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than once horizontal reflection a transformation that reflects a functionβs graph across the y-axis by multiplying the input by β1 horizontal shift input a transformation that shifts a functionβs graph left or right by adding a positive or negative constant to the horizontal stretch 0 < b < 1 a transformation that stretches a functionβs graph horizontally by multiplying the input by a constant increasing function a function is increasing in some open interval if f (b) > f (a) for any two input values a and b in the given interval where b > a independent variable an input variable input each object or value in a domain that relates to another object or value by a relationship known as a function interval notation a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion inverse function for any one-to-one function f (x), the inverse is a function f β1(x) such that f β1 β β f (x)β β = x for all x in the domain of f ; this also implies that f β β β f β1 (x) β = x for all x in the domain of f β1 local extrema collectively, all of a function's local maxima and minima local maximum a value of the input where a function changes from increasing to decreasing as the input value increases. local minimum a value of the input where a function changes from decreasing to increasing as the input value increases. odd function a function whose graph is unchanged by combined horizontal and vertical reflection, f (x) = β f ( β x), and is symmetric about the origin one-to-one function a function for which each value of the output is associated with a unique input value 380 output each object or value in the range that is produced when an input value is entered into a function Chapter 3 Functions piecewise function a function in which more than one formula is used to define the output range the set of output values that result from the input values in a relation rate of change the change of an output quantity relative to the change of the input quantity relation a set of ordered pairs set-builder notation a method of describing a set by a rule that all of its members obey; it takes the form {x| statement about x} vertical compression a function transformation that compresses
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the functionβs graph vertically by multiplying the output by a constant 0 < a < 1 vertical line test a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once vertical reflection a transformation that reflects a functionβs graph across the x-axis by multiplying the output by β1 vertical shift a transformation that shifts a functionβs graph up or down by adding a positive or negative constant to the output vertical stretch a > 1 a transformation that stretches a functionβs graph vertically by multiplying the output by a constant KEY EQUATIONS Constant function f (x) = c, where c is a constant Identity function f (x) = x Absolute value function f (x) = |x| Quadratic function Cubic function Reciprocal function Reciprocal squared function f (x) = x2 f (x) = x3 f (x) = 1 x f (x) = 1 x2 Square root function f (x) = x Cube root function f (x) = x3 Average rate of change Ξy Ξx = f (x2) β f (x1) x2 β x1 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 3 Functions 381 Composite function β β f β gβ β (x) = f β βg(x)β β Vertical shift g(x) = f (x) + k (up for k > 0 ) Horizontal shift g(x) = f (x β h) (right for h > 0 ) Vertical reflection g(x) = β f (x) Horizontal reflection g(x) = f ( β x) Vertical stretch g(x) = a f (x) ( a > 0 ) Vertical compression g(x) = a f (x) (0 < a < 1) Horizontal stretch g(x) = f (bx) (0 < b < 1) Horizontal compression. g(x) = f (bx) ( b > 1 ) KEY CONCEPTS 3.1 Functions and Function Notation β’ A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See Example 3.1 and Example 3.2
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. β’ Function notation is a shorthand method for relating the input to the output in the form y = f (x). See Example 3.3 and Example 3.4. β’ In tabular form, a function can be represented by rows or columns that relate to input and output values. See Example 3.5. β’ To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See Example 3.6 and Example 3.7. β’ To solve for a specific function value, we determine the input values that yield the specific output value. See Example 3.8. β’ An algebraic form of a function can be written from an equation. See Example 3.9 and Example 3.10. β’ Input and output values of a function can be identified from a table. See Example 3.11. β’ Relating input values to output values on a graph is another way to evaluate a function. See Example 3.12. β’ A function is one-to-one if each output value corresponds to only one input value. See Example 3.13. β’ A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See Example 3.14. β’ The graph of a one-to-one function passes the horizontal line test. See Example 3.15. 382 Chapter 3 Functions 3.2 Domain and Range β’ The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number. β’ The domain of a function can be determined by listing the input values of a set of ordered pairs. See Example 3.16. β’ The domain of a function can also be determined by identifying the input values of a function written as an equation. See Example 3.17, Example 3.18, and Example 3.19. β’ Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See Example 3.20. β’ For many functions, the domain and range can be determined from a graph. See Example 3.21 and Example 3.22. β’ An understanding of toolkit functions can be used to find the domain and range of related functions. See Example 3.23, Example 3.24, and Example 3.25. β’ A piecewise function is described by more
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than one formula. See Example 3.26 and Example 3.27. β’ A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See Example 3.28. 3.3 Rates of Change and Behavior of Graphs β’ A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. See Example 3.29. β’ Identifying points that mark the interval on a graph can be used to find the average rate of change. See Example 3.30. β’ Comparing pairs of input and output values in a table can also be used to find the average rate of change. See Example 3.31. β’ An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See Example 3.32 and Example 3.33. β’ The average rate of change can sometimes be determined as an expression. See Example 3.34. β’ A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See Example 3.35. β’ A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values. β’ A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values. β’ Minima and maxima are also called extrema. β’ We can find local extrema from a graph. See Example 3.36 and Example 3.37. β’ The highest and lowest points on a graph indicate the maxima and minima. See Example 3.38. 3.4 Composition of Functions β’ We can perform algebraic operations on functions. See Example 3.39. β’ When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function. β’ The function produced by combining two functions is a composite function. See Example 3.40 and Example 3.41. β’ The order of function composition must be considered when interpreting the meaning of composite functions. See Example 3.42. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 3 Functions 383 β’ A composite function can be evaluated by evaluating the inner function using the given input value and
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then evaluating the outer function taking as its input the output of the inner function. β’ A composite function can be evaluated from a table. See Example 3.43. β’ A composite function can be evaluated from a graph. See Example 3.44. β’ A composite function can be evaluated from a formula. See Example 3.45. β’ The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See Example 3.46 and Example 3.47. β’ Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions. β’ Functions can often be decomposed in more than one way. See Example 3.48. 3.5 Transformation of Functions β’ A function can be shifted vertically by adding a constant to the output. See Example 3.49 and Example 3.50. β’ A function can be shifted horizontally by adding a constant to the input. See Example 3.51, Example 3.52, and Example 3.53. β’ Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See Example 3.54. β’ Vertical and horizontal shifts are often combined. See Example 3.55 and Example 3.56. β’ A vertical reflection reflects a graph about the x- axis. A graph can be reflected vertically by multiplying the output by β1. β’ A horizontal reflection reflects a graph about the y- axis. A graph can be reflected horizontally by multiplying the input by β1. β’ A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See Example 3.57. β’ A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See Example 3.58. β’ A function presented as an equation can be reflected by applying transformations one at a time. See Example 3.59. β’ Even functions are symmetric about the y- axis, whereas odd functions are symmetric about the origin. β’ Even functions satisfy the condition f (x) = f ( β x). β’ Odd functions satisfy the condition f (x) = β f ( β x). β’ A function can be odd, even, or neither. See Example 3.60. β’ A function can be compressed or stretched vertically by multiplying the output by a constant. See Example 3.61, Example 3.62
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, and Example 3.63. β’ A function can be compressed or stretched horizontally by multiplying the input by a constant. See Example 3.64, Example 3.65, and Example 3.66. β’ The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See Example 3.67 and Example 3.68. 3.6 Absolute Value Functions β’ Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See Example 3.69. β’ The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See Example 3.70. 384 β’ β’ In an absolute value equation, an unknown variable is the input of an absolute value function. If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See Example 3.71. Chapter 3 Functions 3.7 Inverse Functions β’ If g(x) is the inverse of f (x), then g( f (x)) = f (g(x)) = x. See Example 3.72, Example 3.73, and Example 3.74. β’ Only some of the toolkit functions have an inverse. See Example 3.75. β’ For a function to have an inverse, it must be one-to-one (pass the horizontal line test). β’ A function that is not one-to-one over its entire domain may be one-to-one on part of its domain. β’ For a tabular function, exchange the input and output rows to obtain the inverse. See Example 3.76. β’ The inverse of a function can be determined at specific points on its graph. See Example 3.77. β’ To find the inverse of a formula, solve the equation y = f (x) for x as a function of y. Then exchange the labels x and y. See Example 3.78, Example 3.79, and Example 3.80. β’ The graph of an inverse function is the reflection of the graph of the original function across the line y = x. See Example 3.81. CHAPTER 3 REVIEW EXERCISES Functions and Function Notation 473. f (x) = 2|3x β 1| For the following exercises, determine whether the relation is a function. 468. {(
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a, b), (c, d), (e, d)} the following exercises, determine whether For functions are one-to-one. the 469. β§ β¨(5, 2), (6, 1), (6, 2), (4, 8)β« β¬ β β© 474. f (x) = β 3x + 5 475. f (x) = |x β 3| 470. y2 + 4 = x, for x the independent variable and y the dependent variable 471. Is the graph in Figure 3.120 a function? For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function. 476. Figure 3.120 For the following exercises, evaluate the function at the indicated values: f ( β 3); f (2); f ( β a); β f (a); f (a + h). 472. f (x) = β 2x2 + 3x 477. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 3 Functions 385 482. f (β2) 483. If f (x) = β2, then solve for x. 484. If f (x) = 1, then solve for x. the following For exercises, function h(t) = β 16t 2 + 80t to find the values in simplest form. use the 478. 485. h(2) β h(1) 2 β 1 486. h(a) β h(1) a β 1 Domain and Range For the following exercises, find the domain of each function, expressing answers using interval notation. 487. f (x) = 2 3x + 2 488. f (x) = x β 3 x2 β 4x β 12 489. f (x) = x β 6 x β 4 490. f (x) = Graph this 2x β 3 x β₯ β 2 β© piecewise function: For the following exercises, graph the functions. 479. f (x) = |x + 1| 480. f (x) = x2 β 2 Rates of Change and Behavior of Graphs For the following exercises, find the average rate of change of the functions from x = 1 to x = 2. 491. f (x) = 4x β 3 the following
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exercises, use Figure 3.121 to For approximate the values. 492. f (x) = 10x2 + x 493. f (x) = β 2 x2 For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant. 494. Figure 3.121 481. f (2) 386 Chapter 3 Functions 499. For the graph in Figure 3.122, the domain of the function is [β3, 3]. The range is [β10, 10]. Find the absolute minimum of the function on this interval. 500. Find the absolute maximum of the function graphed in Figure 3.122. 495. 496. Figure 3.122 Composition of Functions For the following exercises, find ( f β g)(x) and (g β f )(x) for each pair of functions. 501. f (x) = 4 β x, g(x) = β 4x 502. f (x) = 3x + 2, g(x) = 5 β 6x 503. f (x) = x2 + 2x, g(x) = 5x + 1 504. f (x) = x + 2, g(x) = 1 x 497. Find the local minimum of the function graphed in Exercise 3.494. 498. Find the local extrema for the function graphed in Exercise 3.495. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 3 Functions 387 505. f (x) = x + 3 2, g(x) = 1 β x For the following exercises, sketch the graph of the function g if the graph of the function f is shown in Figure 3.123. For the following exercises, find β β f β gβ β and the domain for β β f β gβ β (x) for each pair of functions. 506. f (x(x) = 1 x 507. f (x) = 1 x + 3, g(x) = 1 x β 9 508. f (x) = 1 x, g(x) = x 509. f (x) = 1 x2 β 1, g(x) = x + 1 For the following exercises, express each function H as functions
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f and g where a composition H(x) = ( f β g)(x). two of 510. H(x) = 2x β 1 3x + 4 511. H(x) = 1 (3x2 β 4)β3 Transformation of Functions For the following exercises, sketch a graph of the given function. 512. f (x) = (x β 3)2 513. f (x) = (x + 4)3 514. f (x) = x + 5 515. f (x) = β x3 516. f (x) = βx3 517. f (x) = 5 βx β 4 518. f (x) = 4[|x β 2| β 6] 519. f (x) = β (x + 2)2 β 1 Figure 3.123 520. g(x) = f (x β 1) 521. g(x) = 3 f (x) For the following exercises, write the equation for the standard function represented by each of the graphs below. 522. 523. the following exercises, determine whether each For function below is even, odd, or neither. 524. f (x) = 3x4 525. g(x) = x 388 Chapter 3 Functions 526. h(x) = 1 x + 3x For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither. 527. 528. 529. 531. 532. For the following exercises, graph the absolute value function. 533. f (x) = |x β 5| 534. f (x) = β |x β 3| 535. f (x) = |2x β 4| Inverse Functions For the following exercises, find f β1(x) for each function. 536. f (x) = 9 + 10x 537. f (x) = x x + 2 Absolute Value Functions For the following exercises, write an equation for the transformation of f (x) = |x|. 530. For the following exercise, find a domain on which the function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain. This content is available for free at https://cnx.org/content/col11758/
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1.5 Chapter 3 Functions 538. f (x) = x2 + 1 389 539. Given f (x) = x3 β 5 and g(x) = x + 5 a. Find f (g(x)) and g( f (x)). b. What does the answer relationship between f (x) and g(x)? : 3 tell us about the For the following exercises, use a graphing utility to determine whether each function is one-to-one. 540. f (x) = 1 x 541. f (x) = β 3x2 + x 542. If f (5) = 2, find f β1(2). If f (1) = 4, 543. CHAPTER 3 PRACTICE TEST find f β1(4). For the following exercises, determine whether each of the following relations is a function. 544. y = 2x + 8 545. {(2, 1), (3, 2), ( β 1, 1), (0, β 2)} the For following exercises, f (x) = β 3x2 + 2x at the given input. evaluate the function 546. f (β2) 547. f (a) 548. Show that the function f (x) = β 2(x β 1)2 + 3 is not one-to-one. 552. Find the average rate of change of the function f (x) = 3 β 2x2 + x by finding f (b) β f (a) form. in simplest b β a the following For functions exercises, f (x) = 3 β 2x2 + x and g(x) = x to find the composite functions. use the 553. 554. β βg β f β β (x) β βg β f β β (1) 555. Express H(x) = 5x2 β 3x functions, f and g, where β β f β gβ 3 β (x) = H(x). as a composition of two 549. Write the domain of the function f (x) = 3 β x in interval notation. the following exercises, graph the functions by stretching, and/or compressing a toolkit For translating, function. 550. Given f (x) =
