 f x 10 1 3 x 3 333202_0203.qxd 12/7/05 9:23 AM Page 153 2.3 Polynomial and Synthetic Division Section 2.3 Polynomial and Synthetic Division 153 What you should learn • Use long division to divide polynomials by other polynomials. • Use synthetic division to divide polynomials by binomials of x k the form . • Use the Remainder Theorem and the Factor Theorem. Why you should learn it Synthetic division can help you evaluate polynomial functions. For instance, in Exercise 73 on page 160, you will use synthetic division to determine the number of U.S. military personnel in 2008. © Kevin Fleming/Corbis Long Division of Polynomials In this section, you will study two procedures for dividing polynomials. These procedures are especially valuable in factoring and finding the zeros of polynomial functions. To begin, suppose you are given the graph of fx 6x3 19x 2 16x 4. f, f you know that Notice that a zero of occurs at is a zero of exists a second-degree polynomial fx x 2 qx. x 2, x 2 qx as shown in Figure 2.27. Because x 2 This means that there fx. is a factor of such that To find qx, you can use long division, as illustrated in Example 1. Example 1 Long Division of Polynomials 6x3 19x 2 16x 4 Divide mial completely. Solution by x 2, and use the result to factor the polyno- 6x2. Think Think Think 6x3 x 7x2 x 2x x 2. 7x. 6x2 7x 2 x 2 ) 6x3 19x2 16x 4 6x3 12x2 7x2 16x 7x2 14x 2x 4 2x 4 0 Multiply: 6x2x 2. Subtract. Multiply: 7xx 2. Subtract. Multiply: 2x 2. Subtract. y 1 ( 1 2 ) , 0 ( , 02 From this division, you can conclude that 6x3 19x 2 16x 4 x 26x2 7x 2 and by factoring the quadratic 6x2 7x 2, you have 6x3 19x 2 16x 4 x 22x 13x 2. f(x) = 6x3 − 19x2 + 16x − 4 Note that this factorization agrees with the graph shown in Figure 2.27 in that the x 2, x 1 2, three -intercepts occur at x 2 3. and x FIGURE 2.27 Now try Exercise 5. 333202_0203.qxd 12/7/05 9:23 AM Page 154 154 Chapter 2 Polynomial and Rational Functions x 2 In Example 1, 6x3 19x 2 16x 4, is a factor of the polynomial and the long division process produces a remainder of zero. Often, long division will produce a nonzero remainder. For instance, if you divide by x 1, you obtain the following. x2 3x 5 Divisor x 2 x 1 ) x2 3x 5 x2 x Quotient Dividend 2x 5 2x 2 3 Remainder In fractional form, you can write this result as follows. Remainder Dividend Quotient x 2 3x 5 x 1 x 2 3 x 1 Divisor Divisor This implies that x 2 3x 5 x 1(x 2 3 Multiply each side by x 1. which illustrates the following theorem, called the Division Algorithm. and f x dx The Division Algorithm If less than or equal to the degree of and rx f x dxqx rx such that are polynomials such that dx 0, and the degree of there exist unique polynomials dx is qx f x, Dividend Quotient Divisor Remainder where remainder rx 0 rx or the degree of dx is zero, rx divides evenly into f x. is less than the degree of dx. If the The Division Algorithm can also be written as f x dx qx rx dx. In the Division Algorithm, the rational expression the degree of hand, the rational expression than the degree of dx. rxdx f x is greater than or equal to the degree of f xdx is proper because the degree of is improper because On the other is less dx. rx 333202_0203.qxd 12/7/05 9:23 AM Page 155 Section 2.3 Polynomial and Synthetic Division 155 Before you apply the Division Algorithm, follow these steps. 1. Write the dividend and divisor in descending powers of the variable. 2. Insert placeholders with zero coefficients for missing powers of the variable. Example 2 Long Division of Polynomials Divide x3 1 by x 1. Solution Because there is no -term or -term in the dividend, you need to line up the subtraction by using zero coefficients (or leaving spaces) for the missing terms. x2 x x2 x 1 x 1 ) x 3 0x2 0x 1 x 3 x2 x2 0x x2 x x 1 x 1 0 x3 1, So, divides evenly into and you can write x 1 x3 1 x 1 x2 x 1, x 1. Now try Exercise 13. You can check the result of Example 2 by multiplying. x 1x2 x 1 x3 x2 x x2 x 1 x3 1 Example 3 Long Division of Polynomials Divide 2x4 4x3 5x 2 3x 2 by x 2 2x 3. Solution 1 x2 2x 3 ) 2x 4 4x 3 5x2 3x 2 2x2 2x 4 4x 3 6x2 x2 3x 2 x2 2x 3 x 1 Note that the first subtraction eliminated two terms from the dividend. When this happens, the quotient skips a term. You can write the result as 2x4 4x3 5x 2 3x 2 x 2 2x 3 2x 2 1 x 1 x 2 2x 3 . Now try Exercise 15. 333202_0203.qxd 12/7/05 9:23 AM Page 156 156 Chapter 2 Polynomial and Rational Functions Synthetic Division There is a nice shortcut for long division of polynomials when dividing by This shortcut is called synthetic division. The pattern divisors of the form for synthetic division of a cubic polynomial is summarized as follows. (The pattern for higher-degree polynomials is similar.) x k. Synthetic Division (for a Cubic Polynomial) To divide ax3 bx2 cx d x k, by use the following pattern. k a b ka c d Coefficients of dividend a r Remainder Coefficients of quotient Vertical pattern: Add terms. Diagonal pattern: Multiply by k. Synthetic division works only for divisors of the form x k x k. [Remember ] You cannot use synthetic division to divide a polynomial x k. that by a quadratic such as x 2 3. Example 4 Using Synthetic Division Use synthetic division to divide x 4 10x 2 2x 4 by x 3. Solution You should set up the array as follows. Note that a zero is included for the missing x3 -term in the dividend. 3 1 0 10 2 4 Then, use the synthetic division pattern by adding terms in columns and multiplying the results by 3. Divisor: x 3 Dividend: x 4 10x2 2x 4 3 1 1 0 3 3 10 9 1 2 3 1 4 3 1 Quotient: x3 3x2 x 1 Remainder: 1 So, you have x4 10x2 2x 4 x 3 x3 3x2 x 1 1 x 3 . Now try Exercise 19. 333202_0203.qxd 12/7/05 9:23 AM Page 157 Section 2.3 Polynomial and Synthetic Division 157 The Remainder and Factor Theorems The remainder obtained in the synthetic division process has an important interpretation, as described in the Remainder Theorem. The Remainder Theorem If a polynomial r f k. f x is divided by x k, the remainder is For a proof of the Remainder Theorem, see Proofs in Mathematics on page 213. The Remainder Theorem tells you that synthetic division can be used to f x as illustrated in evaluate a polynomial function. That is, to evaluate a polynomial function x k, when Example 5. The remainder will be x k. divide f k, f x by Example 5 Using the Remainder Theorem Use the Remainder Theorem to evaluate the following function at x 2. f x 3x3 8x2 5x 7 Solution Using synthetic division, you obtain the following. 2 3 3 8 6 2 Because the remainder is f 2 9. This means that substituting x 2 2, r f k you can conclude that is a point on the graph of You can check this by f. in the original function. Check f 2 323 822 52 7 38 84 10 7 9 Now try Exercise 45. Another important theorem is the Factor Theorem, stated below. This theoas a factor rem states that you can test to see whether a polynomial has by evaluating the polynomial at If the result is 0, is a factor. x k x k x k. The Factor Theorem A polynomial f x has a factor x k if and only if f k 0. For a proof of the Factor Theorem, see Proofs in Mathematics on page 213. 333202_0203.qxd 12/7/05 9:23 AM Page 158 158 Chapter 2 Polynomial and Rational Functions Example 6 Factoring a Polynomial: Repeated Division Show that x 2 and x 3 are factors of fx 2x4 7x3 4x2 27x 18. Then find the remaining factors of f x. Solution Using synthetic division with the factor 18 18 27 36 4 22 7 4 2 2 x 2, you obtain the following. 2 11 18 9 0 0 remainder, so x 2 is a factor. f 2 0 and Take the result of this division and perform synthetic division again using the factor x 3. f(x) = 2x 4 + 7x3 − 4x 2 − 27x − 18 y 3 2 2 11 6 5 18 15 9 9 3 0 0 remainder, so x 3 and is a factor. f 3 0 40 30 20 10 2( 3 − , 0 ( −4 −1 (−1, 0) (−3, 0) −20 −30 −40 Because the resulting quadratic expression factors as (2, 0) 1 3 4 2x 2 5x 3 2x 3x 1 x the complete factorization of fx is fx x 2x 32x 3x 1. Note that this factorization implies that has four real zeros: x 2, x 3, x 3 2, and f x 1. FIGURE 2.28 This is confirmed by the graph of which is shown in Figure 2.28. f, Now try Exercise 57. Uses of the Remainder in Synthetic Division f x The remainder obtained in the synthetic division of provides the following information. r, by x k, 1. The remainder gives the value of at f x k. That is, r f k. r r 0, x k r 0, k, 0 2. If 3. If is a factor of f x. is an -intercept of the graph of f. x Throughout this text, the importance of developing several problem-solving strategies is emphasized. In the exercises for this section, try using more than one x k strategy to solve several of the exercises. For instance, if you find that f. (with no remainder), try sketching the graph of You divides evenly into should find that is an -intercept of the graph. k, 0 f x x 333202_0203.qxd 12/7/05 9:23 AM Page 159 2.3 Exercises Section 2.3 Polynomial and Synthetic Division 159 VOCABULARY CHECK: 1. Two forms of the Division Algorithm are shown below. Identify and label each term or function. qx rx dx f x dxqx rx f x dx In Exercises 2–5, fill in the blanks. 2. The rational expression pxqx is called ________ if the degree of the numerator is greater than or equal to that of the denominator,
and is called ________ if the degree of the numerator is less than that of the denominator. 3. An alternative method to long division of polynomials is called ________ ________, in which the divisor must be of the form x k. 4. The ________ Theorem states that a polynomial f x has a factor x k if and only if f k 0. 5. The ________ Theorem states that if a polynomial f x is divided by x k, the remainder is r f k. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. Analytical Analysis to verify that y1 y2. In Exercises 1 and 2, use long division 1. y1 x2 , x 2 y2 x 2 4 2. y1 x4 3x 2 1 , x2 5 y2 x 2 x 2 8 39 x2 5 Graphical Analysis In Exercises 3 and 4, (a) use a graphing utility to graph the two equations in the same viewing window, (b) use the graphs to verify that the expressions are equivalent, and (c) use long division to verify the results algebraically. 3. y1 4. y1 , y2 x5 3x3 x2 1 x3 2x2 5 , x2 x 1 x3 4x 4x x2 1 x 3 2x 4 x2 x 1 y2 In Exercises 5 –18, use long division to divide. 5. 6. 7. 8. 9. 10. 11. 13. 14. 15. 16. 2x2 10x 12 x 3 5x2 17x 12 x 4 4x3 7x 2 11x 5 4x 5 6x3 16x 2 17x 6 3x 2 x4 5x3 6x2 x 2 x 2 x3 4x 2 3x 12 x 3 7x 3 x 2 12. 6x3 10x2 x 8 2x2 1 x3 9 x2 1 x4 3x2 1 x2 2x 3 x5 7 x3 1 8x 5 2x 1 17. x 4 x 13 18. 2x3 4x2 15x 5 x 12 In Exercises 19 –36, use synthetic division to divide. 19. 20. 21. 22. 23. 24. 25. 26. 27. 29. 31. 33. 35. 3x3 17x2 15x 25 x 5 5x3 18x2 7x 6 x 3 4x3 9x 8x2 18 x 2 9x3 16x 18x2 32 x 2 x3 75x 250 x 10 3x3 16x2 72 x 6 5x3 6x2 8 x 4 5x3 6x 8 x 2 10x4 50x3 800 x 6 28. x5 13x4 120x 80 x 3 x3 512 x 8 3x4 x 2 180x x4 x 6 4x3 16x2 23x 15 x 1 2 30. 32. 34. 36. x 3 729 x 9 3x 4 x 2 5 3x 2x2 x3 x 1 3x3 4x2 5 x 3 2 In Exercises 37– 44, write the function in the form f x x kqx r for the given value of and demonstrate that f k r. k, Function fx x3 x2 14x 11 fx x3 5x2 11x 8 37. 38. Value of k k 4 k 2 333202_0203.qxd 12/7/05 9:23 AM Page 160 160 Chapter 2 Polynomial and Rational Functions Function fx 15x 4 10x3 6x2 14 fx 10x3 22x2 3x 4 fx x3 3x2 2x 14 fx x 3 2x2 5x 4 fx 4x3 6x2 12x 4 fx 3x3 8x2 10x 8 39. 40. 41. 42. 43. 44. Value of In Exercises 45–48, use synthetic division to find each function value. Verify your answers using another method. 45. f x 4x3 13x 10 f 2 f 1 (b) (a) 46. gx x6 4x4 3x2 2 (c) f 1 2 (d) f 8 47. g2 g4 (a) (b) hx 3x3 5x2 10x 1 (b) h3 h 1 (a) 3 (c) (c) g3 (d) g1 h2 (d) h5 Function f x 6x3 41x2 9x 14 f x 10x3 11x2 72x 45 f x 2x3 x2 10x 5 f x x3 3x2 48x 144 61. 62. 63. 64. Factors 2x 1, 2x 5, 2x 1, x 43, 3x 2 5x 3 x5 x 3 Graphical Analysis In Exercises 65–68, (a) use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places, (b) determine one of the exact zeros, and (c) use synthetic division to verify your result from part (b), and then factor the polynomial completely. 65. 66. 67. 68. f x x3 2x2 5x 10 gx x3 4x2 2x 8 ht t3 2t 2 7t 2 f s s3 12s2 40s 24 48. f x 0.4x4 1.6x3 0.7x2 2 (a) f 1 (b) f 2 (c) f 5 (d) f 10 In Exercises 49–56, use synthetic division to show that is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. x In Exercises 69–72, simplify the rational expression by using long division or synthetic division. 69. 71. 4x3 8x2 x 3 2x 3 x4 6x3 11x 2 6x x2 3x 2 70. 72. x3 x2 64x 64 x 8 x4 9x3 5x 2 36x 4 x2 4 Polynomial Equation 49. 50. 51. 52. 53. 54. 55. 56. x3 7x 6 0 x3 28x 48 0 2x3 15x 2 27x 10 0 48x3 80x 2 41x 6 0 x3 2x 2 3x 6 0 x3 2x 2 2x 4 0 x3 3x 2 2 0 x3 x 2 13x 3 0 Value of Model It 73. Data Analysis: Military Personnel The numbers M (in thousands) of United States military personnel on active duty for the years 1993 through 2003 are shown t 3 in the table, where corresponding to 1993. (Source: U.S. Department of Defense) represents the year, with t f, In Exercises 57– 64, (a) verify the given factors of the funcf, (c) use your results (b) find the remaining factors of tion f, to write the complete factorization of (d) list all real zeros of and (e) confirm your results by using a graphing utility to graph the function. f, 57. 58. 59. 60. Function f x 2x3 x2 5x 2 f x 3x3 2x2 19x 6 f x x 4 4x3 15x2 58x 40 f x 8x 4 14x3 71x2 10x 24 Factors x 2, x 3, x 5, x 1 x 2 x 4 x 2, x 4 Year, t Military personnel, M 3 4 5 6 7 8 9 10 11 12 13 1705 1611 1518 1472 1439 1407 1386 1384 1385 1412 1434 333202_0203.qxd 12/7/05 9:23 AM Page 161 Model It (co n t i n u e d ) (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of the graphing utility to find a cubic model for the data. Graph the model in the same viewing window as the scatter plot. (c) Use the model to create a table of estimated values of M. Compare the model with the original data. (d) Use synthetic division to evaluate the model for the year 2008. Even though the model is relatively accurate for estimating the given data, would you use this model to predict the number of military personnel in the future? Explain. R 74. Data Analysis: Cable Television The average monthly (in dollars) for cable television in the United basic rates States for the years 1992 through 2002 are shown in the table, where represents the year, with corresponding to 1992. t 2 (Source: Kagan Research LLC) t Year, t Basic rate 10 11 12 19.08 19.39 21.62 23.07 24.41 26.48 27.81 28.92 30.37 32.87 34.71 (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of the graphing utility to find a cubic model for the data. Then graph the model in the same viewing window as the scatter plot. Compare the model with the data. (c) Use synthetic division to evaluate the model for the year 2008. Synthesis True or False? statement is true or false. Justify your answer. In Exercises 75–77, determine whether the 75. If 7x 4 is a zero of f. 4 7 is a factor of some polynomial function f, then Section 2.3 Polynomial and Synthetic Division 161 76. 2x 1 is a factor of the polynomial 6x 6 x 5 92x 4 45x 3 184x2 4x 48. 77. The rational expression x3 2x2 13x 10 x2 4x 12 is improper. 78. Exploration Use the form f x x kqx r to create a cubic function that (a) passes through the point 2, 5 and rises to the right, and (b) passes through the point 3, 1 and falls to the right. (There are many correct answers.) Think About It division by assuming that n is a positive integer. In Exercises 79 and 80, perform the 79. x3n 9x2n 27xn 27 xn 3 80. x3n 3x2n 5xn 6 x n 2 81. Writing Briefly explain what it means for a divisor to divide evenly into a dividend. 82. Writing Briefly explain how to check polynomial divi- sion, and justify your reasoning. Give an example. Exploration In Exercises 83 and 84, find the constant c such that the denominator will divide evenly into the numerator. 83. x3 4x2 3x c x 5 84. x5 2x2 x c x 2 Think About It questions about the division f x x 32x 3x 13. In Exercises 85 and 86, answer the where f x x k, 85. What is the remainder when k 3? Explain. 86. If it is necessary to find f2, function directly or to use synthetic division? Explain. is it easier to evaluate the Skills Review In Exercises 87–92, use any method to solve the quadratic equation. 87. 89. 91. 9x2 25 0 5x2 3x 14 0 2x2 6x 3 0 88. 90. 92. 16x2 21 0 8x2 22x 15 0 x2 3x 3 0 In Exercises 93– 96, find a polynomial function that has the given zeros. (There are many correct answers.) 93. 95. 0, 3, 4 3, 1 2, 1 2 94. 96. 6, 1 1, 2, 2 3, 2 3 333202_0204.qxd 12/7/05 9:30 AM Page 162 162 Chapter 2 Polynomial and Rational Functions 2.4 Complex Numbers What you should learn • Use the imaginary unit i to write complex numbers. • Add, subtract, and multiply complex numbers. • Use complex conjugates to write the quotient of two complex numbers in standard form. • Find complex solutions of quadratic equations. Why you should learn it You can use complex numbers to model and solve real-life problems in electronics. For instance, in Exercise 83 on page 168, you will learn how to use complex numbers to find the impedance of an electrical circuit. The Imaginary Unit i x 2 1 0 You have learned that some quadratic equations have no real solutions. For instance, the quadratic equation has no real solution because there is x To overcome this deficienno real number cy, mathematicians created an expanded system of numbers using the imaginary i, unit defined as i 1 that can be squared to produce Imaginary unit 1. i 2 1. where By adding real numbers to real multiples of this imaginary unit, the set of complex numbers is obtained. Each complex number can be written in the standard form For instance, the standard form of the complex num5 9 is ber 5 9 5 321 5 31 5 3i. a bi. 5 3i because In the standard form a bi, bi complex number the imaginary part of the complex number. and the number the real number a bi, a (where is called the real part of the is a real number) is called b a b Definition of a Complex Number If and are real numbers, the number is said to be written in standard form. If b 0, a bi a real number. If bi, A number of the form where the number b 0, a bi b 0, is a complex number, and it the number is is called an imaginary number. is called a pure imaginary number. a bi a The set of real numbers is a subset of the set of complex numbers, as shown in Figure 2.29. This is true because every real number can be written as a complex a a 0i. number using That is, for every real number you can write b 0. a a, © Richard Megna/Fundamental Photographs Real numbers Imaginary numbers FIGURE 2.29 Complex numbers Equality of Complex Numbers Two complex numbers equal to each other a bi and c di, written in standard form, are a bi c di Equality of two complex numbers if and only if a c and b d. 333202_0204.qxd 12/7/05 9:30 AM Page 163 Section 2.4 Complex Numbers 163 Operations with Complex Numbers To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately. Addition and Su
btraction of Complex Numbers If their sum and difference are defined as follows. are two complex numbers written in standard form, a bi c di and a bi c di a c b di Sum: Difference: a bi c di a c b di The additive identity in the complex number system is zero (the same as in the real number system). Furthermore, the additive inverse of the complex number a bi is (a bi) a bi. Additive inverse So, you have a bi a bi 0 0i 0. Example 1 Adding and Subtracting Complex Numbers a. 4 7i 1 6i 4 7i 1 6i (4 1) (7i 6i) 5 i b. (1 2i) 4 2i 1 2i 4 2i 1 4 2i 2i 3 0 3 Remove parentheses. Group like terms. Write in standard form. Remove parentheses. Group like terms. Simplify. Write in standard form. c. 3i 2 3i 2 5i 3i 2 3i 2 5i 2 2 3i 3i 5i 0 5i 5i d. 3 2i 4 i 7 i 3 2i 4 i 7 i 3 4 7 2i i i 0 0i 0 Now try Exercise 17. Note in Examples 1(b) and 1(d) that the sum of two complex numbers can be a real number. 333202_0204.qxd 12/7/05 9:30 AM Page 164 164 Chapter 2 Polynomial and Rational Functions Exploration Complete the following. i1 i i 2 1 i3 i10 i11 i12 What pattern do you see? Write a brief description of how you i would find raised to any positive integer power. Many of the properties of real numbers are valid for complex numbers as well. Here are some examples. Associative Properties of Addition and Multiplication Commutative Properties of Addition and Multiplication Distributive Property of Multiplication Over Addition Notice below how these properties are used when two complex numbers are multiplied. a bic di ac di bic di ac adi bci bdi 2 ac adi bci bd1 ac bd adi bci ac bd ad bci Distributive Property Distributive Property i2 1 Commutative Property Associative Property Rather than trying to memorize this multiplication rule, you should simply remember how the Distributive Property is used to multiply two complex numbers. The procedure described above is similar to multiplying two polynomials and combining like terms, as in the FOIL Method shown in Appendix A.3. For instance, you can use the FOIL Method to multiply the two complex numbers from Example 2(b). F L 2 i4 3i 8 6i 4i 3i2 O I Example 2 Multiplying Complex Numbers a. 42 3i 42 43i 8 12i b. 2 i4 3i 24 3i i4 3i 8 6i 4i 3i 2 8 6i 4i 31 8 3 6i 4i 11 2i (3 2i)(3 2i) 33 2i 2i3 2i c. 9 6i 6i 4i 2 9 6i 6i 41 9 4 13 d. 3 2i2 3 2i3 2i 33 2i 2i3 2i 9 6i 6i 4i 2 9 6i 6i 41 9 12i 4 5 12i Now try Exercise 27. Distributive Property Simplify. Distributive Property Distributive Property i2 1 Group like terms. Write in standard form. Distributive Property Distributive Property i2 1 Simplify. Write in standard form. Square of a binomial Distributive Property Distributive Property i2 1 Simplify. Write in standard form. 333202_0204.qxd 12/7/05 9:30 AM Page 165 Section 2.4 Complex Numbers 165 Complex Conjugates Notice in Example 2(c) that the product of two complex numbers can be a real number. This occurs with pairs of complex numbers of the form and a bi, called complex conjugates. a bi a bia bi a2 abi abi b2i 2 a2 b21 a2 b2 Example 3 Multiplying Conjugates Multiply each complex number by its complex conjugate. a. 1 i b. 4 3i Solution a. The complex conjugate of 1 i1 i 12 i 2 b. The complex conjugate of is 4 3i4 3i 42 3i2 4 3i 16 91 25 1 i is 1 i. 1 1 4 3i. 16 9i 2 2 Note that when you multiply the numerator and denominator of a quotient of complex numbers by c di c di you are actually multiplying the quotient by a form of 1. You are not changing the original expression, you are only creating an expression that is equivalent to the original expression. Now try Exercise 37. To write the quotient of d are not both zero, multiply the numerator and denominator by the complex conjugate of the denominator to obtain in standard form, where and and c a bi c di a bi c di c di a bi c di c di ac bd bc adi c 2 d 2 . Standard form Example 4 Writing a Quotient of Complex Numbers in Standard Form 2 3i 4 2i 2 3i 4 2i 4 2i 4 2i Multiply numerator and denominator by complex conjugate of denominator. 8 4i 12i 6i 2 16 4i 2 8 6 16i 16 4 2 16i 20 4 5 1 10 i Now try Exercise 49. Expand. i2 1 Simplify. Write in standard form. 333202_0204.qxd 12/7/05 9:30 AM Page 166 166 Chapter 2 Polynomial and Rational Functions Complex Solutions of Quadratic Equations When using the Quadratic Formula to solve a quadratic equation, you often obtain a result such as which you know is not a real number. By factoring out you can write this number in standard form. 3, i 1, 3 31 31 3i The number 3i is called the principal square root of 3. Principal Square Root of a Negative Number a If a is a positive number, the principal square root of the negative number is defined as a ai. The definition of principal square root uses the rule ab ab and a > 0 b < 0. a This rule b and are for is not valid if both negative. For example, 55 5151 5i5i 25i 2 5i 2 5 whereas 55 25 5. To avoid problems with square roots of negative numbers, be sure to convert complex numbers to standard form before multiplying. Example 5 Writing Complex Numbers in Standard Form a. b. c. 312 3 i12 i 36 i 2 61 6 48 27 48i 27 i 43i 33i 3 i 1 32 1 3i2 12 23i 32i 2 1 23i 31 2 23i Now try Exercise 59. Example 6 Complex Solutions of a Quadratic Equation Solve (a) x2 4 0 and (b) 3x 2 2x 5 0. Solution a. x2 4 0 x2 4 x ±2i 3x2 2x 5 0 b. x 2 ± 22 435 23 2 ± 56 6 2 ± 214i 6 1 3 ± 14 3 i Now try Exercise 65. Write original equation. Subtract 4 from each side. Extract square roots. Write original equation. Quadratic Formula Simplify. Write 56 in standard form. Write in standard form. 333202_0204.qxd 12/7/05 9:30 AM Page 167 Section 2.4 Complex Numbers 167 2.4 Exercises VOCABULARY CHECK: 1. Match the type of complex number with its definition. (a) Real Number (b) Imaginary number (c) Pure imaginary number (i) (ii) (iii) a bi, a bi, a bi, a 0, a 0, b 0 b 0 b 0 In Exercises 2–5, fill in the blanks. 2. The imaginary unit i is defined as i ________, where i2 ________. 3. If a is a positive number, the ________ ________ root of the negative number a is defined as 4. The numbers a bi and a bi are called ________ ________, and their product is a real number a a i. a2 b2. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, find real numbers and equation is true. a b such that the 1. 3. 4. a bi 10 6i a 1 b 3i 5 8i a 6 2bi 6 5i 2. a bi 13 4i In Exercises 5–16, write the complex number in standard form. 5. 7. 9. 4 9 2 27 75 11. 8 13. 15. 6i i 2 0.09 6. 8. 10. 3 16 1 8 4 12. 45 14. 16. 4i 2 2i 0.0004 In Exercises 17–26, perform the addition or subtraction and write the result in standard form. 17. 19. 21. 22. 23. 25. 26. 18. 5 i 6 2i 8 i 4 i 20. 2 8 5 50 8 18 4 32 i 13i 14 7i 3 2i 5 5 3 1.6 3.2i 5.8 4.3i 3 i 11 24. 2 13 2i 5 6i 3 2i 6 13i 22 5 8i 10i 27. In Exercises 27–36, perform the operation and write the result in standard form. 1 i3 2i 6i5 2i 14 10 i14 10 i 6 2i2 3i 8i9 4i 31. 29. 28. 30. 32. 33. 35. 3 15 i3 15 i 4 5i2 2 3i2 2 3i2 34. 36. 2 3i2 1 2i2 1 2i2 In Exercises 37– 44, write the complex conjugate of the complex number.Then multiply the number by its complex conjugate. 37. 39. 41. 43. 6 3i 1 5 i 20 8 38. 40. 42. 44. 7 12i 3 2 i 15 1 8 In Exercises 45–54, write the quotient in standard form. 45. 47. 49. 51. 53. 5 i 2 4 5i 3 i 3 i 6 5i i 3i 4 5i 2 46. 48. 50. 52. 54. 14 2i 5 1 i 6 7i 1 2i 8 16i 2i 5i 2 3i2 In Exercises 55–58, perform the operation and write the result in standard form. 55. 57. 3 1 i 2i 2 1 i i 3 2i 3 8i 56. 58. 2i 333202_0204.qxd 12/7/05 2:51 PM Page 168 168 Chapter 2 Polynomial and Rational Functions In Exercises 59–64, write the complex number in standard form. 59. 61. 63. 6 2 102 3 57 10 60. 62. 64. 5 10 752 2 62 84. Cube each complex number. (a) 2 (b) 1 3 i (c) 1 3 i 85. Raise each complex number to the fourth power. (a) 2 (b) 2 (c) 2i 86. Write each of the powers of as 2i (d) i, i, 1, (d) i 67 or 1. i i 50 In Exercises 65–74, use the Quadratic Formula to solve the quadratic equation. Synthesis (a) i 40 (b) i 25 (c) 65. 67. 69. 71. 73. x 2 2x 2 0 4x 2 16x 17 0 4x 2 16x 15 0 3 2 x2 6x 9 0 1.4x2 2x 10 0 66. 68. 70. 72. 74. x 2 6x 10 0 9x 2 6x 37 0 16t 2 4t 3 0 7 8 x2 3 4x 5 0 4.5x2 3x 12 0 16 In Exercises 75–82, simplify the complex number and write it in standard form. 75. 77. 79. 81. 6i 3 i 2 5i 5 753 1 i 3 76. 78. 80. 82. 4i 2 2i 3 i 3 26 1 2i 3 Model It 83. Impedance The opposition to current in an electrical circuit is called its impedance. The impedance in a parallel circuit with two pathways satisfies the equation z 1 z 1 z1 1 z 2 z1 is the impedance of pathway 2. is the impedance (in ohms) of pathway 1 and where z2 (a) The impedance of each pathway in a parallel circuit is found by adding the impedances of all compoz2. nents in the pathway. Use the table to find and z1 (b) Find the impedance z. Resistor Inductor Capacitor Symbol Impedance aΩ a bΩ bi cΩ ci 1 16 Ω 2 20 Ω 9 Ω 10 Ω True or False? statement is true or false. Justify your answer. In Exercises 87– 89, determine whether the 87. There is no complex number that is equal to its complex 88. conjugate. i6 i 44 i 150 i 74 i 109 i61 1 90. Error Analysis Describe the error. is a solution of 89. x 4 x2 14 56. 66 66 36 6 91. Proof Prove that the complex conjugate of the product is the b1i of two complex numbers product of their complex conjugates. b2i and a2 a1 92. Proof Prove that the complex conjugate of the sum of b1i is the sum of b2i and a2 a1 two complex numbers their complex conjugates. Skills Review In Exercises 93–96, perform the operation and write the result in standard form. 93. 94. 95. 4 3x 8 6x x 2 x3 3x2 6 2x 4x 2 3x 1 x 4 2 96. 2x 52 In Exercises 97–100, solve the equation and check your solution. 97. 99. 100. x 12 19 45x 6 36x 1 0 5x 3x 11 20x 15 98. 8 3x 34 101. Volume of an Oblate Spheroid a Solve for : V 4 3 102. Newton’s Law of Universal Gravitation a2b r Solve for : F m1m2 r 2 103. Mixture Problem A five-liter container contains a mixture with a concentration of 50%. How much of this mixture must be withdrawn and
replaced by 100% concentrate to bring the mixture up to 60% concentration? 333202_0205.qxd 12/7/05 9:36 AM Page 169 2.5 Zeros of Polynomial Functions Section 2.5 Zeros of Polynomial Functions 169 What you should learn • Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. • Find rational zeros of polyno- mial functions. • Find conjugate pairs of com- plex zeros. • Find zeros of polynomials by factoring. • Use Descartes’s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. Why you should learn it Finding zeros of polynomial functions is an important part of solving real-life problems. For instance, in Exercise 112 on page 182, the zeros of a polynomial function can help you analyze the attendance at women’s college basketball games. set f x, Recall that in order to find the f x zeros of a function equal to 0 and solve the resulting equation for For instance, the function in Example 1(a) has a zero at because x. x 2 x 2 0 x 2. The Fundamental Theorem of Algebra n th-degree polynomial can have at most You know that an real zeros. In the complex number system, this statement can be improved. That is, in the complex number system, every th-degree polynomial function has precisely zeros. This important result is derived from the Fundamental Theorem of Algebra, first proved by the German mathematician Carl Friedrich Gauss (1777–1855). n n n The Fundamental Theorem of Algebra n, n > 0, If is a polynomial of degree where zero in the complex number system. f x then has at least one f Using the Fundamental Theorem of Algebra and the equivalence of zeros and factors, you obtain the Linear Factorization Theorem. f x is a polynomial of degree where Linear Factorization Theorem If linear factors f x an n, x c1 c1, c2, . . . , cn . . . x cn x c2 are complex numbers. where n > 0, then has precisely f n For a proof of the Linear Factorization Theorem, see Proofs in Mathematics on page 214. Note that the Fundamental Theorem of Algebra and the Linear Factorization Theorem tell you only that the zeros or factors of a polynomial exist, not how to find them. Such theorems are called existence theorems. Example 1 Zeros of Polynomial Functions a. The first-degree polynomial b. Counting multiplicity, the second-degree polynomial function has exactly one zero: f x x 2 x 2. f x x 2 6x 9 x 3x 3 x 3 x 3. has exactly two zeros: and (This is called a repeated zero.) c. The third-degree polynomial function f x x3 4x xx 2 4 xx 2ix 2i x 0, x 2i, d. The fourth-degree polynomial function has exactly three zeros: x 2i. and f x x4 1 x 1x 1x ix i has exactly four zeros: x 1, x 1, x i, and x i. Now try Exercise 1. 333202_0205.qxd 12/7/05 9:36 AM Page 170 170 Chapter 2 Polynomial and Rational Functions The Rational Zero Test The Rational Zero Test relates the possible rational zeros of a polynomial (having integer coefficients) to the leading coefficient and to the constant term of the polynomial Historical Note Although they were not contemporaries, Jean Le Rond d’Alembert (1717–1783) worked independently of Carl Gauss in trying to prove the Fundamental Theorem of Algebra. His efforts were such that, in France, the Fundamental Theorem of Algebra is frequently known as the Theorem of d’Alembert. The Rational Zero Test If the polynomial has integer coefficients, every rational zero of has the form f x anx n an1x n1 . . . a 2 x 2 f a1x a0 Rational zero p q where and have no common factors other than 1, and q p p a factor of the constant term a0 q a factor of the leading coefficient an. To use the Rational Zero Test, you should first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. Possible rational zeros factors of constant term factors of leading coefficient Having formed this list of possible rational zeros, use a trial-and-error method to determine which, if any, are actual zeros of the polynomial. Note that when the leading coefficient is 1, the possible rational zeros are simply the factors of the constant term. Example 2 Rational Zero Test with Leading Coefficient of 1 Find the rational zeros of f x x3 x 1. f(x) = x3 + x + 1 y Solution Because the leading coefficient is 1, the possible rational zeros are the factors of the constant term. By testing these possible zeros, you can see that neither works. ±1, f 1 13 1 1 3 f 1 13 1 1 1 2 3 x 1 So, you can conclude that the given polynomial has no rational zeros. Note from the graph of and 0. However, by the Rational Zero Test, you know that this real zero is not a rational number. in Figure 2.30 that does have one real zero between 1 f f −3 −2 3 2 1 −1 −2 −3 FIGURE 2.30 Now try Exercise 7. 333202_0205.qxd 12/7/05 9:36 AM Page 171 When the list of possible rational zeros is small, as in Example 2, it may be quicker to test the zeros by evaluating the function. When the list of possible rational zeros is large, as in Example 3, it may be quicker to use a different approach to test the zeros, such as using synthetic division or sketching a graph. Section 2.5 Zeros of Polynomial Functions 171 Example 3 Rational Zero Test with Leading Coefficient of 1 Find the rational zeros of f x x4 x3 x 2 3x 6. Solution Because the leading coefficient is 1, the possible rational zeros are the factors of the constant term. Possible rational zeros: ±1, ±2, ±3, ±6 By applying synthetic division successively, you can determine that x 2 1 are the only two rational zeros and 1 2 3 6 0 0 remainder, so x 1 is a zero So, factors as f x f x x 1x 2x 2 3. 0 remainder, so x 2 is a zero. Because the factor x 1 x 2 and x 2 3 produces no real zeros, you can conclude that are the only real zeros of which is verified in Figure 2.31. f, 0) (2, 0) −8 −6 −4 −2 4 6 8 x −6 −8 FIGURE 2.31 Now try Exercise 11. If the leading coefficient of a polynomial is not 1, the list of possible rational zeros can increase dramatically. In such cases, the search can be shortened in several ways: (1) a programmable calculator can be used to speed up the calculations; (2) a graph, drawn either by hand or with a graphing utility, can give a good estimate of the locations of the zeros; (3) the Intermediate Value Theorem along with a table generated by a graphing utility can give approximations of zeros; and (4) synthetic division can be used to test the possible rational zeros. Finding the first zero is often the most difficult part. After that, the search is simplified by working with the lower-degree polynomial obtained in synthetic division, as shown in Example 3. 333202_0205.qxd 12/7/05 9:36 AM Page 172 172 Chapter 2 Polynomial and Rational Functions Remember that when you try to find the rational zeros of a polynomial function with many possible rational zeros, as in Example 4, you must use trial and error. There is no quick algebraic method to determine which of the possibilities is an actual zero; however, sketching a graph may be helpful. y 15 10 5 −5 −10 3 f x ( ) = 10 + 15 + 16 x x 2 − − x 12 FIGURE 2.32 Example 4 Using the Rational Zero Test Find the rational zeros of f x 2x3 3x 2 8x 3. Solution The leading coefficient is 2 and the constant term is 3. Possible rational zeros: Factors of 3 Factors of 2 By synthetic division, you can determine that ±1, ±3 ±1, ±2 x 1 ±1, ±3, ± 1 2 , ± 3 2 is a rational zero So, factors as f x f x x 12x 2 5x 3 x 12x 1x 3 and you can conclude that the rational zeros of are f x 1, x 1 2, and x 3. Now try Exercise 17. Recall from Section 2.2 that if x a is a solution of the polynomial equation x a then f x 0. is a zero of the polynomial function f, Solution The leading coefficient is 10 Possible rational solutions: 12. and the constant term is Factors of 12 Factors of 10 ±1, ±2, ±3, ±4, ±6, ±12 ±1, ±2, ±5, ±10 With so many possibilities (32, in fact), it is worth your time to stop and sketch a graph. From Figure 2.32, it looks like three reasonable solutions would be x 6 x 2 5, is the only rational solution. So, you have x 210x2 5x 6 0. Testing these by synthetic division shows that x 1 2, x 2. and Using the Quadratic Formula for the second factor, you find that the two additional solutions are irrational numbers. x 5 265 20 1.0639 and x 5 265 20 0.5639 Now try Exercise 23. Example 5 Solving a Polynomial Equation 1 x Find all the real solutions of 10x3 15x2 16x 12 0. 333202_0205.qxd 12/7/05 9:36 AM Page 173 Section 2.5 Zeros of Polynomial Functions 173 Conjugate Pairs In Example 1(c) and (d), note that the pairs of complex zeros are conjugates. That is, they are of the form a bi. a bi and Complex Zeros Occur in Conjugate Pairs f x Let b 0, function. is a zero of the function, the conjugate a bi be a polynomial function that has real coefficients. If a bi, where is also a zero of the Be sure you see that this result is true only if the polynomial function has real f x coefficients. For instance, the result applies to the function given by x2 1 but not to the function given by gx x i. Example 6 Finding a Polynomial with Given Zeros Find a fourth-degree polynomial function with real coefficients that has and as zeros. 3i 1, 1, Solution 3i Because know that the conjugate f x Theorem, can be written as is a zero and the polynomial is stated to have real coefficients, you must also be a zero. So, from the Linear Factorization 3i f x ax 1x 1x 3ix 3i. For simplicity, let to obtain f x x2 2x 1x2 9 a 1 x4 2x3 10x 2 18x 9. Now try Exercise 37. Factoring a Polynomial The Linear Factorization Theorem shows that you can write any n linear factors. polynomial as the product of x c3 x c2 . . . x cn f x an x c1 n th-degree However, this result includes the possibility that some of the values of are complex. The following theorem says that even if you do not want to get involved with “complex factors,” you can still write as the product of linear and/or quadratic factors. For a proof of this theorem, see Proofs in Mathematics on page 214. f x ci Factors of a Polynomial with real coe