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2x2 β 5x, find f (a + 1) β f (1) in simplest form. 551. Graph the function f (x) = x + 1 if β2 < x < 3 β§ β¨ β x if β© x β₯ 3 556. f (x) = x + 6 β 1 557. f (x) = 1 x + 2 β 1 the following exercises, determine whether For functions are even, odd, or neither. the 558. f (x) = β 5 x2 + 9x6 390 Chapter 3 Functions 559. f (x) = β 5 x3 + 9x5 560. f (x) = 1 x 561. the Graph f (x) = β 2|x β 1| + 3. absolute value function For the following exercises, find the inverse of the function. 562. f (x) = 3x β 5 563. f (x) = 4 x + 7 Figure 3.125 568. Find f (2). 569. Find f (β2). For the following exercises, use the graph of g shown in Figure 3.124. 570. Write an equation for the piecewise function. For the following exercises, use the values listed in Table 3.48(x) 1 3 5 7 9 11 13 15 17 Table 3.48 Figure 3.124 564. On what intervals is the function increasing? 565. On what intervals is the function decreasing? 566. Approximate the local minimum of the function. Express the answer as an ordered pair. 567. Approximate the local maximum of the function. Express the answer as an ordered pair. For the following exercises, use the graph of the piecewise function shown in Figure 3.125. 571. Find F(6). 572. Solve the equation F(x) = 5. 573. Is the graph increasing or decreasing on its domain? This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 3 Functions 391 574. Is the function represented by the graph one-to-one? 575. Find F β1(15). 576. Given f (x) = β 2x + 11, find f β1(x). 392 Chapter 3 Functions This content is available for free at https://cnx.org/content/col11758/1.5
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Chapter 4 Linear Functions 393 4 | LINEAR FUNCTIONS Chapter Outline 4.1 Linear Functions 4.2 Modeling with Linear Functions 4.3 Fitting Linear Models to Data Introduction Figure 4.1 A bamboo forest in China (credit: "JFXie"/Flickr) Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour. [1] In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible 3 feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function. Recall from Functions and Function Notation that a function is a relation that assigns to every element in the domain exactly one element in the range. Linear functions are a specific type of function that can be used to model many real-world applications, such as plant growth over time. In this chapter, we will explore linear functions, their graphs, and how to relate them to data. 1. http://www.guinnessworldrecords.com/records-3000/fastest-growing-plant/ 394 Chapter 4 Linear Functions 4.1 | Linear Functions Learning Objectives In this section you will: 4.1.1 Represent a linear function. 4.1.2 Determine whether a linear function is increasing, decreasing, or constant. 4.1.3 Interpret slope as a rate of change. 4.1.4 Write and interpret an equation for a linear function. 4.1.5 Graph linear functions. 4.1.6 Determine whether lines are parallel or perpendicular. 4.1.7 Write the equation of a line parallel or perpendicular to a given line. Figure 4.2 Shanghai MagLev Train (credit: "kanegen"/Flickr) Just as with the growth of a bamboo plant, there are many situations that involve constant change over time. Consider, for example, the first commercial maglev train in the world, the Shanghai MagLev Train (Figure 4.2). It carries passengers comfortably for a 30-kilometer trip from the airport to the subway station in only eight minutes[2]. Suppose a maglev train travels a long distance, and maintains a constant speed of 83 meters per second for
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a period of time once it is 250 meters from the station. How can we analyze the trainβs distance from the station as a function of time? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-world situations such as the trainβs distance from the station at a given point in time. Representing Linear Functions The function describing the trainβs motion is a linear function, which is defined as a function with a constant rate of change. This is a polynomial of degree 1. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. We will describe the trainβs motion as a function using each method. Representing a Linear Function in Word Form Letβs begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship. β’ The trainβs distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed. The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station. 2. http://www.chinahighlights.com/shanghai/transportation/maglev-train.htm This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 395 Representing a Linear Function in Function Notation Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the slope-intercept form of a line, where x is the input value, m is the rate of change, and b is the initial value of the dependent variable. Equation form Function notation y = mx + b f (x) = mx + b In the example of the train, we might use the notation D(t) where the total distance D is a function of the time t. The rate, m, is 83 meters
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per second. The initial value of the dependent variable b is the original distance from the station, 250 meters. We can write a generalized equation to represent the motion of the train. Representing a Linear Function in Tabular Form D(t) = 83t + 250 A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in Figure 4.3. From the table, we can see that the distance changes by 83 meters for every 1 second increase in time. Figure 4.3 Tabular representation of the function D showing selected input and output values Can the input in the previous example be any real number? No. The input represents time so while nonnegative rational and irrational numbers are possible, negative real numbers are not possible for this example. The input consists of non-negative real numbers. Representing a Linear Function in Graphical Form Another way to represent linear functions is visually, using a graph. We can use the function relationship from above, D(t) = 83t + 250, to draw a graph as represented in Figure 4.4. Notice the graph is a line. When we plot a linear function, the graph is always a line. The rate of change, which is constant, determines the slant, or slope of the line. The point at which the input value is zero is the vertical intercept, or y-intercept, of the line. We can see from the graph that the y-intercept in the train example we just saw is (0, 250) and represents the distance of the train from the station when it began moving at a constant speed. Figure 4.4 The graph of D(t) = 83t + 250. Graphs of linear functions are lines because the rate of change is constant. 396 Chapter 4 Linear Functions Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line f (x) = 2x + 1. Ask yourself what numbers can be input to the function. In other words, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product. Linear Function A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line f (x) = mx + b where b is the initial or starting value of the function (when input
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, x = 0 ), and m is the constant rate of change, or slope of the function. The y-intercept is at (0, b). Example 4.1 Using a Linear Function to Find the Pressure on a Diver The pressure, P, water surface, d, this function in words. in pounds per square inch (PSI) on the diver in Figure 4.5 depends upon her depth below the in feet. This relationship may be modeled by the equation, P(d) = 0.434d + 14.696. Restate Figure 4.5 (credit: Ilse Reijs and Jan-Noud Hutten) Solution To restate the function in words, we need to describe each part of the equation. The pressure as a function of depth equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths. Analysis The initial value, 14.696, is the pressure in PSI on the diver at a depth of 0 feet, which is the surface of the water. The rate of change, or slope, is 0.434 PSI per foot. This tells us that the pressure on the diver increases 0.434 PSI for each foot her depth increases. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 397 Determining Whether a Linear Function Is Increasing, Decreasing, or Constant The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an increasing function, as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in Figure 4.6(a). For a decreasing function, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in Figure 4.6(b). If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in Figure 4.6(c). Figure 4.6 Increasing and Decreasing Functions The slope determines if the function is an increasing linear function, a decreasing linear function, or a constant function. f (x) = mx + b is an increasing function if m
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> 0. f (x) = mx + b is a decreasing function if m < 0. f (x) = mx + b is a constant function if m = 0. Example 4.2 Deciding Whether a Function Is Increasing, Decreasing, or Constant Some recent studies suggest that a teenager sends an average of 60 texts per day[3]. For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant. a. The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent. b. A teen has a limit of 500 texts per month in his or her data plan. The input is the number of days, and output is the total number of texts remaining for the month. c. A teen has an unlimited number of texts in his or her data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month. 3. http://www.cbsnews.com/8301-501465_162-57400228-501465/teens-are-sending-60-texts-a-day-study-says/ 398 Chapter 4 Linear Functions Solution Analyze each function. a. The function can be represented as f (x) = 60x where x is the number of days. The slope, 60, is positive so the function is increasing. This makes sense because the total number of texts increases with each day. b. The function can be represented as f (x) = 500 β 60x where x is the number of days. In this case, the slope is negative so the function is decreasing. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after x days. c. The cost function can be represented as f (x) = 50 because the number of days does not affect the total cost. The slope is 0 so the function is constant. Interpreting Slope as a Rate of Change In the examples we have seen so far, the slope was provided to us. However, we often need to calculate the slope given input and output values. Recall that given two values for the input, x1 and x2, and two
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corresponding values for the output, y1 and y2 βwhich can be represented by a set of points, (x1, y1) and (x2, y2) βwe can calculate the slope m. m = change in output (rise) change in input (run) = Ξy Ξx = y2 β y1 x2 β x1 Note that in function notation we can obtain two corresponding values for the output y1 and y2 for the function f, y1 = f β β , so we could equivalently write β and y2 = f β βx2 βx1 β β x2 x2 β x1 βx1 β β Figure 4.7 indicates how the slope of the line between the points, (x1, y1) and (x2, y2), is calculated. Recall that the slope measures steepness, or slant. The greater the absolute value of the slope, the steeper the slant is. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 399 Figure 4.7 The slope of a function is calculated by the change in y divided by the change in x. It does not matter which coordinate is used as the (x2, y2) and which is the (x1, y1), as long as each calculation is started with the elements from the same coordinate pair. Are the units for slope always units for the output units for the input? Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input. Calculate Slope The slope, or rate of change, of a function m can be calculated according to the following: m = change in output (rise) change in input (run) = Ξy Ξx = y2 β y1 x2 β x1 where x1 and x2 are input values, y1 and y2 are output values. Given two points from a linear function, calculate and interpret the slope. 1. Determine the units for output and input values. 2. Calculate the change of output values and change of input values. 3. Interpret the slope as the change in output values
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per unit of the input value. 400 Chapter 4 Linear Functions Example 4.3 Finding the Slope of a Linear Function If f (x) is a linear function, and (3, β2) and (8, 1) are points on the line, find the slope. Is this function increasing or decreasing? Solution The coordinate pairs are (3, β2) and (8, 1). To find the rate of change, we divide the change in output by the change in input. m = change in output change in input = 1 β (β2) 8 β 3 = 3 5 We could also write the slope as m = 0.6. The function is increasing because m > 0. Analysis As noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or y-coordinate, used corresponds with the first input value, or x-coordinate, used. Note that if we had reversed them, we would have obtained the same slope. m = (β2) β (1) 3 β 8 = β3 β5 = 3 5 4.1 If f (x) is a linear function, and (2, 3) and (0, 4) are points on the line, find the slope. Is this function increasing or decreasing? Example 4.4 Finding the Population Change from a Linear Function The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012. Solution The rate of change relates the change in population to the change in time. The population increased by 27, 800 β 23, 400 = 4400 people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years. So the population increased by 1,100 people per year. 4,400 people 4 years = 1,100 people year Analysis Because we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable. The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of 4.2 population per year if we assume the change was constant from 2009 to 2012. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 401 Writing and Interpre
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ion for a linear function given a graph of f shown in Figure 4.9. Figure 4.9 Solution Identify two points on the line, such as (0, 2) and (β2, β4). Use the points to calculate the slope. m = y2 β y1 x2 β x1 = β4 β 2 β2 β 0 = β6 β2 = 3 Substitute the slope and the coordinates of one of the points into the point-slope form. y β y1 = m(x β x1) y β (β4) = 3(x β (β2)) y + 4 = 3(x + 2) We can use algebra to rewrite the equation in the slope-intercept form. y + 4 = 3(x + 2) y + 4 = 3x + 6 y = 3x + 2 Analysis This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 403 This makes sense because we can see from Figure 4.10 that the line crosses the y-axis at the point (0, 2), which is the y-intercept, so b = 2. Figure 4.10 Example 4.6 Writing an Equation for a Linear Cost Function Suppose Ben starts a company in which he incurs a fixed cost of $1,250 per month for the overhead, which includes his office rent. His production costs are $37.50 per item. Write a linear function C where C(x) is the cost for x items produced in a given month. Solution The fixed cost is present every month, $1,250. The costs that can vary include the cost to produce each item, which is $37.50. The variable cost, called the marginal cost, is represented by 37.5. The cost Ben incurs is the sum of these two costs, represented by C(x) = 1250 + 37.5x. Analysis If Ben produces 100 items in a month, his monthly cost is found by substituting 100 for x. C(100) = 1250 + 37.5(100) = 5000 So his monthly cost would be $5,000. Example 4.7 Writing an Equation for a Linear Function Given Two Points If f is a linear function, with f (3) = β2, and f (8) = 1, find an equation for the function in slope-intercept form. 404 Chapter 4 Linear Functions Solution We
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can write the given points using coordinates. We can then use the points to calculate the slope. f (3) = β2 β (3, β2) f (8) = 1 β (8, 1) m = = y2 β y1 x2 β x1 1 β (β2) 8 β 3 = 3 5 Substitute the slope and the coordinates of one of the points into the point-slope form. y β y1 = m(x β x1) (x β 3) y β (β2) = 3 5 We can use algebra to rewrite the equation in the slope-intercept formx β 3) x β 9 5 x β 19 5 4.3 If f (x) is a linear function, with f (2) = β11, and f (4) = β25, write an equation for the function in slope-intercept form. Modeling Real-World Problems with Linear Functions In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems. Given a linear function f and the initial value and rate of change, evaluate f(c). 1. Determine the initial value and the rate of change (slope). 2. Substitute the values into f (x) = mx + b. 3. Evaluate the function at x = c. Example 4.8 Using a Linear Function to Determine the Number of Songs in a Music Collection This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 405 Marcus currently has 200 songs in his music collection. Every month, he adds 15 new songs. Write a formula for the number of songs, N, the number of months. How many songs will he own at the end of one year? in his collection as a function of time, t, Solution The initial value for this function is 200 because he currently owns 200 songs, so N(0) = 200, which means that b = 200. The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know that m = 15. We can substitute the initial
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value and the rate of change into the slope-intercept form of a line. We can write the formula N(t) = 15t + 200. With this formula, we can then predict how many songs Marcus will have at the end of one year (12 months). In other words, we can evaluate the function at t = 12. N(12) = 15(12) + 200 = 180 + 200 = 380 Marcus will have 380 songs in 12 months. Analysis Notice that N is an increasing linear function. As the input (the number of months) increases, the output (number of songs) increases as well. Example 4.9 Using a Linear Function to Calculate Salary Based on Commission Working as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilyaβs weekly income I, depends on the number of new policies, n, he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies and earned $920. Find an equation for I(n), and interpret the meaning of the components of the equation. Solution The given information gives us two input-output pairs: (3, 760) and (5, 920). We start by finding the rate of change. m = 920 β 760 = 5 β 3 $160 2 policies = $80 per policy 406 Chapter 4 Linear Functions Keeping track of units can help us interpret this quantity. Income increased by $160 when the number of policies increased by 2, so the rate of change is $80 per policy. Therefore, Ilya earns a commission of $80 for each policy sold during the week. We can then solve for the initial value. I(n) = 80n + b 760 = 80(3) + b when n = 3, I(3) = 760 760 β 80(3) = b 520 = b The value of b is the starting value for the function and represents Ilyaβs income when n = 0, or when no new policies are sold. We can interpret this as Ilyaβs base salary for the week, which does not depend upon the number of policies sold. We can now write the final equation. I(n) = 80n + 520 Our final interpretation is that Ilyaβs base salary is $520 per week and he earns an additional $80 commission for each policy sold. Example 4.10 Using Tabular Form
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to Write an Equation for a Linear Function Table 4.1 relates the number of rats in a population to time, in weeks. Use the table to write a linear equation. number of weeks, w 0 2 4 6 number of rats, P(w) 1000 1080 1160 1240 Table 4.1 Solution We can see from the table that the initial value for the number of rats is 1000, so b = 1000. Rather than solving for m, we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week. P(w) = 40w + 1000 If we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. For example, using (2, 1080) and (6, 1240) m = 1240 β 1080 6 β 2 = 160 4 = 40 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 407 Is the initial value always provided in a table of values like Table 4.1? No. Sometimes the initial value is provided in a table of values, but sometimes it is not. If you see an input of 0, then the initial value would be the corresponding output. If the initial value is not provided because there is no value of input on the table equal to 0, find the slope, substitute one coordinate pair and the slope into f (x) = mx + b, and solve for b. 4.4 A new plant food was introduced to a young tree to test its effect on the height of the tree. Table 4.2 shows the height of the tree, in feet, x months since the measurements began. Write a linear function, H(x), where x is the number of months since the start of the experiment. x 0 2 4 8 12 H(x) 12.5 13.5 14.5 16.5 18.5 Table 4.2 Graphing Linear Functions Now that weβve seen and interpreted graphs of linear functions, letβs take a look at how to create the graphs. There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the y-intercept and slope. And the third method is by using
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transformations of the identity function f (x) = x. Graphing a Function by Plotting Points To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. The input values and corresponding output values form coordinate pairs. We then plot the coordinate pairs on a grid. In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph. For example, given the function, f (x) = 2x, we might use the input values 1 and 2. Evaluating the function for an input value of 1 yields an output value of 2, which is represented by the point (1, 2). Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point (2, 4). Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error. Given a linear function, graph by plotting points. 1. Choose a minimum of two input values. 2. Evaluate the function at each input value. 3. Use the resulting output values to identify coordinate pairs. 4. Plot the coordinate pairs on a grid. 5. Draw a line through the points. Example 4.11 Graphing by Plotting Points Graph f (x) = β 2 3 x + 5 by plotting points. 408 Chapter 4 Linear Functions Solution Begin by choosing input values. This function includes a fraction with a denominator of 3, so letβs choose multiples of 3 as input values. We will choose 0, 3, and 6. Evaluate the function at each input value, and use the output value to identify coordinate pairs0) = β 2 3 f (3) = β 2 3 f (6) = β 2 3 (0) + 5 = 5 β (0, 5) (3) + 5 = 3 β (3, 3) (6) + 5 = 1 β (6, 1) Plot the coordinate pairs and draw a line through the points. Figure 4.11 represents the graph of the function f (x) = β 2 3 x + 5. Figure 4.11 The graph of the linear function f (x) = β 2 3 x + 5. Analysis The graph of the function is a line as expected for a linear function. In addition, the graph has a downward slant, which indicates a negative slope. This
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is also expected from the negative, constant rate of change in the equation for the function. 4.5 Graph f (x) = β 3 4 x + 6 by plotting points. Graphing a Function Using y-intercept and Slope Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its y-intercept, which is the point at which the input value is zero. To find the y-intercept, we can set x = 0 in the equation. The other characteristic of the linear function is its slope. Letβs consider the following function. f (x) = 1 2 x + 1 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 409 The slope is 1 2. Because the slope is positive, we know the graph will slant upward from left to right. The y-intercept is the point on the graph when x = 0. The graph crosses the y-axis at (0, 1). Now we know the slope and the y-intercept. We can begin graphing by plotting the point (0, 1). We know that the slope is the change in the y-coordinate over the change in the x-coordinate. This is commonly referred to as rise over run, m = rise, which means run. From our example, we have m = 1 2 that the rise is 1 and the run is 2. So starting from our y-intercept (0, 1), we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in Figure 4.12. Figure 4.12 Graphical Interpretation of a Linear Function In the equation f (x) = mx + b β’ b is the y-intercept of the graph and indicates the point (0, b) at which the graph crosses the y-axis. β’ m is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope: m = change in output (rise) change in input (run) = Ξy Ξx = y2 β y1 x2 β x1 Do all linear functions have y-intercepts? Yes. All linear functions cross the y-axis and
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therefore have y-intercepts. (Note: A vertical line is parallel to the y-axis does not have a y-intercept, but it is not a function.) Given the equation for a linear function, graph the function using the y-intercept and slope. 1. Evaluate the function at an input value of zero to find the y-intercept. 2. Identify the slope as the rate of change of the input value. 3. Plot the point represented by the y-intercept. 4. Use rise run to determine at least two more points on the line. 5. Sketch the line that passes through the points. 410 Chapter 4 Linear Functions Example 4.12 Graphing by Using the y-intercept and Slope Graph f (x) = β 2 3 x + 5 using the y-intercept and slope. Solution Evaluate the function at x = 0 to find the y-intercept. The output value when x = 0 is 5, so the graph will cross the y-axis at (0, 5). According to the equation for the function, the slope of the line is β 2 3. This tells us that for each vertical decrease in the βriseβ of β 2 units, the βrunβ increases by 3 units in the horizontal direction. We can now graph the function by first plotting the y-intercept on the graph in Figure 4.13. From the initial value (0, 5) we move down 2 units and to the right 3 units. We can extend the line to the left and right by repeating, and then drawing a line through the points. Figure 4.13 Graph of f (x) = β2/3x + 5 and shows how to calculate the rise over run for the slope. Analysis The graph slants downward from left to right, which means it has a negative slope as expected. 4.6 Find a point on the graph we drew in Example 4.14 that has a negative x-value. Graphing a Function Using Transformations Another option for graphing is to use a transformation of the identity function f (x) = x. A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression. Vertical Stretch or Compression the m is acting as the vertical stretch or compression of the identity function. When m is In the equation f (x) = mx, negative, there
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is also a vertical reflection of the graph. Notice in Figure 4.14 that multiplying the equation of f (x) = x by m stretches the graph of f by a factor of m units if m > 1 and compresses the graph of f by a factor of m units if 0 < m < 1. This means the larger the absolute value of m, the steeper the slope. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 411 Figure 4.14 Vertical stretches and compressions and reflections on the function f (x) = x Vertical Shift the b acts as the vertical shift, moving the graph up and down without affecting the slope of the line. In f (x) = mx + b, Notice in Figure 4.15 that adding a value of b to the equation of f (x) = x shifts the graph of f a total of b units up if b is positive and |b| units down if b is negative. 412 Chapter 4 Linear Functions Figure 4.15 This graph illustrates vertical shifts of the function f (x) = x. Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method. Given the equation of a linear function, use transformations to graph the linear function in the form f(x) = mx + b. 1. Graph f (x) = x. 2. Vertically stretch or compress the graph by a factor m. 3. Shift the graph up or down b units. Example 4.13 Graphing by Using Transformations This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 413 Graph f (x) = 1 2 x β 3 using transformations. Solution The equation for the function shows that m = 1 2 so the identity function is vertically compressed by 1 2. The equation for the function also shows that b = β 3 so the identity function is vertically shifted down 3 units. First, graph the identity function, and show the vertical compression as in Figure 4.16. Figure 4.16 The function, y = x, compressed by a factor of 1 2 Then show the vertical shift as in Figure 4.17. Figure 4.17 The function y = 1 2 x, shifted down 3 units 4.7
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Graph f (x) = 4 + 2x using transformations. 414 Chapter 4 Linear Functions In Example 4.15, could we have sketched the graph by reversing the order of the transformations? No. The order of the transformations follows the order of operations. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. This is why we performed the compression first. For example, following the order: Let the input be 2. (2) β 3 f (22 Writing the Equation for a Function from the Graph of a Line Earlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at Figure 4.18. We can see right away that the graph crosses the y-axis at the point (0, 4) so this is the y-intercept. Figure 4.18 Then we can calculate the slope by finding the rise and run. We can choose any two points, but letβs look at the point ( β 2, 0). To get from this point to the y-intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be m = rise run = 4 2 = 2 Substituting the slope and y-intercept into the slope-intercept form of a line gives y = 2x + 4 Given a graph of linear function, find the equation to describe the function. 1. Identify the y-intercept of an equation. 2. Choose two points to determine the slope. 3. Substitute the y-intercept and slope into the slope-intercept form of a line. Example 4.14 Matching Linear Functions to Their Graphs Match each equation of the linear functions with one of the lines in Figure 4.19. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 415 a. b. c. d. f (x) = 2x + 3 g(x) = 2x β 3 h(x) = β2x + 3 j(x) = 1 2 x + 3 Figure 4.19 Solution Analyze the information for each function. a. This function has a slope of 2 and a y-intercept of 3. It must pass through the point (0, 3) and slant
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upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function g has the same slope, but a different y-intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through (0, 3) so f must be represented by line I. b. This function also has a slope of 2, but a y-intercept of β3. It must pass through the point (0, β3) and slant upward from left to right. It must be represented by line III. c. This function has a slope of β2 and a y-intercept of 3. This is the only function listed with a negative slope, so it must be represented by line IV because it slants downward from left to right. d. This function has a slope of 1 2 and a y-intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. Lines I and II pass through (0, 3), but the slope of j is less than the slope of f so the line for j must be flatter. This function is represented by Line II. Now we can re-label the lines as in Figure 4.20. 416 Chapter 4 Linear Functions Figure 4.20 Finding the x-intercept of a Line So far we have been finding the y-intercepts of a function: the point at which the graph of the function crosses the y-axis. Recall that a function may also have an x-intercept, which is the x-coordinate of the point where the graph of the function crosses the x-axis. In other words, it is the input value when the output value is zero. To find the x-intercept, set a function f (x) equal to zero and solve for the value of x. For example, consider the function shown. Set the function equal to 0 and solve for x. f (x) = 3x β 6 0 = 3x β 6 6 = 3x 2 = x x = 2 The graph of the function crosses the x-axis at the point (2, 0). This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 417 Do all linear functions have x-intercepts? No. However, linear functions of the form y = c,