fficients can be written as Every polynomial of degree the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. n > 0 333202_0205.qxd 12/7/05 9:36 AM Page 174 174 Chapter 2 Polynomial and Rational Functions A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Be sure you see that this is not the same as being irreducible over the rationals. For example, the quadratic is irreducible over the reals (and therefore over the rationals). On the other hand, the quadratic x2 2 x 2 x 2 is irreducible over the rationals but reducible over the reals. x2 1 x ix i Example 7 Finding the Zeros of a Polynomial Function Find all the zeros of zero of f. f x x4 3x3 6x2 2x 60 given that 1 3i is a Algebraic Solution Because complex zeros occur in conjugate pairs, you know that 1 3i is also a zero of This means that both f. x 1 3i and x 1 3i are factors of Multiplying these two factors produces f. x 1 3ix 1 3i x 1 3ix 1 3i Graphical Solution Because complex zeros always occur in conjugate pairs, you know that is also a zero of f. Because the polynomial is a fourth-degree polynomial, you know that there are at most two other zeros of the function. Use a graphing utility to graph 1 3i y x4 3x3 6x2 2x 60 as shown in Figure 2.33. x 12 9i 2 x 2 2x 10. x 2 2x 10 into Using long division, you can divide the following. x2 x 6 x2 2x 10 ) x 4 3x3 6x2 2x 60 x 4 2x3 10x2 x3 4x2 2x x3 2x2 10x 6x2 12x 60 6x2 12x 60 0 So, you have f x x2 2x 10x2 x 6 x2 2x 10x 3x 2 and you can conclude that the zeros of x 1 3i, and x 2. x 3, f are x 1 3i, Now try Exercise 47. f to obtain y = x4 − 3x3 + 6x2 + 2x − 60 80 −80 −4 FIGURE 2.33 5 2 You can see that and 3 appear to be zeros of the graph of the function. Use the zero or root feature or the zoom and trace features of the and graphing utility to confirm that x 3 are zeros of the graph. So, you can x 1 3i, conclude that the zeros of x 1 3i, and x 2. x 2 x 3, are f In Example 7, if you were not told that is a zero of you could still 2 find all zeros of the function by using synthetic division to find the real zeros x 2x 3x2 2x 10. and 3. Then you could factor the polynomial as Finally, by using the Quadratic Formula, you could determine that the zeros are x 2, and x 1 3i. x 1 3i, x 3, f, 1 3i 333202_0205.qxd 12/7/05 9:36 AM Page 175 In Example 8, the fifth-degree polynomial function has three real zeros. In such cases, you can use the zoom and trace features or the zero or root feature of a graphing utility to approximate the real zeros. You can then use these real zeros to determine the complex zeros algebraically. Section 2.5 Zeros of Polynomial Functions 175 Example 8 shows how to find all the zeros of a polynomial function, including complex zeros. Example 8 Finding the Zeros of a Polynomial Function f x x5 x3 2x2 12x 8 Write all of its zeros. as the product of linear factors, and list Solution The possible rational zeros are the following. ±1, ±2, ±4, and ±8. Synthetic division produces 12 is a zero. 2 is a zero. f(x) = x5 + x3 + 2x2 −12x + 8 f x x5 x3 2x2 12x 8 So, you have y 10 5 (−2, 0) −4 FIGURE 2.34 x 1x 2x3 x2 4x 4. You can factor x3 x2 4x 4 as x 1x2 4, and by factoring x 2 4 as x 2 4 x 4x 4 x 2ix 2i you obtain f x x 1x 1x 2x 2ix 2i (1, 0) 2 4 x which gives the following five zeros of f. x 1, x 1, x 2, x 2i, and x 2i From the graph of only ones that appear as -intercepts. Note that shown in Figure 2.34, you can see that the real zeros are the x 1 is a repeated zero. x f Now try Exercise 63. Te c h n o l o g y You can use the table feature of a graphing utility to help you determine which of the possible rational zeros are zeros of the polynomial in Example 8. The table should be set to ask mode. Then enter each of the possible rational zeros in the table. When you do this, you will see that there are two rational zeros, right. and 1, as shown at the 2 333202_0205.qxd 12/7/05 9:36 AM Page 176 176 Chapter 2 Polynomial and Rational Functions Other Tests for Zeros of Polynomials n You know that an th-degree polynomial function can have at most real zeros. n th-degree polynomials do not have that many real zeros. For Of course, many f x x2 1 has only one real instance, zero. The following theorem, called Descartes’s Rule of Signs, sheds more light on the number of real zeros of a polynomial. has no real zeros, and f x x3 1 n Descartes’s Rule of Signs Let real coefficients and 0. a0 f (x) anxn an1xn1 . . . a2x2 a1x a0 be a polynomial with 1. The number of positive real zeros of f is either equal to the number of variations in sign of or less than that number by an even integer. f x 2. The number of negative real zeros of variations in sign of f x is either equal to the number of or less than that number by an even integer. f A variation in sign means that two consecutive coefficients have opposite signs. When using Descartes’s Rule of Signs, a zero of multiplicity counted as zeros. For instance, the polynomial in sign, and so has either two positive or no positive real zeros. Because k x 3 3x 2 k should be has two variations x3 3x 2 x 1x 1x 2 you can see that the two positive real zeros are x 1 of multiplicity 2. Example 9 Using Descartes’s Rule of Signs Describe the possible real zeros of f x 3x3 5x2 6x 4. Solution The original polynomial has three variations in sign. to to f x 3x3 5x2 6x 4 to The polynomial f(x) = 3x3 − 5x2 + 6x − 4 y 3 2 1 −1 −2 −3 −3 −2 −1 f x 3x3 5x2 6x 4 3x3 5x 2 6x 4 x 2 3 has no variations in sign. So, from Descartes’s Rule of Signs, the polynomial f x 3x3 5x 2 6x 4 has either three positive real zeros or one positive real zero, and has no negative real zeros. From the graph in Figure 2.35, you can x 1 ). see that the function has only one real zero (it is a positive number, near FIGURE 2.35 Now try Exercise 79. 333202_0205.qxd 12/7/05 9:36 AM Page 177 Section 2.5 Zeros of Polynomial Functions 177 Another test for zeros of a polynomial function is related to the sign pattern in the last row of the synthetic division array. This test can give you an upper or is an upper bound for the lower bound of the real zeros of A real number is a lower bound if no f real zeros of f real zeros of are less than f. if no zeros are greater than Similarly, b. b. b b Upper and Lower Bound Rules f x Let cient. Suppose is divided by x c, fx be a polynomial with real coefficients and a positive leading coeffi- using synthetic division. 1. If c > 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f. 2. If c < 0 and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), for the real zeros of f. c is a lower bound Example 10 Finding the Zeros of a Polynomial Function Find the real zeros of f x 6x 3 4x 2 3x 2. Solution The possible real zeros are as follows. Factors of 2 Factors of 6 ±1, ±2 ±1, ±2, ±3, ±6 f x ±12 The original polynomial has three variations in sign. The polynomial f x 6x3 4x2 3x 2 6x3 4x2 3x 2 has no variations in sign. As a result of these two findings, you can apply Descartes’s Rule of Signs to conclude that there are three positive real zeros or one positive real zero, and no negative zeros. Trying produces the following So, that zeros between 0 and 1. By trial and error, you can determine that So, is not a zero, but because the last row has all positive entries, you know is an upper bound for the real zeros. So, you can restrict the search to is a zero. x 2 3 f x x 2 3 6x2 3. Because 6x 2 3 has no real zeros, it follows that x 2 3 is the only real zero. Now try Exercise 87. 333202_0205.qxd 12/7/05 9:36 AM Page 178 178 Chapter 2 Polynomial and Rational Functions Before concluding this section, here are two additional hints that can help you find the real zeros of a polynomial. f x 1. If the terms of have a common monomial factor, it should be factored out before applying the tests in this section. For instance, by writing f x x4 5x3 3x 2 x xx3 5x 2 3x 1 x 0 f you can see that obtained by analyzing the cubic factor. is a zero of and that the remaining zeros can be 2. If you are able to find all but two zeros of you can always use the Quadratic Formula on the remaining quadratic factor. For instance, if you succeeded in writing f x, f x x4 5x3 3x 2 x xx 1x 2 4x 1 you can apply the Quadratic Formula to remaining zeros are x 2 5 and x2 4x 1 x 2 5. to conclude that the two Example 11 Using a Polynomial Model You are designing candle-making kits. Each kit contains 25 cubic inches of candle wax and a mold for making a pyramid-shaped candle. You want the height of the candle to be 2 inches less than the length of each side of the candle’s square base. What should the dimensions of your candle mold be? V 1 x2 B 3 Bh, where and the height is is the x 2. So, the volume of the Substituting 25 for the volume yields the following. h is the area of the base and 3 x2x 2. Solution The volume of a pyramid is height. The area of the base is V 1 pyramid is 25 1 3 75 x3 2x2 0 x3 2x2 75 x2x 2 Substitute 25 for V. Multiply each side by 3. Write in general form. Use The possible rational solutions are synthetic division to test some of the possible solutions. Note that in this case, it makes sense to test only positive -values. Using synthetic division, you can determine that is a solution. x ±15, ±25, ±75. ±5, ±3, x ±1, x 5 2 5 3 0 15 15 75 75 0 5 1 1 are imaginary and The other two solutions, which satisfy can be discarded. You can conclude that the base of the candle mold should be 5 inches by 5 inches and the height of the mold should be 5 2 3 inches. x2 3x 15 0, Now try Exercise 107. 333202_0205.qxd 12/7/05 9:36 AM Page 179 Section 2.5 Zeros of Polynomial Functions 179 2.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The ________ ________ of ________ states that if at least one zero in the complex number system. f x is a polynomial of degree n n > 0, f then has 2. The ________ ________ ________ st
ates that if n linear factors f x an x c1 x c2 f x . . . x cn is a polynomial of degree where c1, c2, . . . , cn n n > 0, then has precisely f are complex numbers. 3. The test that gives a list of the possible rational zeros of a polynomial function is called the ________ ________ Test. 4. If a bi is a complex zero of a polynomial with real coefficients, then so is its ________, a bi. 5. A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be ________ over the ________. 6. The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called ________ ________ of ________. 7. A real number b bound if no real zeros are greater than is a(n) ________ bound for the real zeros of b. f if no real zeros are less than b, and is a(n) ________ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, find all the zeros of the function. 9. f x 2x4 17x3 35x 2 9x 45 1. 2. 3. 4. 5. 6. f x xx 62 f x x2x 3x2 1 gx) x 2x 43 f x x 5x 82 f x x 6x ix i ht t 3t 2t 3i t 3i In Exercises 7–10, use the Rational Zero Test to list f all possible rational zeros of Verify that the zeros of shown on the graph are contained in the list. f. 7. f x x3 3x 2 x 3 y 4 2 −4 −2 2 x −4 8. f x x3 4x 2 4x 16 y 18 −6 6 12 x y −8 −4 x 8 −20 −30 −40 10. f x 4x5 8x4 5x3 10x 2 x 2 y 2 −4 −2 x 4 In Exercises 11–20, find all the rational zeros of the function. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. f x x3 6x 2 11x 6 f x x3 7x 6 gx x3 4x 2 x 4 hx x3 9x 2 20x 12 ht t 3 12t 2 21t 10 px x3 9x 2 27x 27 Cx 2x3 3x 2 1 f x 3x3 19x 2 33x 9 f x 9x4 9x3 58x 2 4x 24 f x 2x4 15x3 23x 2 15x 25 333202_0205.qxd 12/7/05 9:36 AM Page 180 180 Chapter 2 Polynomial and Rational Functions In Exercises 21–24, find all real solutions of the polynomial equation. 21. 22. 23. 24. z4 z3 2z 4 0 x 4 13x 2 12x 0 2y4 7y 3 26y 2 23y 6 0 x5 x4 3x3 5x 2 2x 0 f, In Exercises 25–28, (a) list the possible rational zeros of (b) sketch the graph of so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f. f 25. 26. 27. 28. f x x3 x 2 4x 4 f x 3x3 20x 2 36x 16 f x 4x3 15x 2 8x 3 f x 4x3 12x 2 x 15 f, In Exercises 29–32, (a) list the possible rational zeros of so that some of the (b) use a graphing utility to graph possible zeros in part (a) can be disregarded, and then f. (c) determine all real zeros of f 29. 30. 31. 32. f x 2x4 13x 3 21x 2 2x 8 f x 4x4 17x 2 4 f x 32x3 52x 2 17x 3 f x 4x3 7x 2 11x 18 Graphical Analysis In Exercises 33–36, (a) use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places, (b) determine one of the exact zeros (use synthetic division to verify your result), and (c) factor the polynomial completely. 33. 35. 36. f x x 4 3x 2 2 hx x5 7x4 10x3 14x 2 24x gx 6x4 11x3 51x 2 99x 27 34. Pt t 4 7t 2 12 In Exercises 37– 42, find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 37. 39. 41. 1, 5i, 5i 6, 5 2i, 5 2i 2 3, 1, 3 2 i 38. 40. 42. 4, 3i, 3i 2, 4 i, 4 i 5, 5, 1 3 i In Exercises 43– 46, write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. 43. 44. f x x 4 6x 2 27 f x x 4 2x 3 3x 2 12x 18 (Hint: One factor is x 2 6. ) 45. 46. f x x 4 4x 3 5x 2 2x 6 x 2 2x 2. (Hint: One factor is ) f x x 4 3x 3 x 2 12x 20 (Hint: One factor is x 2 4. ) In Exercises 47–54, use the given zero to find all the zeros of the function. Function 47. 48. 49. 50. 51. 52. 53. 54. f x 2x 3 3x 2 50x 75 f x x3 x 2 9x 9 f x 2x 4 x 3 7x 2 4x 4 g x x 3 7x 2 x 87 g x 4x 3 23x 2 34x 10 h x 3x 3 4x 2 8x 8 f x x 4 3x 3 5x 2 21x 22 f x x 3 4x 2 14x 20 Zero 5i 3i 2i 5 2i 3 i 1 3 i 3 2 i 1 3i In Exercises 55–72, find all the zeros of the function and write the polynomial as a product of linear factors. f x x 2 x 56 gx x 2 10x 23 55. 57. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 58. 56. f x x 2 25 hx x 2 4x 1 f x x 4 81 f y y4 625 f z z 2 2z 2 h(x) x 3 3x 2 4x 2 gx x3 6x2 13x 10 f x x 3 2x 2 11x 52 h x x3 x 6 h x x3 9x2 27x 35 f x 5x 3 9x 2 28x 6 gx 3x3 4x2 8x 8 gx x 4 4x3 8x2 16x 16 h x x 4 6x3 10x2 6x 9 f x x 4 10x 2 9 72. f x x 4 29x 2 100 In Exercises 73–78, find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to discard any rational zeros that are obviously not zeros of the function. 73. 74. 75. 76. f x x3 24x 2 214x 740 f s 2s3 5s2 12s 5 f x 16x 3 20x 2 4x 15 f x 9x 3 15x 2 11x 5 f x 2x4 5x 3 4x 2 5x 2 77. 78. gx x 5 8x 4 28x 3 56x 2 64x 32 333202_0205.qxd 12/7/05 9:36 AM Page 181 In Exercises 79– 86, use Descartes’s Rule of Signs to determine the possible numbers of positive and negative zeros of the function. 80. 82. hx 4x2 8x 3 hx 2x4 3x 2 79. 81. 83. 84. 85. 86. gx 5x5 10x hx 3x4 2x 2 1 gx 2x3 3x 2 3 f x 4x3 3x 2 2x 1 f x 5x3 x 2 x 5 f x 3x3 2x 2 x 3 In Exercises 87– 90, use synthetic division to verify the upper and lower bounds of the real zeros of f. (b) Lower: (b) Lower: 87. 88. 89. 90. f x x4 4x3 15 x 4 (a) Upper: f x 2x3 3x 2 12x 8 x 4 (a) Upper: f x x4 4x3 16x 16 x 5 (a) Upper: f x 2x4 8x 3 (a) Upper: x 3 (b) Lower: (b) Lower: x 1 x 3 x 3 x 4 In Exercises 91–94, find all the real zeros of the function. 91. 92. 93. 94. f x 4x3 3x 1 f z 12z3 4z 2 27z 9 f y 4y3 3y 2 8y 6 gx 3x3 2x 2 15x 10 In Exercises 95–98, find all the rational zeros of the polynomial function. 95. 96. 97. 98. 4 Px x 4 25 f x x3 3 f x x3 1 f z z3 11 x2 23 4 x 2 x 1 6 z2 1 2z 1 4x4 25x 2 36 2x33x 2 23x 12 4x3 x 2 4x 1 6z311z2 3z 2 3 4 In Exercises 99–102, match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: 0; irrational zeros: 1 (b) Rational zeros: 3; irrational zeros: 0 (c) Rational zeros: 1; irrational zeros: 2 (d) Rational zeros: 1; irrational zeros: 0 99. 101. f x x3 1 f x x3 x 100. 102. f x x3 2 f x x3 2x 103. Geometry An open box is to be made from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. Section 2.5 Zeros of Polynomial Functions 181 (a) Let x represent the length of the sides of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume of the box as a function of Determine the domain of the function. x. V (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of such that Which of these values is a physical impossibility in the construction of the box? Explain. x V 56. 104. Geometry A rectangular package to be sent by a delivery service (see figure) can have a maximum combined length and girth (perimeter of a cross section) of 120 inches. x x y (a) Show that the volume of the package is Vx 4x 230 x. (b) Use a graphing utility to graph the function and approximate the dimensions of the package that will yield a maximum volume. (c) Find values of Which of these values is a physical impossibility in the construction of the package? Explain. such that V 13,500. x 105. Advertising Cost A company that produces MP3 (in dollars) for selling a P players estimates that the profit particular model is given by P 76x 3 4830x 2 320,000, 0 ≤ x ≤ 60 x is the advertising expense (in tens of thousands where of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of $2,500,000. 106. Advertising Cost A company that manufactures bicy(in dollars) for selling a P cles estimates that the profit particular model is given by P 45x 3 2500x 2 275,000, 0 ≤ x ≤ 50 x is the advertising expense (in tens of thousands where of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of $800,000. 333202_0205.qxd 12/7/05 9:36 AM Page 182 182 Chapter 2 Polynomial and Rational Functions 107. Geometry A bulk food storage bin with dimensions 2 feet by 3 feet by 4 feet needs to be increased in size to hold five times as much food as the current bin. (Assume each dimension is increased by the same amount.) (a) Write a function that represents the volume V of the new bin. (b) Find the dimensions of the new bin. 108. Geometry A rancher wants to enlarge an existing rectangular corral such that the total area of the new corral is 1.5 times that of the original corral. The current corral’s dimensions are 250 feet by 160 feet. The rancher wants to increase each dimension by the same amount. (a) Write a function that represents the area of the new A corral. (b) Find the dimensions of the new corral. (c) A rancher wants to add a length to the sides of the corral that are 160 feet, and twice the length to the sides that are 250 feet, such that the total area of the new corral is 1.5 times that of the original corral. Repeat parts (a) and (b). Explain your results. 109. Cost The ordering and transportation cost (in thousands of dollars) for the components used in manufacturing a product is given by C 100200 x 2 x x 30 x ≥ 1 , C x is the order size (in hundreds). In calculus, it can where be shown that the cost is a minimum when 3x3 40x 2 2400x 36,000 0. Use a calculator to approximate the optimal order size to the nearest hundred units. 110. Height of a Baseball A baseball is thrown upward from a height of 6 feet with an initial velocity of 48 feet per second, and its height (in feet) is ht 16t 2 48t 6, 0 ≤ t ≤ 3 h t is the time (in seconds). You are told the ball where reaches a height of 64 feet. Is this possible? p 111. Profit The demand equation for a certain product is where is the unit price (in dollars) is the number of units produc
ed and is is the total cost (in dollars) is the number of units produced. The total profit p 140 0.0001x, x of the product and sold. The cost equation C 80x 150,000, C and obtained by producing and selling units is the product where for x x P R C xp C. You are working in the marketing department of the company that produces this product, and you are asked to determine a price that will yield a profit of 9 million dollars. Is this possible? Explain. p Model It 112. Athletics The attendance A (in millions) at NCAA women’s college basketball games for the years 1997 through 2003 is shown in the table, where represents the year, with to 1997. (Source: National Collegiate Athletic Association) t corresponding t 7 Year, t Attendance, A 7 8 9 10 11 12 13 6.7 7.4 8.0 8.7 8.8 9.5 10.2 (a) Use the regression feature of a graphing utility to find a cubic model for the data. (b) Use the graphing utility to create a scatter plot of the data. Then graph the model and the scatter plot in the same viewing window. How do they compare? (c) According to the model found in part (a), in what year did attendance reach 8.5 million? (d) According to the model found in part (a), in what year did attendance reach 9 million? (e) According to the right-hand behavior of the model, will the attendance continue to increase? Explain. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 113 and 114, decide whether 113. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros. 114. If x i is a zero of the function given by f x x 3 ix2 ix 1 then x i must also be a zero of f. Think About It ble) the zeros of the function x r1, x r2, x r3. and In Exercises 115–120, determine (if possiif the function has zeros at f g 115. gx f x 116. gx 3f x 333202_0205.qxd 12/7/05 9:36 AM Page 183 117. 119. gx f x 5 gx 3 f x 118. 120. gx f 2x gx f x 121. Exploration Use a graphing utility to graph the funcf x x 4 4x 2 k for different values of tion given by k. satisfy the specified characteristics. (Some parts do not have unique answers.) such that the zeros of Find values of k f (a) Four real zeros (b) Two real zeros, each of multiplicity 2 (c) Two real zeros and two complex zeros (d) Four complex zeros 122. Think About It Will the answers to Exercise 121 g? change for the function gx f x 2 (a) (b) gx f 2x 123. Think About It A third-degree polynomial function f and 3, and its leading coefficient is has real zeros f. negative. Write an equation for Sketch the graph of How many different polynomial functions are possible for f ? 2,2, f. 1 124. Think About It Sketch the graph of a fifth-degree polynomial function whose leading coefficient is positive x 3 and that has one zero at 125. Writing Compile a list of all the various techniques for factoring a polynomial that have been covered so far in the text. Give an example illustrating each technique, and write a paragraph discussing when the use of each technique is appropriate. of multiplicity 2. 126. Use the information in the table to answer each question. Interval , 2 2, 1 1, 4 4, Value of f x Positive Negative Negative Positive (a) What are the three real zeros of the polynomial func- tion f ? (b) What can be said about the behavior of the graph of f at x 1? (c) What is the least possible degree of the degree of ever be odd? Explain. f f ? Explain. Can (d) Is the leading coefficient of f positive or negative? Explain. Section 2.5 Zeros of Polynomial Functions 183 (e) Write an equation for f. (There are many correct answers.) (f) Sketch a graph of the equation you wrote in part (e). 127. (a) Find a quadratic function f as zeros. Assume that (with integer coefficients) b is a positive ± b i that has integer. (b) Find a quadratic function a ± bi f as zeros. Assume that (with integer coefficients) b is a positive that has integer. 128. Graphical Reasoning The graph of one of the following functions is shown below. Identify the function shown in the graph. Explain why each of the others is not the correct function. Use a graphing utility to verify your result. (a) (b) (c) (d) f x x 2x 2)x 3.5 g x x 2)x 3.5 h x x 2)x 3.5x 2 1 k x x 1)x 2x 3.5 x 2 4 y 10 –20 –30 –40 Skills Review In Exercises 129–132, perform the operation and simplify. 129. 130. 131. 132. 3 6i 8 3i 12 5i 16i 6 2i1 7i 9 5i9 5i In Exercises 133–138, use the graph of f to sketch the graph of g. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 135. 134. 133. gx f x 2 gx f x 2 gx 2 f x gx f x gx f 2x 137. 2x 138. gx f 1 136. y 5 4 (4, 4) (0, 2) f (2, 2) (−2, 0) 21 43 x 333202_0206.qxd 12/7/05 9:56 AM Page 184 184 Chapter 2 Polynomial and Rational Functions 2.6 Rational Functions What you should learn • Find the domains of rational functions. • Find the horizontal and vertical asymptotes of graphs of rational functions. • Analyze and sketch graphs of rational functions. • Sketch graphs of rational functions that have slant asymptotes. • Use rational functions to model and solve real-life problems. Why you should learn it Rational functions can be used to model and solve real-life problems relating to business. For instance, in Exercise 79 on page 196, a rational function is used to model average speed over a distance. Introduction A rational function can be written in the form f x N(x) D(x) where Nx and Dx are polynomials and Dx is not the zero polynomial. In general, the domain of a rational function of includes all real numbers x -values that make the denominator zero. Much of the discussion of except rational functions will focus on their graphical behavior near the -values excluded from the domain. x x Example 1 Finding the Domain of a Rational Function Find the domain of x -values. f x 1 x and discuss the behavior of f near any excluded Solution Because the denominator is zero when except f x to the left and right of x 0, x 0. is all real numbers To determine the behavior of near this excluded value, evaluate the domain of x 0, f f as indicated in the following tables. x f x 1 1 0.5 2 0.1 10 0.01 100 0.001 1000 0 x f x 0 0.001 0.01 0.1 0.5 1000 100 10 2 1 1 Note that as x contrast, as graph of f is shown in Figure 2.36. x approaches 0 from the left, approaches 0 from the right, f x f x decreases without bound. In increases without bound. The Mike Powell/Getty Images Note that the rational function f x 1x given by is also referred to as the reciprocal function discussed in Section 1.6. f x1 −1 FIGURE 2.36 Now try Exercise 1. 333202_0206.qxd 12/7/05 9:56 AM Page 185 Vertical asymptote: x = 0 2 1 −2 −1 −1 Section 2.6 Rational Functions 185 y f(x) = 1 x Horizontal and Vertical Asymptotes In Example 1, the behavior of near f f x as x 0 x 0 f x is denoted as follows. x as 0 1 2 Horizontal asymptote: y = 0 x f x decreases without bound x as approaches 0 from the left. f x x increases without bound as approaches 0 from the right. x 0 is a vertical asymptote of the graph of The line as shown in Figure 2.37. From this figure, you can see that the graph of also has a horizontal asymptote— x the line increases or decreases without bound. This means that the values of approach zero as f x 1x y 0. f, f FIGURE 2.37 f x 0 as x f x 0 as x x approaches 0 as f x decreases without bound. approaches 0 as f x x increases without bound. Definitions of Vertical and Horizontal Asymptotes x a 1. The line f x is a vertical asymptote of the graph of f x or f if either from the right or from the left. is a horizontal asymptote of the graph of f if a, y b as x 2. The line f x as x b x or . Eventually (as ), the distance between the horizontal asymptote and the points on the graph must approach zero. Figure 2.38 shows the horizontal and vertical asymptotes of the graphs of three rational functions. x x or f(x) = 2x + 1 x + 1 Vertical asymptote: x = −1 y 4 3 2 1 Horizontal asymptote: y = 2 f(x) = 4 x + 12 y 3 2 1 Horizontal asymptotex) = 2 (x −1)2 Vertical asymptote: x = 1 Horizontal asymptote: y = 0 −3 −2 −1 x 1 −2 −1 1 2 x (a) FIGURE 2.38 (b) 1 2 3 x −1 (c) The graphs of f x 1x in Figure 2.37 and f x 2x 1x 1 in Figure 2.38(a) are hyperbolas. You will study hyperbolas in Section 10.4. 333202_0206.qxd 12/7/05 9:56 AM Page 186 186 Chapter 2 Polynomial and Rational Functions Asymptotes of a Rational Function Let be the rational function given by f f x Nx Dx where Nx and Dx anx n an1x n1 . . . a1x a 0 bmx m bm1x m1 . . . b1x b0 have no common factors. 1. The graph of has vertical asymptotes at the zeros of f Dx. 2. The graph of has one or no horizontal asymptote determined by f comparing the degrees of Nx and Dx. a. If n < m, asymptote. n m, b. If the graph of the graph of f f has the line y 0 (the -axis) as a horizontal x has the line y an bm (ratio of the leading coefficients) as a horizontal asymptote. c. If n > m, the graph of f has no horizontal asymptote. Example 2 Finding Horizontal and Vertical Asymptotes Find all horizontal and vertical asymptotes of the graph of each rational function. a. f x 2x2 x2 1 b. f x x2 x 2 x2 x 6 Solution a. For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of the numerator is 2 and the leadas a ing coefficient of the denominator is 1, so the graph has the line horizontal asymptote. To find any vertical asymptotes, set the denominator equal to zero and solve the resulting equation for y 2 x. x2 1 0 x 1x 1 0 x 1 0 x 1 0 Set denominator equal to zero. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. x 1 x 1 x 1 This equation has two real solutions x 1 lines shown in Figure 2.39. x 1 and so the graph has the as vertical asymptotes. The graph of the function is and x 1, b. For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of both the numerator and denomy 1 inator is 1, so the graph has the line as a horizontal asymptote. To find any vertical asymptotes, first factor the numerator and den