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where c is a nonzero real number are the only examples of linear functions with no x-intercept. For example, y = 5 is a horizontal line 5 units above the x-axis. This function has no x-intercepts, as shown in Figure 4.21. Figure 4.21 x-intercept The x-intercept of the function is value of x when f (x) = 0. It can be solved by the equation 0 = mx + b. Example 4.15 Finding an x-intercept Find the x-intercept of f (x) = 1 2 x β 3. Solution Set the function equal to zero to solve for x The graph crosses the x-axis at the point (6, 0). Analysis A graph of the function is shown in Figure 4.22. We can see that the x-intercept is (6, 0) as we expected. 418 Chapter 4 Linear Functions Figure 4.22 4.8 Find the x-intercept of f (x) = 1 4 x β 4. Describing Horizontal and Vertical Lines There are two special cases of lines on a graphβhorizontal and vertical lines. A horizontal line indicates a constant output, or y-value. In Figure 4.23, we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use m = 0 in the equation f (x) = mx + b, the equation simplifies to f (x) = b. In other words, the value of the function is a constant. This graph represents the function f (x) = 2. Figure 4.23 A horizontal line representing the function f (x) = 2 A vertical line indicates a constant input, or x-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 419 function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined. Figure 4.24 Example of how a line
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has a vertical slope. 0 in the denominator of the slope. A vertical line, such as the one in Figure 4.25, has an x-intercept, but no y-intercept unless itβs the line x = 0. This graph represents the line x = 2. Figure 4.25 The vertical line, x = 2, which does not represent a function Horizontal and Vertical Lines Lines can be horizontal or vertical. A horizontal line is a line defined by an equation in the form f (x) = b. A vertical line is a line defined by an equation in the form x = a. Example 4.16 Writing the Equation of a Horizontal Line Write the equation of the line graphed in Figure 4.26. 420 Chapter 4 Linear Functions Figure 4.26 Solution For any x-value, the y-value is β 4, so the equation is y = β 4. Example 4.17 Writing the Equation of a Vertical Line Write the equation of the line graphed in Figure 4.27. Figure 4.27 Solution The constant x-value is 7, so the equation is x = 7. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 421 Determining Whether Lines are Parallel or Perpendicular The two lines in Figure 4.28 are parallel lines: they will never intersect. They have exactly the same steepness, which means their slopes are identical. The only difference between the two lines is the y-intercept. If we shifted one line vertically toward the other, they would become coincident. Figure 4.28 Parallel lines We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the y-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel. f (x) = β 2x + 6 β« β¬ parallel f (x) = β 2x β 4 β f (x) = 3x + 2 β« β¬ not parallel f (x) = 2x + 2 β Unlike parallel lines, perpendicular lines do intersect. Their intersection forms a right, or 90-degree, angle. The two lines in Figure 4.29 are perpendicular. Figure 4.29 Perpendicular lines Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one
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another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line. The product of a number and its reciprocal is 1. So, if m1 and m2 are negative reciprocals of one another, they can be multiplied together to yield β1. 422 Chapter 4 Linear Functions m1 m2 = β1 To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is 1 8, and the reciprocal of 1 8 is 8. To find the negative reciprocal, first find the reciprocal and then change the sign. As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor vertical. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular. f (x) = 1 4 x + 2 f (x) = β4x + 3 negative reciprocal of 1 4 is β4 negative reciprocal of β4 is 1 4 The product of the slopes is β1. β Parallel and Perpendicular Lines Two lines are parallel lines if they do not intersect. The slopes of the lines are the same. f (x) = m1 x + b1 and g(x) = m2 x + b2 are parallel if and only if m1 = m2 If and only if b1 = b2 and m1 = m2, we say the lines coincide. Coincident lines are the same line. Two lines are perpendicular lines if they intersect to form a right angle. f (x) = m1 x + b1 and g(x) = m2 x + b2 are perpendicular if and only if m1 m2 = β 1, so m2 = β 1 m1 Example 4.18 Identifying Parallel and Perpendicular Lines Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines. f (x) = 2x + 3 g(x) = 1 2 x β 4 h(x) = β2x + 2 j(x) = 2x β 6 Solution Parallel lines have the same slope. Because the functions f (x) = 2x + 3 and j(x) = 2x β 6 each have a slope of 2, they represent parallel lines. Perpendicular lines have negative reciprocal slopes. Because β2 and 1 2 are negative reciprocals, the functions g(x) = 1 2 x β 4 and
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h(x) = β2x + 2 represent perpendicular lines. Analysis A graph of the lines is shown in Figure 4.30. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 423 Figure 4.30 The graph shows that the lines f (x) = 2x + 3 and j(x) = 2x β 6 are parallel, and the lines g(x) = 1 2 x β 4 and h(x) = β 2x + 2 are perpendicular. Writing the Equation of a Line Parallel or Perpendicular to a Given Line If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line. Writing Equations of Parallel Lines Suppose for example, we are given the equation shown. f (x) = 3x + 1 We know that the slope of the line formed by the function is 3. We also know that the y-intercept is (0, 1). Any other line with a slope of 3 will be parallel to f (x). So the lines formed by all of the following functions will be parallel to f (x). g(x) = 3x + 6 h(x) = 3x + 1 p(x) = 3x + 2 3 Suppose then we want to write the equation of a line that is parallel to f and passes through the point (1, 7). This type of problem is often described as a point-slope problem because we have a point and a slope. In our example, we know that the slope is 3. We need to determine which value of b will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form. y β y1 = m(x β x1) y β 7 = 3(x β 1) y β 7 = 3x β 3 y = 3x + 4 So g(x) = 3x + 4 is parallel to f (x) = 3x + 1 and passes through the point (1, 7). 424 Chapter 4 Linear Functions Given the equation of a function and a point through which its graph passes, write the equation of a line parallel to the given line that passes through the given point. 1. Find the slope of the function. 2. Substitute the given values into
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either the general point-slope equation or the slope-intercept equation for a line. 3. Simplify. Example 4.19 Finding a Line Parallel to a Given Line Find a line parallel to the graph of f (x) = 3x + 6 that passes through the point (3, 0). Solution The slope of the given line is 3. If we choose the slope-intercept form, we can substitute m = 3, x = 3, and f (x) = 0 into the slope-intercept form to find the y-intercept. g(x) = 3x + b 0 = 3(3) + b b = β9 The line parallel to f (x) that passes through (3, 0) is g(x) = 3x β 9. Analysis We can confirm that the two lines are parallel by graphing them. Figure 4.31 shows that the two lines will never intersect. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 425 Figure 4.31 Writing Equations of Perpendicular Lines We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown. f (x) = 2x + 4 The slope of the line is 2, and its negative reciprocal is β 1 2. Any function with a slope of β 1 2 f (x). So the lines formed by all of the following functions will be perpendicular to f (x). will be perpendicular to 426 Chapter 4 Linear Functions g(x) = β 1 2 h(x) = β 1 2 p(x As before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given point. Suppose then we want to write the equation of a line that is perpendicular to f (x) and passes through the point (4, 0). We already know that the slope is β 1 2. Now we can use the point to find the y-intercept by substituting the given values into the slope-intercept form of a line and solving for b. (4) + b g(x) = mx + b 0 = β 1 2 0 = β2 + b 2 = b b = 2 The equation for the function with a slope of β 1 2 and a y-inter
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cept of 2 is g(x) = β 1 2 x + 2 So g(x) = β 1 2 x + 2 is perpendicular to f (x) = 2x + 4 and passes through the point (4, 0). Be aware that perpendicular lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature. A horizontal line has a slope of zero and a vertical line has an undefined slope. These two lines are perpendicular, but the product of their slopes is not β1. Doesnβt this fact contradict the definition of perpendicular lines? No. For two perpendicular linear functions, the product of their slopes is β1. However, a vertical line is not a function so the definition is not contradicted. Given the equation of a function and a point through which its graph passes, write the equation of a line perpendicular to the given line. 1. Find the slope of the function. 2. Determine the negative reciprocal of the slope. 3. Substitute the new slope and the values for x and y from the coordinate pair provided into g(x) = mx + b. 4. Solve for b. 5. Write the equation of the line. Example 4.20 Finding the Equation of a Perpendicular Line Find the equation of a line perpendicular to f (x) = 3x + 3 that passes through the point (3, 0). Solution This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 427 The original line has slope m = 3, so the slope of the perpendicular line will be its negative reciprocal, or β 1 3. Using this slope and the given point, we can find the equation of the line. x + b (3) + b g(x The line perpendicular to f (x) that passes through (3, 0) is g(x) = β 1 3 x + 1. Analysis A graph of the two lines is shown in Figure 4.32. Figure 4.32 Note that that if we graph perpendicular lines on a graphing calculator using standard zoom, the lines may not appear to be perpendicular. Adjusting the window will make it possible to zoom in further to see the intersection more closely. 428 Chapter 4 Linear Functions 4.9 Given the function h(x) = 2x β 4, write an equation for the line passing through (0, 0) that is a. parallel to h(x) b. perpendicular
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to h(x) Given two points on a line and a third point, write the equation of the perpendicular line that passes through the point. 1. Determine the slope of the line passing through the points. 2. Find the negative reciprocal of the slope. 3. Use the slope-intercept form or point-slope form to write the equation by substituting the known values. 4. Simplify. Example 4.21 Finding the Equation of a Line Perpendicular to a Given Line Passing through a Point A line passes through the points (β2, 6) and (4, 5). Find the equation of a perpendicular line that passes through the point (4, 5). Solution From the two points of the given line, we can calculate the slope of that line. Find the negative reciprocal of the slope. m1 = 5 β 6 4 β (β2) = β1 6 = β 1 6 m2 = β1 β 1 6 β ββ 6 = β1 1 β β = 6 We can then solve for the y-intercept of the line passing through the point (4, 5). g(x) = 6x + b 5 = 6(4) + b 5 = 24 + b β19 = b b = β19 The equation for the line that is perpendicular to the line passing through the two given points and also passes through point (4, 5) is y = 6x β 19 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 429 4.10 A line passes through the points, (β2, β15) and (2, β3). Find the equation of a perpendicular line that passes through the point, (6, 4). Access this online resource for additional instruction and practice with linear functions. β’ Linear Functions (http://Openstaxcollege.org/l/linearfunctions) β’ Finding Input of Function from the Output and Graph (http://Openstaxcollege.org/l/ findinginput) β’ Graphing Functions using Tables (http://Openstaxcollege.org/l/graphwithtable) 430 Chapter 4 Linear Functions 4.1 EXERCISES Verbal in feet 1. Terry is skiing down a steep hill. Terry's elevation, after t seconds E(t), by E(t) = 3000 β 70t. Write a complete sentence describing Terryβs starting elevation and how it
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is changing over time. given is Jessica is walking home from a friendβs house. After 2 2. minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home. What is her rate in miles per hour? A boat is 100 miles away from the marina, sailing 3. directly toward it at 10 miles per hour. Write an equation for the distance of the boat from the marina after t hours. If the graphs of two linear functions are perpendicular, 4. describe the relationship between the slopes and the yintercepts. If a horizontal line has the equation f (x) = a and a 5. vertical line has the equation x = a, what is the point of intersection? Explain why what you found is the point of intersection. Algebraic the following exercises, determine whether For equation of the curve can be written as a linear function. the 6. 7. 8. y = 1 4 x + 6 y = 3x β 5 y = 3x2 β 2 9. 3x + 5y = 15 10. 11. 12. 13. 3x2 + 5y = 15 3x + 5y2 = 15 β2x2 + 3y2 = 6 β x β 3 5 = 2y 16. a(x) = 5 β 2x 17. b(x) = 8 β 3x 18. h(x) = β2x + 4 19. k(x) = β4x + 1 20. 21. 22. 23. j(x) = 1 2 x β 3 p(x) = 1 4 x β 5 n(x) = β 1 3 x β 2 m(x) = β 3 8 x + 3 For the following exercises, find the slope of the line that passes through the two given points. 24. (2, 4) and (4, 10) 25. (1, 5) and (4, 11) 26. (β1, 4) and (5, 2) 27. (8, β2) and (4, 6) 28. (6, 11) and (β4, 3) For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. 29. f ( β 5) = β4, and f (5) = 2 30. f (β1) = 4, and f (5) = 1 31. Passes through (2
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, 4) and (4, 10) 32. Passes through (1, 5) and (4, 11) 33. Passes through (β1, 4) and (5, 2) 34. Passes through (β2, 8) and (4, 6) the following exercises, determine whether each For function is increasing or decreasing. 35. x intercept at (β2, 0) and y intercept at (0, β3) 14. f (x) = 4x + 3 15. g(x) = 5x + 6 36. x intercept at (β5, 0) and y intercept at (0, 4) For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 431 Write an equation for a line parallel to f (x) = β 5x β 3 and passing through the point (2, β12). Write an equation for a line parallel to g(x) = 3x β 1 53. and passing through the point (4, 9). Write an equation for a line perpendicular 54. h(t) = β2t + 4 and passing through the point (β4, β1). to Write an equation for a line perpendicular 55. p(t) = 3t + 4 and passing through the point (3, 1). to Graphical the following exercises, For line graphed. find the slope of the 56. 37. 38. 39. 40. 4x β 7y = 10 7x + 4y = 1 3y + x = 12 βy = 8x + 1 3y + 4x = 12 β6y = 8x + 1 6x β 9y = 10 3x + 2y = 1 For the following exercises, find the x- and y-intercepts of each equation. 41. f (x) = β x + 2 42. g(x) = 2x + 4 43. h(x) = 3x β 5 44. k(x) = β5x + 1 45. β2x + 5y = 20 46. 7x + 2y = 56 For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither? 47. Line 1: Passes
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through (0, 6) and (3, β24) Line 2: Passes through (β1, 19) and (8, β71) 48. Line 1: Passes through (β8, β55) and (10, 89) 57. Line 2: Passes through (9, β 44) and (4, β 14) 49. Line 1: Passes through (2, 3) and (4, β1) Line 2: Passes through (6, 3) and (8, 5) 50. Line 1: Passes through (1, 7) and (5, 5) Line 2: Passes through (β1, β3) and (1, 1) 51. Line 1: Passes through (2, 5) and (5, β 1) Line 2: Passes through (β3, 7) and (3, β5) For the following exercises, write an equation for the line described. 52. 432 Chapter 4 Linear Functions For the following exercises, write an equation for the line graphed. 58. 59. 60. 61. This content is available for free at https://cnx.org/content/col11758/1.5 62. 63. Chapter 4 Linear Functions 433 the following exercises, match the given linear For equation with its graph in Figure 4.33. 77. f (x) = β3x + 2 78. 79. f (x) = 1 3 x + 2 f (x) = 2 3 x β 3 80. f (t) = 3 + 2t 81. p(t) = β2 + 3t 82. x = 3 83. x = β2 84. r(x) = 4 For the following exercises, write the equation of the line shown in the graph. 85. Figure 4.33 64. f (x) = β x β 1 65. f (x) = β2x β 1 66. f (x) = β 1 2 x β 1 67. f (x) = 2 68. f (x) = 2 + x 69. f (x) = 3x + 2 For the following exercises, sketch a line with the given features. 70. An x-intercept of (β4, 0) and y-intercept of (0, β2) 71. An x-intercept (β2, 0) and y-intercept of (0, 4) 86. 72. 73.
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A y-intercept of (0, 7) and slope β 3 2 A y-intercept of (0, 3) and slope 2 5 74. Passing through the points (β6, β2) and (6, β6) 75. Passing through the points (β3, β4) and (3, 0) For the following exercises, sketch the graph of each equation. 76. f (x) = β2x β 1 434 Chapter 4 Linear Functions x g(x) x h(x) 0 5 0 5 5 10 15 β10 β25 β40 5 10 15 30 105 230 x 0 5 10 15 f (x) β5 20 45 70 x 5 10 20 25 k(x) 13 28 58 73 x g(x) 0 6 2 4 6 β19 β44 β69 x 2 4 8 10 h(x) 13 23 43 53 x 2 4 6 8 f (x) β4 16 36 56 89. 90. 91. 92. 93. 94. 95. 96. 87. 88. Numeric For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 435 x k(x) 0 6 2 6 8 31 106 231 Technology For the following exercises, use a calculator or graphing technology to complete the task. If f is 97. f (0.1) = 11.5, for the function. a function, and f (0.4) = β5.9, find an equation linear a the Graph domain function f on 98. of [β10, 10] : f (x) = 0.02x β 0.01. Enter the function in a graphing utility. For the viewing window, set the minimum value of x to be β10 and the maximum value of x to be 10. intercept is 31 16 β10 and 10.. Label the points for the input values of linear Graph the function f on a domain of 103. [β0.1, 0.1] for the function whose slope is 75 and yintercept is β22.5. Label the points for the input values of β0.1 and 0.1. Graph the linear function f where f (x) = ax + b on 104. the same set of axes
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on a domain of [β4, 4] for the following values of a and b. i. a = 2; b = 3 ii. a = 2; b = 4 iii. a = 2; b = β4 iv. a = 2; b = β5 Extensions Graph function f on 99. [β10, 10] : f x) = 2, 500x + 4, 000 the a domain of Find the value of x if a linear function goes through the following slope: 105. the following points and has (x, 2), (β4, 6), m = 3 Table 4.3 shows the input, w, and output, k, for a 100. linear function k. a. Fill in the missing values of the table. b. Write the linear function k, round to 3 decimal places. Find the value of y if a linear function goes through the following slope: 106. the following points and has (10, y), (25, 100), m = β5 w k β10 5.5 67.5 b Find the equation of the line that passes through the 107. following points: 30 β26 a β44 (a, b) and (a, b + 1) Table 4.3 Table 4.4 shows the input, p, and output, q, for a 101. linear function q. a. Fill in the missing values of the table. b. Write the linear function k. p q 0.5 0.8 12 b 400 700 a 1,000,000 Table 4.4 102. Graph the linear function f on a domain of [β10, 10] for the function whose slope is 1 8 and y- Find the equation of the line that passes through the 108. following points: (2a, b) and (a, b + 1) Find the equation of the line that passes through the 109. following points: (a, 0) and (c, d) Find the equation of the line parallel to the line 110. g(x) = β0.01x+2.01 through the point (1, 2). Find the equation of the line perpendicular to the line 111. g(x) = β0.01x+2.01 through the point (1, 2). the For f (x) = β0.1x+200 and g(x) = 20x + 0.1. exercises, following use the functions 112. Find the point of intersection
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of the lines f and g. 113. 436 Chapter 4 Linear Functions c. Each year in the decade of the 1990s, average annual income increased by $1,054. d. Average annual income rose to a level of $23,286 by the end of 1999. When temperature temperature the 122. the Celsius 32. When Fahrenheit temperature corresponding Fahrenheit temperature is 212. Express the Fahrenheit temperature as a linear function of C, is 0 degrees Celsius, is the the Celsius temperature, F(C). 100, is a. Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius. b. Find and interpret F(28). c. Find and interpret F(β40). Where is f (x) greater than g(x)? Where is g(x) greater than f (x)? Real-World Applications 114. At noon, a barista notices that she has $20 in her tip jar. If she makes an average of $0.50 from each customer, how much will she have in her tip jar if she serves n more customers during her shift? A gym membership with two personal 115. training sessions costs $125, while gym membership with five personal training sessions costs $260. What is cost per session? it can charge per shirt. In particular, historical data A clothing business finds there is a linear relationship it can sell and the price, 116. between the number of shirts, n, p, shows that 1,000 shirts can be sold at a price of $30, while 3,000 shirts can be sold at a price of $22. Find a linear equation in the form p(n) = mn + b that gives the price p they can charge for n shirts. 117. A phone company charges for service according to the formula: C(n) = 24 + 0.1n, where n is the number of minutes talked, and C(n) is the monthly charge, in dollars. Find and interpret the rate of change and initial value. 118. A farmer finds there is a linear relationship between the number of bean stalks, n, she plants and the yield, y, each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationships in the form y = mn + b that gives the yield when n stalks are planted. A cityβs population in the year 1960 was 287,
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500. In 119. 1989 the population was 275,900. Compute the rate of growth of the population and make a statement about the population rate of change in people per year. A townβs population has been growing linearly. In 120. 2003, the population was 45,000, and the population has been growing by 1,700 people each year. Write an equation, P(t), for the population t years after 2003. 121. Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function: I(x) = 1054x + 23, 286, where x is the number of years after 1990. Which of the following interprets the slope in the context of the problem? a. As of 1990, average annual income was $23,286. b. In the ten-year period from 1990β1999, average annual income increased by a total of $1,054. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 437 4.2 | Modeling with Linear Functions Learning Objectives In this section you will: 4.2.1 Build linear models from verbal descriptions. 4.2.2 Model a set of data with a linear function. Figure 4.34 (credit: EEK Photography/Flickr) Emily is a college student who plans to spend a summer in Seattle. She has saved $3,500 for her trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent her situation? What would be the x-intercept, and what can she learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models. Building Linear Models from Verbal Descriptions When building linear models to solve problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function. Letβs briefly review them: 1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system. 2. Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as
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slope and initial value. 3. Carefully read the problem to determine what we are trying to find, identify, solve, or interpret. 4. Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem. 5. When needed, write a formula for the function. 6. Solve or evaluate the function using the formula. 7. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically. 8. Clearly convey your result using appropriate units, and answer in full sentences when necessary. Now letβs take a look at the student in Seattle. In her situation, there are two changing quantities: time and money. The amount of money she has remaining while on vacation depends on how long she stays. We can use this information to define our variables, including units. Output: M, money remaining, in dollars Input: t, time, in weeks 438 Chapter 4 Linear Functions So, the amount of money remaining depends on the number of weeks: M(t). We can also identify the initial value and the rate of change. Initial Value: She saved $3,500, so $3,500 is the initial value for M. Rate of Change: She anticipates spending $400 each week, so β $400 per week is the rate of change, or slope. Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week. The rate of change is constant, so we can start with the linear model M(t) = mt + b. Then we can substitute the intercept and slope provided. To find the x-intercept, we set the output to zero, and solve for the input. 0 = β400t + 3500 t = 3500 400 = 8.75 The x-intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8.75 weeks. When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be validβalmost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesnβt make sense
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to talk about input values less than zero. A negative input value could refer to a number of weeks before she saved $3,500, but the scenario discussed poses the question once she saved $3,500 because this is when her trip and subsequent spending starts. It is also likely that this model is not valid after the x-intercept, unless Emily uses a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is 0 β€ t β€ 8.75. In this example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model. Using a Given Intercept to Build a Model Some real-world problems provide the y-intercept, which is the constant or initial value. Once the y-intercept is known, the x-intercept can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The y-intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model. Now we can set the function equal to 0, and solve for x to find the x-intercept. f (x) = mx + b = β250x + 1000 0 = β250x + 1000 1000 = 250x 4 = x x = 4 The x-intercept is the number of months it takes her to reach a balance of $0. The x-intercept is 4 months, so it will take Hannah four months to pay off her loan. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 439 Using a Given Input and Output to Build a Model Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given
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input or set the equation of the linear model equal to a specified output. Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem. 1. Identify the input and output values. 2. Convert the data to two coordinate pairs. 3. Find the slope. 4. Write the linear model. 5. Use the model to make a prediction by evaluating the function at a given x-value. 6. Use the model to identify an x-value that results in a given y-value. 7. Answer the question posed. Example 4.22 Using a Linear Model to Investigate a Townβs Population A townβs population has been growing linearly. In 2004, the population was 6,200. By 2009, the population had grown to 8,100. Assume this trend continues. a. Predict the population in 2013. b. Identify the year in which the population will reach 15,000. Solution The two changing quantities are the population size and time. While we could use the actual year value as the input quantity, doing so tends to lead to very cumbersome equations because the y-intercept would correspond to the year 0, more than 2000 years ago! To make computation a little nicer, we will define our input as the number of years since 2004. Input: t, years since 2004 Output: P(t), the townβs population To predict the population in 2013 ( t = 9 ), we would first need an equation for the population. Likewise, to find when the population would reach 15,000, we would need to solve for the input that would provide an output of 15,000. To write an equation, we need the initial value and the rate of change, or slope. To determine the rate of change, we will use the change in output per change in input. m = change in output change in input The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would correspond to t = 0, giving the point (0, 6200). Notice that through our clever choice of variable definition, we have βgivenβ ourselves the y-intercept of the function. The year 2009 would correspond to t = 5, giving the point (5, 8100). The two coordinate pairs are (0, 6200) and (5, 8100). Recall that we encountered examples in which we were provided two points earlier in the chapter. We can use
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these values to calculate the slope. 440 Chapter 4 Linear Functions m = 8100 β 6200 5 β 0 = 1900 5 = 380 people per year We already know the y-intercept of the line, so we can immediately write the equation: To predict the population in 2013, we evaluate our function at t = 9. P(t) = 380t + 6200 P(9) = 380(9) + 6,200 = 9,620 If the trend continues, our model predicts a population of 9,620 in 2013. To find when the population will reach 15,000, we can set P(t) = 15000 and solve for t. 15000 = 380t + 6200 8800 = 380t t β 23.158 Our model predicts the population will reach 15,000 in a little more than 23 years after 2004, or somewhere around the year 2027. A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It 4.11 costs $0.25 to produce each doughnut. a. Write a linear model to represent the cost C of the company as a function of x, the number of doughnuts produced. b. Find and interpret the y-intercept. A cityβs population has been growing linearly. In 2008, the population was 28,200. By 2012, the 4.12 population was 36,800. Assume this trend continues. a. Predict the population in 2014. b. Identify the year in which the population will reach 54,000. Using a Diagram to Build a Model It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful. Example 4.23 Using a Diagram to Model Distance Walked This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 441 Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south
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at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact? Solution In essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart, which leads us to ask a new question: "How long will it take them to be 2 miles apart"? In this problem, our changing quantities are time and position, but ultimately we need to know how long will it take for them to be 2 miles apart. We can see that time will be our input variable, so weβll define our input and output variables. Input: t, time in hours. Output: A(t), distance in miles, and E(t), distance in miles Because it is not obvious how to define our output variable, weβll start by drawing a picture such as Figure 4.35. Figure 4.35 Initial Value: They both start at the same intersection so when t = 0, also be 0. Thus the initial value for each is 0. the distance traveled by each person should Rate of Change: Anna is walking 4 miles per hour and Emanuel is walking 3 miles per hour, which are both rates of change. The slope for A is 4 and the slope for E is 3. Using those values, we can write formulas for the distance each person has walked. A(t) = 4t E(t) = 3t For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the βstarting pointβ at the intersection where they both started. Then we can use the variable, A, which we introduced above, to represent Annaβs position, and define it to be a measurement from the starting point in the eastward direction. Likewise, can use the variable, E, to represent Emanuelβs position, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure. We can then define a third variable, D, Showing the variables on the diagram is often helpful, as we can see from Figure 4.36. to be the measurement of the distance between Anna and Emanuel. 442 Chapter 4 Linear Functions Recall that we need to know how long it takes for D, for any given input t, the outputs A(t), E(