ominator as follows. f x x2 x 2 x2 x 6 x 1x 2 x 2x By setting the denominator you can determine that the graph has the line (of the simplified function) equal to zero, as a vertical asymptote. x 3 Now try Exercise 9. y 4 3 2 1 f(x) = 2x 2 x 2 − 1 Horizontal asymptote: y = 2 −4 −3 −2 −1 1 2 3 4 x Vertical asymptote: x = −1 FIGURE 2.39 Vertical asymptote: x = 1 333202_0206.qxd 12/7/05 9:56 AM Page 187 Section 2.6 Rational Functions 187 Analyzing Graphs of Rational Functions To sketch the graph of a rational function, use the following guidelines. You may also want to test for symmetry when graphing rational functions, especially for simple rational functions. Recall from Section 1.6 that the graph of f x 1 x is symmetric with respect to the origin. Guidelines for Analyzing Graphs of Rational Functions and Let f x NxDx, are polynomials. where Dx Nx 1. Simplify f, if possible. 2. Find and plot the -intercept (if any) by evaluating y f 0. 3. Find the zeros of the numerator (if any) by solving the equation Then plot the corresponding -intercepts. Nx 0. x 4. Find the zeros of the denominator (if any) by solving the equation Dx 0. Then sketch the corresponding vertical asymptotes. 5. Find and sketch the horizontal asymptote (if any) by using the rule for finding the horizontal asymptote of a rational function. 6. Plot at least one point between and one point beyond each -intercept and x vertical asymptote. 7. Use smooth curves to complete the graph between and beyond the vertical asymptotes. Te c h n o l o g y Some graphing utilities have difficulty graphing rational functions that have vertical asymptotes. Often, the utility will connect parts of the graph that are not supposed to be connected. For instance, the top screen on the right shows the graph of f x 1 x 2 . x 2 and the other to the right of Notice that the graph should consist of two unconnected portions—one to the left of problem, you can try changing the mode of the graphing utility to dot mode. The problem with this is that the graph is then represented as a collection of dots (as shown in the bottom screen on the right) rather than as a smooth curve. To eliminate this x 2. −5 −5 5 −5 5 −5 5 5 The concept of test intervals from Section 2.2 can be extended to graphing of rational functions. To do this, use the fact that a rational function can change signs only at its zeros and its undefined values (the -values for which its denominator is zero). Between two consecutive zeros of the numerator and the denominator, a rational function must be entirely positive or entirely negative. This means that when the zeros of the numerator and the denominator of a rational function are put in order, they divide the real number line into test intervals in which the function has no sign changes. A representative -value is chosen to determine if the value of the rational function is positive (the graph lies above the -axis) or negative (the graph lies below the -axis). x x x x 333202_0206.qxd 12/7/05 9:56 AM Page 188 188 Chapter 2 Polynomial and Rational Functions You can use transformations to help you sketch graphs of rational functions. For instance, the graph of in Example 3 is a vertical stretch and a right shift of the graph of because f x 1x g gx 3 x 2 3 1 x 2 3f x 2. Horizontal asymptote: y = 0 y 4 2 −2 −4 g(x Vertical asymptote: x = 2 FIGURE 2.40 y 3 2 1 −1 Horizontal asymptote2 f x( ) = 2 x − x 1 −1 −4 −3 −2 Vertical asymptote: = 0 x Example 3 Sketching the Graph of a Rational Function Sketch the graph of gx 3 x 2 and state its domain. Solution y-intercept: x-intercept: Vertical asymptote: Horizontal asymptote: Additional points: g0 3 2 because 2 , 0, 3 None, because x 2, y 0, 3 0 zero of denominator because degree of Nx < degree of Dx Test interval , 2 2, Representative x-value Value of g Sign 4 3 g4 0.5 g3 3 Negative Positive Point on graph 4, 0.5 3, 3 By plotting the intercepts, asymptotes, and a few additional points, you can obtain x 2. g the graph shown in Figure 2.40. The domain of is all real numbers except x Now try Exercise 27. Example 4 Sketching the Graph of a Rational Function Sketch the graph of f x 2x 1 x and state its domain. Solution y-intercept: x-intercept: Vertical asymptote: Horizontal asymptote: Additional points: is not in the domain because x 0 2x 1 0 zero of denominator None, because 2, 0, 1 x 0, y 2, because degree of Nx degree of Dx Test interval , 0 0, 1 2, 1 2 Representative Value of f Sign x-value .75 4 Positive Negative Positive Point on graph 1, 3 4, 2 1 4, 1.75 By plotting the intercepts, asymptotes, and a few additional points, you can obtain the graph shown in Figure 2.41. The domain of except x 0. is all real numbers x f FIGURE 2.41 Now try Exercise 31. 333202_0206.qxd 12/7/05 9:56 AM Page 189 Section 2.6 Rational Functions 189 Example 5 Sketching the Graph of a Rational Function Sketch the graph of f x xx2 x 2. Solution Factoring the denominator, you have f x x x 1x 2. because f 0 0 y-intercept: x-intercept: Vertical asymptotes: Horizontal asymptote: Additional points: 0, 0, 0, 0 x 1, y 0, x 2, zeros of denominator because degree of Nx < degree of Dx Test interval , 1 1, 0 0, 2 2, Representative x-value Value of f Sign 3 0.5 1 3 f 3 0.3 f 0.5 0.4 f 1 0.5 f 3 0.75 Negative Positive Negative Positive Point on graph 3, 0.3 0.5, 0.4 1, 0.5 3, 0.75 Vertical asymptote: x = −1 Vertical asymptote: x = 2 y Horizontal asymptote: y = 0 3 2 1 −1 −2 −3 − 1 x 2 3 f(x) = x x2 − x − 2 FIGURE 2.42 The graph is shown in Figure 2.42. Now try Exercise 35. If you are unsure of the shape of a portion of the graph of a rational function, plot some additional points. Also note that when the numerator and the denominator of a rational function have a common factor, the graph of the function has a hole at the zero of the common factor (see Example 6). f(x) = x2 − 9 x2 − 2x − 3 Horizontal asymptote: y = 1 y 3 2 1 −4 −3 −1 1 2 3 4 5 6 x Vertical asymptote: x = −1 −2 −3 −4 −5 Example 6 A Rational Function with Common Factors f x x2 9x2 2x 3. Sketch the graph of Solution By factoring the numerator and denominator, you have 2x 3 y-intercept: because , x 3. x 3x 3 x 3x 1 0, 3, 3, 0, x 1, y 1, because f 3 0 zero of (simplified) denominator because degree of Nx degree of Dx x-intercept: Vertical asymptote: Horizontal asymptote: Additional points: Test interval , 3 3, 1 1, Representative x-value Value of f Sign 4 2 2 f 4 0.33 f 2 1 f 2 1.67 Positive Negative Positive Point on graph 4, 0.33 2, 1 2, 1.67 The graph is shown in Figure 2.43. Notice that there is a hole in the graph at x 3 because the function is not defined when x 3. FIGURE 2.43 HOLE AT x 3 Now try Exercise 41. 333202_0206.qxd 12/7/05 9:56 AM Page 190 190 Chapter 2 Polynomial and Rational Functions f x Vertical asymptote: x − = 1 − 8 −6 −4 2 −2 −2 −4 FIGURE 2.44 8 4 6 Slant asymptote: y = x − 2 Slant Asymptotes Consider a rational function whose denominator is of degree 1 or greater. If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote. For example, the graph of x f x x 2 x x 1 has a slant asymptote, as shown in Figure 2.44. To find the equation of a slant asymptote, use long division. For instance, by dividing you obtain x2 x, x 1 into . Slant asymptote y x 2 increases or decreases without bound, the remainder term f approaches the line y x 2, x As approaches 0, so the graph of Figure 2.44. 2x 1 as shown in Example 7 A Rational Function with a Slant Asymptote Sketch the graph of f x x2 x 2x 1. Solution Factoring the numerator as x -intercepts. Using long division x 2x 1 allows you to recognize the allows you to recognize that the line y x y-intercept: x-intercepts: Vertical asymptote: Slant asymptote: Additional points: is a slant asymptote of the graph. f 0 2 because and 2, 0 0, 2, 1, 0 x 1, y x zero of denominator Test interval , 1 1, 1 1, 2 2, Representative x-value Value of f Sign 2 0.5 1.5 3 f 2 1.33 f 0.5 4.5 f 1.5 2.5 f 3 2 Negative Positive Negative Positive Point on graph 2, 1.33 0.5, 4.5 1.5, 2.5 3, 2 The graph is shown in Figure 2.45. Slant asymptote3 −2 11 3 4 5 x −2 −3 Vertical asymptote: x = 1 f(x) = x2 − x − 2 x − 1 FIGURE 2.45 Now try Exercise 61. 333202_0206.qxd 12/7/05 9:56 AM Page 191 Section 2.6 Rational Functions 191 Applications There are many examples of asymptotic behavior in real life. For instance, Example 8 shows how a vertical asymptote can be used to analyze the cost of removing pollutants from smokestack emissions. Example 8 Cost-Benefit Model A utility company burns coal to generate electricity. The cost removing % of the smokestack pollutants is given by p C 80,000p 100 p C (in dollars) of 0 ≤ p < 100. for Sketch the graph of this function. You are a member of a state legislature considering a law that would require utility companies to remove 90% of the pollutants from their smokestack emissions. The current law requires 85% removal. How much additional cost would the utility company incur as a result of the new law? Solution The graph of this function is shown in Figure 2.46. Note that the graph has a Because the current law requires 85% removal, vertical asymptote at the current cost to the utility company is p 100. C 80,000(85) 100 85 $453,333. Evaluate when C p 85. If the new law increases the percent removal to 90%, the cost will be C 80,000(90) 100 90 $720,000. Evaluate when C p 90. So, the new law would require the utility company to spend an additional 720,000 453,333 $266,667. Subtract 85% removal cost from 90% removal cost Smokestack Emissions 90% 85% C = 80,000 p 100 − p C 1000 800 600 400 200 20 40 60 80 100 Percent of pollutants removed p FIGURE 2.46 Now try Exercise 73. 333202_0206.qxd_pg 192 1/9/06 8:55 AM Page 192 192 Chapter 2 Polynomial and Rational Functions Example 9 Finding a Minimum Area A rectangular page is designed to contain 48 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side in
ches wide. What should the dimensions of the page be so that the least are amount of paper is used? 11 2 y 11 in. 2 1 in. x 1 in. 11 in. 2 FIGURE 2.47 Graphical Solution Let be the area to be minimized. From Figure 2.47, you can write A Numerical Solution A Let be the area to be minimized. From Figure 2.47, you can write A x 3y 2. A x 3 y 2. The printed area inside the margins is modeled by 48 xy y 48x. To find the minimum area, or A rewrite the equation for in terms of just one variable y. for by substituting 2 48x A x 348 x x 348 2x x , x > 0 x The graph of this rational function is shown in Figure 2.48. Because represents the width of the printed area, you need consider only the portion of the graph is positive. Using a graphing utility, you for which to occur can approximate the minimum value of x 8.5 when is 488.5 5.6 y inches. The corresponding value of inches. So, the dimensions should be A x x 3 11.5 inches by y 2 7.6 inches. 200 A = (x + 3)(48 + 2x) x , x > 0 The printed area inside the margins is modeled by y 48x. A or To find the minimum area, rewrite the equation for 48x for y. 48 xy in terms of just one variable by substituting 2 A x 348 x x 348 2x x , x > 0 Use the table feature of a graphing utility to create a table of values for the function y1 x 348 2x x x 1. y1 occurs when From the table, you can see that the minibeginning at x mum value of is somewhere between 8 and 9, as shown in Figure 2.49. To approximate the minimum value y1 to one decimal place, change the table so that it starts at of x 8 and increases by 0.1. The minimum value of occurs x 8.5, when as shown in Figure 2.50. The corresponding y inches. So, the dimensions should value of is x 3 11.5 y 2 7.6 inches. be 488.5 5.6 inches by y1 0 0 FIGURE 2.48 24 Now try Exercise 77. FIGURE 2.49 FIGURE 2.50 If you go on to take a course in calculus, you will learn an analytic technique that produces a minimum area. In this case, that x for finding the exact value of value is x 62 8.485. 333202_0206.qxd 12/7/05 9:56 AM Page 193 2.6 Exercises Section 2.6 Rational Functions 193 VOCABULARY CHECK: Fill in the blanks. 1. Functions of the form f x NxDx, where Nx and Dx are polynomials and Dx is not the zero polynomial, are called ________ ________. 2. If 3. If x → a as x → ±, f x → ± f x → b as from the left or the right, then x a y b is a ________ ________ of the graph of f x NxDx, 4. For the rational function given by f Nx then the graph of has a ________ (or oblique) ________. if the degree of degree of Dx, then f. is a ________ ________ of the graph of f. is exactly one more than the PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, (a) complete each table for the function, (b) determine the vertical and horizontal asymptotes of the graph of the function, and (c) find the domain of the function. f x x 0.5 0.9 0.99 0.999 12 −4 −4 −2 3. f x 3x .5 1.1 1.01 1.001 f x x 5 10 100 1000 2. f x 5x x 1 y 12 8 −8 −4 −4 x 4 8 4. f x 4x x2 1 y 8 4 x 4 8 −8 x 4 8 −8 −4 −4 −8 In Exercises 5 –12, find the domain of the function and identify any horizontal and vertical asymptotes. 5. f x 1 x 2 6. f x 4 x 23 7. 9. 11 3x 2 1 x 2 x 9 8. f x 1 5x 1 2x 10. 12. f x 2x 2 x 1 f x 3x 2 x 5 x 2 1 In Exercises 13 –16, match the rational function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) y (b) 4 2 −2 −4 x 6 −8 −6 −4 −2 (c) y (d) y 4 2 −2 −4 y 4 2 −2 x 4 6 −4 −2 4 2 −2 −4 x x 13. 15 14. 16 In Exercises 17–20, find the zeros (if any) of the rational function. 17. 19. gx 18. hx 2 5 x 2 2 20. gx x3 8 x 2 1 333202_0206.qxd 12/7/05 9:56 AM Page 194 194 194 Chapter 2 Chapter 2 Polynomial and Rational Functions Polynomial and Rational Functions In Exercises 21– 26, find the domain of the function and identify any horizontal and vertical asymptotes. f x x 3 x2 9 f x x2 4 f x x 4 x2 16 f x x2 1 21. 23. 24. 22. x2 3x 2 x2 2x 3 f x x2 3x 4 2x2 x 1 25. 26. f x 6x2 11x 3 6x2 7x 3 In Exercises 27–46, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. 27. 29. 31. 33. 35. 37. 39. 40. 41. 43. 45. 28. 30. 32. 34. 36. 38. f x 1 x 3 gx 1 3 x Px 1 3x 1 x f t 1 2t t f x 1 x 22 gx x2 2x 8 x2 9 f x 1 hx x 2 1 x 2 Cx 5 2x 1 x f x x 2 x 2 9 gs s s 2 1 hx x2 5x 4 x2 4 f x 2x2 5x 3 x 3 2x2 x 2 x2 x 2 x 3 2x2 5x 6 f x f x x2 3x x2 x 6 f x 2x2 5x 2 2x2 x 6 f t t2 1 t 1 42. 44. 46. f x 5x 4 x2 x 12 f x 3x2 8x 4 2x2 3x 2 f x x2 16 x 4 Analytical, Numerical, and Graphical Analysis 47– 50, do the following. In Exercises (a) Determine the domains of and f g. (b) Simplify graph of f f. and find any vertical asymptotes of the (c) Compare the functions by completing the table. (d) Use a graphing utility to graph and f g in the same viewing window. (e) Explain why the graphing utility may not show the difference in the domains of and f g. 47. f x x 2 1 x 1 , gx x 1 3 2 1.5 1 0.5 0 1 x f x gx 48. f x x 2x 2 x 2 2x , gx x 1 0 1 1.5 2 2.5 3 x f x gx 49. f x x 2 x 2 2x , gx 1 x 0.5 0 0.5 1 1.5 2 3 x f x gx 50. f x 2x 6 x 2 7x 12 , gx gx In Exercises 51–64, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gx x2 5 hx x2 4 51. 52. x x 53. 55. 57. 59. 61. f x 2x2 1 x gx x2 54. 56. 58. 60. 62. f x 1 x2 x hx x2 x 1 f x x2 3x 1 gx x 3 2x 2 8 f x 2x 2 5x 5 x 2 333202_0206.qxd 12/7/05 9:56 AM Page 195 63. 64. f x 2x3 x2 2x 1 x2 3x 2 f x 2x3 x2 8x 4 x2 3x 2 In Exercises 65– 68, use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. 65. 66. 67. 68. f x x 2 5x 8 x 3 f x 2x 2 x x 1 gx 1 3x 2 x 3 x 2 hx 12 2x x 2 24 x Graphical Reasoning In Exercises 69–72, (a) use the graph x to determine any -intercepts of the graph of the rational and solve the resulting equation function and (b) set to confirm your result in part (a). y 0 69. y x 1 x 3 y 70. y 2x x 3 y 6 4 2 −2 −4 x 4 6 8 6 4 2 −2 −4 x 2 4 6 8 71. y 1 x x 724 −2 −4 y 8 4 x 4 −8 −4 −4 x 4 8 73. Pollution The cost (in millions of dollars) of removing of the industrial and municipal pollutants discharged C p% into a river is given by C 255p 100 p , 0 ≤ p < 100. (a) Use a graphing utility to graph the cost function. Section 2.6 Rational Functions 195 (b) Find the costs of removing 10%, 40%, and 75% of the pollutants. (c) According to this model, would it be possible to remove 100% of the pollutants? Explain. 74. Recycling In a pilot project, a rural township is given recycling bins for separating and storing recyclable p% products. The cost of the population is given by (in dollars) for supplying bins to C C 25,000p 100 p , 0 ≤ p < 100. (a) Use a graphing utility to graph the cost function. (b) Find the costs of supplying bins to 15%, 50%, and 90% of the population. (c) According to this model, would it be possible to supply bins to 100% of the residents? Explain. 75. Population Growth The game commission introduces 100 deer into newly acquired state game lands. The popuN lation N 205 3t 1 0.04t of the herd is modeled by t ≥ 0 , where t is the time in years (see figure). N 1400 1200 1000 800 600 400 200 100 150 200 50 Time (in years) (a) Find the populations when t 25. (b) What is the limiting size of the herd as time increases? t 10, t 5, and 76. Concentration of a Mixture A 1000-liter tank contains liters of a 75% 50 liters of a 25% brine solution. You add brine solution to the tank. x (a) Show that the concentration C , the proportion of brine to total solution, in the final mixture is C 3x 50 4x 50. (b) Determine the domain of the function based on the physical constraints of the problem. (c) Sketch a graph of the concentration function. (d) As the tank is filled, what happens to the rate at which the concentration of brine is increasing? What percent does the concentration of brine appear to approach? 333202_0206.qxd 12/7/05 9:56 AM Page 196 196 196 Chapter 2 Chapter 2 Polynomial and Rational Functions Polynomial and Rational Functions 77. Page Design A page that is inches high contains 30 square inches of print. The top and bottom margins are 1 inch deep and the margins on each side are 2 inches wide (see figure). inches wide and y x 2 in. 1 in. 1 in. x 2 in. y (a) Show that the total area on the page is A A 2xx 11 x 4 . (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper will be used. Verify your answer numerically using the table feature of the graphing utility. 78. Page Design A rectangular page is designed to contain 64 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on inches wide. What should the dimensions each side are of the page be so that the least amount of paper is used? 11 2 Model It 80. Sales The sales (in millions of dollars) for the Yankee Candle Company in the years 1998 through 2003 are shown in the table. (Source: The Yankee Candle Company) S 1998 184.5 2001 379.8 1999 256.6 2002 444.8 2000 338.8 2003 508.6 A model for these data is given by S 5.816t2 130.68 0.004t2 1.00 t represents the year, with , 8 ≤ t ≤ 13 where 1998. t 8 corresponding to (a) Use a graphing utility to plot the data and graph the model in the same viewing window. How well does the model fit the data? (b) Use the model to estimate the sales for the Yankee Candle Company in 2008. (c) Would this model be useful for estimating sales after 2008? Explain. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 81 and 82, determine whether 81. A polynomial can have infinitely man
y vertical asymptotes. 82. The graph of a rational function can never cross one of its asymptotes. Think About It In Exercises 83 and 84, write a rational function that has the specified characteristics. (There are many correct answers.) f 83. Vertical asymptote: None 84. Vertical asymptote: Horizontal asymptote: y 2 x 2, Horizontal asymptote: None x 1 79. Average Speed A driver averaged 50 miles per hour on the round trip between Akron, Ohio, and Columbus, Ohio, 100 miles away. The average speeds for going and returning were and miles per hour, respectively. (a) Show that y x y 25x x 25 . Skills Review (b) Determine the vertical and horizontal asymptotes of the graph of the function. (c) Use a graphing utility to graph the function. (d) Complete the table. 30 35 40 45 50 55 60 x y (e) Are the results in the table what you expected? Explain. (f) Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? Explain. In Exercises 85– 88, completely factor the expression. 85. 87. x2 15x 56 x 3 5x2 4x 20 86. 88. 3x2 23x 36 x 3 6x2 2x 12 In Exercises 93–96, solve the inequality and graph the solution on the real number line. 89. 91. 10 3x ≤ 0 4x 2 < 20 90. 92. 5 2x > 5x 1 22x 3 ≥ 5 1 93. Make a Decision To work an extended application analyzing the total manpower of the Department of Defense, visit this text’s website at college.hmco.com. (Data Source: U.S. Census Bureau) 333202_0207.qxd 12/7/05 9:40 AM Page 197 2.7 Nonlinear Inequalities Section 2.7 Nonlinear Inequalities 197 What you should learn • Solve polynomial inequalities. • Solve rational inequalities. • Use inequalities to model and solve real-life problems. Why you should learn it Inequalities can be used to model and solve real-life problems. For instance, in Exercise 73 on page 205, a polynomial inequality is used to model the percent of households that own a television and have cable in the United States. Polynomial Inequalities x2 2x 3 < 0, x To solve a polynomial inequality such as you can use the fact that a polynomial can change signs only at its zeros (the -values that make the polynomial equal to zero). Between two consecutive zeros, a polynomial must be entirely positive or entirely negative. This means that when the real zeros of a polynomial are put in order, they divide the real number line into intervals in which the polynomial has no sign changes. These zeros are the critical numbers of the inequality, and the resulting intervals are the test intervals for the inequality. For instance, the polynomial above factors as x2 2x 3 x 1x 3 x 1 and has two zeros, into three test intervals: , 1, 1, 3, and x 3. These zeros divide the real number line and 3, . (See Figure 2.51.) you need only test one value from So, to solve the inequality each of these test intervals to determine whether the value satisfies the original inequality. If so, you can conclude that the interval is a solution of the inequality. x2 2x 3 < 0, Zero x − = 1 Zero x = 3 © Jose Luis Pelaez, Inc./Corbis Test Interval − ( − , 1) Test Interval −1 , 3) ( Test Interval (3, ) x −4 −3 −2 −1 0 1 2 3 4 5 FIGURE 2.51 Three test intervals for x2 2x 3 You can use the same basic approach to determine the test intervals for any polynomial. Finding Test Intervals for a Polynomial To determine the intervals on which the values of a polynomial are entirely negative or entirely positive, use the following steps. 1. Find all real zeros of the polynomial, and arrange the zeros in increasing order (from smallest to largest). These zeros are the critical numbers of the polynomial. 2. Use the critical numbers of the polynomial to determine its test intervals. x 3. Choose one representative -value in each test interval and evaluate the polynomial at that value. If the value of the polynomial is negative, the polynomial will have negative values for every -value in the interval. If the value of the polynomial is positive, the polynomial will have positive values for every -value in the interval. x x 333202_0207.qxd 12/7/05 9:40 AM Page 198 198 Chapter 2 Polynomial and Rational Functions Example 1 Solving a Polynomial Inequality Solve x2 x 6 < 0. Solution By factoring the polynomial as x2 x 6 x 2x 3 you can see that the critical numbers are polynomial’s test intervals are x 2 and x 3. So, the , 2, 2, 3, and 3, . Test intervals In each test interval, choose a representative -value and evaluate the polynomial. x Test Interval , 2 2, 3 3, x-Value x 3 x 0 x 4 Polynomial Value 32 3 6 6 02 0 6 6 42 4 6 6 Conclusion Positive Negative Positive This implies that the solution of the inequality 2, 3, From this you can conclude that the inequality is satisfied for all -values in 2, 3. is the interval as shown in Figure 2.52. Note that the original inequality contains a less than symbol. This means that the solution set does not contain the endpoints of the test interval x2 x 6 < 0 2, 3. x − x Choose = 3. − x x 3) > 0 ( + 2)( Choose = x x ( + 2)( x − 4. 3) > 0 −6 −5 −4 −3 −2 −4 −3 −1 1 2 4 5 x −2 −3 −6 −7 FIGURE 2.53 y = 2 − − x x 6 Choose = x x ( + 2)( x − 0. 3) < 0 FIGURE 2.52 Now try Exercise 13. As with linear inequalities, you can check the reasonableness of a solution by substituting -values into the original inequality. For instance, to check the solution found in Example 1, try substituting several -values from the interval 2, 3 into the inequality x x x2 x 6 < 0. Regardless of which -values you choose, the inequality should be satisfied. x You can also use a graph to check the result of Example 1. Sketch the graph as shown in Figure 2.53. Notice that the graph is below the y x2 x 6, of -axis on the interval 2, 3. x 333202_0207.qxd 12/7/05 9:40 AM Page 199 Section 2.7 Nonlinear Inequalities 199 In Example 1, the polynomial inequality was given in general form (with the polynomial on one side and zero on the other). Whenever this is not the case, you should begin the solution process by writing the inequality in general form. Example 2 Solving a Polynomial Inequality Solve 2x3 3x2 32x > 48. Solution Begin by writing the inequality in general form. 2x 3 3x2 32x > 48 Write original inequality. 2x 3 3x2 32x 48 > 0 x 4x 42x 3 > 0 Write in general form. Factor. You may find it easier to determine the sign of a polynomial from its factored form. For instance, in Example 2, if the test value is substituted into the factored form x 2 x 4x 42x 3 you can see that the sign pattern of the factors is which yields a negative result. Try using the factored forms of the polynomials to determine the signs of the polynomials in the test intervals of the other examples in this section. 2 , 3 The critical numbers are 2, 4, , 4, 4, 3 x-Value x 5 x 0 x 2 x 5 Test Interval , 4 4, 3 3 2, 4 4, 2 x 4, x 3 2, 4, . and and x 4, and the test intervals are Polynomial Value 253 352 325 48 203 302 320 48 223 322 322 48 253 352 325 48 Conclusion Negative Positive Negative Positive From this you can conclude that the inequality is satisfied on the open intervals 4, 3 Therefore, the solution set consists of all real numbers in the 4, , intervals as shown in Figure 2.54. and 4, 3 4, . and 2 2 x Choose = x 4)( + 4)(2 x − ( x 0. − 3) > 0 − ( x x Choose = 5. − 4)( + 4)(2 x x 3) > 0 −7 −6 −5 −4 −3 −2 − Choose = x 4)( + 4)(2 x − ( x −5. − 3) < 0 − ( x FIGURE 2.54 Now try Exercise 21. x Choose = 2 − 4)( + 4)(2 x x . 3) < 0 When solving a polynomial inequality, be sure you have accounted for the particular type of inequality symbol given in the inequality. For instance, in Example 2, note that the original inequality contained a “greater than” symbol and the solution consisted of two open intervals. If the original inequality had been 2x3 3x2 32x ≥ 48 the solution would have consisted of the closed interval 4, . 4, 3 2 and the interval 333202_0207.qxd 12/7/05 9:40 AM Page 200 200 Chapter 2 Polynomial and Rational Functions Each of the polynomial inequalities in Examples 1 and 2 has a solution set that consists of a single interval or the union of two intervals. When solving the exercises for this section, watch for unusual solution sets, as illustrated in Example 3. Example 3 Unusual Solution Sets a. The solution set of the following inequality consists of the entire set of real x2 2x 4 In other words, the value of the quadratic , . numbers, is positive for every real value of x. x2 2x 4 > 0 b. The solution set of the following inequality consists of the single real number x2 2x 1 has only one critical number, because the quadratic and it is the only value that satisfies the inequality. 1, x 1, x2 2x 1 ≤ 0 c. The solution set of the following inequality is empty. In other words, the quadis not less than zero for any value of x2 3x 5 ratic x. x2 3x 5 < 0 d. The solution set of the following inequality consists of all real numbers this solution set can be written as In interval notation, x 2. except , 2 2, . x2 4x 4 > 0 Now try Exercise 25. Exploration You can use a graphing utility to verify the results in Example 3. For instance, the graph of y -values are greater than 0 for all values of Use the graphing utility to graph the following: y x 2 3x 5 is shown below. Notice that the as stated in Example 3(a). y x 2 4x 4 y x 2 2x 1 y x 2 2x 4 x, Explain how you can use the graphs to verify the results of parts (b), (c), and (d) of Example 3. 10 −2 9 −9 333202_0207.qxd 12/7/05 9:40 AM Page 201 Section 2.7 Nonlinear Inequalities 201 Rational Inequalities The concepts of critical numbers and test intervals can be extended to rational inequalities. To do this, use the fact that the value of a rational expression can change sign only at its zeros (the -values for which its numerator is zero) and its undefined values (the -values for which its denominator is zero). These two types of numbers make up the critical numbers of a rational inequality. When solving a rational inequality, begin by writing the inequality in general form with the rational expression on the left and zero on the right. x x Example 4 Solvi
ng a Rational Inequality Solve 2x 7 x 5 ≤ 3. Solution 2x 7 x 5 ≤ 3 3 ≤ 0 2x 7 x 5 2x 7 3x 15 x 5 Write original inequality. Write in general form. ≤ 0 ≤ 0 Find the LCD and add fractions. Simplify. x 8 x 5 Critical numbers: Test intervals: Test: x 5, x 8 , 5, 5, 8, 8, Zeros and undefined values of rational expression Is x 8 x 5 ≤ 0? After testing these intervals, as shown in Figure 2.55, you can see that the (, 5) inequality is satisfied on the open intervals Moreover, x 8, you can conclude that the solution because , 5 8, . set consists of all real numbers in the intervals (Be sure to use a closed interval to indicate that can equal 8.) x 8x 5 0 8, . when and x x Choose = 6. − Choose = 4. x −x + 8 x − 5 < 0 FIGURE 2.55 Now try Exercise 39. x Choose = 9. −x + 8 x − 5 < 0 333202_0207.qxd 12/7/05 9:40 AM Page 202 202 Chapter 2 Polynomial and Rational Functions Applications One common application of inequalities comes from business and involves profit, revenue, and cost. The formula that relates these three quantities is Profit Revenue Cost P R C. Example 5 Increasing the Profit for a Product Calculators R The marketing department of a calculator manufacturer has determined that the demand for a new model of calculator is p 100 0.00001x, 0 ≤ x ≤ 10,000,000 Demand equation 250 200 150 100 50 ) FIGURE 2.56 x 0 2 4 6 8 10 Number of units sold (in millions) p is the price per calculator (in dollars) and where represents the number of calculators sold. (If this model is accurate, no one would be willing to pay $100 for the calculator. At the other extreme, the company couldn’t sell more than 10 million calculators.) The revenue for selling calculators is x x R xp x100 0.00001x Revenue equation as shown in Figure 2.56. The total cost of producing calculator plus a development cost of $2,500,000. So, the total cost is x calculators is $10 per C 10x 2,500,000. Cost equation What price should the company charge per calculator to obtain a profit of at least $190,000,000? Solution Verbal Model: Equation: Profit Revenue Cost P R C P 100x 0.00001x2 10x 2,500,000 P 0.00001x 2 90x 2,500,000 Calculators P To answer the question, solve the inequality P ≥ 190,000,000 0.00001x2 90x 2,500,000 ≥ 190,000,000. When you write the inequality in general form, find the critical numbers and the test intervals, and then test a value in each test interval, you can find the solution to be x 3,500,000 ≤ x ≤ 5,500,000 as shown in Figure 2.57. Substituting the -values in the original price equation shows that prices of 200 150 100 50 0 −50 −100 0 2 4 6 8 10 Number of units sold (in millions) $45.00 ≤ p ≤ $65.00 will yield a profit of at least $190,000,000. FIGURE 2.57 Now try Exercise 71. 333202_0207.qxd 12/7/05 9:40 AM Page 203 Section 2.7 Nonlinear Inequalities 203 Another common application of inequalities is finding the domain of an expression that involves a square root, as shown in Example 6. Example 6 Finding the Domain of an Expression Find the domain of 64 4x2. Algebraic Solution Remember that the domain of an expression is the set of all -values 64 4x2 for which the expression is defined. Because is defined is nonnegative, the domain is (has real values) only if given by 64 4x2 x 64 4x2 ≥ 0. 64 4x2 ≥ 0 16 x2 ≥ 0 4 x4 x ≥ 0 Write in general form. Divide each side by 4. Write in factored form. So, the inequality has two critical numbers: can use these two numbers to test the inequality as follows. x 4, x 4 , 4, 4, 4, 4, Critical numbers: Test intervals: and x 4 x 4. You Test: For what values of x is 64 4x2 ≥ 0? A test shows that the inequality is satisfied in the closed interval 4, 4. is the interval 4, 4. So, the domain of the expression 64 4x2 Graphical Solution Begin by sketching the graph of the equation y 64 4x2, as shown in Figure 2.58. From the graph, you can determine that the -values 4 and 4). So, extend from to 4 (including the domain of the expression is the interval 4 64 4x2 4, 4. x y = 64 − 4x 2 y 10 6 4 2 −6 −4 −2 2 4 6 −2 x FIGURE 2.58 Now try Exercise 55. Complex Number Nonnegative Radicand Complex Number −4 FIGURE 2.59 4 x To analyze a test interval, choose a representative -value in the interval and evaluate the expression at that value. For instance, in Example 6, if you substitute any number from the interval you will obtain a nonnegative number under the radical symbol that simplifies to a real number. If you substitute any number from the intervals you will obtain a complex number. It might be helpful to draw a visual representation of the intervals as shown in Figure 2.59. into the expression 64 4x2 , 4 4, 4 4, and W RITING ABOUT MATHEMATICS Profit Analysis Consider the relationship P R C described on page 202. Write a paragraph discussing why it might be beneficial to solve illustrate your reasoning. if you owned a business. Use the situation described in Example 5 to P < 0 333202_0207.qxd 12/7/05 9:40 AM Page 204 204 Chapter 2 Polynomial and Rational Functions 2.7 Exercises VOCABULARY CHECK: Fill in the blanks. 1. To solve a polynomial inequality, find the ________ numbers of the polynomial, and use these numbers to create ________ ________ for the inequality. 2. The critical numbers of a rational expression are its ________ and its ________ ________. 3. The formula that relates cost, revenue, and profit is ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, determine whether each value of solution of the inequality. x is a In Exercises 27–32, solve the inequality and write the solution set in interval notation. Inequality 1. x2 3 < 0 2. x2 x 12 ≥ 0 3. 4. x 2 x 4 ≥ 3 3x2 x2 4 < 1 Values (a) (c) (a) (c) (a) (c) (a) (cb) (d) (b) (d) (b) (d) (b) (d In Exercises 5–8, find the critical numbers of the expression. 5. 7. 2x2 x 6 2 3 x 5 6. 8. 9x3 25x 2 2 x x 2 x 1 27. 29. 31. 4x3 6x2 < 0 x3 4x ≥ 0 x 12x 23 ≥ 0 28. 30. 32. 4x3 12x 2 > 0 2x3 x 4 ≤ 0 x4x 3 ≤ 0 Graphical Analysis In Exercises 33–36, use a graphing utility to graph the equation. Use the graph to approximate that satisfy each inequality. the values of x Equation Inequalities 33. 34. 35. 36. y x 2 2x 3 y 1 2x 2 2x 1 8x3 1 y 1 2x y x3 x 2 16x 16 (a) (a) (a) (ab) (b) (b) (b ≥ 36 In Exercises 37–50, solve the inequality and graph the solution on the real number line. In Exercises 9–26, solve the inequality and graph the solution on the real number line. 10. 12. 14. 16. x2 < 36 x 32 ≥ 1 x2 6x 9 < 16 x2 2x > 3 9. 11. 13. 15. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. x2 ≤ 9 x 22 < 25 x2 4x 4 ≥ 9 x2 x < 6 x2 2x 3 < 0 x2 4x 1 > 0 x2 8x 5 ≥ 0 2x2 6x 15 ≤ 0 x3 3x2 x 3 > 0 x3 2x2 4x 8 ≤ 0 x3 2x2 9x 2 ≥ 20 2x3 13x2 8x 46 ≥ 6 4x2 4x 1 ≤ 0 x2 3x 8 > 0 37. 39. 41. 43. 45. 47. 48. 49. 50. 4 < 0 38. 40. 42. 44. 46. 1 x x 12 x 2 5 7x 1 2x 3x 5 x 5 4 x 5 1 x 3 x2 2x x2 9 x2 x 6 x > ≤ ≤ 0 1 2x 3 9 4x 3 ≥ 0 5 x 1 3x x 1 2x x 1 x x 4 ≤ < 1 3 333202_0207.qxd 12/7/05 9:40 AM Page 205 Graphical Analysis In Exercises 51–54, use a graphing utility to graph the equation. Use the graph to approximate that satisfy each inequality. the values of x 71. Cost, Revenue, and Profit The revenue and cost equations for a product are R x75 0.0005x and C 30x 250,000 Section 2.7 Nonlinear Inequalities 205 Equation y 3x x 2 y 2x 2 x 1 y 2x2 x2 4 y 5x x2 4 51. 52. 53. 54. Inequalities (a) y ≤ 0 (b) y ≥ 6 (a) y ≤ 0 (b) y ≥ 8 (a) y ≥ 1 (b) y ≤ 2 (a) y ≥ 1 (b) y ≤ 0 In Exercises 55–60, find the domain of Use a graphing utility to verify your result. x in the expression. 55. 57. 59. 4 x2 x2 7x 12 x x2 2x 35 56. 58. 60. x2 4 144 9x2 x x2 9 In Exercises 61–66, solve the inequality. (Round your answers to two decimal places.) 61. 62. 63. 64. 65. 66. 0.4x2 5.26 < 10.2 1.3x2 3.78 > 2.12 0.5x2 12.5x 1.6 > 0 1.2x2 4.8x 3.1 < 5.3 1 2.3x 5.2 2 3.1x 3.7 > 3.4 > 5.8 67. Height of a Projectile A projectile is fired straight upward from ground level with an initial velocity of 160 feet per second. (a) At what instant will it be back at ground level? (b) When will the height exceed 384 feet? 68. Height of a Projectile A projectile is fired straight upward from ground level with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet? 69. Geometry A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie? 70. Geometry A rectangular parking lot with a perimeter of 440 feet is to have an area of at least 8000 square feet. Within what bounds must the length of the rectangle lie? R C and represents the are measured in dollars and where number of units sold. How many units must be sold to obtain a profit of at least $750,000? What is the price per unit? x 72. Cost, Revenue, and Profit The revenue and cost equations for a product are R x50 0.0002x and C 12x 150,000 R C and where represents the are measured in dollars and number of units sold. How many units must be sold to obtain a profit of at least $1,650,000? What is the price per unit? x Model It 73. Cable Television The percents of households in the United States that owned a television and had cable from 1980 to 2003 can be modeled by C C 0.0031t3 0.216t2 5.54t 19.1, 0 ≤ t ≤ 23 t where is the year, with (Source: Nielsen Media Research) t 0 corresponding to 1980. (a) Use a graphing utility to graph the equation. (b) Complete the table to determine the year in which the percent of households that own a television and have cable will exceed 75%. 24 26 28 30 32 34 t C (c) Use the trace feature of a graphing utility to verify your answer to part (b). (d) Complete the table to determine the years during which the percent of households that own a television and have cable will be between 85% and 100%. 36 37 38 39 40 41 42 43 t C (e) Use the trace feature of a graphing utility to verify your answer to part (d). (f) Explain why the model may give values greater than 100% even though such values are not reasona