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t), and D(t) represent distances. the distance between them, to equal 2 miles. Notice that Figure 4.36 Figure 4.35 shows us that we can use the Pythagorean Theorem because we have drawn a right angle. Using the Pythagorean Theorem, we get: D(t)2 = A(t)2 + E(t)2 = (4t)2 + (3t)2 = 16t 2 + 9t 2 = 25t 2 D(t) = Β± 25t 2 = Β± 5|t| Solve for D(t) using the square root. In this scenario we are considering only positive values of t, so our distance D(t) will always be positive. We can simplify this answer to D(t) = 5t. This means that the distance between Anna and Emanuel is also a linear function. Because D is a linear function, we can now answer the question of when the distance between them will reach 2 miles. We will set the output D(t) = 2 and solve for t. D(t) = 2 5t = 2 t = 2 5 = 0.4 They will fall out of radio contact in 0.4 hour, or 24 minutes. Should I draw diagrams when given information based on a geometric shape? Yes. Sketch the figure and label the quantities and unknowns on the sketch. Example 4.24 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 443 Using a Diagram to Model Distance Between Cities There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Partway down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough. If the town of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from Westborough? Solution It might help here to draw a picture of the situation. See Figure 4.37. It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates (30, 10), and Eastborough at (20, 0). Figure 4.37 Using this point along with the origin, we can find the slope of the line from Westborough to Agritown. Now we can write an equation to describe the
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road from Westborough to Agritown. m = 10 β 0 30 β 0 = 1 3 W(x) = 1 3 x From this, we can determine the perpendicular road to Eastborough will have slope m = β 3. Because the town of Eastborough is at the point (20, 0), we can find the equation. E(x) = β3x + b 0 = β3(20) + b b = 60 E(x) = β3x + 60 Substitute (20, 0)into the equation. We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal, 1 3 10 3 x = β3x + 60 x = 60 10x = 180 x = 18 y = W(18) = 1 3 = 6 (18) Substituting this back into W(x). The roads intersect at the point (18, 6). Using the distance formula, we can now find the distance from Westborough to the junction. 444 Chapter 4 Linear Functions distance = (x2 β x1)2 + (y2 β y1)2 = (18 β 0)2 + (6 β 0)2 β 18.974 miles Analysis One nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This means real-world applications discussing maps need linear functions to model the distances between reference points. 4.13 There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of Timpson, how far is the road junction from Timpson? Modeling a Set of Data with Linear Functions Real-world situations including two or more linear functions may be modeled with a system of linear equations. Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in Figure 4.38. Figure 4.38 Given a situation that represents a system of linear equations, write the system of equations and identify the solution. 1. 2. Identify the input and output of each linear model. Identify the slope and y-intercept of each linear model. 3. Find the solution by setting the two linear functions
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equal to another and solving for x, or find the point of intersection on a graph. Example 4.25 Building a System of Linear Models to Choose a Truck Rental Company This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 445 Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile[4]. When will Keep on Trucking, Inc. be the better choice for Jamal? Solution The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions in Table 4.5. Input d, distance driven in miles Outputs K(d) : cost, in dollars, for renting from Keep on Trucking M(d) cost, in dollars, for renting from Move It Your Way Initial Value Up-front fee: K(0) = 20 and M(0) = 16 Rate of Change K(d) = $0.59 /mile and P(d) = $0.63 /mile Table 4.5 A linear function is of the form f (x) = mx + b. Using the rates of change and initial charges, we can write the equations K(d) = 0.59d + 20 M(d) = 0.63d + 16 Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when K(d) < M(d). The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the K(d) function is smaller. These graphs are sketched in Figure 4.39, with K(d) in blue. 4. Rates retrieved Aug 2, 2010 from http://www.budgettruck.com and http://www.uhaul.com/ 446 Chapter 4 Linear Functions Figure 4.39 To find the intersection, we set the equations equal and solve: K(d) = M(d) 0.59d + 20 = 0.63d + 16 4 = 0.
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04d 100 = d d = 100 This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that K(d) is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is d > 100. Access this online resource for additional instruction and practice with linear function models. β’ Interpreting a Linear Function (http://Openstaxcollege.org/l/interpretlinear) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 447 4.2 EXERCISES Verbal Explain how to find the input variable in a word 123. problem that uses a linear function. Find the linear function that models the townβs population P as a function of the year, t, where t is the number of years since the model began. Explain how to find the output variable in a word 124. problem that uses a linear function. 136. P. Find a reasonable domain and range for the function Explain how to interpret the initial value in a word 125. problem that uses a linear function. If the function P is graphed, find and interpret the x- 137. and y-intercepts. Explain how to determine the slope in a word problem 126. that uses a linear function. If the function P is graphed, find and interpret the 138. slope of the function. Algebraic 139. When will the population reach 100,000? Find the area of a parallelogram bounded by the ythe line f (x) = 1 + 2x, and the line 127. axis, the line x = 3, parallel to f (x) passing through (2, 7). 128. line f (x) = 12 β 1 3 Find the area of a triangle bounded by the x-axis, the x, and the line perpendicular to f (x) that passes through the origin. What is the population in the year 12 years from the 140. onset of the model? For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained onehalf pound a month for its first year. Find the linear function that models the babyβs weight 141. W as a function of the age of the baby, in months, t. 129. line f (x
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) = 9 β 6 7 Find the area of a triangle bounded by the y-axis, the x, and the line perpendicular to f (x) 142. W. Find a reasonable domain and range for the function that passes through the origin. Find the area of a parallelogram bounded by the xthe line f (x) = 3x, and the line 130. axis, the line g(x) = 2, parallel to f (x) passing through (6, 1). For the following exercises, consider this scenario: A townβs population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped 4,700. Assume this trend continues. 131. Predict the population in 2016. 132. Identify the year in which the population will reach 0. For the following exercises, consider this scenario: A townβs population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues. 133. Predict the population in 2016. If the function W is graphed, find and interpret the x- 143. and y-intercepts. If the function W is graphed, find and interpret the 144. slope of the function. 145. When did the baby weight 10.4 pounds? 146. What is the output when the input is 6.2? For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted. Find the linear function that models the number of 147. people inflicted with the common cold C as a function of the year, t. Find a reasonable domain and range for the function 148. C. Identify the year in which the population will reach 134. 75,000. For the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years. If the function C is graphed, find and interpret the x- 149. and y-intercepts. If the function C is graphed, find and interpret the 150. slope of the function. 135. 151. When will the output reach 0? 448 Chapter 4 Linear Functions 152. In what year will the number of people be 9,700? Numeric Graphical For the
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following exercises, use the graph in Figure 4.40, which shows the profit, y, in thousands of dollars, of a company in a given year, t, where t represents the number of years since 1980. Figure 4.40 153. Find the linear function y, where y depends on t, the number of years since 1980. 154. Find and interpret the y-intercept. 155. Find and interpret the x-intercept. 156. Find and interpret the slope. For the following exercises, use the graph in Figure 4.41, which shows the profit, y, in thousands of dollars, of a company in a given year, t, where t represents the number of years since 1980. Figure 4.41 For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in Table 4.6. Assume that the house values are changing linearly. Year Mississippi Hawaii 1950 $25,200 $74,400 2000 $71,400 $272,700 Table 4.6 In which state have home values increased at a higher 161. rate? If these trends were to continue, what would be the 162. median home value in Mississippi in 2010? If we assume the linear trend existed before 1950 and 163. continues after 2000, the two statesβ median house values will be (or were) equal in what year? (The answer might be absurd.) For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in Table 4.7. Assume that the house values are changing linearly. Year Indiana Alabama 1950 $37,700 $27,100 2000 $94,300 $85,100 Table 4.7 In which state have home values increased at a higher 164. rate? If these trends were to continue, what would be the 165. median home value in Indiana in 2010? If we assume the linear trend existed before 1950 and 166. continues after 2000, the two statesβ median house values will be (or were) equal in what year? (The answer might be absurd.) 157. Find the linear function y, where y depends on t, the number of years since 1980. Real-World Applications 158. Find and interpret the y-intercept. 159. Find and interpret the x-intercept. 160. Find and interpret the slope. In 2004, a school population was 1001. By 2008 the 167. population had grown to 1697. Assume the population is changing lin
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early. a. How much did the population grow between the year 2004 and 2008? This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 449 b. How long did it take the population to grow from a. Find a formula for the moose population, P since 1001 students to 1697 students? 1990. c. What is the average population growth per year? b. What does your model predict the moose d. What was the population in the year 2000? e. Find an equation for the population, P, of the school t years after 2000. f. Using your equation, predict the population of the school in 2011. In 2003, a townβs population was 1431. By 2007 the 168. population had grown to 2134. Assume the population is changing linearly. a. How much did the population grow between the year 2003 and 2007? b. How long did it take the population to grow from 1431 people to 2134 people? c. What is the average population growth per year? d. What was the population in the year 2000? e. Find an equation for the population, P, of the town t years after 2000. f. Using your equation, predict the population of the town in 2014. A phone company has a monthly cellular plan where a 169. customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be $71.50. If the customer uses 720 minutes, the monthly cost will be $118. a. Find a linear equation for the monthly cost of the cell plan as a function of x, the number of monthly minutes used. b. Interpret the slope and y-intercept of the equation. c. Use your equation to find the total monthly cost if 687 minutes are used. A phone company has a monthly cellular data plan 170. where a customer pays a flat monthly fee of $10 and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly cost will be $17.80. a. Find a linear equation for the monthly cost of the the number of MB data plan as a function of x, used. b. Interpret the slope and y-intercept of the equation
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. c. Use your equation to find the total monthly cost if 250 MB are used. In 1991, the moose population in a park was 171. measured to be 4,360. By 1999, the population was measured again to be 5,880. Assume the population continues to change linearly. population to be in 2003? In 2003, the owl population in a park was measured to 172. be 340. By 2007, the population was measured again to be 285. The population changes linearly. Let the input be years since 1990. a. Find a formula for the owl population, P. Let the input be years since 2003. b. What does your model predict the owl population to be in 2012? The Federal Helium Reserve held about 16 billion 173. cubic feet of helium in 2010 and is being depleted by about 2.1 billion cubic feet each year. a. Give a linear equation for the remaining federal the number of in terms of t, helium reserves, R, years since 2010. b. c. In 2015, what will the helium reserves be? If the rate of depletion doesnβt change, in what year will the Federal Helium Reserve be depleted? Suppose the worldβs oil reserves in 2014 are 1,820 reserves are 174. billion barrels. decreasing by 25 billion barrels of oil each year: If, on average, the total a. Give a linear equation for reserves, R, since now. in terms of t, the remaining oil the number of years b. Seven years from now, what will the oil reserves be? c. If the rate at which the reserves are decreasing is constant, when will the worldβs oil reserves be depleted? You are choosing between two different prepaid cell 175. phone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of $19.95 plus 11 cents per minute. How many minutes would you have to use in a month in order for the second plan to be preferable? You are choosing between two different window 176. washing companies. The first charges $5 per window. The second charges a base fee of $40 plus $3 per window. How many windows would you need to have for the second company to be preferable? When hired at a new job selling jewelry, you are given 177. two pay options: Option A: Base salary of $17,000 a year with a commission of 12% of your sales Option B: Base salary of $20,000 a