ble. 333202_0207.qxd 12/7/05 9:40 AM Page 206 206 Chapter 2 Polynomial and Rational Functions 74. Safe Load The maximum safe load uniformly distributed over a one-foot section of a two-inch-wide wooden beam Load 168.5d 2 472.1, is approximated by the model where is the depth of the beam. d (a) Evaluate the model for d 4, d 6, d 8, d 10, and d 12. Use the results to create a bar graph. (b) Determine the minimum depth of the beam that will safely support a load of 2000 pounds. 75. Resistors When two resistors of resistances R1 and R2 connected in parallel (see figure), the total resistance satisfies the equation are R Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 77 and 78, determine whether 77. The zeros of the polynomial x3 2x2 11x 12 ≥ 0 divide the real number line into four test intervals. 78. The solution set of the inequality 3 2x2 3x 6 ≥ 0 is the entire set of real numbers. Exploration In Exercises 79–82, find the interval for such that the equation has at least one real solution. b 1 R 1 R1 1 R2 . for a parallel circuit in which R1 Find must be at least 1 ohm. R2 2 ohms and R 79. 80. 81. 82. x2 bx 4 0 x2 bx 4 0 3x2 bx 10 0 2x2 bx 5 0 + _ E R1 R2 76. Education The numbers N (in thousands) of master’s degrees earned by women in the United States from 1990 to 2002 are approximated by the model N 0.03t2 9.6t 172 represents the year, with corresponding to (Source: U.S. National Center for t 0 t where 1990 (see figure). Education Statistics) 83. (a) Write a conjecture about the intervals for Exercises 79–82. Explain your reasoning. b in (b) What is the center of each interval for b in Exercises 79–82? 84. Consider the polynomial line shown below. x ax b and the real number a b x (a) Identify the points on the line at which the polynomial is zero. (b) In each of the three subintervals of the line, write the sign of each factor and the sign of the product. (c) For what -values does the polynomial change signs? x Skills Review In Exercises 85–88, factor the expression completely ( 320 300 280 260 240 220 200 180 160 140 2 4 85. 86. 87. 88. 4x2 20x 25 x 32 16 x2x 3 4x 3 2x 4 54x t 14 16 18 In Exercises 89 and 90, write an expression for the area of the region. 8 10 6 Year (0 ↔ 1990) 12 (a) According to the model, during what year did the number of master’s degrees earned by women exceed 220,000? (b) Use the graph to verify the result of part (a). (c) According to the model, during what year will the number of master’s degrees earned by women exceed 320,000? (d) Use the graph to verify the result of part (c). 89. x x2 + 1 90. 3 + 2 b b 333202_020R.qxd 12/7/05 9:43 AM Page 207 2 Chapter Summary What did you learn? Section 2.1 Analyze graphs of quadratic functions (p. 128). Write quadratic functions in standard form and use the results to sketch graphs of functions (p. 131). Use quadratic functions to model and solve real-life problems (p. 133). Section 2.2 Use transformations to sketch graphs of polynomial functions (p. 139). Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions (p. 141). Find and use zeros of polynomial functions as sketching aids (p. 142). Use the Intermediate Value Theorem to help locate zeros of polynomial functions (p. 146). Section 2.3 Use long division to divide polynomials by other polynomials (p. 153). Use synthetic division to divide polynomials by binomials of the form x k (p. 156). Use the Remainder Theorem and the Factor Theorem (p. 157). Section 2.4 Use the imaginary unit to write complex numbers (p. 162). Add, subtract, and multiply complex numbers (p. 163). Use complex conjugates to write the quotient of two complex numbers i in standard form (p. 165). Find complex solutions of quadratic equations (p. 166). Section 2.5 Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions (p. 169). Find rational zeros of polynomial functions (p. 170). Find conjugate pairs of complex zeros (p. 173). Use factoring (p. 173), Descartes’s Rule of Signs (p. 176), and the Upper and Lower Bound Rules (p. 177), to find zeros of polynomials. Section 2.6 Find the domains of rational functions (p. 184). Find the horizontal and vertical asymptotes of graphs of rational functions (p. 185). Analyze and sketch graphs of rational functions (p. 187). Sketch graphs of rational functions that have slant asymptotes (p. 190). Use rational functions to model and solve real-life problems (p. 191). Section 2.7 Solve polynomial inequalities (p. 197), and rational inequalities (p. 201). Use inequalities to model and solve real-life problems (p. 202). Chapter Summary 207 Review Exercises 1, 2 3–18 19–22 23–28 29–32 33–42 43–46 47–52 53–60 61–64 65–68 69–74 75–78 79–82 83–88 89–96 97, 98 99 –110 111–114 115–118 119–130 131–134 135–138 139–146 147, 148 333202_020R.qxd 12/7/05 9:43 AM Page 208 208 Chapter 2 Polynomial and Rational Functions 2 Review Exercises In Exercises 1 and 2, graph each function. Compare 2.1 the graph of each function with the graph of y x2. 1. (a) (b) (c) (d) 2. (a) (b) (c) (d) f x 2x 2 gx 2x 2 hx x 2 2 kx x 22 f x x 2 4 gx 4 x 2 hx x 32 kx 1 2x 2 1 In Exercises 3–14, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and -intercept(s). x 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. gx x2 2x f x 6x x2 f x x2 8x 10 hx 3 4x x2 f t 2t 2 4t 1 f x x2 8x 12 hx 4x2 4x 13 f x x2 6x 1 hx x2 5x 4 f x 4x 2 4x 5 f x 1 x2 5x 4 3 6x2 24x 22 f x 1 2 In Exercises 15–18, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. 15. y 16. y 2 −2 −4 −6 17. Vertex: 18. Vertex: (4, 1) 4 (2, −1) x 8 6 2 (0, 3) (2, 2) −2 2 4 6 x 1, 4; 2, 3; point: 2, 3 point: 1, 6 19. Geometry The perimeter of a rectangle is 200 meters. (a) Draw a diagram that gives a visual representation of y, the problem. Label the length and width as respectively. and x x. (b) Write as a function of Use the result to write the y area as a function of x. (c) Of all possible rectangles with perimeters of 200 meters, find the dimensions of the one with the maximum area. 20. Maximum Revenue The total revenue R earned (in dollars) from producing a gift box of candles is given by R p 10p2 800p where p is the price per unit (in dollars). (a) Find the revenues when the prices per box are $20, $25, and $30. (b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results. 21. Minimum Cost A soft-drink manufacturer has daily production costs of C 70,000 120x 0.055x 2 C is the total cost (in dollars) and where is the number of units produced. How many units should be produced each day to yield a minimum cost? x 22. Sociology The average age of the groom at a first marriage for a given age of the bride can be approximated by the model y 0.107x2 5.68x 48.5, 20 ≤ x ≤ 25 y is the age of the groom and is the age of the where bride. Sketch a graph of the model. For what age of the bride is the average age of the groom 26? (Source: U.S. Census Bureau) x In Exercises 23–28, sketch the graphs of 2.2 the transformation. y x n and 23. 24. 25. 26. 27. 28. y x3, y x3, y x4, y x 4, y x5, y x5, f x x 43 f x 4x3 f x 2 x 4 f x 2x 24 f x x 35 f x 1 2x5 3 333202_020R.qxd 12/7/05 9:43 AM Page 209 In Exercises 29–32, describe the right-hand and left-hand behavior of the graph of the polynomial function. 29. 30. 31. 32. f x x 2 6x 9 f x 1 2 x3 2x x4 3x 2 2 gx 3 4 hx x5 7x 2 10x In Exercises 33–38, find all the real zeros of the polynomial function. Determine the multiplicity of each zero and the number of turning points of the graph of the function. Use a graphing utility to verify your answers. 33. 35. 37. f x 2x2 11x 21 f t t 3 3t f x 12x3 20x2 34. 36. 38. f x xx 32 f x x3 8x2 gx x4 x3 2x2 In Exercises 39– 42, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. 39. 40. 41. 42. f x x3 x2 2 gx 2x3 4x2 f x xx3 x2 5x 3 hx 3x2 x4 In Exercises 43– 46, (a) use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. (b) Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results. 43. 44. 45. 46. f x 3x 3 x2 3 f x 0.25x 3 3.65x 6.12 f x x 4 5x 1 f x 7x 4 3x 3 8x2 2 2.3 In Exercises 47–52, use long division to divide. 47. 48. 49. 50. 51. 52. 24x 2 x 8 3x 2 4x 7 3x 2 5x 3 13x 2 x 2 x2 3x 1 3x4 x 2 1 x4 3x3 4x 2 6x 3 x2 2 6x4 10x3 13x 2 5x 2 2x2 1 Review Exercises 209 In Exercises 53–56, use synthetic division to divide. 53. 54. 55. 56. 6x4 4x3 27x 2 18x x 2 0.1x3 0.3x 2 0.5 x 5 2x3 19x 2 38x 24 x 4 3x3 20x2 29x 12 x 3 In Exercises 57 and 58, use synthetic division to determine x whether the given values of are zeros of the function. 57. 58. x 1 f x 20x4 9x3 14x 2 3x (a) f x 3x3 8x 2 20x 16 (a) x 4 x 4 x 3 4 (b) (b) (c) (c) x 0 x 2 3 (d) x 1 (d) x 1 In Exercises 59 and 60, use synthetic division to find each function value. 59. 60. f 3 f x x4 10x3 24x 2 20x 44 (a) (b) gt 2t 5 5t 4 8t 20 (a) (b) g2 g4 f 1 f, In Exercises 61– 64, (a) verify the given factor(s) of the funcf, (b) find the remaining factors of (c) use your results tion f, to write the complete factorization of (d) list all real zeros of and (e) confirm your results by using a graphing utility to graph the function. f, Function 61. 62. 63. 64. f x x3 4x2 25x 28 f x 2x3 11x2 21x 90 f x x 4 4x 3 7x2 22x 24 f x x4 11x3 41x2 61x 30 Factor(s) x 4 x 6 x 2x 3 x 2x 5 In Exercises 65– 68, write the complex number in 2.4 standard form. 65. 67. 6 4 i2 3i 66. 68. 3 25 5i i2 In Exercises 69–74, perform the operation and write the result in standard form. 69. 70. 71. 73. 7 5i 4 2i i 2 2 2 2 2 2 5i13 8i 10 8
i2 3i i 2 2 1 6i5 2i 72. 74. i6 i3 2i 333202_020R.qxd 12/7/05 9:43 AM Page 210 210 Chapter 2 Polynomial and Rational Functions In Exercises 75 and 76, write the quotient in standard form. 75. 6 i 4 i 76. 3 2i 5 i In Exercises 77 and 78, perform the operation and write the result in standard form. 77. 4 2 3i 2 1 i 78. 1 2 i 5 1 4i In Exercises 79– 82, find all solutions of the equation. 79. 81. 3x2 1 0 x2 2x 10 0 80. 82. 2 8x2 0 6x2 3x 27 0 2.5 83. 84. 85. 86. 87. 88. In Exercises 83–88, find all the zeros of the function. f x 3xx 22 f x x 4x 92 f x x2 9x 8 f x x 3 6x f x x 4x 6x 2ix 2i f x x 8x 52x 3 ix 3 i In Exercises 89 and 90, use the Rational Zero Test to list all possible rational zeros of f. 89. 90. f x 4x3 8x 2 3x 15 f x 3x4 4x3 5x 2 8 In Exercises 91–96, find all the rational zeros of the function. 113. f x 91. 92. 93. 94. 95. 96. f x x3 2x2 21x 18 f x 3x3 20x2 7x 30 f x x3 10x2 17x 8 f x x3 9x2 24x 20 f x x 4 x3 11x2 x 12 f x 25x4 25x3 154x2 4x 24 In Exercises 97 and 98, find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 97. 2 3, 4, 3 i 98. 2, 3, 1 2i In Exercises 99–102, use the given zero to find all the zeros of the function. Function 99. 100. 101. 102. f x x3 4x2 x 4 h x x3 2x2 16x 32 gx 2x 4 3x 3 13x2 37x 15 f x 4x4 11x3 14x2 6x Zero i 4i 2 i 1 i In Exercises 103–106, find all the zeros of the function and write the polynomial as a product of linear factors. 103. 104. 105. 106. f x x3 4x2 5x gx x3 7x2 36 gx x4 4x3 3x2 40x 208 f x x4 8x3 8x2 72x 153 In Exercises 107 and 108, use Descartes’s Rule of Signs to determine the possible numbers of positive and negative zeros of the function. 107. 108. gx 5x3 3x 2 6x 9 hx 2x5 4x3 2x 2 5 In Exercises 109 and 110, use synthetic division to verify the upper and lower bounds of the real zeros of f. 109. 110. f x 4x3 3x2 4x 3 x 1 (a) Upper: x 1 (b) Lower: 4 f x 2x3 5x2 14x 8 x 8 (a) Upper: x 4 (b) Lower: In Exercises 111–114, find the domain of the rational 2.6 function. 111. f x 5x x 12 8 x2 10x 24 112. 114. f x 3x2 1 3x f x x2 x 2 x2 4 In Exercises 115–118, identify any horizontal or vertical asymptotes. 115. 117. f x 4 x 3 hx 2x 10 x2 2x 15 116. 118. f x 2x2 5x 3 x2 2 hx x3 4x2 x2 3x 2 In Exercises 119–130, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. 119. f x 5 x 2 121. 123. 125. gx 2 x 1 x px 120. 122. 124. f x 4 x hx x 3 x 2 f x 2x 126. hx x 2 4 4 x 12 333202_020R.qxd 12/7/05 9:43 AM Page 211 Review Exercises 211 127. 129. f x 6x2 x 2 1 f x 6x2 11x 3 3x2 x 128. 130. y 2x 2 x 2 4 f x 6x2 7x 2 4x2 1 In Exercises 131–134, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x x2 1 x 1 f x 2x3 x2 1 131. 132. 133. 134. f x 3x3 2x2 3x 2 3x2 x 4 f x 3x3 4x2 12x 16 3x2 5x 2 135. Average Cost A business has a production cost of for producing units of a product. The x C 0.5x 500 average cost per unit, 0.5x 500 x C C x C, is given by , x > 0. x Determine the average cost per unit as bound. (Find the horizontal asymptote.) increases without 136. Seizure of Illegal Drugs The cost C dollars) for the federal government to seize illegal drug as it enters the country is given by C 528p 100 p 0 ≤ p < 100. , (in millions of p% of an (a) Use a graphing utility to graph the cost function. (b) Find the costs of seizing 25%, 50%, and 75% of the drug. (c) According to this model, would it be possible to seize 100% of the drug? 137. Page Design A page that is inches high contains 30 square inches of print. The top and bottom margins are 2 inches deep and the margins on each side are 2 inches wide. inches wide and x y (a) Draw a diagram that gives a visual representation of the problem. (b) Show that the total area on the page is A A 2x2x 7 x 4 . (c) Determine the domain of the function based on the physical constraints of the problem. (d) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper will be used. Verify your answer numerically using the table feature of the graphing utility. 138. Photosynthesis The amount y of CO2 uptake (in milligrams per square decimeter per hour) at optimal temperatures and with the natural supply of is approximated by the model y 18.47x 2.96 0.23x 1 x > 0 CO2 , x where is the light intensity (in watts per square meter). Use a graphing utility to graph the function and determine the limiting amount of uptake. CO2 2.7 In Exercises 139–146, solve the inequality. 139. 141. 143. 145. 6x2 5x < 4 x3 16x ≥ 0 3 2 x 1 x 1 x2 7x 12 x ≤ ≥ 0 140. 142. 144. 146. 2x2 x ≥ 15 12x3 20x2 < > 147. Investment P dollars invested at interest rate r compounded annually increases to an amount A P1 r2 in 2 years. An investment of $5000 is to increase to an amount greater than $5500 in 2 years. The interest rate must be greater than what percent? 148. Population of a Species A biologist introduces 200 of the P ladybugs into a crop field. The population ladybugs is approximated by the model P 10001 3t 5 t t is the time in days. Find the time required for the where population to increase to at least 2000 ladybugs. Synthesis True or False? In Exercises 149 and 150, determine whether the statement is true or false. Justify your answer. 149. A fourth-degree polynomial with real coefficients can have 5, 8i, 4i, and 5 as its zeros. 150. The domain of a rational function can never be the set of all real numbers. 151. Writing Explain how to determine the maximum or minimum value of a quadratic function. 152. Writing Explain the connections among factors of a polynomial, zeros of a polynomial function, and solutions of a polynomial equation. 153. Writing Describe what is meant by an asymptote of a graph. 333202_020R.qxd 12/7/05 9:43 AM Page 212 212 Chapter 2 Polynomial and Rational Functions 2 y 6 4 2 −4 −2 −4 −6 Chapter Test (0, 3) x 2 4 6 8 (3, −6) FIGURE FOR 2 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Describe how the graph of differs from the graph of g f x x 2. (a) gx 2 x 2 (b) gx x 3 2 2 2. Find an equation of the parabola shown in the figure at the left. 3. The path of a ball is given by x the ball and y 1 20 x 2 3x 5, where y is the height (in feet) of is the horizontal distance (in feet) from where the ball was thrown. (a) Find the maximum height of the ball. (b) Which number determines the height at which the ball was thrown? Does changing this value change the coordinates of the maximum height of the ball? Explain. 4. Determine the right-hand and left-hand behavior of the graph of the function h t 3 4t 5 2t 2. Then sketch its graph. 5. Divide using long division. 6. Divide using synthetic division. 3x 3 4x 1 x 2 1 2x4 5x 2 3 x 2 7. Use synthetic division to show that f x 4x3 x 2 12x 3. x 3 is a zero of the function given by Use the result to factor the polynomial function completely and list all the real zeros of the function. 8. Perform each operation and write the result in standard form. 10i 3 25 (a) 2 3 i2 3 i (b) 9. Write the quotient in standard form: 5 2 i . In Exercises 10 and 11, find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 10. 0, 3, 3 i, 3 i 11. 1 3 i, 1 3 i, 2, 2 In Exercises 12 and 13, find all the zeros of the function. 12. f x x3 2x2 5x 10 13. f x x4 9x2 22x 24 In Exercises 14–16, identify any intercepts and asymptotes of the graph the function. Then sketch a graph of the function. 14. hx 4 x 2 1 15. f x 2x2 5x 12 x2 16 16. gx x 2 2 x 1 In Exercises 17 and 18, solve the inequality. Sketch the solution set on the real number line. 17. 2x2 5x > 12 18. 2 x > 5 x 6 333202_020R.qxd 12/7/05 9:43 AM Page 213 Proofs in Mathematics These two pages contain proofs of four important theorems about polynomial functions. The first two theorems are from Section 2.3, and the second two theorems are from Section 2.5. The Remainder Theorem (p. 157) If a polynomial r f k. is divided by x k, f x the remainder is Proof From the Division Algorithm, you have f x x kqx rx and because either rx you know that you have at x k, rx 0 must be a constant. That is, or the degree of rx is less than the degree of rx r. Now, by evaluating x k, f x f k k kqk r 0qk r r. To be successful in algebra, it is important that you understand the connection among factors of a polynomial, zeros of a polynomial function, and solutions or roots of a polynomial equation. The Factor Theorem is the basis for this connection. The Factor Theorem (p. 157) A polynomial has a factor x k f x if and only if f k 0. Proof Using the Division Algorithm with the factor x k, you have f x x kqx rx. By the Remainder Theorem, rx r f k, and you have f x x kqx fk where qx is a polynomial of lesser degree than f x. If f k 0, then f x x kqx x k f x by and you see that f x, division of Theorem, you have f k 0. is a factor of x k is a factor of yields a remainder of 0. So, by the Remainder Conversely, if x k f x. 213 333202_020R.qxd 12/7/05 9:43 AM Page 214 Proofs in Mathematics The Fundamental Theorem of Algebra The Linear Factorization Theorem is closely related to the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra has a long and interesting history. In the early work with polynomial equations, The Fundamental Theorem of Algebra was thought to have been not true, because imaginary solutions were not considered. In fact, in the very early work by mathematicians such as Abu al-Khwarizmi (c. 800 A.D.), negative solutions were also not considered. Once imaginary numbers were accepted, several mathematicians attempted to give a general proof of the Fundamental Theorem of Algebra. These included Gottfried von Leib
niz (1702), Jean d’Alembert (1746), Leonhard Euler (1749), JosephLouis Lagrange (1772), and Pierre Simon Laplace (1795). The mathematician usually credited with the first correct proof of the Fundamental Theorem of Algebra is Carl Friedrich Gauss, who published the proof in his doctoral thesis in 1799. f x Linear Factorization Theorem (p. 169) n > 0, is a polynomial of degree where If linear factors f x an n, x c1 c1, c2, . . . , cn . . . x cn x c2 are complex numbers. where then has precisely f n Proof Using the Fundamental Theorem of Algebra, you know that must have at least one zero, and you have is a factor of f x, f x c1 c1. f x x c1 f1 Consequently, f1 x x. If the degree of Theorem to conclude that must have a zero which implies that is greater than zero, you again apply the Fundamental f x x c1 f1 x c2 f2 f1 It is clear that the degree of n that you can repeatedly apply the Fundamental Theorem x. x n 1, c2, is that the degree of f2 x and times until you obtain n 2, is x c1 x c2 . . . x cn f x an an where is the leading coefficient of the polynomial f x. Factors of a Polynomial with real coefficients can be written as Every polynomial of degree the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. (p. 173) n > 0 Proof To begin, you use the Linear Factorization Theorem to conclude that completely factored in the form f x can be f x dx c1 x c2 x c3 . . . x cn . ci If each is real, there is nothing more to prove. If any b 0, then, because the coefficients of a bi cj gate obtain a bi, are real, you know that the conjuis also a zero. By multiplying the corresponding factors, you is complex f x ci ci x ci x cj x a bix a bi x2 2ax a2 b2 where each coefficient is real. 214 333202_020R.qxd 12/7/05 9:43 AM Page 215 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. Show that if r ak3 bk2 ck d f x ax3 bx2 cx d f k r, then using long division. In where other words, verify the Remainder Theorem for a thirddegree polynomial function. 2. In 2000 B.C., the Babylonians solved polynomial equations by referring to tables of values. One such table gave the values of To be able to use this table, the Babylonians sometimes had to manipulate the equation as shown below. y3 y2. ax3 bx2 c Original equation a3 x3 b3 3 a2 x2 b2 2 ax b ax b a2 c b3 a2 c b3 Multiply each side by a2 b3. Rewrite. a2cb3 Then they would find column of the table. Because they knew that the corresponding -value was equal to y x bya. 2, 3, . . . , 10. Record the they could conclude that y 1, axb, (a) Calculate in the for y3 y2 y3 y2 values in a table. Use the table from part (a) and the method above to solve each equation. (b) (c) (d) (e) (f) (g) x3 x2 252 x3 2x2 288 3x3 x2 90 2x3 5x2 2500 7x3 6x2 1728 10x3 3x2 297 Using the methods from this chapter, verify your solution to each equation. 3. At a glassware factory, molten cobalt glass is poured into molds to make paperweights. Each mold is a rectangular prism whose height is 3 inches greater than the length of each side of the square base. A machine pours 20 cubic inches of liquid glass into each mold. What are the dimensions of the mold? 4. Determine whether the statement is true or false. If false, provide one or more reasons why the statement is false and f x ax3 bx2 cx d, correct the statement. Let a 0, f x x 1 qx 2 f 2 1. and let x 1 Then where qx is a second-degree polynomial. y ax2 bx c. 5. The parabola shown in the figure has an equation of the form Find the equation of this parabola by the following methods. (a) Find the equation analytically. (b) Use the regression feature of a graphing utility to find the equation. x y 2 (2, 2) (4, 0) −4 −2 −4 −6 6 8 (1, 0) (0, − 4) (6, − 10) 6. One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this on the graph of the can be done, consider the point quadratic function f x x2. 2, 4 (2, 4) y 5 4 3 2 1 −3 −2 −1 1 2 3 x (a) Find the slope of the line joining 2, 4 slope of the tangent line at than the slope of the line through (b) Find the slope of the line joining 2, 4 slope of the tangent line at than the slope of the line through and and and 2, 4 3, 9? 3, 9. Is the greater than or less 2, 4 2, 4 1, 1. Is the greater than or less 2, 4 2, 4 2, 4 1, 1? 2.1, 4.41. and greater than or 2, 4 and and (c) Find the slope of the line joining Is the slope of the tangent line at less than the slope of the line through 2.1, 4.41? (d) Find the slope of the line joining 2 h, f 2 h in terms of the nonzero number 2, 4 and h. (e) Evaluate the slope formula from part (d) for h 1, 1, and 0.1. Compare these values with those in parts (a)–(c). (f) What can you conclude the slope of the tangent line at 2, 4 to be? Explain your answer. 215 333202_020R.qxd 12/7/05 9:43 AM Page 216 7. Use the form f x x kqx r function that (a) passes through the point the right and (b) passes through the point to the right. (There are many correct answers.) to create a cubic 2, 5 and rises to 3, 1 and falls 8. The multiplicative inverse of z zm 1. such that each complex number. z zm Find the multiplicative inverse of is a complex number (a) z 1 i (b) z 3 i (c) z 2 8i 9. Prove that the product of a complex number a bi and its complex conjugate is a real number. 10. Match the graph of the rational function given by (b) Determine the effect on the graph of f if a 0 and b is varied. 12. The endpoints of the interval over which distinct vision is possible is called the near point and far point of the eye (see figure). With increasing age, these points normally change. The table shows the approximate near points (in inches) for various ages (in years). y x Object blurry Object clear Object blurry Near point Far point f x ax b cx d with the given conditions. (a) y (b) y FIGURE FOR 12 (c) y x x (d) y x x (i) a > 0 (ii) a > 0 (iii) a < 0 (iv 11. Consider the function given by f x ax x b2. (a) Determine the effect on the graph of a is varied. Consider cases in which negative. f if b 0 and a is positive and a is 216 Age, x Near point, y 16 32 44 50 60 3.0 4.7 9.8 19.7 39.4 (a) Use the regression feature of a graphing utility to find a quadratic model for the data. Use a graphing utility to plot the data and graph the model in the same viewing window. (b) Find a rational model for the data. Take the reciprocals of the near points to generate the points Use the regression feature of a graphing utility to find a linear model for the data. The resulting line has the form x, 1y. 1 y ax b. Solve for Use a graphing utility to plot the data and graph the model in the same viewing window. y. (c) Use the table feature of a graphing utility to create a table showing the predicted near point based on each model for each of the ages in the original table. How well do the models fit the original data? (d) Use both models to estimate the near point for a person who is 25 years old. Which model is a better fit? (e) Do you think either model can be used to predict the near point for a person who is 70 years old? Explain. 333202_0300.qxd 12/7/05 10:24 AM Page 217 Exponential and Logarithmic Functions 33 Logarithmic Functions and Their Graphs Exponential Functions and Their Graphs Properties of Logarithms 3.3 3.1 3.2 3.4 3.5 Exponential and Logarithmic Equations Exponential and Logarithmic Models Carbon dating is a method used to determine the ages of archeological artifacts up to 50,000 years old. For example, archeologists are using carbon dating to determine the ages of the great pyramids of Egypt AT I O N S Exponential and logarithmic functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Computer Virus, Exercise 65, page 227 • Galloping Speeds of Animals, • IQ Scores, Exercise 85, page 244 Exercise 47, page 266 • Data Analysis: Meteorology, • Average Heights, • Forensics, Exercise 70, page 228 Exercise 115, page 255 Exercise 63, page 268 • Sound Intensity, Exercise 90, page 238 • Carbon Dating, Exercise 41, page 266 • Compound Interest, Exercise 135, page 273 217 333202_0301.qxd 12/7/05 10:25 AM Page 218 218 Chapter 3 Exponential and Logarithmic Functions 3.1 Exponential Functions and Their Graphs What you should learn • Recognize and evaluate exponential functions with base a. • Graph exponential functions and use the One-to-One Property. • Recognize, evaluate, and graph exponential functions with base e. • Use exponential functions to model and solve real-life problems. Why you should learn it Exponential functions can be used to model and solve real-life problems. For instance, in Exercise 70 on page 228, an exponential function is used to model the atmospheric pressure at different altitudes. © Comstock Images/Alamy Exponential Functions So far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions. In this chapter, you will study two types of nonalgebraic functions—exponential functions and logarithmic functions. These functions are examples of transcendental functions. Definition of Exponential Function The exponential function with base a f is denoted by f x ax where a > 0, a 1, and x is any real number. a 1 The base function, not an exponential function. is excluded because it yields f x 1x 1. This is a constant You have evaluated 43 64 and for integer and rational values of For example, you x, know that you need to interpret forms with irrational exponents. For the purposes of this text, it is sufficient to think of However, to evaluate for any real number ax 412 2. 4x x. a2 (where 2 1.41421356 ) as the number that has the successively closer approximations a1.4, a1.41, a1.414, a1.4142, a1.41421, . . . . Example 1 Evaluating Exponential Functions Use a calculator to evaluate each function at the indicated value of x. Function a. b. c. f x 2x f x 2x f x 0
.6x Solution Value x 3.1 x x 3 2 Function Value f 3.1 23.1 f 2 f 3 0.632 2 a. b. c. Graphing Calculator Keystrokes Display 2 2 .6 > > 3.1 ENTER ENTER > 3 2 ENTER 0.1166291 0.1133147 0.4647580 The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. When evaluating exponential functions with a calculator, remember to enclose fractional exponents in parentheses. Because the calculator follows the order of operations, parentheses are crucial in order to obtain the correct result. Now try Exercise 1. 333202_0301.qxd 12/7/05 10:25 AM Page 219 Section 3.1 Exponential Functions and Their Graphs 219 Graphs of Exponential Functions The graphs of all exponential functions have similar characteristics, as shown in Examples 2, 3, and 5. Example 2 Graphs of y ax In the same coordinate plane, sketch the graph of each function. a. f x 2x b. gx 4x Solution The table below lists some values for each function, and Figure 3.1 shows the graphs of the two functions. Note that both graphs are increasing. Moreover, the graph of is increasing more rapidly than the graph of f x 2x. gx 4x x 2x 4x 3 1 8 1 64 2 1 4 1 16 16 Now try Exercise 11. Exploration Note that an exponential function f x ax is a constant raised to a variable power, whereas a power function is a variable raised to a constant power. Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs. gx xn a. y1 b. y1 2x, y2 3x, y2 x2 x3 y g(x) = 4x 16 14 12 10 8 6 4 2 The table in Example 2 was evaluated by hand. You could, of course, use a graphing utility to construct tables with even more values. f(x) = 2x Example 3 Graphs of y a –x − 4 −3 −2 −1 −2 1 2 3 4 FIGURE 3.1 G(x) = 4−x y 16 14 12 10 8 6 4 F(x) = 2−x − 4 −3 −2 −1 −2 1 2 3 4 FIGURE 3.2 x x In the same coordinate plane, sketch the graph of each function. G x 4x F x 2x b. a. Solution The table below lists some values for each function, and Figure 3.2 shows the graphs of the two functions. Note that both graphs are decreasing. Moreover, the graph of is decreasing more rapidly than the graph of F x 2x. Gx 4x x 2x 4x 2 1 4 16 16 3 1 8 1 64 Now try Exercise 13. tions In Example 3, note that by using one of the properties of exponents, the funcFx 2x and Fx 2x 1 2x can be rewritten with positive exponents. and Gx 4x 1 4x Gx 4x x 1 2 1 4 x 333202_0301.qxd 12/7/05 3:30 PM Page 220 220 Chapter 3 Exponential and Logarithmic Functions Comparing the functions in Examples 2 and 3, observe that Fx 2x f x Gx 4x gx. and F g G is a reflection (in the -axis) of the graph of The Consequently, the graph of and have the same relationship. The graphs in Figures 3.1 and 3.2 graphs of y ax are typical of the exponential functions They have one y x -intercept and one horizontal asymptote (the -axis), and they are continuous. The basic characteristics of these exponential functions are summarized in Figures 3.3 and 3.4. y ax. and f. y y Graph of y ax, a > 1 , • Domain: Notice that the range of an exponential function is which means that values of x. ax > 0 0, , for all y = ax (0, 1) FIGURE 3.3 y y = a −x (0, 1) FIGURE 3.4 x x • Range: • Intercept: 0, 0, 1 • Increasing • x -axis is a horizontal asymptote ax → 0 x→ • Continuous as Graph of • Domain: y ax, a > 1 , • Range: • Intercept: 0, 0, 1 • Decreasing • x -axis is a horizontal asymptote ax→ 0 x→ • Continuous as From Figures 3.3 and 3.4, you can see that the graph of an exponential function is always increasing or always decreasing. As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one-to-one functions. You can use the following One-to-One Property to solve simple exponential equations. For a > 0 and a 1, ax ay if and only if x y. One-to-One Property Example 4 Using the One-to-One Property a. b. 9 3x1 32 3x1 2 x 1 1 x 1 2 x 8 ⇒ 2x 23 ⇒ x 3 Now try Exercise 45. Original equation 9 32 One-to-One Property Solve for x. 333202_0301.qxd 12/7/05 10:25 AM Page 221 Section 3.1 Exponential Functions and Their Graphs 221 In the following example, notice how the graph of y a x can be used to sketch the graphs of functions of the form f x b ± axc. Example 5 Transformations of Graphs of Exponential Functions Each of the following graphs is a transformation of the graph of f x 3x. a. Because gx 3x1 f x 1, the graph of can be obtained by shifting g the graph of f one unit to the left, as shown in Figure 3.5. b. Because shifting the graph of hx 3x 2 f x 2, f the graph of downward two units, as shown in Figure 3.6. k the graph of kx 3x f x, h can be obtained by reflecting can be obtained by c. Because the graph of f in the -axis, as shown in Figure 3.7. x d. Because graph of jx 3x f x, f y in the -axis, as shown in Figure 3.8. the graph of can be obtained by reflecting the j g(x) = 3x + 1 y 3 2 1 f(x) = 3 x x x −2 −1 1 FIGURE 3.5 Horizontal shift y 2 1 −1 −2 f(x) = 3x 1 2 k(x) = −3x −2 y 2 1 f(x) = 3x − (x) = 3 x − 2 FIGURE 3.6 Vertical shift y 4 3 2 1 j(x) = 3−x f(x) = 3x −2 −1 1 2 x x FIGURE 3.7 Reflection in x-axis FIGURE 3.8 Reflection in y-axis Now try Exercise 17. Notice that the transformations in Figures 3.5, 3.7, and 3.8 keep the -axis as a horizontal asymptote, but the transformation in Figure 3.6 yields a new horizontal asymptote of Also, be sure to note how the -intercept is affected by each transformation. y 2. y x 333202_0301.qxd 12/7/05 10:25 AM Page 222 222 Chapter 3 Exponential and Logarithmic Functions y 3 2 The Natural Base e (1, e) In many applications, the most convenient choice for a base is the irrational number e 2.718281828 . . . . f(x) = ex (−1, e−1) (0, 1) (−2, e−2) −2 −1 FIGURE 3.9 x 1 f(x) = 2e0.24x 3 − 2 −1 1 2 3 4 FIGURE 3.10 (x) = e−0.58x 1 2 − 4 −3 − 2 −1 1 2 3 4 FIGURE 3.11 This number is called the natural base. The function given by is called the natural exponential function. Its graph is shown in Figure 3.9. Be sure is the constant you see that for the exponential function 2.718281828 . . . , whereas is the variable. f x ex, e x f x e x Exploration Use a graphing utility to graph viewing window. Using the trace feature, explain what happens to the graph of 1 1xx in the same increases. e and as y1 y2 x y1 Example 6 Evaluating the Natural Exponential Function Use a calculator to evaluate the function given by value of a. x. x 2 x 0.25 x 1 b. c. f x ex at each indicated d. x 0.3 Solution Function Value f 2 e2 f 1 e1 f 0.25 e0.25 f 0.3 e0.3 a. b. c. d. Graphing Calculator Keystrokes 2 ENTER ex ex ex ex 1 ENTER 0.25 ENTER 0.3 ENTER Display 0.1353353 0.3678794 1.2840254 0.7408182 x x Now try Exercise 27. Example 7 Graphing Natural Exponential Functions Sketch the graph of each natural exponential function. a. f x 2e0.24x b. gx 1 2e0.58x Solution To sketch these two graphs, you can use a graphing utility to construct a table of values, as shown below. After constructing the table, plot the points and connect them with smooth curves, as shown in Figures 3.10 and 3.11. Note that the graph in Figure 3.10 is increasing, whereas the graph in Figure 3.11 is decreasing. x f x gx 3 2 1 0 1 2 3 0.974 1.238 1.573 2.000 2.542 3.232 4.109 2.849 1.595 0.893 0.500 0.280 0.157 0.088 Now try Exercise 35. 333202_0301.qxd 12/7/05 10:25 AM Page 223 Exploration Use the formula A P1 r n nt P $3000, years, and to calculate the amount in an account when r 6%, t 10 compounding is done (a) by the day, (b) by the hour, (c) by the minute, and (d) by the second. Does increasing the number of compoundings per year result in unlimited growth of the amount in the account? Explain. m 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 1 1 m m 2 2.59374246 2.704813829 2.716923932 2.718145927 2.718268237 2.718280469 2.718281693 e Section 3.1 Exponential Functions and Their Graphs 223 Applications One of the most familiar examples of exponential growth is that of an investment earning continuously compounded interest. Using exponential functions, you can develop a formula for interest compounded times per year and show how it leads to continuous compounding. n Suppose a principal compounded once a year. If the interest is added to the principal at the end of the year, the new balance is invested at an annual interest rate r, P P1 P1 is P Pr P1 r. This pattern of multiplying the previous principal by successive year, as shown below. 1 r is then repeated each Year Balance After Each Compounding P1 P2 P3 . . . Pt P1 r P1 P2 P1 rt 1 r P1 r1 r P1 r2 1 r P1 r21 r P1 r3 To accommodate more frequent (quarterly, monthly, or daily) compounding of interest, let be the number of compoundings per year and let be the number of years. Then the rate per compounding is and the account balance after t years is rn n t A P1 r n nt . n Amount (balance) with compoundings per year If you let the number of compoundings approaches what is called continuous compounding. In the formula for compoundings per year, let increase without bound, the process n This produces m nr. n mrt nt A P1 r n P1 r mr mrt P1 1 m P1 1 m mrt . Amount with compoundings per year n Substitute mr for n. Simplify. Property of exponents increases without bound, the table at the left shows that 1 1mm → e From this, you can conclude that the formula for continuous m As m → . as compounding is A Pert. Substitute e for 1 1mm. 333202_0301.qxd 12/7/05 10:25 AM Page 224 224 Chapter 3 Exponential and Logarithmic Functions Be sure you see that the annual interest rate must be written in decimal form. For instance, 6% should be written as 0.06. Formulas for Compound Interest t After years, the balance r interest rate A 1. For compoundings per year: n (in decimal form) is given by the following formulas. A P1 r n nt in an account with principal P and annual 2. For continuous compounding: A Pe rt Example 8 Compound Interest A total of $12,000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded a. quarterly. b. monthly. c. continuously. Solution a.