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year with a commission of 5% of your sales 450 Chapter 4 Linear Functions How much jewelry would you need to sell for option A to produce a larger income? When hired at a new job selling electronics, you are 178. given two pay options: Option A: Base salary of $14,000 a year with a commission of 10% of your sales Option B: Base salary of $19,000 a year with a commission of 4% of your sales How much electronics would you need to sell for option A to produce a larger income? When hired at a new job selling electronics, you are 179. given two pay options: Option A: Base salary of $20,000 a year with a commission of 12% of your sales Option B: Base salary of $26,000 a year with a commission of 3% of your sales How much electronics would you need to sell for option A to produce a larger income? When hired at a new job selling electronics, you are 180. given two pay options: Option A: Base salary of $10,000 a year with a commission of 9% of your sales Option B: Base salary of $20,000 a year with a commission of 4% of your sales How much electronics would you need to sell for option A to produce a larger income? This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 451 4.3 | Fitting Linear Models to Data Learning Objectives In this section you will: 4.3.1 Draw and interpret scatter diagrams. 4.3.2 Use a graphing utility to find the line of best fit. 4.3.3 Distinguish between linear and nonlinear relations. 4.3.4 Fit a regression line to a set of data and use the linear model to make predictions. A professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relates the age of each student to the exam score received. In this section, we will examine one such diagram known as a scatter plot. Drawing and Interpreting Scatter Plots A scatter plot is a graph of plotted points that may show a relationship between two sets of data. If the relationship is from a linear model, or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear functions
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. Figure 4.42 shows a sample scatter plot. Figure 4.42 A scatter plot of age and final exam score variables Notice this scatter plot does not indicate a linear relationship. The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam. Example 4.26 Using a Scatter Plot to Investigate Cricket Chirps Table 4.8 shows the number of cricket chirps in 15 seconds, for several different air temperatures, in degrees Fahrenheit[5]. Plot this data, and determine whether the data appears to be linearly related. 5. Selected data from http://classic.globe.gov/fsl/scientistsblog/2007/10/. Retrieved Aug 3, 2010 452 Chapter 4 Linear Functions Chirps 44 35 20.4 33 31 35 18.5 37 26 Temperature 80.5 70.5 57 66 68 72 52 73.5 53 Table 4.8 Cricket Chirps vs Air Temperature Solution Plotting this data, as depicted in Figure 4.43 suggests that there may be a trend. We can see from the trend in the data that the number of chirps increases as the temperature increases. The trend appears to be roughly linear, though certainly not perfectly so. Figure 4.43 Finding the Line of Best Fit Once we recognize a need for a linear function to model that data, the natural follow-up question is βwhat is that linear function?β One way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can extend the line until we can verify the y-intercept. We can approximate the slope of the line by extending it until we can estimate the rise run. Example 4.27 Finding a Line of Best Fit Find a linear function that fits the data in Table 4.8 by βeyeballingβ a line that seems to fit. Solution On a graph, we could try sketching a line. Using the starting and ending points of our hand drawn line, points (0, 30) and (50, 90), this graph has a slope of This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 453 m = 60 50 = 1.2 and a y-intercept at 30. This gives an equation of T(c) = 1.2c + 30 where c is the
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number of chirps in 15 seconds, and T(c) is the temperature in degrees Fahrenheit. The resulting equation is represented in Figure 4.44. Figure 4.44 Analysis This linear equation can then be used to approximate answers to various questions we might ask about the trend. Recognizing Interpolation or Extrapolation While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as interpolation when we predict a value inside the domain and range of the data. The process of extrapolation is used when we predict a value outside the domain and range of the data. Figure 4.45 compares the two processes for the cricket-chirp data addressed in Example 4.27. We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44. Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44. There is a difference between making predictions inside the domain and range of values for which we have data and outside that domain and range. Predicting a value outside of the domain and range has its limitations. When our model no longer applies after a certain point, it is sometimes called model breakdown. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. But if we try to extrapolate a cost when x = 50, that is in 50 years, the model would not apply because we could not account for factors fifty years in the future. 454 Chapter 4 Linear Functions Figure 4.45 Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside. Interpolation and Extrapolation Different methods of making predictions are used to analyze data. The method of interpolation involves predicting a value inside the domain and/or range of the data. The method of extrapolation involves predicting a value outside the domain and/or range of the data. Model breakdown occurs at the point when the model no longer applies. Example 4.28 Understanding Interpolation and Extrapolation Use the cricket data from Table 4.8 to answer the following questions: a. Would predicting the temperature when crickets are chirping 30 times in 15 seconds be interpolation or
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extrapolation? Make the prediction, and discuss whether it is reasonable. b. Would predicting the number of chirps crickets will make at 40 degrees be interpolation or extrapolation? Make the prediction, and discuss whether it is reasonable. Solution a. The number of chirps in the data provided varied from 18.5 to 44. A prediction at 30 chirps per 15 seconds is inside the domain of our data, so would be interpolation. Using our model: T (30) = 30 + 1.2(30) = 66 degrees Based on the data we have, this value seems reasonable. b. The temperature values varied from 52 to 80.5. Predicting the number of chirps at 40 degrees is extrapolation because 40 is outside the range of our data. Using our model: 40 = 30 + 1.2c 10 = 1.2c c β 8.33 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 455 We can compare the regions of interpolation and extrapolation using Figure 4.46. Figure 4.46 Analysis Our model predicts the crickets would chirp 8.33 times in 15 seconds. While this might be possible, we have no reason to believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether below around 50 degrees. According to the data from Table 4.8, what temperature can we predict it is if we counted 20 chirps in 15 4.14 seconds? Finding the Line of Best Fit Using a Graphing Utility While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data values[6]. One such technique is called least squares regression and can be computed by many graphing calculators, spreadsheet software, statistical software, and many web-based calculators[7]. Least squares regression is one means to determine the line that best fits the data, and here we will refer to this method as linear regression. Given data of input and corresponding outputs from a linear function, find the best fit line using linear regression. 1. Enter the input in List 1 (L1). 2. Enter the output in List 2 (L2). 3. On a graphing utility, select Linear Regression (LinReg). Example 4.29 Finding a Least Squares Regression Line 6. Technically,
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the method minimizes the sum of the squared differences in the vertical direction between the line and the data values. 7. For example, http://www.shodor.org/unchem/math/lls/leastsq.html 456 Chapter 4 Linear Functions Find the least squares regression line using the cricket-chirp data in Table 4.9. Solution 1. Enter the input (chirps) in List 1 (L1). 2. Enter the output (temperature) in List 2 (L2). See Table 4.9. L1 44 35 20.4 33 31 35 18.5 37 26 L2 80.5 70.5 57 66 68 72 52 73.5 53 Table 4.9 3. On a graphing utility, select Linear Regression (LinReg). Using the cricket chirp data from earlier, with technology we obtain the equation: T(c) = 30.281 + 1.143c Analysis Notice that this line is quite similar to the equation we βeyeballedβ but should fit the data better. Notice also that using this equation would change our prediction for the temperature when hearing 30 chirps in 15 seconds from 66 degrees to: T(30) = 30.281 + 1.143(30) = 64.571 β 64.6 degrees The graph of the scatter plot with the least squares regression line is shown in Figure 4.47. Figure 4.47 Will there ever be a case where two different lines will serve as the best fit for the data? No. There is only one best fit line. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 457 Distinguishing Between Linear and Nonlinear Models As we saw above with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the correlation coefficient, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a βdiagnostic onβ selection to find the correlation coefficient, which mathematicians label as r The correlation coefficient provides an easy way to get an idea of how close to a line the data falls. We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear.
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If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense for the relationship between the value of r and the graph of the data, Figure 4.48 shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output. Figure 4.48 Plotted data and related correlation coefficients. (credit: βDenisBoigelot,β Wikimedia Commons) Correlation Coefficient The correlation coefficient is a value, r, between β1 and 1. β’ β’ r > 0 suggests a positive (increasing) relationship r < 0 suggests a negative (decreasing) relationship β’ The closer the value is to 0, the more scattered the data. β’ The closer the value is to 1 or β1, the less scattered the data is. Example 4.30 Finding a Correlation Coefficient Calculate the correlation coefficient for cricket-chirp data in Table 4.8. Solution 458 Chapter 4 Linear Functions Because the data appear to follow a linear pattern, we can use technology to calculate r Enter the inputs and corresponding outputs and select the Linear Regression. The calculator will also provide you with the correlation coefficient, r = 0.9509. This value is very close to 1, which suggests a strong increasing linear relationship. Note: For some calculators, the Diagnostics must be turned "on" in order to get the correlation coefficient when linear regression is performed: [2nd]>[0]>[alpha][xβ1], then scroll to DIAGNOSTICSON. Fitting a Regression Line to a Set of Data Once we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions. As we learned above, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data. Example 4.31 Using a Regression Line to Make Predictions Gasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in Table 4.10.[8] Determine whether the trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008. Year '94 '95 '96 '97 '98 '99 '00 '01 '02 '03 '04 Consumption (billions of gallons) Table 4.10 113 116 118 119 123 125
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126 128 131 133 136 The scatter plot of the data, including the least squares regression line, is shown in Figure 4.49. Figure 4.49 8. http://www.bts.gov/publications/national_transportation_statistics/2005/html/table_04_10.html This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 459 Solution We can introduce new input variable, t, representing years since 1994. The least squares regression equation is: C(t) = 113.318 + 2.209t Using technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear trend. Using this to predict consumption in 2008 (t = 14), The model predicts 144.244 billion gallons of gasoline consumption in 2008. C(14) = 113.318 + 2.209(14) = 144.244 Use the model we created using technology in Example 4.31 to predict the gas consumption in 2011. Is 4.15 this an interpolation or an extrapolation? Access these online resources for additional instruction and practice with fitting linear models to data. β’ Introduction to Regression Analysis (http://Openstaxcollege.org/l/introregress) β’ Linear Regression (http://Openstaxcollege.org/l/linearregress) 460 Chapter 4 Linear Functions 4.3 EXERCISES Verbal 190. Describe what it means if there is a model breakdown 181. when using a linear model. 100 250 300 450 600 750 182. What is interpolation when using a linear model? 12 12.6 13.1 14 14.5 15.2 183. What is extrapolation when using a linear model? Explain the difference between a positive and a 184. negative correlation coefficient. 191. Explain how to interpret 185. correlation coefficient. the absolute value of a 1 1 3 9 5 7 9 11 28 65 125 216 Algebraic 186. A regression was run to determine whether there is a relationship between hours of TV watched per day (x) and number of sit-ups a person can do (y). The results of the regression are given below. Use this to predict the number of sit-ups a person who watches 11 hours of TV can do. y = ax + b a = β1.341 b = 32.234 r = β0.896 A regression was run to determine whether there is a in
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inches) 187. relationship between the diameter of a tree ( x, and the treeβs age ( y, regression are given below. Use this to predict the age of a tree with diameter 10 inches. in years). The results of the y = ax + b a = 6.301 b = β1.044 r = β0.970 For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related? 0 2 4 6 8 10 β22 β19 β15 β11 β6 β2 188. 189. For the following data, draw a scatter plot. If we 192. wanted to know when the population would reach 15,000, would the answer involve interpolation or extrapolation? Eyeball the line, and estimate the answer. Year Population 1990 11,500 1995 12,100 2000 12,700 2005 13,000 2010 13,750 For the following data, draw a scatter plot. If we 193. wanted to know when the temperature would reach 28Β°F, would the answer involve interpolation or extrapolation? Eyeball the line and estimate the answer. Temperature,Β°F 16 18 20 25 30 Time, seconds 46 50 54 55 62 1 2 3 4 5 6 Graphical 46 50 59 75 100 136 For the following exercises, match each scatterplot with one of the four specified correlations in Figure 4.50 and Figure 4.51. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 461 Figure 4.50 Figure 4.51 194. r = 0.95 195. r = β0.89 196. r = β0.26 197. r = β0.39 For the following exercises, draw a best-fit line for the plotted data. 198. 462 Chapter 4 Linear Functions Numeric 202. The U.S. Census tracks the percentage of persons 25 years or older who are college graduates. That data for several years is given in Table 4.11.[9] Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the percentage exceed 35%? Year Percent Graduates 199. 200. 201. 1990 21.3 1992 21.4 1994 22.2 1996 23.6 1998 24.4 2000 25.6 2002 26.7 2004 27.7 2006 28 2008 29.4 Table 4.11 The U.S. import of wine (
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in hectoliters) for several 203. years is given in Table 4.12. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will imports exceed 12,000 hectoliters? 9. Based on data from http://www.census.gov/hhes/socdemo/education/data/cps/historical/index.html. Accessed 5/1/ 2014. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 463 Year Imports Year Number Unemployed 1992 2665 1994 2688 1996 3565 1998 4129 2000 4584 2002 5655 2004 6549 2006 7950 2008 8487 2009 9462 1990 750 1992 670 1994 650 1996 605 1998 550 2000 510 2002 460 2004 420 2006 380 2008 320 Table 4.12 Table 4.13 204. Table 4.13 shows the year and the number of people unemployed in a particular city for several years. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the number of unemployed reach 5? Technology For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. 8 15 26 31 56 23 41 53 72 103 5 4 7 10 12 15 12 17 22 24 x y x y 205. 206. 207. 464 Chapter 4 Linear Functions x y x y 21.9 10 18.54 22.22 11 15.76 x y 21 25 30 31 40 50 17 11 2 β1 β18 β40 3 4 5 6 7 8 9 22.74 12 13.68 210. 22.26 13 14.1 20.78 14 14.02 17.6 15 11.94 16.52 16 12.76 x y 100 2000 80 60 55 40 20 1798 1589 1580 1390 1202 208. x y 4 5 6 7 8 9 44.8 43.1 38.8 39 38 32.7 10 30.1 11 29.3 12 27 13 25.8 209. This content is available for free at https://cnx.org/content/col11758/1.5 211. x y 900 988 1000 1010 1200 1205 70 80 82 84 105 108 Extensions Graph f (x) = 0.5x + 10. Pick a set of five ordered linear 212