For quarterly compounding, you have balance is n 4. So, in 5 years at 9%, the nt A P1 r n 12,0001 0.09 4 4(5) Formula for compound interest Substitute for r,P, n, and t. $18,726.11. Use a calculator. b. For monthly compounding, you have n 12. So, in 5 years at 9%, the balance is nt A P1 r n 12,0001 0.09 12 12(5) Formula for compound interest Substitute for r,P, n, and t. $18,788.17. Use a calculator. c. For continuous compounding, the balance is A Pert 12,000e0.09(5) $18,819.75. Formula for continuous compounding Substitute for r,P, and t. Use a calculator. Now try Exercise 53. In Example 8, note that continuous compounding yields more than quarterly or monthly compounding. This is typical of the two types of compounding. That is, for a given principal, interest rate, and time, continuous compounding will always yield a larger balance than compounding times a year. n 333202_0301.qxd 12/7/05 10:25 AM Page 225 ) 10 9 8 7 6 5 4 3 2 1 FIGURE 3.12 Section 3.1 Exponential Functions and Their Graphs 225 Example 9 Radioactive Decay Radioactive Decay P ( ( P = 10 t/24,100 1 2 In 1986, a nuclear reactor accident occurred in Chernobyl in what was then the Soviet Union. The explosion spread highly toxic radioactive chemicals, such as plutonium, over hundreds of square miles, and the government evacuated the city and the surrounding area. To see why the city is now uninhabited, consider the model (24,100, 5) (100,000, 0.564) t 50,000 100,000 Years of decay P 101 2 t24,100 which represents the amount of plutonium that remains (from an initial amount t of 10 pounds) after years. Sketch the graph of this function over the interval t 100,000, represents 1986. How much of the 10 from pounds will remain in the year 2010? How much of the 10 pounds will remain after 100,000 years? where t 0 t 0 to P Solution The graph of this function is shown in Figure 3.12. Note from this graph that plutonium has a half-life of about 24,100 years. That is, after 24,100 years, half of the original amount will remain. After another 24,100 years, one-quarter of the original amount will remain, and so on. In the year 2010 there will still be t 24, P 101 22424,100 101 20.0009959 9.993 pounds of plutonium remaining. After 100,000 years, there will still be P 101 2100,00024,100 101 24.1494 0.564 pound of plutonium remaining. Now try Exercise 67. W RITING ABOUT MATHEMATICS Identifying Exponential Functions Which of the following functions generated the two tables below? Discuss how you were able to decide. What do these functions have in common? Are any of them the same? If so, explain why. a. d. x 2(x3) f1 x 1 x 7 f4 2 x x 8 1 x 7 2x 2 b. f2 e. f5 (x3) x 1 x 82x 2 c. f3 f. f6 x gx 1 7.5 0 8 1 9 2 3 11 15 x hx 2 1 32 16 0 8 1 4 2 2 Create two different exponential functions of the forms with y-intercepts of 0, 3. y abx and y cx d 333202_0301.qxd 12/7/05 10:25 AM Page 226 226 Chapter 3 Exponential and Logarithmic Functions 3.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. Polynomials and rational functions are examples of ________ functions. 2. Exponential and logarithmic functions are examples of nonalgebraic functions, also called ________ functions. 3. The exponential function given by f x ex is called the ________ ________ function, and the base e is called the ________ base. 4. To find the amount A t in an account after years with principal P and an annual interest rate compounded r n times per year, you can use the formula ________. 5. To find the amount A t in an account after years with principal P and an annual interest rate compounded r continuously, you can use the formula ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, evaluate the function at the indicated value of Round your result to three decimal places. x. Function 1. 2. 3. 4. 5. 6. f x 3.4x f x 2.3x f x 5x f x 2 3 gx 50002x f x 2001.212x 5x Value x 5.6 x 3 2 x x 3 10 x 1.5 x 24 In Exercises 7–10, match the exponential function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) y 6 4 −4 − 2 −2 x 2 4 y 6 4 (b) y 6 4 2 −2 −2 (d) 4 6 2 y 6 4 2 −4 −2 −2 7. 9. f x 2x f x 2x x 2 4 −4 −2 −2 2 4 8. 10. f x 2x 1 f x 2x2 x x In Exercises 11–16, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. x f x 1 f x 6x f x 2 x1 f x 1 2 f x 6x f x 4x3 3 x 16. 14. 15. 11. 13. 12. 2 In Exercises 17–22, use the graph of transformation that yields the graph of f g. to describe the 17. 18. 19. 20. 21. 22. f x 3 x, f x 4x, f x 2x, f x 10 x, f x 7 x , f x 0.3x, 2 gx 3x4 gx 4x 1 gx 5 2 x gx 10 x3 gx 7 2 gx 0.3x 5 x6 In Exercises 23–26, use a graphing utility to graph the exponential function. 23. 25. y 2x 2 y 3x2 1 24. 26. y 3x y 4x1 2 In Exercises 27–32, evaluate the function at the indicated value of Round your result to three decimal places. x. Function 27. 28. 29. 30. 31. 32. hx ex f x ex f x 2e5x f x 1.5ex2 f x 5000e0.06x f x 250e0.05x Value x 3 4 x 3.2 x 10 x 240 x 6 x 20 333202_0301.qxd 12/7/05 10:25 AM Page 227 Section 3.1 Exponential Functions and Their Graphs 227 In Exercises 33–38, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 33. 35. 37. f x ex f x 3ex4 f x 2ex2 4 34. 36. 38. f x ex f x 2e0.5x f x 2 e x5 In Exercises 39– 44, use a graphing utility to graph the exponential function. 39. 41. 43. y 1.085x st 2e0.12t gx 1 ex 40. 42. 44. y 1.085x st 3e0.2t hx e x2 In Exercise 45–52, use the One-to-One Property to solve the equation for x. 45. 47. 49. 51. 3x1 27 2x2 1 32 e3x2 e3 ex23 e2x 46. 48. 50. 52. 125 2x3 16 x1 1 5 e2x1 e4 ex 26 e5x Compound Interest A table to determine the balance rate for years and compounded In Exercises 53–56, complete the for dollars invested at n times per year. P r t 1 2 4 12 365 Continuous n A 53. 54. 55. 56. years P $2500, r 2.5%, t 10 P $1000, r 4%, t 10 P $2500, r 3%, t 20 P $1000, r 6%, t 40 years years years Compound Interest table to determine the balance rate for years, compounded continuously. In Exercises 57– 60, complete the for $12,000 invested at A r t 62. Trust Fund A deposit of $5000 is made in a trust fund that pays 7.5% interest, compounded continuously. It is specified that the balance will be given to the college from which the donor graduated after the money has earned interest for 50 years. How much will the college receive? 63. Inflation If the annual rate of inflation averages 4% over the next 10 years, the approximate costs of goods or services during any year in that decade will be modeled by Ct P1.04t, is the present cost. The price of an oil change for your car is presently $23.95. Estimate the price 10 years from now. is the time in years and where C P t 64. Demand The demand equation for a product is given by p 50001 4 4 e0.002x where p is the price and x is the number of units. (a) Use a graphing utility to graph the demand function for x > 0 and p > 0. (b) Find the price p for a demand of x 500 units. (c) Use the graph in part (a) to approximate the greatest price that will still yield a demand of at least 600 units. 65. Computer Virus The number of computers infected by a computer virus increases according to the model Vt 100e4.6052t, V1, and (c) (b) where is the time in hours. Find (a) V2. V1.5, V t P 66. Population The population (in millions) of Russia from 1996 to 2004 can be approximated by the model P 152.26e0.0039t, t 6 corresponding (Source: Census Bureau, International Data Base) where represents the year, with t to 1996. (a) According to the model, is the population of Russia increasing or decreasing? Explain. (b) Find the population of Russia in 1998 and 2000. (c) Use the model to predict the population of Russia in 2010. Q 67. Radioactive Decay Let 226Ra represent a mass of radioactive (in grams), whose half-life is 1599 years. years is radium The quantity of radium present after Q 251 (a) Determine the initial quantity (when t1599. t 0 ). t 2 10 20 30 40 50 t A (b) Determine the quantity present after 1000 years. (c) Use a graphing utility to graph the function over the interval t 0 to t 5000. 57. 59. r 4% r 6.5% 58. 60. r 6% r 3.5% 61. Trust Fund On the day of a child’s birth, a deposit of $25,000 is made in a trust fund that pays 8.75% interest, compounded continuously. Determine the balance in this account on the child’s 25th birthday. 68. Radioactive Decay Let represent a mass of carbon (in grams), whose half-life is 5715 years. The quan- 14 14C tity of carbon 14 present after years is t5715. Q t (a) Determine the initial quantity (when (b) Determine the quantity present after 2000 years. (c) Sketch the graph of this function over the interval t 0 to t 10,000. Q 101 t 0 ). 2 333202_0301.qxd 12/7/05 10:25 AM Page 228 228 Chapter 3 Exponential and Logarithmic Functions Model It Synthesis 69. Data Analysis: Biology To estimate the amount of defoliation caused by the gypsy moth during a given year, a forester counts the number of egg masses on 1 of an acre (circle of radius 18.6 feet) in the fall. The 40 percent of defoliation the next spring is shown in the table. y (Source: USDA, Forest Service) x Egg masses, x Percent of defoliation, y 0 25 50 75 100 12 44 81 96 99 A model for the data is given by y 100 1 7e0.069x . (a) Use a graphing utility to create a scatter plot of the data and graph the model in the same viewing window. (b) Create a table that compares the model with the sample data. (c) Estimate the percent of defoliation if 36 egg masses are counted on 1 40 acre. 2 3 (d) You observe that of a forest is defoliated the following spring. Use the graph in part (a) to estimate the number of egg masses per acre. 1 40 70. Data Analysis: Meteorology A meteorologist measures P (in (in pascals) at altitude the atmospheric pres
sure kilometers). The data are shown in the table. h True or False? the statement is true or false. Justify your answer. In Exercises 71 and 72, determine whether 71. The line y 2 is an asymptote for the graph of f x 10 x 2. e 271,801 . 99,990 72. Think About It nents to determine which functions (if any) are the same. In Exercises 73–76, use properties of expo- 73. 75. 3x f x 3x2 gx 3x 9 hx 1 9 f x 164x gx 1 x2 4 hx 1622x 74. 76. f x 4x 12 gx 22x6 hx 644x f x ex 3 gx e3x hx ex3 77. Graph the functions given by y 3x and y 4x and use the graphs to solve each inequality. (a) 4x < 3x (b) 4x > 3x 78. Use a graphing utility to graph each function. Use the graph to find where the function is increasing and decreasing, and approximate any relative maximum or minimum values. f x x 2ex gx x23x (b) (a) 79. Graphical Analysis Use a graphing utility to graph f x 1 0.5 x x and gx e0.5 in the same viewing window. What is the relationship between increases and decreases without bound? and as g x f 80. Think About It Which functions are exponential? (a) 3x (b) 3x 2 (c) 3x (d) 2x Altitude, h Pressure, P Skills Review 0 5 10 15 20 101,293 54,735 23,294 12,157 5,069 A model for the data is given by P 107,428e 0.150h. (a) Sketch a scatter plot of the data and graph the model on the same set of axes. (b) Estimate the atmospheric pressure at a height of 8 kilometers. In Exercises 81 and 82, solve for y . 81. x 2 y 2 25 82. x y 2 In Exercises 83 and 84, sketch the graph of the function. 83. f x 2 9 x 84. f x 7 x 85. Make a Decision To work an extended application analyzing the population per square mile of the United (Data States, visit this text’s website at college.hmco.com. Source: U.S. Census Bureau) 333202_0302.qxd 12/7/05 10:28 AM Page 229 Section 3.2 Logarithmic Functions and Their Graphs 229 3.2 Logarithmic Functions and Their Graphs What you should learn • Recognize and evaluate logarithmic functions with base a. • Graph logarithmic functions. • Recognize, evaluate, and graph natural logarithmic functions. • Use logarithmic functions to model and solve real-life problems. Why you should learn it Logarithmic functions are often used to model scientific observations. For instance, in Exercise 89 on page 238, a logarithmic function is used to model human memory. © Ariel Skelley/Corbis Remember that a logarithm is an exponent. So, to evaluate the loga x, logarithmic expression you need to ask the question, “To what power must be raised to obtain ”x? a Logarithmic Functions In Section 1.9, you studied the concept of an inverse function. There, you learned that if a function is one-to-one—that is, if the function has the property that no horizontal line intersects the graph of the function more than once—the function must have an inverse function. By looking back at the graphs of the exponential functions introduced in Section 3.1, you will see that every function of the form f x ax passes the Horizontal Line Test and therefore must have an inverse function. This inverse function is called the logarithmic function with base a. Definition of Logarithmic Function with Base a For a 1, and if and only if x ay. a > 0, x > 0, y loga x The function given by f x loga x x. Read as “log base of ” a is called the logarithmic function with base a. The equations y loga x and x a y are equivalent. The first equation is in logarithmic form and the second is in can be exponential form. For example, the logarithmic equation can rewritten in exponential form as be rewritten in logarithmic form as The exponential equation log5 125 3. When evaluating logarithms, remember that a logarithm is an exponent. is the exponent to which must be raised to obtain For 2 log3 9 53 125 9 32. x. a because 2 must be raised to the third power to get 8. This means that instance, log2 8 3 loga x Example 1 Evaluating Logarithms Use the definition of logarithmic function to evaluate each logarithm at the indicated value of x. f x log2 x, f x log4 x, x 32 x 2 b. d. f x log3 x, f x log10 x, x 1 x 1 100 Solution a. f 32 log2 32 5 f 1 log3 1 0 f 2 log4 2 1 f 1 log10 1 100 100 2 2 because because because because 25 32. 30 1. 412 4 2. 102 1 10 2 1 100. Now try Exercise 17. a. c. b. c. d. 333202_0302.qxd 12/7/05 10:28 AM Page 230 230 Chapter 3 Exponential and Logarithmic Functions Exploration Complete the table for f x 10 x. 2 1 0 1 2 x f x Complete the table for f x log x. 1 100 1 10 1 10 100 x f x Compare the two tables. What is the relationship between f x 10 x and f x log x? The logarithmic function with base 10 is called the common logarithmic function. It is denoted by or simply by log. On most calculators, this log10 . Example 2 shows how to use a calculator to evaluate function is denoted by common logarithmic functions. You will learn how to use a calculator to calculate logarithms to any base in the next section. LOG Example 2 Evaluating Common Logarithms on a Calculator Use a calculator to evaluate the function given by x 1 x 2.5 3 x 10 b. a. c. f x log x d. at each value of x 2 x. Solution Function Value f 10 log 10 log f 1 f 2.5 log 2.5 f 2 log2 1 3 3 a. b. c. d. Graphing Calculator Keystrokes Display LOG 10 ENTER LOG LOG LOG 1 3 ENTER 2.5 ENTER 2 ENTER 1 0.4771213 0.3979400 ERROR Note that the calculator displays an error message (or a complex number) when The reason for this is that there is no real number you try to evaluate power to which 10 can be raised to obtain log2. 2. Now try Exercise 23. The following properties follow directly from the definition of the logarith- mic function with base a. Properties of Logarithms 1. because loga 1 0 loga a 1 loga a x x 2. 3. and loga x loga y, 4. If because a0 1. a1 a. a log a x x Inverse Properties then x y. One-to-One Property Example 3 Using Properties of Logarithms a. Simplify: log4 1 b. Simplify: log7 7 c. Simplify: 6log 620 Solution a. Using Property 1, it follows that log7 7 1. b. Using Property 2, you can conclude that c. Using the Inverse Property (Property 3), it follows that log4 1 0. 6log 620 20. Now try Exercise 27. You can use the One-to-One Property (Property 4) to solve simple logarithmic equations, as shown in Example 4. 333202_0302.qxd 12/7/05 10:28 AM Page 231 Section 3.2 Logarithmic Functions and Their Graphs 231 Example 4 Using the One-to-One Property a. b. c. log3 x log3 12 x 12 Original equation One-to-One Property log2x 1 log x ⇒ 2x 1 x ⇒ x 1 log4 x2 6 log4 10 ⇒ x2 6 10 ⇒ x2 16 ⇒ x ±4 Now try Exercise 79. Graphs of Logarithmic Functions f(x) = 2x y = x g(x) = log 2 x y 10 10 −2 −2 FIGURE 3.13 Vertical asymptote: x = 0 f(x) = log 10 FIGURE 3.14 To sketch the graph of functions are reflections of each other in the line y loga x, y x. you can use the fact that the graphs of inverse Example 5 Graphs of Exponential and Logarithmic Functions In the same coordinate plane, sketch the graph of each function. a. f x 2x b. gx log2 x Solution a. For f x 2x, construct a table of values. By plotting these points and connecting them with a smooth curve, you obtain the graph shown in Figure 3.13. x f x 2x . Because gx log2 x obtained by plotting the points g curve. The graph of shown in Figure 3.13. is the inverse function of f x, x f x 2x, is and connecting them with a smooth as the graph of in the line y x, g f is a reflection of the graph of Now try Exercise 31. Example 6 Sketching the Graph of a Logarithmic Function Sketch the graph of the common logarithmic function vertical asymptote. f x log x. Identify the Solution Begin by constructing a table of values. Note that some of the values can be obtained without a calculator by using the Inverse Property of Logarithms. Others require a calculator. Next, plot the points and connect them with a smooth curve, as shown in Figure 3.14. The vertical asymptote is y ( -axis). x 0 Without calculator With calculator x fx log x 1 100 2 1 10 1 1 0 10 1 2 5 8 0.301 0.699 0.903 Now try Exercise 37. 333202_0302.qxd 12/7/05 10:28 AM Page 232 232 Chapter 3 Exponential and Logarithmic Functions The nature of the graph in Figure 3.14 is typical of functions of the form f x loga x, a > 1. x -intercept and one vertical asymptote. They have one x > 1. Notice how slowly the graph rises for The basic characteristics of logarithmic graphs are summarized in Figure 3.15. y = loga x (1, 0) 1 2 y 1 −1 FIGURE 3.15 Graph of • Domain: • Range: y loga x, a > 1 0, , 1, 0 • x -intercept: • Increasing x • One-to-one, therefore has an • inverse function y -axis is a vertical asymptote loga x → 0. x → as • Continuous • Reflection of graph of about the line y x y a x f x ax and gx loga x. are shown below to illus- The basic characteristics of the graph of f x ax 0, trate the inverse relation between , 0,1 • Domain: -intercept: • Range: y x • • In the next example, the graph of f x b ± loga functions of the form the graph results in a horizontal shift of the vertical asymptote. is used to sketch the graphs of Notice how a horizontal shift of -axis is a horizontal asymptote y loga x x c. ax → 0 as x → . You can use your understanding of transformations to identify vertical asymptotes of logarithmic functions. For instance, in Example 7(a) the graph of gx f x 1 of the vertical asymptote of x 1, the vertical asymptote of the graph of f x. shifts the graph one unit to the right. So, is gx one unit to the right of f x Example 7 Shifting Graphs of Logarithmic Functions The graph of each of the functions is similar to the graph of f x log x. a. Because gx logx 1 f x 1, the graph of g can be obtained by shifting the graph of f one unit to the right, as shown in Figure 3.16. b. Because hx 2 log x 2 f x, the graph of h can be obtained by shifting the graph of f two units upward, as shown in Figure 3.17. y 1 f(x) = log x (1, 0) x 1 (2, 0) −1 FIGURE 3.16 g(x) = log(x − 1) Now try Exercise 39. y 2 1 (1, 2) h(x) = 2 + log x f(x) = log x (1, 0) 2 x FIGURE 3.17 333202_0302.qxd 12/7/05 10:28 AM Page 233 Section 3.2 Logarithmic Functions and Their Graphs 233 The Natural Logarithmic Function By looking back at the
graph of the natural exponential function introduced in Section 3.1 on page 388, you will see that is one-to-one and so has an inverse function. This inverse function is called the natural logarithmic x function and is denoted by the special symbol ln read as “the natural log of ” or “el en of ” Note that the natural logarithm is written without a base. The base is understood to be f x ex e. x, x. The Natural Logarithmic Function The function defined by f x loge x ln x, x > 0 y = x is called the natural logarithmic function. y f(x) = ex (1, e) (e, 1) 3 2 (0, 1) ( ) 1 −1, e −2 −1 −1 −2 x 2 3 (1, 0) ( ) 1 , −1 e g(x) = f −1(x) = ln x Reflection of graph of y x the line FIGURE 3.18 f x ex about Notice that as with every other logarithmic function, the domain of the natural logarithmic function is the set of positive real numbers—be sure you see that ln negative numbers. is not defined for zero or for x The definition above implies that the natural logarithmic function and the natural exponential function are inverse functions of each other. So, every logarithmic equation can be written in an equivalent exponential form and every exponential equation can be written in logarithmic form. That is, and x e y y ln x are equivalent equations. Because the functions given by f x e x and gx ln x tions of each other, their graphs are reflections of each other in the line This reflective property is illustrated in Figure 3.18. are inverse funcy x. On most calculators, the natural logarithm is denoted by LN , as illustrated in Example 8. Example 8 Evaluating the Natural Logarithmic Function Use a calculator to evaluate the function given by x 0.3 x 1 x 2 b. a. c. f x ln x for each value of x. d. x 1 2 Solution Function Value Graphing Calculator Keystrokes Display a. b. c. d. f 2 ln 2 f 0.3 ln 0.3 f 1 ln1 f 1 2 ln1 2 LN LN LN LN Now try Exercise 61. 2 ENTER .3 ENTER 1 1 ENTER 0.6931472 –1.2039728 ERROR 2 ENTER 0.8813736 In Example 8, be sure you see that gives an error message on most calculators. (Some calculators may display a complex number.) This occurs because the domain of ln is the set of positive real numbers (see Figure 3.18). So, ln1 The four properties of logarithms listed on page 230 are also valid for is undefined. x ln1 natural logarithms. 333202_0302.qxd 12/7/05 10:28 AM Page 234 234 Chapter 3 Exponential and Logarithmic Functions Properties of Natural Logarithms 1. because e0 1. e1 e. because ln 1 0 ln e 1 ln e x x 2. 3. and ln x ln y, eln x x then x y. Inverse Properties One-to-One Property 4. If Example 9 Using Properties of Natural Logarithms Use the properties of natural logarithms to simplify each expression. a. ln 1 e b. eln 5 c. ln 1 3 d. 2 ln e Solution 1 ln e a. ln e1 1 Inverse Property b. eln 5 5 Inverse Property c. ln 1 3 0 3 0 Property 1 d. 2 ln e 21) 2 Property 2 Now try Exercise 65. Example 10 Finding the Domains of Logarithmic Functions Find the domain of each function. a. f x lnx 2 b. gx ln2 x c. hx ln x 2 f Solution a. Because is b. Because is c. Because g , 2. ln x 2 real numbers except lnx 2 is defined only if x 2 > 0, it follows that the domain of 2, . The graph of f is shown in Figure 3.19. ln2 x is defined only if g 2 x > 0, is shown in Figure 3.20. The graph of it follows that the domain of is defined only if x 0. x 2 > 0, The graph of it follows that the domain of h is shown in Figure 3.21. h is all f(x) = ln(x − 21 −2 −3 −4 y 2 g(x) = ln(2 − x) −1 x 1 2 −1 −1 FIGURE 3.19 FIGURE 3.20 Now try Exercise 69. h(x) = ln x2 y 4 2 −2 2 4 x −4 FIGURE 3.21 333202_0302.qxd 12/7/05 10:28 AM Page 235 Section 3.2 Logarithmic Functions and Their Graphs 235 Memory Model f t( ) Application Example 11 Human Memory Model f(t) = 75 − 6ln(t + 1) Students participating in a psychology experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model 80 70 60 50 40 30 20 10 FIGURE 3.22 4 10 6 Time (in months) 8 t 12 f t 75 6 lnt 1, 0 ≤ t ≤ 12 where t is the time in months. The graph of f a. What was the average score on the original b. What was the average score at the end of c. What was the average score at the end of t 2 t 6 months? months? is shown in Figure 3.22. t 0 exam? Solution a. The original average score was f 0 75 6 ln0 1 75 6 ln 1 75 60 75. Substitute 0 for t. Simplify. Property of natural logarithms Solution b. After 2 months, the average score was f 2 75 6 ln2 1 75 6 ln 3 75 61.0986 68.4. Substitute 2 for t. Simplify. Use a calculator. Solution c. After 6 months, the average score was f 6 75 6 ln6 1 75 6 ln 7 75 61.9459 63.3. Substitute 6 for t. Simplify. Use a calculator. Solution Now try Exercise 89. W RITING ABOUT MATHEMATICS Analyzing a Human Memory Model Use a graphing utility to determine the time in months when the average score in Example 11 was 60. Explain your method of solving the problem. Describe another way that you can use a graphing utility to determine the answer. 333202_0302.qxd 12/7/05 3:32 PM Page 236 236 Chapter 3 Exponential and Logarithmic Functions 3.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The inverse function of the exponential function given by 2. The common logarithmic function has base ________ . fx ax is called the ________ function with base a. 3. The logarithmic function given by fx ln x is called the ________ logarithmic function and has base ________. 4. The Inverse Property of logarithms and exponentials states that loga ax x and ________. 5. The One-to-One Property of natural logarithms states that if ln x ln y, then ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 8, write the logarithmic equation in exponential form. For example, the exponential form of log5 25 2 52 25. is log4 64 3 log7 1 2 log32 4 2 log36 6 1 49 5 2 1. 3. 5. 7. 2. 4. 6. 8. 3 log3 81 4 log 1 1000 log16 8 3 log8 4 2 4 3 In Exercises 9 –16, write the exponential equation in logarithmic form. For example, the logarithmic form of 23 8 log2 8 3. is 9. 11. 13. 15. 53 125 8114 3 62 1 36 70 1 10. 12. 14. 16. 82 64 932 27 43 1 64 103 0.001 In Exercises 17–22, evaluate the function at the indicated value of without using a calculator. x Function f x log2 x f x log16 x f x log7 x f x log x gx loga x gx logb x 17. 18. 19. 20. 21. 22. Value x 16 x 4 x 1 x 10 x a2 x b3 In Exercises 23–26, use a calculator to evaluate at the indicated value of decimal places. f x log x Round your result to three x. 23. 25. x 4 5 x 12.5 24. 26. x 1 500 x 75.25 In Exercises 27–30, use the properties of logarithms to simplify the expression. 27. 29. log3 34 log 28. 30. log1.5 1 9log915 In Exercises 31–38, find the domain, -intercept, and vertical asymptote of the logarithmic function and sketch its graph. x 31. 33. 35. 37. f x log4 x y log3 x 2 f x log6 y logx 5 x 2 32. 34. 36. gx log6 x hx log4 y log5 x 3 x 1 4 38. y logx to In Exercises 39– 44, use the graph of match the given function with its graph. Then describe the f relationship between the graphs of and [The graphs are labeled (a), (b), (c), (d), (e), and (f).] gx log3 x g. (a) –3 y 3 2 –1 –2 (b) y 3 2 1 x 1 –4 –3 –2 –1 –1 1 –2 (c) y (d) y 4 3 2 1 –1 –2 –1 –1 –2 1 2 3 x x 333202_0302.qxd 12/7/05 3:33 PM Page 237 Section 3.2 Logarithmic Functions and Their Graphs 237 (e) y (f1 –1 –2 1 3 4 x In Exercises 73–78, use a graphing utility to graph the function. Be sure to use an appropriate viewing window. 73. 75. 77. fx logx 1 fx lnx 1 fx ln x 2 74. 76. 78. fx logx 1 fx lnx 2 fx 3 ln x 1 In Exercises 79–86, use the One-to-One Property to solve the equation for x. 39. 41. 43. f x log3 x 2 f x log3 f x log3 1 x x 2 40. 42. 44. f x log3 x f x log3 f x log3 x 1 x 79. 81. 83. 85. x 1 log2 4 log2 log2x 1 log 15 lnx 2 ln 6 lnx2 2 ln 23 82. 80. x 3 log2 9 log2 log5x 3 log 12 lnx 4 ln 2 84. 86. lnx2 x ln 6 In Exercises 45–52, write the logarithmic equation in exponential form. 45. 47. 49. 51. 0.693 . . . ln 1 2 ln 4 1.386 . . . ln 250 5.521 . . . ln 1 0 46. 48. 50. 52. ln 2 0.916 . . . 5 ln 10 2.302 . . . ln 679 6.520 . . . ln e 1 In Exercises 53– 60, write the exponential equation in logarithmic form. 53. 55. 57. 59. e3 20.0855 . . . e12 1.6487 . . . e0.5 0.6065 . . . ex 4 54. 56. 58. 60. e2 7.3890 . . . e13 1.3956 . . . e4.1 0.0165 . . . e2x 3 In Exercises 61–64, use a calculator to evaluate the function x. at the indicated value of Round your result to three decimal places. Function f x ln x f x 3 ln x gx 2 ln x gx ln x 61. 62. 63. 64. Value x 18.42 x 0.32 x 0.75 x 1 2 In Exercises 65– 68, evaluate value of without using a calculator. x gx ln x at the indicated 65. 67. x e3 x e23 66. 68. x e2 x e52 In Exercises 69–72, find the domain, -intercept, and vertical asymptote of the logarithmic function and sketch its graph. x 69. 71. f x lnx 1 gx lnx 70. 72. hx lnx 1 f x ln3 x Model It 87. Monthly Payment The model t 12.542 ln x x 1000 , x > 1000 approximates the length of a home mortgage of $150,000 at 8% in terms of the monthly payment. In the model, is the monthly payment in dollars (see figure). is the length of the mortgage in years and x t t 30 25 20 15 10 ,000 4,000 6,000 8,000 10,000 Monthly payment (in dollars) x (a) Use the model to approximate the lengths of a $150,000 mortgage at 8% when the monthly payment is $1100.65 and when the monthly payment is $1254.68. (b) Approximate the total amounts paid over the term of the mortgage with a monthly payment of $1100.65 and with a monthly payment of $1254.68. (c) Approximate the total interest charges for a monthly payment of $1100.65 and for a monthly payment of $1254.68. (d) What is the vertical asymptote for the model? Interpret its meaning in the context of the problem. 333202_0302.qxd 12/7/05 10:28 AM Page 238 238 Chapter 3 Exponential and Logarithmic Functions 88. Compound Interest A principal 91 2% and K compounded continuously, increases
to an amount times is given by the original principal after years, where t ln K0.095. (a) Complete the table and interpret your results. invested at P, t t 1 2 4 6 8 10 12 K t (b) Sketch a graph of the function. 89. Human Memory Model Students in a mathematics class were given an exam and then retested monthly with an equivalent exam. The average scores for the class are given f t 80 17 logt 1, by the human memory model 0 ≤ t ≤ 12 where (a) Use a graphing utility to graph the model over the is the time in months. t specified domain. (b) What was the average score on the original exam t 0? (c) What was the average score after 4 months? (d) What was the average score after 10 months? 90. Sound Intensity The relationship between the number of in watts per square I decibels and the intensity of a sound meter is 10 log I 1012. (a) Determine the number of decibels of a sound with an intensity of 1 watt per square meter. (b) Determine the number of decibels of a sound with an intensity of 102 watt per square meter. (c) The intensity of the sound in part (a) is 100 times as great as that in part (b). Is the number of decibels 100 times as great? Explain. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 91 and 92, determine whether 91. You can determine the graph of by graphing and reflecting it about the -axis. f x log6 x x gx 6x 92. The graph of f x log3 x contains the point 27, 3. f f 93. g g. In Exercises 93–96, sketch the graph of and and describe the relationship between the graphs of and What is the relationship between the functions and gx log3 x gx log5 x gx ln x gx log x f x 3x, f x 5x, f x ex, f x 10 x, 96. 94. 95. g? f g 97. Graphical Analysis Use a graphing utility to graph f in the same viewing window and determine which and . is increasing at the greater rate as What can you conclude about the rate of growth of the natural logarithmic function? gx x gx 4x f x ln x, f x ln x, approaches (b) (a) x 98. (a) Complete the table for the function given by fx ln x x . 1 5 10 102 104 106 x f x (b) Use the table in part (a) to determine what value f x approaches as x increases without bound. (c) Use a graphing utility to confirm the result of part (b). 99. Think About It The table of values was obtained by evaluating a function. Determine which of the statements may be true and which must be falsea) y is an exponential function of x. (b) y is a logarithmic function of x. (c) x is an exponential function of y. (d) y is a linear function of x. 100. Writing Explain why loga x is defined only for 0 < a < 1 and a > 1. In Exercises 101 and 102, (a) use a graphing utility to graph the function, (b) use the graph to determine the intervals in which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values of the function. f x ln x hx lnx 2 1 101. 102. Skills Review In Exercises 103–108, evaluate f x 3x 2 and f g2 gx x3 1. 103. the function for 105. fg6 107. f g7 104. f g1 0 f g 108. g f 3 106. 333202_0303.qxd 12/7/05 10:29 AM Page 239 3.3 Properties of Logarithms Section 3.3 Properties of Logarithms 239 What you should learn • Use the change-of-base formula to rewrite and evaluate logarithmic expressions. • Use properties of logarithms to evaluate or rewrite logarithmic expressions. • Use properties of logarithms to expand or condense logarithmic expressions. • Use logarithmic functions to model and solve real-life problems. Why you should learn it Logarithmic functions can be used to model and solve real-life problems. For instance, in Exercises 81–83 on page 244, a logarithmic function is used to model the relationship between the number of decibels and the intensity of a sound. Change of Base Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base ). Although common logs and natural logs are the most frequently used, you may occasionally need to evaluate logarithms to other bases. To do this, you can use the following change-of-base formula. e Change-of-Base Formula a, Let loga x b, and be positive real numbers such that can be converted to a different base as follows. x a 1 and b 1. Then Base b loga x logb x logb a Base 10 loga x log x log a Base e loga x ln x ln a One way to look at the change-of-base formula is that logarithms to base a are simply constant multiples of logarithms to base The constant multiplier is 1logba. b. Example 1 Changing Bases Using Common Logarithms a. b. AP Photo/Stephen Chernin log4 25 log 25 log 4 1.39794 0.60206 2.3219 log2 12 log 12 log 2 loga x log x log a Use a calculator. Simplify. 1.07918 0.30103 3.5850 Now try Exercise 1(a). Example 2 Changing Bases Using Natural Logarithms a. b. log4 25 ln 25 ln 4 3.21888 1.38629 2.3219 log2 12 ln 12 ln 2 loga x ln x ln a Use a calculator. Simplify. 2.48491 0.69315 3.5850 Now try Exercise 1(b). 333202_0303.qxd 12/7/05 3:35 PM Page 240 240 Chapter 3 Exponential and Logarithmic Functions Properties of Logarithms a You know from the preceding section that the logarithmic function with base is the inverse function of the exponential function with base So, it makes sense that the properties of exponents should have corresponding properties involving logarithms. For instance, the exponential property has the corresponding logarithmic property a0 1 loga1 0. a. There is no general property that can be used to rewrite u ± v. loga u v loga loga u loga v. Specifically, is not equal to Properties of Logarithms Let be a positive number such that and are positive real numbers, the following properties are true. a 1, a v n and let be a real number. If u 1. Product Property: loga 2. Quotient Property: loga Logarithm with Base a uv loga u loga v u v loga u loga v 3. Power Property: loga un n loga u Natural Logarithm lnuv ln u ln v u ln v ln u ln v ln un n ln u For proofs of the properties listed above, see Proofs in Mathematics on page 278. Example 3 Using Properties of Logarithms Write each logarithm in terms of ln 2 and ln 3. 2 27 a. ln 6 ln b. Solution a. ln 6 ln2 3 b. ln ln 2 ln 3 2 27 ln 2 ln 27 ln 2 ln 33 ln 2 3 ln 3 Rewrite 6 as 2 3. Product Property Quotient Property Rewrite 27 as 33. Power Property Now try Exercise 17. Example 4 Using Properties of Logarithms Find the exact value of each expression without using a calculator. a. log5 35 b. ln e6 ln e2 Solution a. log5 35 log5 513 1 3 log5 5 1 3 1 1 3 b. ln e6 ln e2 ln e6 e2 ln e4 4 ln e 41 4 Now try Exercise 23 Historical Note John Napier, a Scottish mathematician, developed logarithms as a way to simplify some of the tedious calculations of his day. Beginning in 1594, Napier worked about 20 years on the invention of logarithms. Napier was only partially successful in his quest to simplify tedious calculations. Nonetheless, the development of logarithms was a step forward and received immediate recognition. 333202_0303.qxd 12/7/05 10:29 AM Page 241 Section 3.3 Properties of Logarithms 241 Rewriting Logarithmic Expressions The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. This is true because these properties convert complicated products, quotients, and exponential forms into simpler sums, differences, and products, respectively. Example 5 Expanding Logarithmic Expressions Expand each logarithmic expression. a. log4 5x3y b. ln 3x 5 7 Exploration Use a graphing utility to graph the functions given by Solution a. ln x lnx 3 y1 b. and ln y2 x x 3 in the same viewing window. Does the graphing utility show the functions with the same domain? If so, should it? Explain your reasoning. ln 3x 5 7 log4 5x3y log4 5 log4 x3 log4 y log4 5 3 log4 x log4 y 3x 512 7 ln3x 512 ln 7 1 2 ln3x 5 ln 7 ln Product Property Power Property Rewrite using rational exponent. Quotient Property Power Property Now try Exercise 47. In Example 5, the properties of logarithms were used to expand logarithmic expressions. In Example 6, this procedure is reversed and the properties of logarithms are used to condense logarithmic expressions. Example 6 Condensing Logarithmic Expressions Condense each logarithmic expression. a. c. 1 2 log x 3 logx 1 log2 x log2 x 1 1 3 b. 2 lnx 2 ln x Solution 2 log x 3 logx 1 log x12 logx 13 1 a. logxx 13 b. 2 lnx 2 ln x lnx 22 ln x c. 1 3 log2 x log2 ln x 22 x log2 xx 1 x 1 1 3 xx 113 log2 log2 3xx 1 Now try Exercise 69. Power Property Product Property Power Property Quotient Property Product Property Power Property Rewrite with a radical. 333202_0303.qxd 12/7/05 10:29 AM Page 242 242 Chapter 3 Exponential and Logarithmic Functions Application x One method of determining how the - and -values for a set of nonlinear data are related is to take the natural logarithm of each of the - and -values. If the points are graphed and fall on a line, then you can determine that the - and -values are related by the equation ln y m ln x y y x x y where m is the slope of the line. Example 7 Finding a Mathematical Model x y The table shows the mean distance and the period (the time it takes a planet to orbit the sun) for each of the six planets that are closest to the sun. In the table, the mean distance is given in terms of astronomical units (where Earth’s mean distance is defined as 1.0), and the period is given in years. Find an equation that y relates and x. Planet Mean distance, x Period, y Mercury Venus Earth Mars Jupiter Saturn 0.387 0.723 1.000 1.524 5.203 9.537 0.241 0.615 1.000 1.881 11.863 29.447 Solution The points in the table above are plotted in Figure 3.23. From this figure it is not y clear how to find an equation that relates and To solve this problem, take the y natural logarithm of each of the - and -values in the table. This produces the following results. x. x ln x Planet Mercury 0.949 1.423 ln y Venus 0.324 0.486 Earth Mars Jupiter Saturn 0.000 0.421 1.649 2.255 0.000 0.632 2.473 3.383 Now, by plotting the points in the second table, you can see that all six of the points appea
r to lie in a line (see Figure 3.24). Choose any two points to 0, 0, determine the slope of the line. Using the two points you can determine that the slope of the line is 0.421, 0.632 and m 0.632 0 0.421 0 1.5 3 2 . By the point-slope form, the equation of the line is ln y 3 X ln x. You can therefore conclude that 2 ln x. Y 3 2 X, where Y ln y and Now try Exercise 85. Planets Near the Sun y Saturn Mercury Venus Earth Jupiter Mars x 30 25 20 15 10 Mean distance (in astronomical units) 10 FIGURE 3.23 ln y 3 2 1 Earth Venus Mercury FIGURE 3.24 Saturn Jupiter 3 ln y = ln x 2 Mars 1 2 3 ln x 333202_0303.qxd 12/7/05 3:36 PM Page 243 Section 3.3 Properties of Logarithms 243 3.3 Exercises VOCABULARY CHECK: In Exercises 1 and 2, fill in the blanks. 1. To evaluate a logarithm to any base, you can use the ________ formula. 2. The change-of-base formula for base e is given by loga x ________. In Exercises 3–5, match the property of logarithms with its name. uv loga u loga v 3. 4. 5. loga ln un n ln u loga u v loga u loga v (a) Power Property (b) Quotient Property (c) Product Property PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–8, rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. 1. 3. 5. 7. log5 x log15 x logx 3 10 log2.6 x 2. 4. 6. 8. log3 x log13 x logx 3 4 log7.1 x In Exercises 9–16, evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. 9. 11. 13. 15. log3 7 log12 4 log9 0.4 log15 1250 10. 12. 14. 16. log7 4 log14 5 log20 0.125 log3 0.015 17. In Exercises 17–22, use the properties of logarithms to rewrite and simplify the logarithmic expression. 42 34 log2 9 log 300 6 e2 log4 8 log5 1 ln5e6 21. 19. 20. 22. 18. ln 250 32. 3 ln e4 33. 34. 35. 36. 37. 38. ln 1 e ln 4e3 ln e2 ln e5 2 ln e6 ln e5 log5 75 log5 3 log4 2 log4 32 In Exercises 39–60, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 39. log4 5x 41. log8 x 4 log5 5 x ln z ln xyz2 43. 45. 47. 49. ln zz 12, z > 1 In Exercises 23–38, find the exact value of the logarithmic expression without using a calculator. (If this is not possible, state the reason.) 23. 25. 27. 29. 31. log3 9 log2 48 log4 161.2 9 log3 ln e4.5 24. 26. 28. 30. log5 1 125 log6 36 log3 810.2 16 log2 51. 53. 55. 57. 59. , a > 1 a 1 9 log2 ln 3x y x 4y z 5 ln log5 x 2 y 2z3 ln 4x3x2 3 40. 42. log3 10z y 2 log10 44. log6 1 z3 46. 48. 50. 52. 54. 56. 58. ln 3t log 4x2 y lnx 2 1 x3 , x > 1 ln 6 x 2 1 lnx 2 y3 x y4 z4 xy4 z5 log10 log2 60. ln x 2x 2 333202_0303.qxd 12/7/05 3:36 PM Page 244 244 Chapter 3 Exponential and Logarithmic Functions In Exercises 61–78, condense the expression to the logarithm of a single quantity. Model It 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. ln x ln 3 ln y ln t log4 z log4 y log5 8 log5 t x 4 2 log2 z 2 2 3 log7 1 4 log3 5x 4 log6 2x ln x 3 lnx 1 2 ln 8 5 lnz 4 log x 2 log y 3 log z 3 log3 x 4 log3 y 4 log3 z ln x 4lnx 2 lnx 2 4ln z lnz 5 2 lnz 5 2 lnx 3 ln x lnx2 1 1 3 23 ln x lnx 1 lnx 1 log8 y 2 log8 y 4 log8 1 3 log4 1 2 x 1 2 log4 y 1 x 1 6 log4 x In Exercises 79 and 80, compare the logarithmic quantities. If two are equal, explain why. 79. 80. log2 32 log2 4 log7 , log2 32 4 , log2 32 log2 4 70, log7 35, 1 2 log7 10 Sound Intensity In Exercises 81–83, use the following information. The relationship between the number of decibels in watts per square meter is given by and the intensity of a sound I 84. Human Memory Model Students participating in a psychology experiment attended several lectures and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group can be modeled by the human memory model f t 90 15 logt 1, 0 ≤ t ≤ 12 where t is the time in months. (a) Use the properties of logarithms to write the func- tion in another form. (b) What was the average score on the original exam t 0? (c) What was the average score after 4 months? (d) What was the average score after 12 months? (e) Use a graphing utility to graph the function over the specified domain. (f) Use the graph in part (e) to determine when the average score will decrease to 75. (g) Verify your answer to part (f) numerically. 85. Galloping Speeds of Animals Four-legged animals run with two different types of motion: trotting and galloping. An animal that is trotting has at least one foot on the ground at all times, whereas an animal that is galloping has all four feet off the ground at some point in its stride. The number of strides per minute at which an animal breaks from a trot to a gallop depends on the weight of the animal. Use the table to find a logarithmic equation that relates an animal’s weight (in pounds) and its lowest galloping speed (in strides per minute). x y 10 log I 1012. 81. Use the properties of logarithms to write the formula in simpler form, and determine the number of decibels of a 106 sound with an intensity of watt per square meter. 82. Find the difference in loudness between an average office 1.26 107 watt per square meter and watt with an intensity of a broadcast studio with an intensity of per square meter. 3.16 105 83. You and your roommate are playing your stereos at the same time and at the same intensity. How much louder is the music when both stereos are playing compared with just one stereo playing? Weight, x Galloping Speed, y 25 35 50 75 500 1000 191.5 182.7 173.8 164.2 125.9 114.2 333202_0303.qxd 12/7/05 10:29 AM Page 245 78 C 21 C. 86. Comparing Models A cup of water at an initial temperais placed in a room at a constant temperature ture of of The temperature of the water is measured every 5 minutes during a half-hour period. The results are recordwhere is the time (in ed as ordered pairs of the form T minutes) and 0, 78.0, 25, 42.4, is the temperature (in degrees Celsius). 20, 46.3, 5, 66.0, 30, 39.6 15, 51.2, 10, 57.5, t, T, t is given 93. Proof Prove that logb u v logb u logb v. Section 3.3 Properties of Logarithms 245 Synthesis True or False? statement is true or false given that answer. In Exercises 87–92, determine whether the Justify your f x ln x. 87. 88. 89. 90. f 0 0 f ax f a f x, f x 2 f x f 2, f x < 0, then then v u2. 0 < x < 1. 91. If 92. If a > 0, x > 0 x > 2 94. Proof Prove that logb un n logb u. In Exercises 95–100, use the change -of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph both functions in the same viewing window to verify that the functions are equivalent. 95. 97. 99. f x log2 x f x log12 x f x log11.8 x 96. 98. 100. f x log4 x f x log14 x f x log12.4 x 101. Think About It Consider the functions below. f x ln x 2 , gx ln x ln 2 , hx ln x ln 2 Which two functions should have identical graphs? Verify your answer by sketching the graphs of all three functions on the same set of coordinate axes. 102. Exploration For how many integers between 1 and 20 can the natural logarithms be approximated given that ln 2 0.6931, ln 3 1.0986, 1.6094? Approximate these logarithms (do not use a calculator). ln 5 and Skills Review In Exercises 103–106, simplify the expression. 3 2x 2 3y xyx1 y11 24xy2 16x3y 18x3y4318x3y43 106. 105. 103. 104. (a) The graph of the model for the data should be asymptotic with the graph of the temperature of the room. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points t, T (b) An exponential model for the data and t, T 21 t, T 21. by T 21 54.40.964t. Solve for with the plot of the original data. T and graph the model. Compare the result (c) Take the natural logarithms of the revised temperatures. t, lnT 21 Use a graphing utility to plot the points and observe that the points appear to be linear. Use the regression feature of the graphing utility to fit a line to these data. This resulting line has the form lnT 21 at b. T. Use the properties of the logarithms to solve for Verify that the result is equivalent to the model in part (b). (d) Fit a rational model to the data. Take the reciprocals of y the -coordinates of the revised data points to generate the points t, . 1 T 21 Use a graphing utility to graph these points and observe that they appear to be linear. Use the regression feature of a graphing utility to fit a line to these data. The resulting line has the form 1 T 21 at b. Solve for rational function and the original data points. and use a graphing utility to graph the T, (e) Write a short paragraph explaining why the transformations of the data were necessary to obtain each model. Why did taking the logarithms of the temperatures lead to a linear scatter plot? Why did taking the reciprocals of the temperature lead to a linear scatter plot? In Exercises 107–110, solve the equation. 107. 109. 3x2 2x 1 0 x 4 2 3x 1 108. 110. 4x2 5x 1 0 2x 3 5 x 1 333202_0304.qxd 12/7/05 10:31 AM Page 246 246 Chapter 3 Exponential and Logarithmic Functions 3.4 Exponential and Logarithmic Equations What you should learn • Solve simple exponential and logarithmic equations. • Solve more complicated exponential equations. • Solve more complicated logarithmic equations. • Use exponential and logarithmic equations to model and solve real-life problems. Why you should learn it Exponential and logarithmic equations are used to model and solve life science applications. For instance, in Exercise 112, on page 255, a logarithmic function is used to model the number of trees per acre given the average diameter of the trees. © James Marshall/Corbis Introduction So far in this chapter, you have studied the definitions, graphs, and properties of exponential and logarithmic functions. In this section, you will study procedures for solving equations involving these exponential and logarithmic functions. There are two basic strategies
for solving exponential or logarithmic equations. The first is based on the One-to-One Properties and was used to solve simple exponential and logarithmic equations in Sections 3.1 and 3.2. The the following second is based on the Inverse Properties. For properties are true for all and a > 0 and are defined. for which a 1, x y and loga y loga x One-to-One Properties if and only if x y. if and only if a x a y loga x loga y Inverse Properties alog a x x loga a x x x y. Example 1 Solving Simple Equations Original Equation a. b. c. d. e. f. 2x 32 ln x ln 3 0 1 x 9 3 e x 7 ln x 3 log x 1 Rewritten Equation 2x 25 ln x ln 3 3x 32 ln e x ln 7 e ln x e3 10 log x 101 Now try Exercise 13. Solution Property x 5 x 3 x 2 x ln 7 x e3 x 101 1 10 One-to-One One-to-One One-to-One Inverse Inverse Inverse The strategies used in Example 1 are summarized as follows. Strategies for Solving Exponential and Logarithmic Equations 1. Rewrite the original equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions. 2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. 3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions. 333202_0304.qxd 12/7/05 10:31 AM Page 247 Section 3.4 Exponential and Logarithmic Equations 247 Solving Exponential Equations Example 2 Solving Exponential Equations Solve each equation and approximate the result to three decimal places if necessary. a. ex2 e3x4 32 x 42 b. Solution a. ex2 e3x4 x2 3x 4 x2 3x 4 0 x 1x Write original equation. One-to-One Property Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. x 1 and x 4. Check these in the original equation. b. The solutions are 32 x 42 2 x 14 log2 2 x log2 14 x log2 14 x ln 14 ln 2 3.807 Write original equation. Divide each side by 3. Take log (base 2) of each side. Inverse Property Change-of-base formula The solution is x log2 14 3.807. Now try Exercise 25. Check this in the original equation. In Example 2(b), the exact solution is x log2 14 and the approximate An exact answer is preferred when the solution is an solution is intermediate step in a larger problem. For a final answer, an approximate solution is easier to comprehend. x 3.807. Example 3 Solving an Exponential Equation Solve e x 5 60 and approximate the result to three decimal places. Remember that the natural logarithmic function has a base of e. Solution e x 5 60 e x 55 ln ex ln 55 Write original equation. Subtract 5 from each side. Take natural log of each side. x ln 55 4.007 Inverse Property The solution is x ln 55 4.007 . Check this in the original equation. Now try Exercise 51. 333202_0304.qxd 12/7/05 10:31 AM Page 248 248 Chapter 3 Exponential and Logarithmic Functions Example 4 Solving an Exponential Equation Solve 232t5 4 11 and approximate the result to three decimal places. Solution 232t5 4 11 232t5 15 32t5 15 2 log3 32t5 log3 2t 5 log3 15 2 15 2 log3 7.5 2t 5 log3 7.5 t 5 1 2 2 t 3.417 t 5 1 2 Write original equation. Add 4 to each side. Divide each side by 2. Take log (base 3) of each side. Inverse Property Add 5 to each side. Divide each side by 2. Use a calculator. Remember that to evaluate a logarithm such as you need to use the change-of-base formula. log3 7.5, log3 7.5 ln 7.5 ln 3 1.834 The solution is 2 log3 7.5 3.417. Check this in the original equation. Now try Exercise 53. When an equation involves two or more exponential expressions, you can still use a procedure similar to that demonstrated in Examples 2, 3, and 4. However, the algebra is a bit more complicated. Example 5 Solving an Exponential Equation of Quadratic Type Solve e 2x 3e x 2 0. Algebraic Solution e 2x 3e x 2 0 e x2 3e x 2 0 e x 2e x 1 0 e x 2 0 Write original equation. Write in quadratic form. Factor. Set 1st factor equal to 0. x ln 2 Solution Set 2nd factor equal to 0. e x 1 0 x 0 x ln 2 0.693 Solution The solutions are these in the original equation. y e2x 3ex 2. Graphical Solution Use a graphing utility to graph Use the zero or root feature or the zoom and trace features of the graphing utility to approximate the values of for which y 0. In Figure 3.25, you can see that the zeros occur at x 0 and and at x 0.693. So, the solutions are x 0.693. x 0 x y = e2x − 3e x + 2 3 and x 0. Check 3 3 Now try Exercise 67. −1 FIGURE 3.25 333202_0304.qxd 12/7/05 10:31 AM Page 249 Section 3.4 Exponential and Logarithmic Equations 249 Solving Logarithmic Equations To solve a logarithmic equation, you can write it in exponential form. ln x 3 eln x e 3 x e 3 Logarithmic form Exponentiate each side. Exponential form This procedure is called exponentiating each side of an equation. Example 6 Solving Logarithmic Equations a. b. Remember to check your solutions in the original equation when solving equations to verify that the answer is correct and to make sure that the answer lies in the domain of the original equation. ln x 2 e ln x e 2 x e 2 5x 1 log3 5x 1 x 7 4x 8 x 2 log3 x 7 Original equation Exponentiate each side. Inverse Property Original equation One-to-One Property Add x and 1 to each side. Divide each side by 4. Original equation Quotient Property of Logarithms c. log6 3x 14 log6 5 log6 2x log6 2x log63x 14 5 3x 14 5 2x One-to-One Property 3x 14 10x 7x 14 x 2 Now try Exercise 77. Cross multiply. Isolate x. Divide each side by 7. Example 7 Solving a Logarithmic Equation Solve 5 2 ln x 4 and approximate the result to three decimal places. Solution 5 2 ln x 4 2 ln x 1 ln x 1 2 eln x e12 x e12 x 0.607 Now try Exercise 85. Write original equation. Subtract 5 from each side. Divide each side by 2. Exponentiate each side. Inverse Property Use a calculator. 333202_0304.qxd 12/7/05 10:31 AM Page 250 250 Chapter 3 Exponential and Logarithmic Functions Example 8 Solving a Logarithmic Equation Solve 2 log5 3x 4. Solution 2 log5 3x 4 log5 3x 2 5 log5 3x 52 3x 25 x 25 3 x 25 3 . The solution is Write original equation. Divide each side by 2. Exponentiate each side (base 5). Inverse Property Divide each side by 3. Check this in the original equation. Now try Exercise 87. Because the domain of a logarithmic function generally does not include all real numbers, you should be sure to check for extraneous solutions of logarithmic equations. Notice in Example 9 that the logarithmic part of the equation is condensed into a single logarithm before exponentiating each side of the equation. Example 9 Checking for Extraneous Solutions Solve log 5x logx 1 2. Algebraic Solution log 5x logx 1 2 log5xx 1 2 10 log5x 25x 102 5x2 5x 100 x2 x 20 0 x 5x Write original equation. Product Property of Logarithms Exponentiate each side (base 10). Inverse Property Write in general form. Factor. Set 1st factor equal to 0. Solution Set 2nd factor equal to 0. x 4 Solution The solutions appear to be you check these in the original equation, you can see that is the only solution. However, when x 5 and x 4. x 5 Now try Exercise 99. and 2 graphing utility y2 log 5x logx 1 Graphical Solution Use to graph a y1 in the same viewing window. From the graph shown in Figure 3.26, it appears that the graphs intersect at one point. Use the intersect feature or the zoom and trace features to determine that the graphs intersect at x 5. approximately So, Verify that 5 is an exact solution algebraically. the solution is 5, 2. y1 = log 5x + log(x − 1) y2 = 2 9 5 0 −1 FIGURE 3.26 In Example 9, the domain of x > 0 so the domain of the original equation is x > 1, real numbers greater than 1, the solution Figure 3.26 verifies this concept. log 5x is x 4 logx 1 and the domain of x > 1. is Because the domain is all is extraneous. The graph in 333202_0304.qxd 12/7/05 10:31 AM Page 251 Section 3.4 Exponential and Logarithmic Equations 251 Applications Example 10 Doubling an Investment You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? Solution Using the formula for continuous compounding, you can find that the balance in the account is A Pert A 500e0.0675t. A 1000 and solve the Let A 1000. To find the time required for the balance to double, let t. resulting equation for 500e0.0675t 1000 e0.0675t 2 ln e0.0675t ln 2 0.0675t ln 2 t ln 2 0.0675 t 10.27 Divide each side by 0.0675. Divide each side by 500. Take natural log of each side. Use a calculator. Inverse Property The balance in the account will double after approximately 10.27 years. This result is demonstrated graphically in Figure 3.27. Doubling an Investment (10.27, 1000 GTON A = 500e 0.0675t (0, 500) A 1100 ) 900 700 500 300 100 2 4 6 8 Time (in years) t 10 FIGURE 3.27 Now try Exercise 107. In Example 10, an approximate answer of 10.27 years is given. Within the years, does not make ln 20.0675 context of the problem, the exact solution, sense as an answer. Exploration The effective yield of a savings plan is the percent increase in the balance after 1 year. Find the effective yield for each savings plan when $1000 is deposited in a savings account. a. 7% annual interest rate, compounded annually b. 7% annual interest rate, compounded continuously c. 7% annual interest rate, compounded quarterly d. 7.25% annual interest rate, compounded quarterly Which savings plan has the greatest effective yield? Which savings plan will have the highest balance after 5 years? 333202_0304.qxd 12/7/05 10:31 AM Page 252 252 Chapter 3 Exponential and Logarithmic Functions Endangered Animal Species y Example 11 Endangered Animals 450 400 350 300 250 200 10 FIGURE 3.28 The number of endangered animal species in the United States from 1990 to 2002 can be modeled by y y 119 164 ln t, 10 ≤ t ≤ 22 t 10 t where represents the year, with corresponding to 1990 (see Figure 3.28). During which year did the number of endangered animal species reach 357? (Source: U.S. Fish and Wildlife Service) 14 12 20 16 Year (10 ↔ 1990) 18 t 22 Solution 119 164 ln t y 119 164 ln t 357 164 ln t 476 ln t 476 16
4 eln t e476164 t e476164 t 18 Write original equation. Substitute 357 for y. Add 119 to each side. Divide each side by 164. Exponentiate each side. Inverse Property Use a calculator. The solution is number of endangered animals reached 357 in 1998. Because t 18. t 10 represents 1990, it follows that the W RITING ABOUT MATHEMATICS Comparing Mathematical Models The table shows the U.S. Postal Service rates y for sending an express mail package x 5 for selected years from 1985 through 2002, where (Source: U.S. Postal Service) represents 1985. Year, x Rate, y 5 8 11 15 19 21 22 10.75 12.00 13.95 15.00 15.75 16.00 17.85 Now try Exercise 113. a. Create a scatter plot of the data. Find a linear model for the data, and add its graph to your scatter plot. According to this model, when will the rate for sending an express mail package reach $19.00? b. Create a new table showing values for ln x and ln y and create a scatter plot of these transformed data. Use the method illustrated in Example 7 in Section 3.3 to find a model for the transformed data, and add its graph to your scatter plot. According to this model, when will the rate for sending an express mail package reach $19.00? c. Solve the model in part (b) for y, and add its graph to your scatter plot in part (a). Which model better fits the original data? Which model will better predict future rates? Explain. 333202_0304.qxd 12/7/05 10:31 AM Page 253 Section 3.4 Exponential and Logarithmic Equations 253 3.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. To ________ an equation in means to find all values of x x for which the equation is true. 2. To solve exponential and logarithmic equations, you can use the following One-to-One and Inverse Properties. if and only if ________. if and only if ________. (a) (b) (c) (d) ax ay loga x loga y aloga x loga ax ________ ________ 3. An ________ solution does not satisfy the original equation. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–8, determine whether each solution (or an approximate solution) of the equation. -value is a x 2. 23x1 32 x 1 (a) x 2 (b) 1. 3. 4. 5. 6. 7. 8. 42x7 64 x 5 (a) x 2 (b) 3e x2 75 (a) 2 ln 6 (b) (b) (b) (c) log4 (a) x 2 e25 x 2 ln 25 x 1.219 (c) 2e5x2 12 x 1 (a) 5 x ln 6 5 ln 2 x 0.0416 3x 3 x 21.333 x 4 x 64 3 x 3 10 x 1021 x 17 x 102 3 (c) ln2x 3 5.8 (a) (c) log2 (a) x 1 2 x 1 2 x 163.650 (c) lnx 1 3.8 x 1 e3.8 (a) x 45.701 x 1 ln 3.8 (b) (b) (b) (c) 3 ln 5.8 3 e5.8 In Exercises 9–20, solve for x. 9. 11. 13. 15. 17. 19. 4x 16 1 x 32 2 ln x ln 2 0 e x 2 ln x 1 log4 x 3 10. 12. 14. 16. 18. 20. 3x 243 1 x 64 4 ln x ln 5 0 e x 4 ln x 7 log5 x 3 In Exercises 21–24, approximate the point of intersection g. of the graphs of Then solve the equation fx gx algebraically to verify your approximation. and f 21. f x 2x gx 8 22. f x 27x gx 9 y 12 4 −4 −8 −4 g f x 4 8 −8 −4 y g f x 4 8 12 8 4 −4 23. f x log3 x gx 2 y 24. f x lnx 4 gx 0 y 4 g f 4 8 12 x 12 8 4 −4 g f 8 x 12 333202_0304.qxd 12/7/05 10:31 AM Page 254 254 Chapter 3 Exponential and Logarithmic Functions In Exercises 25–66, solve the exponential equation algebraically. Approximate the result to three decimal places. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 45. 47. 49. 51. 53. 55. 57. 59. 61. 63. 65. ex ex22 ex23 ex2 43x 20 2ex 10 ex 9 19 32x 80 5t2 0.20 3x1 27 23x 565 8103x 12 35x1 21 e3x 12 500ex 300 7 2ex 5 623x1 7 9 e2x 4ex 5 0 e2x 3ex 4 0 20 2 500 100 e x2 3000 2 e2x 1 0.065 365 1 0.10 12 365t 4 12t 2 26. 28. 30. 32. 34. 36. 38. 40. 42. 44. 46. 48. 50. 52. 54. 56. 58. 60. 62. 64. 66. e2x ex28 ex2 ex22x 25x 32 4ex 91 6x 10 47 65x 3000 43t 0.10 2x3 32 82x 431 510 x6 7 836x 40 e2x 50 1000e4x 75 14 3ex 11 8462x 13 41 e2x 5ex 6 0 e2x 9ex 36 0 350 7 400 1 ex 119 e6x 14 4 2.471 40 16 0.878 26 9t 3t 30 In Exercises 67–74, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. 67. 69. 71. 73. 6e1x 25 3e3x2 962 e0.09t 3 e 0.125t 8 0 68. 70. 72. 74. 4ex1 15 0 8e2x3 11 e1.8x 7 0 e2.724x 29 In Exercises 75–102, solve the logarithmic equation algebraically. Approximate the result to three decimal places. 75. 77. 79. 81. 83. 85. ln x 3 ln 2x 2.4 log x 6 3 ln 5x 10 lnx 2 1 7 3 ln x 5 76. 78. 80. 82. 84. 86. ln x 2 ln 4x 1 log 3z 2 2 ln x 7 lnx 8 5 2 6 ln x 10 87. 89. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. x 2 11 5 log10 ln x lnx 1 1 90. 88. 0.5x 11 6 log3 ln x lnx 1 2 ln x lnx 2 1 ln x lnx 3 1 lnx 5 lnx 1 lnx 1 lnx 1 lnx 2 ln x 2x 3 log2 log2 logx 6 log2x 1 logx 4 log x logx 2 x 2 log2 log2 x log2 x 1 1 log4 x log4 2 x 8 2 log3 x log3 log 8x log1 x 2 log 4x log12 x 2 x 4 x 6 In Exercises 103–106, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. 103. 105. 7 2x 3 ln x 0 104. 106. 500 1500ex2 10 4 lnx 2 0 Compound Interest In Exercises 107 and 108, $2500 is compounded invested in an account at interest rate continuously. Find the time required for the amount to (a) double and (b) triple. r, 109. Demand The demand equation for a microwave oven is given by p 500 0.5e0.004x. Find the demand p $300. (b) x for a price of (a) p $350 and 110. Demand The demand equation for a hand-held elec- tronic organizer is p 50001 4 4 e0.002x. for a price of (a) p $600 and x Find the demand p $400. (b) 111. Forest Yield The yield t acre) for a forest at age years is given by V (in millions of cubic feet per V 6.7e48.1t. (a) Use a graphing utility to graph the function. (b) Determine the horizontal asymptote of the function. Interpret its meaning in the context of the problem. (c) Find the time necessary to obtain a yield of 1.3 million cubic feet. 21 107. r 0.085 108. r 0.12 333202_0304.qxd 12/7/05 10:31 AM Page 255 Section 3.4 Exponential and Logarithmic Equations 255 112. Trees per Acre The number N of trees of a given species per acre is approximated by the model N 68100.04x, is the average diameter of the trees (in inches) 3 feet above the ground. Use the model to approximate the average diameter of the N 21. trees in a test plot when 5 ≤ x ≤ 40 where x 113. Medicine The number y of hospitals in the United States from 1995 to 2002 can be modeled by y 7312 630.0 ln t, 5 ≤ t ≤ 12 t represents the year, with where corresponding to 1995. During which year did the number of hospitals reach 5800? (Source: Health Forum) t 5 y 114. Sports The number of daily fee golf facilities in the United States from 1995 to 2003 can be modeled by y 4381 1883.6 ln t, represents the year, with corresponding to 1995. During which year did the number of daily fee golf facilities reach 9000? (Source: National Golf Foundation) 5 ≤ t ≤ 13 where t 5 t 115. Average Heights The percent of American males between the ages of 18 and 24 who are no more than x inches tall is modeled by m mx 100 1 e0.6114x69.71 f and the percent of American females between the ages of 18 and 24 who are no more than inches tall is modeled by x f x 100 1 e0.66607x64.51. (Source: U.S. National Center for Health Statistics) (a) Use the graph to determine any horizontal asymptotes of the graphs of the functions. Interpret the meaning in the context of the problem 100 80 60 40 20 f(x) m(x) x 75 55 70 65 60 Height (in inches) (b) What is the average height of each sex? 116. Learning Curve mathematical model for the proportion responses after trials was found to be n In a group project in learning theory, a of correct P P 0.83 1 e0.2n . (a) Use a graphing utility to graph the function. (b) Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of this problem. (c) After how many trials will 60% of the responses be correct? Model It 117. Automobiles Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g’s the crash victims experience. (One g is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 g’s.) In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of g’s experienced during deceleration by crash dummies that were permitted to move meters during impact. The data are shown in the table. x x 0.2 0.4 0.6 0.8 1.0 g’s 158 80 53 40 32 A model for the data is given by y 3.00 11.88 ln x 36.94 x where y is the number of g’s. (a) Complete the table using the model. 0.2 0.4 0.6 0.8 1.0 x y (b) Use a graphing utility to graph the data points and the model in the same viewing window. How do they compare? (c) Use the model to estimate the distance traveled during impact if the passenger deceleration must not exceed 30 g’s. (d) Do you think it is practical to lower the number of g’s experienced during impact to fewer than 23? Explain your reasoning. 333202_0304.qxd 12/7/05 10:31 AM Page 256 256 Chapter 3 Exponential and Logarithmic Functions 118. Data Analysis An object at a temperature of 160 C was removed from a furnace and placed in a room at 20 C. The temperature of the object was measured each hour h and recorded in the table. A model for the data is given The graph of this model is by shown in the figure. T 20 1 72h. T Hour, h Temperature, T 0 1 2 3 4 5 160 90 56 38 29 24 (a) Use the graph to identify the horizontal asymptote of the model and interpret the asymptote in the context of the problem. (b) Use the model to approximate the time when the temperature of the object was 100 C. 122. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers. 123. Think About It Is it possible for a logarithmic equation to have more than one extraneous solution? Explain. 124. Finance You are investing dollars at an annual interest rate of compounded continuously, for years. Which of the
following would result in the highest value of the investment? Explain your reasoning. r, P t (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years. 125. Think About It Are the times required for the investments in Exercises 107 and 108 to quadruple twice as long as the times for them to double? Give a reason for your answer and verify your answer algebraically. 126. Writing Write two or three sentences stating the general guidelines that you follow when solving (a) exponential equations and (b) logarithmic equations. Skills Review In Exercises 127–130, simplify the expression. 127. 128. 129. 130. 48x2y 5 32 225 325 315 3 10 2 In Exercises 131–134, sketch a graph of the function. T 160 140 120 100 80 60 40 20 ( Synthesis 1 2 3 5 4 Hour 6 7 8 h 131. 132. 133. 134 gx 2x, gx x 3, x2 1, x2 4 True or False? In Exercises 119–122, rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. 119. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. 120. The logarithm of the sum of two numbers is equal to the product of the logarithms of the numbers. 121. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers. In Exercises 135–138, evaluate the logarithm using the change-of-base formula. Approximate your result to three decimal places. 135. 136. log6 9 log3 4 137. log34 5 138. log8 22 333202_0305.qxd 12/7/05 10:33 AM Page 257 Section 3.5 Exponential and Logarithmic Models 257 3.5 Exponential and Logarithmic Models What you should learn • Recognize the five most com- mon types of models involving exponential and logarithmic functions. • Use exponential growth and decay functions to model and solve real-life problems. • Use Gaussian functions to model and solve real-life problems. • Use logistic growth functions to model and solve real-life problems. • Use logarithmic functions to model and solve real-life problems. Why you should learn it Exponential growth and decay models are often used to model the population of a country. For instance, in Exercise 36 on page 265, you will use exponential growth and decay models to compare the populations of several countries. Introduction The five most common types of mathematical models involving exponential functions and logarithmic functions are as follows. 1. Exponential growth model: 2. Exponential decay model: 3. Gaussian model: 4. Logistic growth model: y ae bx, b > 0 y aebx, b > 0 y ae(xb) 2c y a 1 berx 5. Logarithmic models: y a b ln x, y a b log x The basic shapes of the graphs of these functions are shown in Figure 3.29. y 4 3 2 1 −1 −1 −2 y = ex y = e−x 1 2 3 x −3 −2 −1 y 4 3 2 1 −1 −2 y 2 y = e−x 2 x 1 −1 x 1 −1 EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL GAUSSIAN MODEL y 3 2 1 −1 −1 y = 3 1 + e−5x x 1 y 2 1 y = 1 + ln x y 2 1 y = 1 + log x −1 x 1 x 1 2 −1 −2 −1 −2 LOGISTIC GROWTH MODEL FIGURE 3.29 NATURAL LOGARITHMIC MODEL COMMON LOGARITHMIC MODEL Alan Becker/Getty Images You can often gain quite a bit of insight into a situation modeled by an exponential or logarithmic function by identifying and interpreting the function’s asymptotes. Use the graphs in Figure 3.29 to identify the asymptotes of the graph of each function. 333202_0305.qxd 12/7/05 10:33 AM Page 258 258 Chapter 3 Exponential and Logarithmic Functions Exponential Growth and Decay Example 1 Digital Television D 100 80 60 40 20 ) FIGURE 3.30 D 100 80 60 40 20 ) FIGURE 3.31 Digital Television Estimates of the numbers (in millions) of U.S. households with digital television from 2003 through 2007 are shown in the table. The scatter plot of the data is shown in Figure 3.30. (Source: eMarketer) t 7 Year 2003 2004 2005 2006 2007 Households 44.2 49.0 55.5 62.5 70.3 4 5 3 Year (3 ↔ 2003) 6 Digital Television D 30.92e0.1171t, 3 ≤ t ≤ 7 An exponential growth model that approximates these data is given by D is the number of households (in millions) and where represents 2003. Compare the values given by the model with the estimates shown in the table. According to this model, when will the number of U.S. households with digital television reach 100 million? t 3 Solution The following table compares the two sets of figures. The graph of the model and the original data are shown in Figure 3.31. t 7 Year 2003 2004 2005 2006 2007 Households Model 44.2 43.9 49.0 49.4 55.5 55.5 62.5 62.4 70.3 70.2 4 5 3 Year (3 ↔ 2003) 6 To find when the number of U.S. households with digital television will reach in the model and solve for 100 million, let D 100 t. Te c h n o l o g y Some graphing utilities have an exponential regression feature that can be used to find exponential models that represent data. If you have such a graphing utility, try using it to find an exponential model for the data given in Example 1. How does your model compare with the model given in Example 1? 30.92e0.1171t D 30.92e0.1171t 100 e0.1171t 3.2342 ln e0.1171t ln 3.2342 0.1171t 1.1738 Write original model. Let D 100. Divide each side by 30.92. Take natural log of each side. Inverse Property t 10.0 Divide each side by 0.1171. According to the model, the number of U.S. households with digital television will reach 100 million in 2010. Now try Exercise 35. 333202_0305.qxd 12/7/05 10:33 AM Page 259 Section 3.5 Exponential and Logarithmic Models 259 In Example 1, you were given the exponential growth model. But suppose this model were not given; how could you find such a model? One technique for doing this is demonstrated in Example 2. Example 2 Modeling Population Growth In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 100 flies, and after 4 days there are 300 flies. How many flies will there be after 5 days? Solution y Let be the number of flies at time From the given information, you know that y 100 . Substituting this information into the model t 2 y aebt and produces t. when y 300 t 4 when 100 ae2b and 300 ae4b. To solve for b, solve for a 100 ae2b in the first equation. a 100 e2b Solve for a in the first equation. Then substitute the result into the second equation. 300 ae4b 300 100 e2be4b e2b 300 100 ln 3 2b ln 3 b 1 2 Write second equation. Substitute 100 e2b for a. Divide each side by 100. Take natural log of each side. Solve for b. Using b 1 and the equation you found for a, you can determine that 2 ln 3 a 100 e212 ln 3 100 e ln 3 100 3 33.33. a 33.33 y 33.33e 0.5493t So, with and Substitute 1 2 ln 3 for b. Simplify. Inverse Property Simplify. b 1 2 ln 3 0.5493, the exponential growth model is as shown in Figure 3.32. This implies that, after 5 days, the population will be y 33.33e 0.54935 520 flies. 600 500 400 300 200 100 Fruit Flies (5, 520) y = 33.33e 0.5493t (4, 300) (2, 100) 1 2 4 3 Time (in days) 5 t FIGURE 3.32 Now try Exercise 37. 333202_0305.qxd 12/7/05 10:33 AM Page 260 260 Chapter 3 Exponential and Logarithmic Functions Carbon Dating R 10−12 t = 0 R = e−t/8223 1 1012 o i 1 t a 2R ( 10−12 ) t = 5,700 t = 19,000 10−13 FIGURE 3.33 t 5,000 15,000 Time (in years) The carbon dating model in Example 3 assumed that the carbon 14 to carbon 12 ratio was one part in 10,000,000,000,000. Suppose an error in measurement occurred and the actual ratio was one part in 8,000,000,000,000. The fossil age corresponding to the actual ratio would then be approximately 17,000 years. Try checking this result. 1012. In living organic material, the ratio of the number of radioactive carbon isotopes (carbon 14) to the number of nonradioactive carbon isotopes (carbon 12) is about 1 to When organic material dies, its carbon 12 content remains fixed, whereas its radioactive carbon 14 begins to decay with a half-life of about 5700 years. To estimate the age of dead organic material, scientists use the following formula, which denotes the ratio of carbon 14 to carbon 12 present at t any time (in years). R 1 Carbon dating model The graph of is shown in Figure 3.33. Note that decreases as R t increases. 1012 et 8223 R Example 3 Carbon Dating Estimate the age of a newly discovered fossil in which the ratio of carbon 14 to carbon 12 is R 1 1013 . Solution In the carbon dating model, substitute the given value of following. R to obtain the 1 1012et 8223 R et 8223 1 1012 1013 et 8223 1 10 ln et 8223 ln 1 10 Write original model. Let R 1 1013 . Multiply each side by 1012. Take natural log of each side. t 8223 2.3026 Inverse Property t 18,934 Multiply each side by 8223. So, to the nearest thousand years, the age of the fossil is about 19,000 years. Now try Exercise 41. b The value of in the exponential decay model determines the decay of radioactive isotopes. For instance, to find how much of an initial 10 grams of isotope with a half-life of 1599 years is left after 500 years, substitute this information into the model y aebt. 226Ra y aebt 10 10eb1599 ln 1 2 Using the value of b found above and 1 2 a 1599b 10, the amount left is b ln 1 2 1599 y 10eln121599500 8.05 grams. 333202_0305.qxd 12/7/05 10:33 AM Page 261 Section 3.5 Exponential and Logarithmic Models 261 Gaussian Models As mentioned at the beginning of this section, Gaussian models are of the form y aexb 2c. This type of model is commonly used in probability and statistics to represent populations that are normally distributed. The graph of a Gaussian model is called a bell-shaped curve. Try graphing the normal distribution with a graphing utility. Can you see why it is called a bell-shaped curve? For standard normal distributions, the model takes the form y 1 ex22. 2 The average value for a population can be found from the bell-shaped curve by observing where the maximum value of the function occurs. The -value corresponding to the maximum value of the function represents the average value of the independent variable—in this case, y- y- x. x Example 4 SAT Scores In 2004, the Scholastic Aptitude Test (SAT) math scores for college-bound sen
iors roughly followed the normal distribution given by y 0.0035ex518 225,992, 200 ≤ x ≤ 800 x where From the graph, estimate the average SAT score. is the SAT score for mathematics. Sketch the graph of this function. (Source: College Board) Solution The graph of the function is shown in Figure 3.34. On this bell-shaped curve, the maximum value of the curve represents the average score. From the graph, you can estimate that the average mathematics score for college-bound seniors in 2004 was 518. 0.003 0.002 0.001 SAT Scores 50% of population x = 518 200 400 600 800 Score x FIGURE 3.34 Now try Exercise 47. 333202_0305.qxd 12/7/05 10:33 AM Page 262 Chapter 3 Exponential and Logarithmic Functions 262 y Decreasing rate of growth Increasing rate of growth x Logistic Growth Models Some populations initially have rapid growth, followed by a declining rate of growth, as indicated by the graph in Figure 3.35. One model for describing this type of growth pattern is the logistic curve given by the function y a 1 ber x y is the population size and is the time. An example is a bacteria culture where that is initially allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. A logistic growth curve is also called a sigmoidal curve. x FIGURE 3.35 Example 5 Spread of a Virus On a college campus of 5000 students, one student returns from vacation with a contagious and long-lasting flu virus. The spread of the virus is modeled by 5000 y 1 4999e0.8t , t ≥ 0 is the total number of students infected after days. The college will where cancel classes when 40% or more of the students are infected. y t a. How many students are infected after 5 days? b. After how many days will the college cancel classes? Solution a. After 5 days, the number of students infected is y 5000 1 4999e0.85 5000 1 4999e4 54. b. Classes are canceled when the number infected is 0.405000 2000. 2000 5000 1 4999e0.8t 1 4999e0.8t 2.5 e0.8t 1.5 4999 ln e0.8t ln 0.8t ln 1.5 4999 1.5 4999 ln 1.5 4999 t 1 0.8 t 10.1 So, after about 10 days, at least 40% of the students will be infected, and the college will cancel classes. The graph of the function is shown in Figure 3.36. Now try Exercise 49. 2500 2000 1500 1000 500 Flu Virus (10.1, 2000) (5, 54) t 2 FIGURE 3.36 6 8 10 4 Time (in days) 12 14 333202_0305.qxd 12/7/05 10:33 AM Page 263 On December 26, 2004, an earthquake of magnitude 9.0 struck northern Sumatra and many other Asian countries. This earthquake caused a deadly tsunami and was the fourth largest earthquake in the world since 1900. Section 3.5 Exponential and Logarithmic Models 263 Logarithmic Models Example 6 Magnitudes of Earthquakes On the Richter scale, the magnitude of an earthquake of intensity R I is given by R log I I0 1 I0 where is the minimum intensity used for comparison. Find the intensities per unit of area for each earthquake. (Intensity is a measure of the wave energy of an earthquake.) a. Northern Sumatra in 2004: b. Southeastern Alaska in 2004: R 9.0 R 6.8 Solution a. Because 1 I0 and I 9.0 log 1 109.0 10log I R 9.0, you have Substitute 1 for I0 and 9.0 for R. Exponentiate each side. I 109.0 100,000,000. Inverse Property b. For R 6.8, you have 6.8 log I 1 106.8 10log I Substitute 1 for I0 and 6.8 for R. Exponentiate each side. I 106.8 6,310,000. Inverse Property Note that an increase of 2.2 units on the Richter scale (from 6.8 to 9.0) represents an increase in intensity by a factor of 1,000,000,000 6,310,000 158. Year Population, P In other words, the intensity of the earthquake in Sumatra was about 158 times greater than that of the earthquake in Alaska. 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 92.23 106.02 123.20 132.16 151.33 179.32 203.30 226.54 248.72 281.42 Now try Exercise 51. W RITING ABOUT MATHEMATICS Comparing Population Models The populations P (in millions) of the United States for the census years from 1910 to 2000 are shown in the table at the left. Least squares regression analysis gives the best quadratic model for these data as P 1.0328t 2 9.607t 81.82, as your conclusion. Which model better fits the data? Describe how you reached and the best exponential model for these data (Source: U.S. Census Bureau) P 82.677e0.124t 10 333202_0305.qxd 12/7/05 10:33 AM Page 264 264 Chapter 3 Exponential and Logarithmic Functions 3.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. An exponential growth model has the form ________ and an exponential decay model has the form ________. 2. A logarithmic model has the form ________ or ________. 3. Gaussian models are commonly used in probability and statistics to represent populations that are ________ ________. 4. The graph of a Gaussian model is ________ shaped, where the ________ ________ is the maximum -value of the graph. y 5. A logistic curve is also called a ________ curve. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, match the function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f ).] (a) y (b) y Compound Interest In Exercises 7–14, complete the table for a savings account in which interest is compounded continuously. 6 4 2 −c) y (d) y 12 8 4 4 2 −2 2 4 6 −8 −4 x 4 8 (e) y (f) y x x Initial Investment 7. $1000 8. $750 9. $750 10. $10,000 11. $500 12. $600 13. 14. Amount After 10 Years Annual % Rate Time to Double 3.5% 101 2% 73 4 yr 12 yr $1505.00 4.5% $2000.00 2% $19,205.00 $10,000.00 that must be invested at rate Compound Interest In Exercises 15 and 16, determine the P principal compounded monthly, so that $500,000 will be available for retirement in years. r 71 r 12%, t 40 2%, t 20 15. 16. r, t 6 4 2 6 −12 −6 x 6 12 −2 −2 2 4 1. 3. y 2e x4 y 6 logx 2 5. y lnx 1 2. 4. y 6ex4 y 3ex2 25 6. y 4 1 e2x Compound Interest In Exercises 17 and 18, determine the time necessary for $1000 to double if it is invested at r interest rate compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. x 17. r 11% 18. r 101 2% 19. Compound Interest Complete the table for the time t necessary for dollars to triple if interest is compounded continuously at rate r. P 2% 4% 6% 8% 10% 12% r t 20. Modeling Data Draw a scatter plot of the data in Exercise 19. Use the regression feature of a graphing utility to find a model for the data. 333202_0305.qxd 12/7/05 10:33 AM Page 265 21. Compound Interest Complete the table for the time t P necessary for dollars to triple if interest is compounded annually at rate r. 2% 4% 6% 8% 10% 12% r t 22. Modeling Data Draw a scatter plot of the data in Exercise 21. Use the regression feature of a graphing utility to find a model for the data. 23. Comparing Models If $1 is invested in an account over a repre10-year period, the amount in the account, where or sents the time in years, is given by A e0.07t depending on whether the account pays simple 71 interest at or continuous compound interest at 7%. 2% Graph each function on the same set of axes. Which grows at a higher rate? (Remember that is the greatest integer function discussed in Section 1.6.) A 1 0.075 t t t 24. Comparing Models 10-year period, the amount in the account, where sents the time in years, is given by If $1 is invested in an account over a repre- t A 1 0.06 t or A 1 0.055 365 365t depending on whether the account pays simple interest at 51 compounded daily. Use a 6% or compound interest at 2% graphing utility to graph each function in the same viewing window. Which grows at a higher rate? Radioactive Decay for the radioactive isotope. In Exercises 25–30, complete the table Isotope 226Ra 226Ra 14C 14C 239Pu 239Pu 25. 26. 27. 28. 29. 30. Half-life (years) Initial Quantity 1599 1599 5715 5715 24,100 24,100 10 g 3 g Amount After 1000 Years 1.5 g 2 g 2.1 g 0.4 g In Exercises 31–34, find the exponential model that fits the points shown in the graph or table. y aebx 31. y 32. y 10 8 6 4 2 (3, 10) (0, 1) x 8 6 4 2 (4, 5) ( )1 20 Section 3.5 Exponential and Logarithmic Models 265 33. x y 0 5 4 1 34. x y 0 1 3 1 4 35. Population The population thousands) of Pittsburgh, Pennsylvania from 2000 through 2003 can be where represents the year, modeled by with (Source: U.S. Census Bureau) P 2430e0.0029t, corresponding to 2000. t 0 (in P t (a) According to the model, was the population of Pittsburgh increasing or decreasing from 2000 to 2003? Explain your reasoning. (b) What were the populations of Pittsburgh in 2000 and 2003? (c) According to the model, when will the population be approximately 2.3 million? Model It 36. Population The table shows the populations (in millions) of five countries in 2000 and the projected populations (in millions) for the year 2010. (Source: U.S. Census Bureau) Country 2000 2010 Bulgaria Canada China United Kingdom United States 7.8 31.3 1268.9 59.5 282.3 7.1 34.3 1347.6 61.2 309.2 or y aebt (a) Find the exponential growth or decay model y aebt for the population of each country by letting correspond to 2000. Use the model to predict the population of each country in 2030. t 0 (b) You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation y aebt is determined by these different growth rates? Discuss the relationship between the different growth rates and the magnitude of the constant. (c) You can see that the population of China is increasing while the population of Bulgaria is y aebt decreasing. What constant in the equation reflects this difference? Explain. 333202_0305.qxd 12/7/05 10:33 AM Page 266 266 Chapter 3 Exponential and Logarithmic Functions 37. Website Growth The number y of hits a new search- engine website receives each month can be modeled by y 4080ekt t represents the number of months the website has where been operating. In the website’s third month, there were 10,000 hits. Find the value of and use this result to predict the number of hits the website will receive after 24 months. k, 38. Value of a Painting The value V (in milli
ons of dollars) of a famous painting can be modeled by V 10ekt t represents the year, with corresponding to where 1990. In 2004, the same painting was sold for $65 million. Find the value of and use this result to predict the value of the painting in 2010. k, t 0 39. Bacteria Growth The number N of bacteria in a culture is modeled by N 100ekt t is the time in hours. If where estimate the time required for the population to double in size. when N 300 t 5, 40. Bacteria Growth The number N of bacteria in a culture is modeled by N 250ekt t is the time in hours. If t 10, where estimate the time required for the population to double in size. N 280 when 41. Carbon Dating (a) The ratio of carbon 14 to carbon 12 in a piece of wood Estimate the age of R 1814. discovered in a cave is the piece of wood. (b) The ratio of carbon 14 to carbon 12 in a piece of paper R 11311. Estimate the age of the buried in a tomb is piece of paper. 42. Radioactive Decay Carbon 14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of 14C absorbed by a tree that grew several centuries ago should be the same as the amount of absorbed by a tree growing today. A piece of ancient charcoal contains only 15% as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of is 5715 years? 14C 14C 43. Depreciation A 2005 Jeep Wrangler that costs $30,788 new has a book value of $18,000 after 2 years. (a) Find the linear model V mt b. (b) Find the exponential model V aekt. (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the vehicle after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller. 44. Depreciation A Dell Inspiron 8600 laptop computer that costs $1150 new has a book value of $550 after 2 years. (a) Find the linear model V mt b. (b) Find the exponential model V aekt. (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages to a buyer and a seller of using each model. 45. Sales The sales S (in thousands of units) of a new CD years are t burner after it has been on the market for modeled by St 1001 ekt. Fifteen thousand units of the new product were sold the first year. (a) Complete the model by solving for k. (b) Sketch the graph of the model. (c) Use the model to estimate the number of units sold after 5 years. 46. Learning Curve The management at a plastics factory has found that the maximum number of units a worker can produce in a day is 30. The learning curve for the number N of units produced per day after a new employee has t worked days is modeled by N 301 ekt . After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of ). k (b) How many days should pass before this employee is producing 25 units per day? 47. IQ Scores The IQ scores from a sample of a class of returning adult students at a small northeastern college roughly follow the normal distribution y 0.0266ex1002450, 70 ≤ x ≤ 115 where x is the IQ score. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average IQ score of an adult student. 333202_0305.qxd 12/7/05 10:33 AM Page 267 48. Education The time (in hours per week) a student utilizes a math-tutoring center roughly follows the normal distribution y 0.7979ex5.420.5, 4 ≤ x ≤ 7 where x is the number of hours. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average num- ber of hours per week a student uses the tutor center. 49. Population Growth A conservation organization releases 100 animals of an endangered species into a game preserve. The organization believes that the preserve has a carrying capacity of 1000 animals and that the growth of the pack will be modeled by the logistic curve pt 1000 1 9e0.1656t where t is measured in months (see figure). 1200 1000 800 600 400 200 2 4 6 8 10 12 14 16 18 Time (in years) t (a) Estimate the population after 5 months. (b) After how many months will the population be 500? (c) Use a graphing utility to graph the function. Use the graph to determine the horizontal asymptotes, and interpret the meaning of the larger -value in the context of the problem. p 50. Sales After discontinuing all advertising for a tool kit in 2000, the manufacturer noted that sales began to drop according to the model S 500,000 1 0.6ekt S t 0 where represents 2000. In 2004, the company sold 300,000 units. represents the number of units sold and (a) Complete the model by solving for k. (b) Estimate sales in 2008. Section 3.5 Exponential and Logarithmic Models 267 Geology In Exercises 51 and 52, use the Richter scale R log I I0 for measuring the magnitudes of earthquakes. 51. Find the intensity of an earthquake measuring 1 Richter scale (let ). I I0 R on the (a) Centeral Alaska in 2002, (b) Hokkaido, Japan in 2003, R 4.2 R (c) Illinois in 2004, R 7.9 R 8.3 52. Find the magnitude of each earthquake of intensity (let I I0 (a) (c) ). 1 I 80,500,000 I 251,200 (b) I 48,275,000 Intensity of Sound In Exercises 53–56, use the following information for determining sound intensity. The level of sound in decibels, with an intensity of , is given by , I 10 log I I0 I0 is an intensity of where watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 53 and 54, find the level of sound . 1012 53. (a) (b) (c) (d) 54. (a) (b) (c) (d) I 1010 I 105 I 108 I 100 I 1011 I 102 I 104 I 102 m2 m2 m2 watt per (quiet room) watt per (busy street corner) watt per (quiet radio) watt per m2 watt per m2 watt per (threshold of pain) m2 (rustle of leaves) (jet at 30 meters) watt per watt per m2 m2 (door slamming) (siren at 30 meters) 55. Due to the installation of noise suppression materials, the noise level in an auditorium was reduced from 93 to 80 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of these materials. 56. Due to the installation of a muffler, the noise level of an engine was reduced from 88 to 72 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of the muffler. pH log [H], pH Levels by hydrogen ion concentration hydrogen per liter) of a solution. In Exercises 57– 62, use the acidity model given where acidity (pH) is a measure of the (measured in moles of [H] H 2.3 105. 57. Find the pH if 58. Find the pH if H 11.3 106. 333202_0305.qxd 12/7/05 10:33 AM Page 268 268 Chapter 3 Exponential and Logarithmic Functions 59. Compute 60. Compute H H for a solution in which pH 5.8. for a solution in which pH 3.2. 61. Apple juice has a pH of 2.9 and drinking water has a pH of 8.0. The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water? 62. The pH of a solution is decreased by one unit. The hydrogen ion concentration is increased by what factor? 63. Forensics At 8:30 A.M., a coroner was called to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person’s temperature twice. At 9:00 A.M. the temperature was 85.7F, 82.8F. and at 11:00 a.m. the temperature was From these two temperatures, the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula t 10 ln T 70 98.6 70 t where is the time in hours elapsed since the person died T is the temperature (in degrees Fahrenheit) of the and person’s body. Assume that the person had a normal body at death, and that the room temperature of 70F. temperature was a constant (This formula is derived from a general cooling principle called Newton’s Law of Cooling.) Use the formula to estimate the time of death of the person. 98.6F 71 2% 64. Home Mortgage A $120,000 home mortgage for 35 years at has a monthly payment of $809.39. Part of the monthly payment is paid toward the interest charge on the unpaid balance, and the remainder of the payment is used to reduce the principal. The amount that is paid toward the interest is u M M Pr 12 1 r 12 12t and the amount that is paid toward the reduction of the principal is v M Pr 12 1 r 12 12t . In these formulas, interest rate, years. M P is the is the size of the mortgage, is the monthly payment, and is the time in r t (a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 35 years of mortgage payments.) (b) In the early years of the mortgage, is the larger part of the monthly payment paid toward the interest or the principal? Approximate the time when the monthly payment is evenly divided between interest and principal reduction. (c) Repeat parts (a) and (b) for a repayment period of 20 years M $966.71. What can you conclude? 65. Home Mortgage The total interest r mortgage of dollars at interest rate P u for years is paid on a home t u P 1 rt 1 1 r12 12t 1. Consider a $120,000 home mortgage at 71 2%. (a) Use a graphing utility to graph the total interest function. (b) Approximate the length of the mortgage for which the total interest paid is the same as the size of the mortgage. Is it possible that some people are paying twice as much in interest charges as the size of the mortgage? 66. Data Analysis The table shows the time (in seconds) required to attain a speed of miles per hour from a standing start for a car. s t Speed, s Time, t 30 40 50 60 70 80 90 3.4 5.0 7.0 9.3 12.0 15.8 20.0 Two models for these data are as follows. 40.757 0.556s 15.817 ln s 1.2259 0.00
23s 2 t1 t2 (a) Use the regression feature of a graphing utility to find a for the data. linear model and an exponential model t4 t3 (b) Use a graphing utility to graph the data and each model in the same viewing window. (c) Create a table comparing the data with estimates obtained from each model. (d) Use the results of part (c) to find the sum of the absolute values of the differences between the data and the estimated values given by each model. Based on the four sums, which model do you think better fits the data? Explain. 333202_0305.qxd 12/7/05 10:33 AM Page 269 Synthesis True or False? statement is true or false. Justify your answer. In Exercises 67–70, determine whether the 67. The domain of a logistic growth function cannot be the set of real numbers. 68. A logistic growth function will always have an -intercept. x 69. The graph of f x 4 1 6e2 x 5 is the graph of gx 4 1 6e2x shifted to the right five units. 70. The graph of a Gaussian model will never have an x -intercept. 71. Identify each model as linear, logarithmic, exponential, logistic, or none of the above. Explain your reasoning. (a) y (bc) y 8 6 4 2 −2 −2 (e) y 12 10 d) y 6 5 4 3 2 1 (f Section 3.5 Exponential and Logarithmic Models 269 72. Writing Use your school’s library, the Internet, or some other reference source to write a paper describing John Napier’s work with logarithms. Skills Review In Exercises 73–78, (a) plot the points, (b) find the distance between the points, (c) find the midpoint of the line segment joining the points, and (d) find the slope of the line passing through the points. 73. 74. 75. 76. 77. 78. 1, 2, 0, 5 4, 3, 6, 1 3, 3, 14, 2 7, 0, 10, 4 4, 0 , 3 1 2, 1 , 2 7 3, 1 3, 1 6 4 3 In Exercises 79–88, sketch the graph of the equation. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. y 10 3x y 4x 1 y 2x2 3 y 2x2 7x 30 3x2 4y 0 x2 8y 0 y 4 1 3x y x2 x 2 x2 y 82 25 x 42 y 7 4 In Exercises 89–92, graph the exponential function. 89. 90. 91. 92. f x 2 x1 5 f x 2x1 1 f x 3x 4 f x 3 x 4 93. Make a Decision To work an extended application analyzing the net sales for Kohl’s Corporation from 1992 to 2004, visit this text’s website at college.hmco.com. (Data Source: Kohl’s Illinois, Inc.) 333202_030R.qxd 12/7/05 10:34 AM Page 270 270 Chapter 3 Exponential and Logarithmic Functions 3 Chapter Summary What did you learn? Section 3.1 Recognize and evaluate exponential functions with base a (p. 218). Graph exponential functions and use the One-to-One Property (p. 219). Recognize, evaluate, and graph exponential functions with base e (p. 222). Use exponential functions to model and solve real-life problems (p. 223). Section 3.2 Recognize and evaluate logarithmic functions with base a (p. 229). Graph logarithmic functions (p. 231). Recognize, evaluate, and graph natural logarithmic functions (p. 233). Use logarithmic functions to model and solve real-life problems (p. 235). Section 3.3 Use the change-of-base formula to rewrite and evaluate logarithmic expressions (p. 239). Use properties of logarithms to evaluate or rewrite logarithmic expressions (p. 240). Use properties of logarithms to expand or condense logarithmic expressions (p. 241). Use logarithmic functions to model and solve real-life problems (p. 242). Section 3.4 Solve simple exponential and logarithmic equations (p. 246). Solve more complicated exponential equations (p. 247). Solve more complicated logarithmic equations (p. 249). Use exponential and logarithmic equations to model and solve real-life problems (p. 251). Section 3.5 Recognize the five most common types of models involving exponential and logarithmic functions (p. 257). Use exponential growth and decay functions to model and solve real-life problems (p. 258). Use Gaussian functions to model and solve real-life problems (p. 261). Use logistic growth functions to model and solve real-life problems (p. 262). Use logarithmic functions to model and solve real-life problems (p. 263). Review Exercises 1–6 7–26 27–34 35–40 41–52 53–58 59–68 69, 70 71–74 75–78 79–94 95, 96 97–104 105–118 119–134 135, 136 137–142 143–148 149 150 151, 152 333202_030R.qxd 12/7/05 10:35 AM Page 271 3 Review Exercises In Exercises 1–6, evaluate the function at the indicated 3.1 value of Round your result to three decimal places. x. Function f x 6.1x f x 30x f x 20.5x f x 1278x5 f x 70.2x f x 145x 1. 2. 3. 4. 5. 6. Value x 2.4 x 3 x x 1 x 11 x 0.8 In Exercises 7–10, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) y 1 −3 −2 −1 1 2 3 −2 −3 −4 −5 (c) y 5 4 3 2 1 −3 −2 −1 1 2 3 x x (b) −3 −2 −1 (d3 −2 −1 21 3 x x 7. 9. f x 4x f x 4x 8. 10. f x 4x f x 4x 1 In Exercises 11–14, use the graph of transformation that yields the graph of f g. to describe the 11. 12. 13. 14. f x 5x, f x 4x gx 5x1 gx 4x 3 gx 1 x2 x gx 8 2 2 3 In Exercises 15–22, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 15. 17. f x 4x 4 f x 2.65x1 16. 18. f x 4x 3 f x 2.65x1 Review Exercises 271 19. 21. f x 5 x2 4 f x 1 x 3 2 20. 22. f x 2 x6 5 f x 1 x2 5 8 In Exercises 23–26, use the One-to-One Property to solve the equation for x. 23. 3x2 1 9 25. e5x7 e15 81 x2 1 3 e82x e3 24. 26. In Exercises 27–30, evaluate the function given by at the indicated value of decimal places. fx ex Round your result to three x. 27. 29. x 8 x 1.7 28. 30. x 5 8 x 0.278 In Exercises 31–34, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 31. 33. hx ex2 f x e x2 32. 34. hx 2 ex2 st 4e2t, t > 0 Compound Interest table to determine the balance rate In Exercises 35 and 36, complete the for dollars invested at n P times per year. for years and compounded A r t 1 2 4 12 365 Continuous n A 35. 36. P $3500, r 6.5%, t 10 years P $2000, r 5%, t 30 years 37. Waiting Times The average time between incoming calls at a switchboard is 3 minutes. The probability of waiting t less than minutes until the next incoming call is approxiFt 1 et 3. mated by the model A call has just come in. Find the probability that the next call will be within F (a) minute. 1 2 (b) 2 minutes. (c) 5 minutes. 38. Depreciation After t years, the value originally cost $14,000 is given by V Vt 14,0003 of a car that t . 4 (a) Use a graphing utility to graph the function. (b) Find the value of the car 2 years after it was purchased. (c) According to the model, when does the car depreciate most rapidly? Is this realistic? Explain. 333202_030R.qxd 12/7/05 3:39 PM Page 272 272 Chapter 3 Exponential and Logarithmic Functions 39. Trust Fund On the day a person is born, a deposit of $50,000 is made in a trust fund that pays 8.75% interest, compounded continuously. (a) Find the balance on the person’s 35th birthday. (b) How much longer would the person have to wait for the balance in the trust fund to double? Q 40. Radioactive Decay Let 241Pu represent a mass of plutonium 241 (in grams), whose half-life is 14.4 years. The quantity of plutonium 241 present after years is given by Q 1001 (a) Determine the initial quantity (when t14.4 . t 0 ). t 2 (b) Determine the quantity present after 10 years. (c) Sketch the graph of this function over the interval t 0 to t 100. In Exercises 41– 44, write the exponential equation in 3.2 logarithmic form. 41. 43. 43 64 e0.8 2.2255 . . . 42. 44. 2532 125 e0 1 In Exercises 45– 48, evaluate the function at the indicated value of without using a calculator. x Function f x log x gx log9 x gx log2 x f x log4 x 45. 46. 47. 48. Value x 1000 x 3 x 1 8 x 1 4 In Exercises 49–52, use the One-to-One Property to solve the equation for x. x 7 log4 14 log4 lnx 9 ln 4 49. 51. 50. 52. 3x 10 log8 5 log8 ln2x 1 ln 11 -intercept, and vertical In Exercises 65–68, find the domain, asymptote of the logarithmic function and sketch its graph. x 65. 67. f x ln x 3 hx lnx2 66. 68. f x lnx 3 f x 1 4 ln x 69. Antler Spread The antler spread (in inches) and h (in inches) of an adult male American elk shoulder height h 116 loga 40 176. are related by the model Approximate the shoulder height of a male American elk with an antler spread of 55 inches. a 70. Snow Removal The number of miles of roads cleared s of snow is approximated by the model s 25 13 lnh12 2 ≤ h ≤ 15 , ln 3 is the depth of the snow in inches. Use this model h where s to find when h 10 inches. 3.3 In Exercises 71–74, evaluate the logarithm using the change-of-base formula. Do each exercise twice, once with common logarithms and once with natural logarithms. Round your the results to three decimal places. 71. 73. log4 9 log12 5 72. 74. log12 200 log3 0.28 In Exercises 75–78, use the properties of logarithms to rewrite and simplify the logarithmic expression. 75. log 18 77. ln 20 76. 78. 1 log2 12 ln3e4 In Exercises 79–86, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) In Exercises 53–58, find the domain, -intercept, and vertical asymptote of the logarithmic function and sketch its graph. x 53. 55. 57. gx log7 x f x logx 3 f x 4 logx 5 54. gx log5 x 56. f x 6 log x 58. f x logx 3 1 79. 81. 83. 85. log3 log5 5x 2 6 3x ln x2y2z lnx 3 xy 80. 82. 84. 86. log7 log 7x4 x 4 ln 3xy2 lny 1 4 2 , y > 1 In Exercises 59–64, use a calculator to evaluate the function given by at the indicated value of Round your result to three decimal places if necessary. f x ln x x. 59. 61. x 22.6 x e12 63. x 7 5 60. 62. x 0.98 x e7 64. x 3 8 In Exercises 87–94, condense the expression to the logarithm of a single quantity. 88. 87. 89. log2 5 log2 x ln x 1 4 ln y x 4 7 log8 y 1 3 log8 2 ln2x 1 2 ln x 1 1 93. 94. 5 ln x 2 ln x 2 3 ln x 90. 92. 91. log6 y 2 log6 z 3 ln x 2 lnx 1 2 log x 5 logx 6 333202_030R.qxd 12/7/05 10:35 AM Page 273 Review Exercises 273 95. Climb Rate The time (in minutes) for a small plane to climb to an altitude of feet is modeled by t h t 50 log 18,000 18,000 h where 18,000 feet is the plane’s absolute ceiling. (a) Determine the domain
of the function in the context of the problem. (b) Use a graphing utility to graph the function and identify any asymptotes. (c) As the plane approaches its absolute ceiling, what can be said about the time required to increase its altitude? (d) Find the time for the plane to climb to an altitude of 4000 feet. 96. Human Memory Model Students in a learning theory study were given an exam and then retested monthly for 6 months with an equivalent exam. The data obtained in t is the study are given as the ordered pairs is the the time in months after the initial exam and average score for the class. Use these data to find a logarithmic equation that relates and 1, 84.2, 2, 78.4, 3, 72.1, 4, 68.5, 5, 67.1, 6, 65.3 where s t, s, s. t 3.4 In Exercises 97–104, solve for x. 97. 99. 101. 103. 8x 512 ex 3 log4 x 2 ln x 4 98. 100. 102. 104. 6x 1 216 ex 6 log6 x 1 ln x 3 In Exercises 105–114, solve the exponential equation algebraically. Approximate your result to three decimal places. 105. 107. 109. 111. 113. ex 12 e4x ex 23 2 x 13 35 45 x 68 e2x 7ex 10 0 106. 108. 110. 112. 114. e3x 25 14e3x2 560 6 x 28 8 212x 190 e2x 6ex 8 0 In Exercises 115–118, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. 115. 117. 20.6x 3x 0 25e0.3x 12 116. 118. 40.2x x 0 4e 1.2 x 9 124. 126. lnx 8 3 ln x ln 5 4 x 2 123. 125. 127. 128. 129. 130. ln x ln 3 2 lnx 1 2 log8 log6 log1 x 1 logx 4 2 x 1 log8 x 2 log6 x log6 x 2 log8 x 5 In Exercises 131–134, use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. 131. 133. 2 lnx 3 3x 8 4 lnx 5 x 10 132. 134. 6 logx2 1 x 0 x 2 logx 4 0 135. Compound Interest You deposit $7550 in an account that pays 7.25% interest, compounded continuously. How long will it take for the money to triple? 136. Meteorology The speed of the wind S (in miles per hour) near the center of a tornado and the distance (in miles) the tornado travels are related by the model S 93 log d 65. On March 18, 1925, a large tornado struck portions of Missouri, Illinois, and Indiana with a wind speed at the center of about 283 miles per hour. Approximate the distance traveled by this tornado. d In Exercises 137–142, match the function with its 3.5 graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) y (b8 −6 −4 −2 −2 x 2 −8 −6 −4 −2 x 2 (c) y (d) y 8 6 4 2 −4 −2 −2 2 4 6 (e) y 10 8 6 4 2 x x −4 −2 2 4 6 y 3 2 −1 −2 −3 1 2 3 (f) x x In Exercises 119–130, solve the logarithmic equation algebraically. Approximate the result to three decimal places. 119. 121. ln 3x 8.2 2 ln 4x 15 120. 122. ln 5x 7.2 4 ln 3x 15 3 2 1 −1 −2 1 2 3 4 5 6 333202_030R.qxd 12/7/05 10:35 AM Page 274 274 Chapter 3 Exponential and Logarithmic Functions 137. 139. y 3e2x3 y lnx 3 141. y 2ex4 23 138. 140. 142. y 4e2x3 y 7 logx 3 6 1 2e2x y In Exercises 143 and 144, find the exponential model y aebx that passes through the points. , 0, 1 4, 3 144. 5, 5 2 143. 0, 2, 145. Population The population P of South Carolina (in thousands) from 1990 through 2003 can be modeled by P 3499e0.0135t, t 0 corresponding to 1990. According to this model, when will the population reach 4.5 million? (Source: U.S. Census Bureau) represents the year, with where t 234U 146. Radioactive Decay The half-life of radioactive uranium II is about 250,000 years. What percent of a present amount of radioactive uranium II will remain after 5000 years? 147. Compound Interest A deposit of $10,000 is made in a savings account for which the interest is compounded continuously. The balance will double in 5 years. (a) What is the annual interest rate for this account? (b) Find the balance after 1 year. 148. Wildlife Population A species of bat is in danger of becoming extinct. Five years ago, the total population of the species was 2000. Two years ago, the total population of the species was 1400. What was the total population of the species one year ago? 149. Test Scores The test scores for a biology test follow a normal distribution modeled by y 0.0499ex71 2128, 40 ≤ x ≤ 100 where x is the test score. (a) Use a graphing utility to graph the equation. (b) From the graph in part (a), estimate the average test score. 150. Typing Speed In a typing class, the average number N of words per minute typed after weeks of lessons was found to be t N 157 1 5.4e0.12t . Find the time necessary to type (a) 50 words per minute and (b) 75 words per minute. 151. Sound Intensity The relationship between the number in watts per and the intensity of a sound I of decibels square centimeter is 10 log 1016. I Determine the intensity of a sound in watts per square centimeter if the decibel level is 125. 152. Geology On the Richter scale, the magnitude R of an earthquake of intensity I is given by R log I I0 1 R. R 8.4 (a) I0 is the minimum intensity used for where comparison. Find the intensity per unit of area for each value of (b) R 6.85 (c) R 9.1 Synthesis True or False? whether the equation is true or false. Justify your answer. In Exercises 153 and 154, determine 153. 154. logb b2x 2x lnx y ln x ln y 155. The graphs of and Which of the four values are negative? Which are are shown where b, c, d. positive? Explain your reasoning. y e kt k a, (a) (c) y 3 2 (0, 1) (b) y 3 y = eat (0, 1) y = ebt −2 −1 −1 1 2 y 3 2 (0, 1) y = ect −2 −1 −1 1 2 x x −2 −1 −1 1 2 (d) y 3 2 (0, 1) −2 −1 −1 y = edt 1 2 x x 333202_030R.qxd 12/7/05 10:35 AM Page 275 3 Chapter Test Chapter Test 275 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1– 4, evaluate the expression. Approximate your result to three decimal places. 1. 12.42.79 2. 432 3. e710 4. e3.1 In Exercises 5–7, construct a table of values. Then sketch the graph of the function. f x 6 x2 f x 1 e2x f x 10x 7. 5. 6. 8. Evaluate (a) log7 70.89 and (b) 4.6 ln e2. In Exercises 9–11, construct a table of values. Then sketch the graph of the function. Identify any asymptotes. f x log x 6 f x 1 lnx 6 f x lnx 4 10. 11. 9. In Exercises 12–14, evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. 12. log7 44 13. log25 0.9 14. log24 68 In Exercises 15–17, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. 15. log2 3a4 16. ln 5x 6 17. log 7x2 yz3 In Exercises 18–20, condense the expression to the logarithm of a single quantity. 18. 20. log3 13 log3 y 2 ln x lnx 5 3 ln y 19. 4 ln x 4 ln y In Exercises 21– 26, solve the equation algebraically. Approximate your result to three decimal places. Exponential Growth y (9, 11,277) (0, 2745) 2 4 6 8 10 t 5x 1 25 5 1025 8 e4x 18 4 ln x 7 21. 23. 25. 22. 3e5x 132 24. ln x 1 2 26. log x log8 5x 2 12,000 10,000 8,000 6,000 4,000 2,000 FIGURE FOR 27 27. Find an exponential growth model for the graph shown in the figure. 28. The half-life of radioactive actinium 227Ac is 21.77 years. What percent of a present amount of radioactive actinium will remain after 19 years? 29. A model that can be used for predicting the height H on his or her age is age of the child in years. H 70.228 5.104x 9.222 ln x, x (Source: Snapshots of Applications in Mathematics) where 1 4 (in centimeters) of a child based is the ≤ x ≤ 6, (a) Construct a table of values. Then sketch the graph of the model. (b) Use the graph from part (a) to estimate the height of a four-year-old child. Then calculate the actual height using the model. 333202_030R.qxd 12/7/05 10:35 AM Page 276 276 Chapter 3 Exponential and Logarithmic Functions 3 Cumulative Test for Chapters 1–3 Take this test to review the material from earlier chapters. When you are finished, check your work against the answers given in the back of the book. 1, 1. segment joining the points and the distance between the points. Find the coordinates of the midpoint of the line 1. Plot the points 3, 4 and x 2 4 In Exercises 2– 4, graph the equation without using a graphing utility. 2. x 3y 12 0 3. y x 2 9 4. y 4 x 5. Find an equation of the line passing through 1 2, 1 and 3, 8. y 4 2 −2 −4 FIGURE FOR 6 6. Explain why the graph at the left does not represent as a function of x. 7. Evaluate (if possible) the function given by for each value. (a) f 6 (b) f 2 (c. Compare the graph of each function with the graph of y 3x. (Note: It is not necessary to sketch the graphs.) (a) r x 1 3x 2 (b) hx 3x 2 (c) gx 3x 2 In Exercises 9 and 10, find (a) is the domain of f x x 3, f/g? 9. gx 4x 1 f gx, (b) f gx, (c) fgx, and (d) f/gx. What 10. f x x 1, gx x2 1 In Exercises 11 and 12, find (a) function. f g and (b) g f. Find the domain of each composite 11. f x 2x2, gx x 6 12. f x x 2, gx x 13. Determine whether hx 5x 2 has an inverse function. If so, find the inverse function. 14. The power produced by a wind turbine is proportional to the cube of the wind speed A wind speed of 27 miles per hour produces a power output of 750 kilowatts. Find P S. the output for a wind speed of 40 miles per hour. 15. Find the quadratic function whose graph has a vertex at the point 4, 7. 8, 5 and passes through In Exercises 16–18, sketch the graph of the function without the aid of a graphing utility. 16. hx x 2 4x 17. f t 1 4tt 22 18. gs s2 4s 10 In Exercises 19–21, find all the zeros of the function and write the function as a product of linear factors. 19. 20. 21. f x x3 2x2 4x 8 f x x4 4x3 21x2 f x 2x4 11x3 30x2 62x 40 333202_030R.qxd 12/7/05 10:35 AM Page 277 Cumulative Test for Chapters 1–3 277 22. Use long division to divide 6x3 4x2 by 2x2 1. 23. Use synthetic division to divide 2x 4 3x3 6x 5 by x 2. 24. Use the Intermediate Value Theorem and a graphing utility to find intervals one unit is guaranteed to have a zero. gx x3 3x2 6 in length in which the function Approximate the real zeros of the function. In Exercises 25–27, sketch the graph of the rational function by hand. Be sure to identify all intercepts and asymptotes. 25. 27. f x 2x x2 9 f x x3 3x2 4
x 12 x2 x 2 26. f x x2 4x 3 x2 2x 3 In Exercises 28 and 29, solve the inequality. Sketch the solution set on the real number line. 28. 3x3 12x ≤ 0 29. 1 x 1 ≥ 1 x 5 In Exercises 30 and 31, use the graph of to describe the transformation that yields the graph of f x 2 gx 2.2x 4 gx 2 f x 2.2x, x3 g. x , 30. 31. f 5 5 In Exercises 32–35, use a calculator to evaluate the expression. Round your result to three decimal places. 32. log 98 33. log6 7 34. ln31 35. ln40 5 36. Use the properties of logarithms to expand lnx2 16 x 4 , where x > 4. 37. Write 2 ln x 1 2 lnx 5 as a logarithm of a single quantity. In Exercises 38–40, solve the equation algebraicially. Approximate the result to three decimal places. 38. 6e2x 72 39. e2x 11ex 24 0 40. lnx 2 3 41. The sales S (in billions of dollars) of lottery tickets in the United States from 1997 through 2003 are shown in the table. (Source: TLF Publications, Inc.) (a) Use a graphing utility to create a scatter plot of the data. Let represent the year, t with t 7 corresponding to 1997. (b) Use the regression feature of the graphing utility to find a quadratic model for the data. (c) Use the graphing utility to graph the model in the same viewing window used for the scatter plot. How well does the model fit the data? (d) Use the model to predict the sales of lottery tickets in 2008. Does your answer Year Sales, S 1997 1998 1999 2000 2001 2002 2003 35.5 35.6 36.0 37.2 38.4 42.0 43.5 TABLE FOR 41 seem reasonable? Explain. 42. The number N of bacteria in a culture is given by the model when t where estimate the time required for the N 420 t 8, N 175ekt, is the time in hours. If population to double in size. 333202_030R.qxd 12/7/05 3:39 PM Page 278 Proofs in Mathematics Each of the following three properties of logarithms can be proved by using properties of exponential functions. Slide Rules The slide rule was invented by William Oughtred (1574–1660) in 1625. The slide rule is a computational device with a sliding portion and a fixed portion. A slide rule enables you to perform multiplication by using the Product Property of Logarithms. There are other slide rules that allow for the calculation of roots and trigonometric functions. Slide rules were used by mathematicians and engineers until the invention of the hand-held calculator in 1972. 278 Properties of Logarithms (p. 240) a 1, Let be a positive number such that and are positive real numbers, the following properties are true. a v n and let be a real number. If u 1. Product Property: 2. Quotient Property: Logarithm with Base a loga uv loga u loga v u v loga u loga v loga 3. Power Property: loga un n loga u Proof Let Natural Logarithm lnuv ln u ln v u ln v ln u ln v ln un n ln u x loga u and y loga v. The corresponding exponential forms of these two equations are ax u and ay v. To prove the Product Property, multiply u and v to obtain uv axay axy. The corresponding logarithmic form of uv axy is loga uv x y. So, loga uv loga u loga v. To prove the Quotient Property, divide u by v to obtain u v ax ay a xy. The corresponding logarithmic form of uv axy is loga uv x y. So, loga u v loga u loga v. To prove the Power Property, substitute follows. ax for u in the expression loga un, as loga un loga axn loga anx nx n loga u So, loga un n loga u. Substitute ax for u. Property of exponents Inverse Property of Logarithms Substitute loga u for x. 333202_030R.qxd 12/7/05 10:35 AM Page 279 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. y x. y1 x, 1. Graph the exponential function given by a 0.5, 1.2, and 2.0. Which of these curves intersects the line y x? for which the curve Determine all positive numbers intersects the line y ax y ax for a y4 y2 y3 x3, ex and x 2. Use a graphing utility to graph x2, x. functions function increases at the greatest rate as approaches and each of the y5 Which ? 3. Use the result of Exercise 2 to make a conjecture about the and is a natural . 4. Use the results of Exercises 2 and 3 to describe what is implied when it is stated that a quantity is growing exponentially. ex rate of growth of x number and approaches y xn, where y1 n 5. Given the exponential function f x ax show that (a) (b) f u v f u f v. f 2x f x2. 6. Given that f x ex ex 2 and gx ex ex 2 show that f x2 gx2 1. 7. Use a graphing utility the function given by function. n. y ex n (read “ n! to compare the graph of with the graph of each given is defined as factorial” (a) y1 (b) y2 (c) y3 1 x 1! 1 x 1! 1 x 1! x2 2! x2 2! x3 3! 8. Identify the pattern of successive polynomials given in Exercise 7. Extend the pattern one more term and compare the graph of the resulting polynomial function with the graph of What do you think this pattern implies? y ex. 9. Graph the function given by f x e x ex. From the graph, the function appears to be one-to-one. Assuming that the function has an inverse function, find f 1x. f 1x if 10. Find a pattern for f x ax 1 ax 1 where a > 0, a 1. 11. By observation, identify the equation that corresponds to the graph. Explain your reasoning. y 8 6 4 −4 −2 −2 x 2 4 (a) (b) (c) y 6ex22 6 1 ex2 y 61 ex 22 y 12. You have two options for investing $500. The first earns 7% compounded annually and the second earns 7% simple interest. The figure shows the growth of each investment over a 30-year period. (a) Identify which graph represents each type of invest- ment. Explain your reasoning ( 4000 3000 2000 1000 20 25 30 t 5 10 15 Year (b) Verify your answer in part (a) by finding the equations that model the investment growth and graphing the models. (c) Which option would you choose? Explain your reasoning. 13. Two different samples of radioactive isotopes are decaying. as well as respectively. Find the time required The isotopes have initial amounts of k2, half-lives of for the samples to decay to equal amounts. and and c2, k1 c1 279 333202_030R.qxd 12/7/05 10:35 AM Page 280 14. A lab culture initially contains 500 bacteria. Two hours later, the number of bacteria has decreased to 200. Find the exponential decay model of the form B B0akt that can be used to approximate the number of bacteria t after hours. 15. The table shows the colonial population estimates of the (Source: U.S. American colonies from 1700 to 1780. Census Bureau) Year Population 1700 1710 1720 1730 1740 1750 1760 1770 1780 250,900 331,700 466,200 629,400 905,600 1,170,800 1,593,600 2,148,100 2,780,400 In each of the following, let the year with t 0 t, corresponding to 1700. y represent the population in (a) Use the regression feature of a graphing utility to find an exponential model for the data. (b) Use the regression feature of the graphing utility to find a quadratic model for the data. (c) Use the graphing utility to plot the data and the models from parts (a) and (b) in the same viewing window. (d) Which model is a better fit for the data? Would you use this model to predict the population of the United States in 2010? Explain your reasoning. 16. Show that 17. Solve logax logab x ln x2 ln x2. 1 loga 1 b . 18. Use a graphing utility to compare the graph of y ln x with the graph of each given (a) the function given by function. y1 y2 y3 b) (c) x 12 x 12 1 3 x 13 280 19. Identify the pattern of successive polynomials given in Exercise 18. Extend the pattern one more term and compare the graph of the resulting polynomial function What do you think the pattern with the graph of implies? y ln x. 20. Using y ab x and y axb take the natural logarithm of each side of each equation. What are the slope and -intercept of the line relating and ln y What are the slope and -intercept of the for and line relating y ab x ? ln x y axb ? ln y for y y x In Exercises 21 and 22, use the model y 80.4 11 ln x, 100 ≤ x ≤ 1500 which approximates the minimum required ventilation rate in terms of the air space per child in a public school classroom. In the model, is the air space per child in cubic feet and is the ventilation rate per child in cubic feet y per minute. x 21. Use a graphing utility to graph the model and approximate the required ventilation rate if there is 300 cubic feet of air space per child. 22. A classroom is designed for 30 students. The air conditioning system in the room has the capacity of moving 450 cubic feet of air per minute. (a) Determine the ventilation rate per child, assuming that the room is filled to capacity. (b) Estimate the air space required per child. (c) Determine the minimum number of square feet of floor space required for the room if the ceiling height is 30 feet. In Exercises 23–26, (a) use a graphing utility to create a scatter plot of the data, (b) decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model, (c) explain why you chose the model you did in part (b), (d) use the regression feature of a graphing utility to find the model you chose in part (b) for the data and graph the model with the scatter plot, and (e) determine how well the model you chose fits the data. 1, 2.0, 1.5, 3.5, 2, 4.0, 4, 5.8, 6, 7.0, 8, 7.8 1, 4.4, 1.5, 4.7, 2, 5.5, 4, 9.9, 6, 18.1, 8, 33.0 1, 7.5, 1.5, 7.0, 2, 6.8, 4, 5.0, 6, 3.5, 8, 2.0 25. 26. 1, 5.0, 1.5, 6.0, 2, 6.4, 4, 7.8, 6, 8.6, 8, 9.0 23. 24. 44 333202_0400.qxd 12/7/05 10:59 AM Page 281 Trigonometry 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry Trigonometric Functions of Any Angle Graphs of Sine and Cosine Functions Graphs of Other Trigonometric Functions Inverse Trigonometric Functions Applications and Models Airport runways are named on the basis of the angles they form with due north, measured in a clockwise direction. These angles are called bearings and can be determined using trigonometry AT I O N S Trigonometric functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Speed of a Bicycle, E
xercise 108, page 293 • Respiratory Cycle, Exercise 73, page 330 • Security Patrol, Exercise 97, page 351 • Machine Shop Calculations, • Data Analysis: Meteorology, • Navigation, Exercise 69, page 310 Exercise 75, page 330 Exercise 29, page 360 • Sales, Exercise 88, page 320 • Predator-Prey Model, Exercise 77, page 341 • Wave Motion, Exercise 60, page 362 281 333202_0401.qxd 12/7/05 11:01 AM Page 282 282 Chapter 4 Trigonometry 4.1 Radian and Degree Measure What you should learn • Describe angles. • Use radian measure. • Use degree measure. • Use angles to model and solve real-life problems. Why you should learn it You can use angles to model and solve real-life problems. For instance, in Exercise 108 on page 293, you are asked to use angles to find the speed of a bicycle. Angles As derived from the Greek language, the word trigonometry means “measurement of triangles.” Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying. With the development of calculus and the physical sciences in the 17th century, a different perspective arose—one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domains. Consequently, the applications of trigonometry expanded to include a vast number of physical phenomena involving rotations and vibrations. These phenomena include sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles. The approach in this text incorporates both perspectives, starting with angles and their measure. e a l si d e r m i n T Initial side Vertex Angle FIGURE 4.1 y Terminal side Initial side x Angle in Standard Position FIGURE 4.2 © Wolfgang Rattay/Reuters/Corbis An angle is determined by rotating a ray (half-line) about its endpoint. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side, as shown in Figure 4.1. The endpoint of the ray is the vertex of the angle. This perception of an angle fits a coordinate system in which the origin is the vertex and the initial side coincides with the positive x -axis. Such an angle is in standard position, as shown in Figure 4.2. Positive angles are generated by counterclockwise rotation, and negative angles by clockwise rotation, as shown in Figure 4.3. Angles are labeled with Greek letters In Figure 4.4, note that angles have the same initial and terminal sides. Such angles are coterminal. (theta), as well as uppercase letters (beta), and (alpha), A, B, and and C. y y y Positive angle (counterclockwise) x Negative angle (clockwise) α β x α x β FIGURE 4.3 FIGURE 4.4 Coterminal Angles The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. 333202_0401.qxd 12/7/05 11:01 AM Page 283 Section 4.1 Radian and Degree Measure 283 y Radian Measure r θ r s = r x The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. One way to measure angles is in radians. This type of measure is especially useful in calculus. To define a radian, you can use a central angle of a circle, one whose vertex is the center of the circle, as shown in Figure 4.5. radius when 1 radian Arc length FIGURE 4.5 Definition of Radian One radian is the measure of a central angle r in length to the radius of the circle. See Figure 4.5. Algebraically, this means that that intercepts an arc equal s s r where is measured in radians. y Because the circumference of a circle is 2r units, it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of 2 radians 1 radian r s 2r. 3 radians r r r r x 6 radians 4 radians r 5 radians FIGURE 4.6 r 2 One revolution around a circle of radius corresponds to an radians because angle of 2r r s r 2 radians. Moreover, because there are just over six radius lengths in a full circle, as shown in Figure 4.6. Because the units of measure for and are the has no units—it is simply a real number. same, the ratio sr s r 2 6.28, Because the radian measure of an angle of one full revolution is 2, you can obtain the following. 1 2 1 4 1 6 revolution revolution revolution 2 2 2 4 2 6 radians 2 3 radians radians These and other common angles are shown in Figure 4.7. π 6 π 2 FIGURE 4.7 π 4 π π 3 π2 Recall that the four quadrants in a coordinate system are numbered I, II, III, lie in each are acute angles and and IV. Figure 4.8 on page 284 shows which angles between 0 and of the four quadrants. Note that angles between 0 and angles between are obtuse angles. 2 2 and 2 333202_0401.qxd 12/7/05 11:01 AM Page 284 284 Chapter 4 Trigonometry θ = π 2 Quadrant II π < < π θ 2 Quadrant Quadrant III < < θ π Quadrant IV The phrase “the terminal side of lies in a quadrant” is often abbreviated by simply saying that “ lies in a quadrant.” The terminal sides of the “quadrant 2, angles” do and not lie within quadrants. 32 , 0, θ = π 3 2 FIGURE 4.8 Two angles are coterminal if they have the same initial and terminal sides. and You can find an angle that is coterminal to a given angle by adding or is coterminal with For instance, the angles 0 and 136. subtracting has infinitely many coterminal angles. For instance, (one revolution), as demonstrated in Example 1. A given angle are coterminal, as are the angles 6 6 2 2 6 where 2n n is an integer. Example 1 Sketching and Finding Coterminal Angles a. For the positive angle 136, subtract 2 to obtain a coterminal angle 13 6 2 . 6 See Figure 4.9. 3 4 b. For the positive angle 2 5 . 4 c. For the negative angle 2 4 . 3 2 3 π 2 θ= π13 6 π π 3 2 π 6 0 π 34, subtract 2 to obtain a coterminal angle See Figure 4.10. 23, add 2 to obtain a coterminal angle See Figure 4.11 FIGURE 4.9 FIGURE 4.10 FIGURE 4.11 Now try Exercise 17. 333202_0401.qxd 12/7/05 11:01 AM Page 285 Section 4.1 Radian and Degree Measure 285 Two positive angles 2. other) if their sum is of each other) if their sum is and are complementary (complements of each Two positive angles are supplementary (supplements . See Figure 4.12. β α β α Complementary Angles FIGURE 4.12 Supplementary Angles Example 2 Complementary and Supplementary Angles If possible, find the complement and the supplement of (a) 25 and (b) 45. Solution a. The complement of 5 2 10 5 2 The supplement of 5 2 5 5 45 b. Because 10 . 25 is 4 10 25 is 2 3 . 5 5 2, is greater than it has no complement. (Remember that complements are positive angles.) The supplement is 4 5 5 5 4 5 . 5 Now try Exercise 21. Degree Measure . A second way to measure angles is in terms of degrees, denoted by the symbol of a complete A measure of one degree (1 ) is equivalent to a rotation of revolution about the vertex. To measure angles, it is convenient to mark degrees on the circumference of a circle, as shown in Figure 4.13. So, a full revolution 360, (counterclockwise) corresponds to a quarter revolution to Because and so on. radians corresponds to one complete revolution, degrees and a half revolution to 90, 2 180, 1 360 x radians are related by the equations 360 2 rad and 180 rad. From the latter equation, you obtain 1 180 rad and 1 rad 180 which lead to the conversion rules at the top of the next page. y 1 ° ° 90 = (360 ) 4 1 (360 )° 60° = 6 1 45° = 8 30° = 0° 360° 330° θ 315° (360 )° 1 (360 )° 12 240° 270° 300° 120° 135° 150° 180° 210° 225° FIGURE 4.13 333202_0401.qxd 12/7/05 11:01 AM Page 286 286 Chapter 4 Trigonometry Conversions Between Degrees and Radians 1. To convert degrees to radians, multiply degrees by 2. To convert radians to degrees, multiply radians by rad 180 . 180 rad . To apply these two conversion rules, use the basic relationship (See Figure 4.14.) rad 180. π 6 30° π 2 90° FIGURE 4.14 π 4 45° π 180° π 3 60° π2 360° When no units of angle measure are specified, radian measure is implied. For instance, if you write 2, you imply that 2 radians. Te c h n o l o g y With calculators it is convenient to use decimal degrees to denote fractional parts of degrees. Historically, however, fractional parts of degrees were expressed in minutes and seconds, using the prime ( ) and double prime ( ) notations, respectively. That is, Example 3 Converting from Degrees to Radians a. b. c. 135 135 deg rad 180 deg 540 540 deg rad 180 deg 270 270 deg rad 180 deg 3 radians 4 3 radians 3 2 radians Now try Exercise 47. Example 4 Converting from Radians to Degrees 1 one minute 1 60 1 one second 1 3600 1 1 64 32 47. Consequently, an angle of 64 degrees, 32 minutes, and 47 seconds is represented by Many calculators have special keys for converting an angle in degrees, minutes, and seconds to decimal degree form, and vice versa. D M S a. b. c. rad180 deg rad rad 2 2 rad180 deg rad 9 9 rad 2 2 360 2 rad 2 rad180 deg rad 90 810 114.59 Multiply by 180. Now try Exercise 51. If you have a calculator with a “radian-to-degree” conversion key, try using it to verify the result shown in part (c) of Example 4. Multiply by 180. Multiply by 180. Multiply by 180. Multiply by 180. Multiply by 180. 333202_0401.qxd 12/7/05 11:01 AM Page 287 Section 4.1 Radian and Degree Measure 287 Applications The radian measure formula, a circle. sr, can be used to measure arc length along Arc Length For a circle of radius by s r r, a central angle intercepts an arc of length given s Length of circular arc is measured in radians. Note that if where measure of equals the arc length. r 1, then s , and the radian Example 5 Finding Arc Length A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of as shown in Figure 4.15. 240, Solution To use the formula s r, 240 240 deg rad 180 deg first convert 4 3 radians 240 to radian measure. Then, using a radius of 44 3 s r r 4 16 3 inches, you can find the arc length to be 16.76 inches. Note that the units for radian measure, which has no units. r are determined by the units for because r is given in
Now try Exercise 87. The formula for the length of a circular arc can be used to analyze the motion of a particle moving at a constant speed along a circular path. Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. the particle is is the length of the arc traveled in time then the linear speed of If t, v s Linear speed v arc length time s t . Moreover, if s, length the particle is is the angle (in radian measure) corresponding to the arc then the angular speed (the lowercase Greek letter omega) of Angular speed central angle time . t s θ = 240° r = 4 FIGURE 4.15 Linear speed measures how fast the particle moves, and angular speed measures how fast the angle changes. By dividing the t, formula for arc length by you can establish a relationship v between linear speed and as shown. angular speed s r r t , s t v r 333202_0401.qxd 12/7/05 11:01 AM Page 288 288 Chapter 4 Trigonometry Example 6 Finding Linear Speed The second hand of a clock is 10.2 centimeters long, as shown in Figure 4.16. Find the linear speed of the tip of this second hand as it passes around the clock face. 10.2 cm Solution In one revolution, the arc length traveled is s 2r 210.2 20.4 centimeters. Substitute for r. FIGURE 4.16 The time required for the second hand to travel this distance is t 1 minute 60 seconds. So, the linear speed of the tip of the second hand is Linear speed s t 20.4 centimeters 60 seconds 1.068 centimeters per second. Now try Exercise 103. 50 ft FIGURE 4.17 Example 7 Finding Angular and Linear Speeds A Ferris wheel with a 50-foot radius (see Figure 4.17) makes 1.5 revolutions per minute. a. Find the angular speed of the Ferris wheel in radians per minute. b. Find the linear speed of the Ferris wheel. Solution a. Because each revolution generates 1.52 3 radians per minute. In other words, the angular speed is 2 radians, it follows that the wheel turns Angular speed t 3 radians 1 minute 3 radians per minute. b. The linear speed is Linear speed s t r t 503 feet 1 minute 471.2 feet per minute. Now try Exercise 105. 333202_0401.qxd 12/7/05 11:01 AM Page 289 Section 4.1 Radian and Degree Measure 289 A sector of a circle is the region bounded by two radii of the circle and their intercepted arc (see Figure 4.18). θ r FIGURE 4.18 Area of a Sector of a Circle A For a circle of radius r, is given by A 1 2 r2 the area of a sector of the circle with central angle where is measured in radians. Example 8 Area of a Sector of a Circle A sprinkler on a golf course fairway is set to spray water over a distance of 70 feet and rotates through an angle of (see Figure 4.19). Find the area of the fairway watered by the sprinkler. 120 120° 70 ft FIGURE 4.19 to radian measure as follows. Multiply by 180. 120 Solution First convert 120 120 deg rad 180 deg 2 3 radians 23 r2 Then, using A 1 2 1 2 4900 3 7022 3 and r 70, the area is Formula for the area of a sector of a circle Substitute for and r . Simplify. Simplify. 5131 square feet. Now try Exercise 107. 333202_0401.qxd 12/7/05 11:01 AM Page 290 290 Chapter 4 Trigonometry 4.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. ________ means “measurement of triangles.” 2. An ________ is determined by rotating a ray about its endpoint. 3. Two angles that have the same initial and terminal sides are ________. 4. One ________ is the measure of a central angle that intercepts an arc equal to the radius of the circle. 2 are ________ angles, and angles that measure between 5. Angles that measure between 0 and 2 and are ________ angles. 6. Two positive angles that have a sum of are ________ angles. have a sum of 2 are ________ angles, whereas two positive angles that 7. The angle measure that is equivalent to 1 360 of a complete revolution about an angle’s vertex is one ________. 8. The ________ speed of a particle is the ratio of the arc length traveled to the time traveled. 9. The ________ speed of a particle is the ratio of the change in the central angle to time. 10. The area of a sector of a circle with radius and central angle where r , is measured in radians, is given by the formula ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, estimate the angle to the nearest one-half radian. 1. 3. 5. 2. 4. 6. In Exercises 7–12, determine the quadrant in which each angle lies. (The angle measure is given in radians.) 18. (a) 8. (a) 11 8 (b) 9 8 7. (a) 9. (a) 5 12 (b) 7 5 (b) 2 10. (a) 1 (b) 11 9 11. (a) 3.5 (b) 2.25 12. (a) 6.02 (b) 4.25 In Exercises 13–16, sketch each angle in standard position. 13. (a) 15. (a) 5 4 11 6 (b) 2 3 14. (a) 7 4 (b) 3 16. (a) 4 (b) 5 2 (b) 7 In Exercises 17–20, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians. 17. (a 19. (a) 20. (a) (b) (b π11 6 θ = − (b) 12 (b) 2 15 333202_0401.qxd 12/7/05 11:01 AM Page 291 In Exercises 21–24, find (if possible) the complement and supplement of each angle. 41. (a) 42. (a) 240 420 (b) (b) 180 230 Section 4.1 Radian and Degree Measure 291 21. (a) 3 23. (a) 1 (b) 3 4 (b) 2 22. (a) 12 (b) 11 12 24. (a) 3 (b) 1.5 In Exercises 25–30, estimate the number of degrees in the angle. 25. 27. 29. 26. 28. 30. In Exercises 31–34, determine the quadrant in which each angle lies. 31. (a) 32. (a) 33. (a) 34. (a) 130 8.3 132 50 260 (b) (b) (b) (b) 285 257 30 336 3.4 In Exercises 35–38, sketch each angle in standard position. 35. (a) 37. (a) 30 405 (b) (b) 150 480 36. (a) 38. (a) 270 750 (b) (b) 120 600 In Exercises 39– 42, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in degrees. 39. (a) (b) θ = −36° θ = 45° 40. (a) (b) θ = 420− ° θ = 120° In Exercises 43– 46, find (if possible) the complement and supplement of each angle. 43. (a) 45. (a) 18 79 (b) (b) 115 150 44. (a) 46. (a) 3 130 (b) (b) 64 170 In Exercises 47–50, rewrite each angle in radian measure as (Do not use a calculator.) a multiple of . 47. (a) 30 20 49. (a) (b) 150 240 (b) 48. (a) 315 270 50. (a) (b) 120 144 (b) In Exercises 51–54, rewrite each angle in degree measure. (Do not use a calculator.) 51. (a) 53. (a) 3 2 7 3 (b) (b) 7 6 11 30 52. (a) 54. (a) 7 12 11 6 (b) (b) 9 34 15 In Exercises 55–62, convert the angle measure from degrees to radians. Round to three decimal places. 55. 57. 59. 61. 115 216.35 532 0.83 56. 58. 60. 62. 87.4 48.27 345 0.54 In Exercises 63–70, convert the angle measure from radians to degrees. Round to three decimal places. 7 15 8 4.2 2 63. 65. 67. 69. 5 11 13 2 4.8 0.57 64. 66. 68. 70. In Exercises 71–74, convert each angle measure to decimal degree form. 71. (a) 72. (a) 73. (a) 74. (a) 54 45 245 10 85 18 30 135 36 (b) (b) (b) (b) 128 30 2 12 330 25 408 16 20 In Exercises 75–78, convert each angle measure to form. D M S 75. (a) 76. (a) 77. (a) 78. (a) 240.6 345.12 2.5 0.355 (b) (b) 145.8 0.45 3.58 (b) (b) 0.7865 333202_0401.qxd 12/7/05 11:01 AM Page 292 292 Chapter 4 Trigonometry In Exercises 79–82, find the angle in radians. City 29 96. San Francisco, California Seattle, Washington Latitude 37 47 36 N 47 37 18 N 97. Difference in Latitudes Assuming that Earth is a sphere of radius 6378 kilometers, what is the difference in the latitudes of Syracuse, New York and Annapolis, Maryland, where Syracuse is 450 kilometers due north of Annapolis? 98. Difference in Latitudes Assuming that Earth is a sphere of radius 6378 kilometers, what is the difference in the latitudes of Lynchburg, Virginia and Myrtle Beach, South Carolina, where Lynchburg is 400 kilometers due north of Myrtle Beach? 99. Instrumentation The pointer on a voltmeter is 6 centimeters in length (see figure). Find the angle through which the pointer rotates when it moves 2.5 centimeters on the scale. 10 in. 79. 6 81. 32 5 7 80. 82. 10 75 60 In Exercises 83–86, find the radian measure of the central angle of a circle of radius that intercepts an arc of length s. r Radius r Arc Length s 83. 27 inches 84. 14 feet 85. 14.5 centimeters 86. 80 kilometers 6 inches 8 feet 25 centimeters 160 kilometers In Exercises 87–90, find the length of the arc on a circle of radius intercepted by a central angle . r Radius r Central Angle 6 cm 2 ft 87. 15 inches 88. 9 feet 89. 3 meters 90. 20 centimeters 180 60 1 radian 4 radian In Exercises 91–94, find the area of the sector of the circle with radius and central angle . r Radius r 91. 4 inches 92. 12 millimeters 93. 2.5 feet 94. 1.4 miles Central Angle 3 4 225 330 Distance Between Cities In Exercises 95 and 96, find the distance between the cities. Assume that Earth is a sphere of radius 4000 miles and that the cities are on the same longitude (one city is due north of the other). City 95. Dallas, Texas Omaha, Nebraska Latitude 32 47 39 N 41 15 50 N Not drawn to scale FIGURE FOR 99 FIGURE FOR 100 100. Electric Hoist An electric hoist is being used to lift a beam (see figure). The diameter of the drum on the hoist is 10 inches, and the beam must be raised 2 feet. Find the number of degrees through which the drum must rotate. 101. Angular Speed A car is moving at a rate of 65 miles per hour, and the diameter of its wheels is 2.5 feet. (a) Find the number of revolutions per minute the wheels are rotating. (b) Find the angular speed of the wheels in radians per minute. 102. Angular Speed A two-inch-diameter pulley on an electric motor that runs at 1700 revolutions per minute is connected by a belt to a four-inch-diameter pulley on a saw arbor. (a) Find the angular speed (in radians per minute) of each pulley. (b) Find the revolutions per minute of the saw. 333202_0401.qxd 12/7/05 11:01 AM Page 293 103. Linear and Angular Speeds A 71 4 -inch circular power saw rotates at 5200 revolutions per minute. (a) Find the angular speed of the saw blade in radians per minute. (b) Find t
he linear speed (in feet per minute) of one of the 24 cutting teeth as they contact the wood being cut. 104. Linear and Angular Speeds A carousel with a 50-foot diameter makes 4 revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed of the platform rim of the carousel. 105. Linear and Angular Speeds The diameter of a DVD is approximately 12 centimeters. The drive motor of the DVD player is controlled to rotate precisely between 200 and 500 revolutions per minute, depending on what track is being read. (a) Find an interval for the angular speed of a DVD as it rotates. (b) Find an interval for the linear speed of a point on the outermost track as the DVD rotates. 106. Area A car’s rear windshield wiper rotates The total length of the wiper mechanism is 25 inches and wipes the windshield over a distance of 14 inches. Find the area covered by the wiper. 125. 107. Area A sprinkler system on a farm is set to spray water over a distance of 35 meters and to rotate through an angle of Draw a diagram that shows the region that can be irrigated with the sprinkler. Find the area of the region. 140. Model It 108. Speed of a Bicycle The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are 4 inches, 2 inches, and 14 inches, respectively. A cyclist is pedaling at a rate of 1 revolution per second. 14 in. 2 in. 4 in. (a) Find the speed of the bicycle in feet per second and miles per hour. (b) Use your result from part (a) to write a function for the distance (in miles) a cyclist travels in terms of the number of revolutions of the pedal sprocket. n d Section 4.1 Radian and Degree Measure 293 Model It (co n t i n u e d ) (c) Write a function for the distance (in miles) a cyclist travels in terms of the time (in seconds). Compare this function with the function from part (b). d t (d) Classify the types of functions you found in parts (b) and (c). Explain your reasoning. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 109–111, determine whether 109. A measurement of 4 radians corresponds to two complete revolutions from the initial side to the terminal side of an angle. 110. The difference between the measures of two coterminal if expressed in degrees radians if expressed in angles is always a multiple of and is always a multiple of radians. 360 2 111. An angle that measures 1260 lies in Quadrant III. 112. Writing In your own words, explain the meanings of (a) an angle in standard position, (b) a negative angle, (c) coterminal angles, and (d) an obtuse angle. 113. Think About It A fan motor turns at a given angular speed. How does the speed of the tips of the blades change if a fan of greater diameter is installed on the motor? Explain. 114. Think About It Is a degree or a radian the larger unit of measure? Explain. 115. Writing If the radius of a circle is increasing and the magnitude of a central angle is held constant, how is the length of the intercepted arc changing? Explain your reasoning. 116. Proof Prove that the area of a circular sector of radius r A 1 is measured in 2 where r2, is with central angle radians. Skills Review In Exercises 117–120, simplify the radical expression. 117. 119. 4 42 22 62 55 210 172 92 118. 120. In Exercises 121–124, sketch the graphs of specified transformation. f x x 25 f x 2 x5 123. 121. 122. 124. f x x5 4 f x x 35 y x5 and the 333202_0402.qxd 12/8/05 8:18 AM Page 294 294 Chapter 4 Trigonometry 4.2 Trigonometric Functions: The Unit Circle What you should learn • Identify a unit circle and describe its relationship to real numbers. • Evaluate trigonometric functions using the unit circle. • Use the domain and period to evaluate sine and cosine functions. • Use a calculator to evaluate trigonometric functions. Why you should learn it Trigonometric functions are used to model the movement of an oscillating weight. For instance, in Exercise 57 on page 300, the displacement from equilibrium of an oscillating weight suspended by a spring is modeled as a function of time. The Unit Circle The two historical perspectives of trigonometry incorporate different methods for introducing the trigonometric functions. Our first introduction to these functions is based on the unit circle. Consider the unit circle given by x2 y 2 1 Unit circle as shown in Figure 4.20. y (0, 1) − ( 1, 0) (1, 0) x (0, 1)− FIGURE 4.20 Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in Figure 4.21 Richard Megna/Fundamental Photographs FIGURE 4.21 θ x (1, 0) (1, 0, y 1, 0. the real number t corresponds to a point corresponds to the point 2, ference of As the real number line is wrapped around the unit circle, each real number on the circle. For example, the real number 0 Moreover, because the unit circle has a circum2 In general, each real number also corresponds to the point t standard position) whose radian measure is With this interpretation of r 1 length formula of the arc intercepted by the angle ) indicates that the real number , (in the arc is the length also corresponds to a central angle given in radians. s r 1, 0. (with t, t. t 333202_0402.qxd 12/7/05 11:02 AM Page 295 Section 4.2 Trigonometric Functions: The Unit Circle 295 The Trigonometric Functions From the preceding discussion, it follows that the coordinates are two t. functions of the real variable You can use these coordinates to define the six trigonometric functions of and t. y x sine cosecant cosine secant tangent cotangent These six functions are normally abbreviated sin, csc, cos, sec, tan, and cot, respectively. Definitions of Trigonometric Functions t Let be a real number and let ding to x, y t. be the point on the unit circle correspon- sin t y cos t x csc t 1 y , y 0 sec t 1 x , x 0 tan t y x cot t x y , , x 0 y 0 In the definitions of the trigonometric functions, note that the tangent and corresponds are undefined. Similarly, For instance, because secant are not defined when to it follows that the cotangent and cosecant are not defined when t 0 y 0. and csc 0 are undefined. x, y 1, 0, cot 0 For instance, because x, y 0, 1, corresponds to sec2 tan2 t 2 x 0. and In Figure 4.22, the unit circle has been divided into eight equal arcs, corre- t sponding to -values of , , 5 4 3 4 , , 0 and 2. Similarly, in Figure 4.23, the unit circle has been divided into 12 equal arcs, corresponding to -values of t 0 , 11 , 6 and 2. To verify the points on the unit circle in Figure 4.22, note that also lies on the line circle produces the following. y x. So, substituting x for y x2 x2 1 2x2 1 Because the point is in the first quadrant, x in the equation of the unit 2 2 2 2 , x2 1 2 2 2 and because x ± 2 2 y x, you also have y 2 2 . You can use similar reasoning to verify the rest of the points in Figure 4.22 and the points in Figure 4.23. x, y Using the coordinates in Figures 4.22 and 4.23, you can easily evaluate t the trigonometric functions for common -values. This procedure is demonstrated in Examples 1 and 2. You should study and learn these exact function values for common -values because they will help you in later sections to perform calculations quickly and easily. t Note in the definition at the right that the functions in the second row are the reciprocals of the corresponding functions in the first row0, 1, 0) x (1, 00, 1)− ( , − 2 2 ) 2 2 FIGURE 4.22 ( − 0, 1, 0) x (1, 00, 1)− FIGURE 4.23 ( 333202_0402.qxd 12/7/05 11:02 AM Page 296 296 Chapter 4 Trigonometry Example 1 Evaluating Trigonometric Functions Evaluate the six trigonometric functions at each real number. a. t 6 b. t 5 4 c. t 0 d. t Solution For each -value, begin by finding the corresponding point circle. Then use the definitions of trigonometric functions listed on page 295. x, y on the unit t a. t 6 corresponds to the point x, y 3 2 , . 1 2 sin 6 y 1 2 cos 6 x tan 6 y x 3 2 12 32 1 3 corresponds to the point 1 y 1 12 2 csc sec 6 6 1 x 2 3 32 12 23 3 3 3 3 x, y cot cos x tan y x 2 2 22 22 5 4 5 4 5 4 csc sec 5 4 5 4 5 4 1 cot 22 22 1 b. t 5 4 sin c. t 0 corresponds to the point x, y 1, 0. sin 0 y 0 cos 0 x 1 tan 0 y x 0 1 0 csc 0 1 y sec 0 1 x cot 0 x y is undefined. 1 1 1 is undefined. d. t corresponds to the point x, y 1, 0. sin y 0 cos x 1 tan y x 0 1 0 Now try Exercise 23. csc 1 y sec 1 x cot x y is undefined. 1 1 1 is undefined. 333202_0402.qxd 12/7/05 11:02 AM Page 297 Exploration With your graphing utility in radian and parametric modes, enter the equations X1T = cos T and Y1T = sin T and use the following settings. Tmin = 0, Tmax = 6.3, Tstep = 0.1 Xmin = -1.5, Xmax = 1.5, Xscl = 1 Ymin = -1, Ymax = 1, Yscl = 1 1. Graph the entered equations and describe the graph. 2. Use the trace feature to move the cursor around the graph. What do the -values represent? What do the x- and values represent? y- t 3. What are the least and greatest values of and y? x y (0, 1) (−1, 0) (1, 0) x −1 ≤ y ≤ 1 (0, −1) −1 ≤ x ≤ 1 FIGURE 4.24 , ... Section 4.2 Trigonometric Functions: The Unit Circle 297 Example 2 Evaluating Trigonometric Functions Evaluate the six trigonometric functions at t . 3 Solution Moving clockwise around the unit circle, it follows that to the point sin x, y 12, 32. csc 3 2 3 2 3 3 23 3 t 3 corresponds cos tan 3 1 2 32 12 3 sec 2 3 3 cot 3 12 32 1 3 3 3 Now try Exercise 25. Domain and Period of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, consider the unit circle shown in Figure 4.24. Because Moreover, x, y 1 ≤ x ≤ 1. because So, the values of sine and cosine also range between and 1 ≤ y ≤ 1 1 is on the unit circle, you know that cos t x. and it follows that sin t y r 1, and 1. 1 1 ≤ ≤ y sin t ≤ ≤ 1 1 and 1 1 ≤ ≤ x cos t ≤ ≤ 1 1 2 t in the interval to each value of completes a second Adding revolution aroun
d the unit circle, as shown in Figure 4.25. The values of sint 2 Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result correspond to those of cost 2 cos t. sin t and and 0, 2 sint 2n sin t and cost 2n cos , ... , ... t. for any integer and real number Functions that behave in such a repetitive (or cyclic) manner are called periodic. n π π t = , 3 , ... x t = 0, 2 , ... , ... 7 t = 4 ππ , ... ππ 7 , 4 π + 4 , ... FIGURE 4.25 Definition of Periodic Function A function is periodic if there exists a positive real number such that c f ft c f t in the domain of The smallest number f. t for all called the period of f. c for which f is periodic is 333202_0402.qxd 12/7/05 11:02 AM Page 298 298 Chapter 4 Trigonometry Recall from Section 1.5 that a function f t f t. if f is even if f t f t, and is odd Even and Odd Trigonometric Functions The cosine and secant functions are even. sect sec t cost cos t The sine, cosecant, tangent, and cotangent functions are odd. sint sin t tant tan t csct csc t cott cot t Example 3 Using the Period to Evaluate the Sine and Cosine From the definition of periodic function, it follows that the sine and cosine functions are peri2. odic and have a period of The other four trigonometric functions are also periodic, and will be discussed further in Section 4.6. Te c h n o l o g y When evaluating trigonometric functions with a calculator, remember to enclose all fractional angle measures in parentheses. For instance, if you want to 6, sin evaluate should enter you for 13 6 7 2 cos 7 2 sin t 4 5 , a. Because 2 b. Because 4 , 6 13 6 sin2 sin 6 6 1 2 . you have sin , 2 you have cos4 cos 2 2 0. c. For because the sine function is odd. sint 4 5 Now try Exercise 31. Evaluating Trigonometric Functions with a Calculator When evaluating a trigonometric function with a calculator, you need to set the calculator to the desired mode of measurement (degree or radian). Most calculators do not have keys for the cosecant, secant, and cotangent key with their respecfunctions. To evaluate these functions, you can use the tive reciprocal functions sine, cosine, and tangent. For example, to evaluate csc8, use the fact that x 1 csc 8 1 sin8 and enter the following keystroke sequence in radian mode. SIN 6 ENTER . SIN 8 x 1 ENTER Display 2.6131259 These keystrokes yield the correct value of 0.5. Note that some calculators automatically place a left parenthesis after trigonometric functions. Check the user’s guide for your calculator for specific keystrokes on how to evaluate trigonometric functions. Example 4 Using a Calculator Function 2 3 sin a. Mode Calculator Keystrokes Display Radian SIN 2 3 ENTER 0.8660254 b. cot 1.5 Radian TAN 1.5 x 1 ENTER 0.0709148 Now try Exercise 45. 333202_0402.qxd 12/7/05 11:02 AM Page 299 Section 4.2 Trigonometric Functions: The Unit Circle 299 4.2 Exercises VOCABULARY CHECK: Fill in the blanks. t 1. Each real number corresponds to a point 2. A function f x, y is ________ if there exists a positive real number such that on the ________ ________. c f t c f t for all t in the domain of f. 3. The smallest number c f is periodic is called the ________ of f. 4. A function f is ________ if and ________ if f t f t. for which a function f t f t PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, determine the exact values of the six trigonometric functions of the angle . 1. ( 8 15 − , 17 17 ( y 2. y θ x 3. y 4. y θ x ( 12 13 , − 5 13 ( ( − 4 5 , − ( 3 5 ( 12 5 , 13 13 ( θ x θ x In Exercises 5–12, find the point corresponds to the real number x, y t. on the unit circle that 5. 7. 9. 11. 8. 10. t 3 t 5 4 t 5 3 12. t 19. 21. t 11 6 t 3 2 20. t 5 3 22. t 2 In Exercises 23–28, evaluate (if possible) the six trigonometric functions of the real number. 23. 25. 27. t 3 4 t 2 t 4 3 24. 26. 28 In Exercises 29–36, evaluate the trigonometric function using its period as an aid. 29. 31. 33. 35. cos sin 5 8 3 cos15 2 sin9 4 30. 32. 34. 36. sin cos 5 9 4 19 6 cos8 3 sin In Exercises 37– 42, use the value of the trigonometric function to evaluate the indicated functions. sint 3 8 sin t (a) 37. 38. In Exercises 13–22, evaluate (if possible) the sine, cosine, and tangent of the real number. 13. t 4 15. 17. t 6 t 7 4 14. t 3 16. 18. t 4 t 4 3 39. 41. sin t 1 3 sint (a) csct (b) cost 1 5 cos t (a) sect (b) sin t 4 5 sin t (a) sint (b) 40. 42. csc t (b) cos t 3 4 cost (a) sect (b) cos t 4 5 cos t (a) (b) cost 333202_0402.qxd 12/7/05 11:02 AM Page 300 300 Chapter 4 Trigonometry In Exercises 43–52, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) sin 4 csc 1.3 cos1.7 csc 0.8 43. 45. 47. 49. 51. sec 22.8 tan 3 cot 1 cos2.5 sec 1.8 sin0.9 44. 46. 48. 50. 52. Estimation In Exercises 53 and 54, use the figure and a straightedge to approximate the value of each trigonometric function. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 53. (a) sin 5 (b) cos 2 54. (a) sin 0.75 (b) cos 2.5 Model It (co n t i n u e d ) (a) Complete the tableb) Use the table feature of a graphing utility to approximate the time when the weight reaches equilibrium. (c) What appears to happen to the displacement as t increases? 58. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring is given by yt 1 is the displacement (in feet) and is the time (in seconds). Find the displacement when (a) t 1 t 0, 2. 4 cos 6t, t 1 4, and (c) where (b) y t Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 59 and 60, determine whether 59. Because sint sin t, negative angle is a negative number. tan a tana 6 it can be said that the sine of a 1.75 1.50 1.25 2.00 1.00 0.75 0.50 0.25 6.25 2.25 2.50 2.75 3.00 3.25 3.50 −0.8 −0.6 −0.4 3.75 4.00 4.25 0.8 0.6 0.4 0.2 −0.2 − 0.2 −0.4 −0.6 −0.8 0.2 0.4 0.6 0.8 1.2 60. 6.00 5.75 5.50 5.25 61. Exploration Let x1, y1 and t t1 circle corresponding to x2, y2 and respectively. x1, y1 . x2, y2 (b) Make a conjecture about any relationship between (a) Identify the symmetry of the points and t t1, be points on the unit sin t1 and . sin t1 4.50 4.75 5.00 FIGURE FOR 53–56 cos t1 and . cos t1 (c) Make a conjecture about any relationship between Estimation In Exercises 55 and 56, use the figure and a straightedge to approximate the solution of each equation, where To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 0 ≤ t < 2. 55. (a) 56. (a) sin t 0.25 sin t 0.75 (b) (b) cos t 0.25 cos t 0.75 62. Use the unit circle to verify that the cosine and secant functions are even and that the sine, cosecant, tangent, and cotangent functions are odd. Skills Review In Exercises 63– 66, find the inverse function one-to-one function 3x 2 64. 63. f x 1 2 f. f 1 of the Model It 65. f x x2 4, x ≥ 2 66. 57. Harmonic Motion The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is y t 1 given by is the displacement (in feet) and where is the time (in seconds). 4et cos 6t t y In Exercises 67–70, sketch the graph of the rational function by hand. Show all asymptotes. 67. 69. f x 2x x 3 f x x2 3x 10 2x2 8 68. 70. f x 5x x2 x 6 f x x3 6x2 x 1 2x2 5x 8 333202_0403.qxd 12/7/05 11:03 AM Page 301 4.3 Right Triangle Trigonometry Section 4.3 Right Triangle Trigonometry 301 What you should learn • Evaluate trigonometric functions of acute angles. • Use the fundamental trigonometric identities. • Use a calculator to evaluate trigonometric functions. • Use trigonometric functions to model and solve real-life problems. Why you should learn it Trigonometric functions are often used to analyze real-life situations. For instance, in Exercise 71 on page 311, you can use trigonometric functions to find the height of a helium-filled balloon. The Six Trigonometric Functions Our second look at the trigonometric functions is from a right triangle perspective. Consider a right triangle, with one acute angle labeled as shown in Figure 4.26. the three sides of the triangle are the hypotenuse, the Relative to the angle opposite side (the side opposite the angle ), and the adjacent side (the side adjacent to the angle ). , , Hypotenuse Side adjacent to θ FIGURE 4.26 Using the lengths of these three sides, you can form six ratios that define the six trigonometric functions of the acute angle . sine cosecant cosine secant tangent cotangent lies in the In the following definitions, it is important to see that first quadrant) and that for such angles the value of each trigonometric function is positive. 0 < < 90 Right Triangle Definitions of Trigonometric Functions Let be an acute angle of a right triangle. The six trigonometric functions of the angle are defined as follows. (Note that the functions in the second row are the reciprocals of the corresponding functions in the first row.) Joseph Sohm; Chromosohm sin opp hyp csc hyp opp cos adj hyp sec hyp adj tan opp adj cot adj opp The abbreviations opp, adj, and hyp represent the lengths of the three sides of a right triangle. opp the length of the side opposite adj the length of the side adjacent to hyp the length of the hypotenuse 333202_0403.qxd 12/7/05 11:03 AM Page 302 302 Chapter 4 Trigonometry Hypotenuse 4 θ 3 FIGURE 4.27 Example 1 Evaluating Trigonometric Functions Use the triangle in Figure 4.27 to find the values of the six trigonometric functions of . hyp2 opp2 adj2, it follows that Solution By the Pythagorean Theorem, hyp 42 32 25 5. So, the six trigonometric functions of are csc hyp opp sin opp hyp 4 5 5 4 cos adj hyp 3 5 tan opp adj 4 3 sec hyp adj 5 3 cot adj opp 3 4 . Now try Exercise 3. Historical Note Georg Joachim Rhaeticus (1514–1576) was the leading Teutonic mathematical astronomer of the 16th century. He was the first to define the trigonometric functi

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.