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. pairs using inputs x = β2, 1, 5, 6, 9 and use regression to verify that the function is a good fit for the data. Graph f (x) = β 2x β 10. Pick a set of five ordered linear 213. pairs using inputs x = β2, 1, 5, 6, 9 and use regression to verify the function. For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years: (46, 1, 600), (48, 1, 550), (50, 1, 505), (52, 1, 540), (54, 1, 495). Chapter 4 Linear Functions 465 Use linear regression to determine a function P 214. where the profit in thousands of dollars depends on the number of units sold in hundreds. 215. Find to the nearest tenth and interpret the x-intercept. 216. Find to the nearest tenth and interpret the y-intercept. Real-World Applications For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years: (2500, 2000), (2650, 2001), (3000, 2003), (3500, 2006), (4200, 2010) 217. Use linear regression to determine a function y, where the year depends on the population. Round to three decimal places of accuracy. 218. Predict when the population will hit 8,000. For the following exercises, consider this scenario: The profit of a company increased steadily over a ten-year span. The following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten year span, (number of units sold, profit) for specific recorded years: (46, 250), (48, 305), (50, 350), (52, 390), (54, 410). 219. Use linear regression to determine a function y, where the profit in thousands of dollars depends on the number of units sold in hundreds. Predict when the profit will exceed one million 220. dollars. For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs show
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dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span (number of units sold, profit) for specific recorded years: (46, 250), (48, 225), (50, 205), (52, 180), (54, 165). 221. Use linear regression to determine a function y, where the profit in thousands of dollars depends on the number of units sold in hundreds. Predict when the profit will dip below the $25,000 222. threshold. 466 Chapter 4 Linear Functions CHAPTER 4 REVIEW KEY TERMS correlation coefficient a value, r, between β1 and 1 that indicates the degree of linear correlation of variables, or how closely a regression line fits a data set. decreasing linear function a function with a negative slope: If f (x) = mx + b, then m < 0. extrapolation predicting a value outside the domain and range of the data horizontal line a line defined by f (x) = b, where b is a real number. The slope of a horizontal line is 0. increasing linear function a function with a positive slope: If f (x) = mx + b, then m > 0. interpolation predicting a value inside the domain and range of the data least squares regression the line and data values a statistical technique for fitting a line to data in a way that minimizes the differences between linear function line a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight model breakdown when a model no longer applies after a certain point parallel lines two or more lines with the same slope perpendicular lines two lines that intersect at right angles and have slopes that are negative reciprocals of each other point-slope form the equation for a line that represents a linear function of the form y β y1 = m(x β x1) slope the ratio of the change in output values to the change in input values; a measure of the steepness of a line slope-intercept form the equation for a line that represents a linear function in the form f (x) = mx + b vertical line a line defined by x = a, where a is a real number. The slope of a vertical line is undefined. KEY CONCEPTS 4.1 Linear Functions β’ Linear functions can be represented in words, function notation, tabular form, and graphical form. See Example 4.1. β’ An increasing linear function results in a graph that slants upward from left
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to right and has a positive slope. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line. See Example 4.2. β’ Slope is a rate of change. The slope of a linear function can be calculated by dividing the difference between yvalues by the difference in corresponding x-values of any two points on the line. See Example 4.3 and Example 4.4. β’ An equation for a linear function can be written from a graph. See Example 4.5. β’ The equation for a linear function can be written if the slope m and initial value b are known. See Example 4.6 and Example 4.7. β’ A linear function can be used to solve real-world problems given information in different forms. See Example 4.8, Example 4.9, and Example 4.10. β’ Linear functions can be graphed by plotting points or by using the y-intercept and slope. See Example 4.11 and Example 4.12. β’ Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See Example 4.13. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 467 β’ The equation for a linear function can be written by interpreting the graph. See Example 4.14. β’ The x-intercept is the point at which the graph of a linear function crosses the x-axis. See Example 4.15. β’ Horizontal lines are written in the form, f (x) = b. See Example 4.16. β’ Vertical lines are written in the form, x = b. See Example 4.17. β’ Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See Example 4.18. β’ A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x- and y-values of the given point into the equation, f (x) = mx + b, and using the b that results. Similarly, the point-slope form of an equation can also be used. See Example 4.19. β’ A line perpendicular to another line, passing through a given point, may be found in the same
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manner, with the exception of using the negative reciprocal slope. See Example 4.20 and Example 4.21. 4.2 Modeling with Linear Functions β’ We can use the same problem strategies that we would use for any type of function. β’ When modeling and solving a problem, identify the variables and look for key values, including the slope and y- intercept. See Example 4.22. β’ Draw a diagram, where appropriate. See Example 4.23 and Example 4.24. β’ Check for reasonableness of the answer. β’ Linear models may be built by identifying or calculating the slope and using the y-intercept. β¦ The x-intercept may be found by setting y = 0, which is setting the expression mx + b equal to 0. β¦ The point of intersection of a system of linear equations is the point where the x- and y-values are the same. See Example 4.25. β¦ A graph of the system may be used to identify the points where one line falls below (or above) the other line. 4.3 Fitting Linear Models to Data β’ Scatter plots show the relationship between two sets of data. See Example 4.26. β’ Scatter plots may represent linear or non-linear models. β’ The line of best fit may be estimated or calculated, using a calculator or statistical software. See Example 4.27. β’ Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. See Example 4.28. β’ The correlation coefficient, r, indicates the degree of linear relationship between data. See Example 4.29. β’ A regression line best fits the data. See Example 4.30. β’ The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. See Example 4.31. CHAPTER 4 REVIEW EXERCISES Linear Functions 223. Determine whether the algebraic equation is linear. 2x + 3y = 7 225. Determine whether the function is increasing or decreasing. f (x) = 7x β 2 224. Determine whether the algebraic equation is linear. 6x2 β y = 5 226. Determine whether the function is increasing or decreasing. g(x) = β x + 2 468 Chapter 4 Linear Functions 227. Given each set of
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information, find a linear equation that satisfies the given conditions, if possible. Passes through (7, 5) and (3, 17) 228. Given each set of information, find a linear equation that satisfies the given conditions, if possible. x-intercept at (6, 0) and y-intercept at (0, 10) 229. Find the slope of the line shown in the graph. 230. Find the slope of the line graphed. 232. Does the following table represent a linear function? If so, find the linear equation that models the data. x β4 0 2 10 g(x) 18 β2 β12 β52 233. Does the following table represent a linear function? If so, find the linear equation that models the data. x 6 8 12 26 g(x) β8 β12 β18 β46 234. On June 1st, a company has $4,000,000 profit. If the company then loses 150,000 dollars per day thereafter in the month of June, what is the companyβs profit nth day after June 1st? For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular: 235. 2x β 6y = 12 βx + 3y = 1 231. Write an equation in slope-intercept form for the line shown. 236. x β 2 y = 1 3 3x + y = β 9 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 469 Find the linear function that models the number of people afflicted with the common cold C as a function of the year, t. When will no one be afflicted? For the following exercises, use the graph in Figure 4.52 showing the profit, y, in thousands of dollars, of a company in a given year, x, where x represents years since 1980. For the following exercises, find the x- and y- intercepts of the given equation 237. 7x + 9y = β 63 238. f (x) = 2x β 1 For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither? 239. Line 1: Passes through (5, 11) and (10, 1) Line 2: Passes through (β1, 3) and (β5,
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11) 240. Line 1: Passes through (8, β10) and (0, β26) Line 2: Passes through (2, 5) and (4, 4) 241. Write an equation for a line perpendicular f (x) = 5x β 1 and passing through the point (5, 20). to Figure 4.52 242. Find the equation of a line with a y- intercept of (0, 2) and slope β 1 2. 243. Sketch a graph of the linear function f (t) = 2t β 5. 244. Find the point of intersection for the 2 linear functions: x = y + 6 2x β y = 13. 245. A car rental company offers two plans for renting a car. Plan A: 25 dollars per day and 10 cents per mile Plan B: 50 dollars per day with free unlimited mileage How many miles would you need to drive for plan B to save you money? Modeling with Linear Functions 246. Find the area of a triangle bounded by the y axis, the line f (x) = 10 β 2x, and the line perpendicular to f that passes through the origin. 249. Find the linear function y, where y depends on x, number of years since 1980. the 250. Find and interpret the y-intercept. For the following exercise, consider this scenario: In 2004, a school population was 1,700. By 2012 the population had grown to 2,500. 251. Assume the population is changing linearly. a. How much did the population grow between the year 2004 and 2012? b. What year? c. Find an equation for the population, P, of the school t years after 2004. is the average population growth per For the following exercises, consider this scenario: In 2000, the moose population in a park was measured to be 6,500. the population was measured to be 12,500. By 2010, Assume the population continues to change linearly. 252. Find a formula for the moose population, P. 253. What does your model predict the moose population to be in 2020? 247. A townβs population increases at a constant rate. In 2010 the population was 55,000. By 2012 the population had increased to 76,000. If this trend continues, predict the population in 2016. For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in
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Table 4.14. Assume that the house values are changing linearly. 248. The number of people afflicted with the common cold in the winter months dropped steadily by 50 each year since 2004 until 2010. In 2004, 875 people were inflicted. 470 Chapter 4 Linear Functions Year Pima Central East Valley Predicted Actual 1970 2010 Table 4.14 32,000 120,250 85,000 150,000 In which subdivision have home values increased at 254. a higher rate? If these trends were to continue, what would be the 255. median home value in Pima Central in 2015? Fitting Linear Models to Data 256. Draw a scatter plot for the data in Table 4.15. Then determine whether the data appears to be linearly related. 0 2 β105 β50 4 1 6 8 10 55 105 160 6 7 7 8 7 9 10 10 6 7 8 8 9 10 10 9 Table 4.15 Table 4.17 257. Draw a scatter plot for the data in Table 4.16. If we wanted to know when the population would reach 15,000, would the answer involve interpolation or extrapolation? 259. Draw a best-fit line for the plotted data. Year Population 1990 5,600 1995 5,950 2000 6,300 2005 6,600 2010 6,900 Table 4.16 For the following exercises, consider the data in Table 4.18, which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year. Year 2000 2002 2005 2007 2010 258. Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in Table 4.17. Plot the points, then sketch a line that fits the data. Percent Graduates Table 4.18 6.5 7.0 7.4 8.2 9.0 260. Determine whether the trend appears to be linear. If so, and assuming the trend continues, find a linear This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 471 regression model to predict the percent of unemployed in a given year to three decimal places. 261. In what year will the percentage exceed 12%? 262. Based on the set of data given in Table 4.19, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places. x y 17 20 23 26 29 15 25 31 37 40
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Table 4.19 263. Based on the set of data given in Table 4.20, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places. x y 10 12 15 18 20 36 34 30 28 22 Table 4.20 For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years: (3,600, 2000); (4,000, 2001); (4,700, 2003); (6,000, 2006) 264. Use linear regression to determine a function y, where the year depends on the population, to three decimal places of accuracy. 265. Predict when the population will hit 12,000. 266. What is the correlation coefficient for this model to three decimal places of accuracy? 267. According to the model, what is the population in 2014? CHAPTER 4 PRACTICE TEST 268. Determine whether the following algebraic equation can be written as a linear function. 2x + 3y = 7 269. Determine whether increasing or decreasing. f (x) = β 2x + 5 the following function is 472 Chapter 4 Linear Functions 270. Determine whether increasing or decreasing. f (x) = 7x + 9 the following function is x β6 0 2 4 271. Find a linear equation that passes through (5, 1) and (3, β9), if possible. g(x) 14 32 38 44 272. Find a linear equation, that has an x intercept at (β4, 0) and a y-intercept at (0, β6), if possible. Table 4.21 273. Find the slope of the line in Figure 4.53. 276. Does Table 4.22 represent a linear function? If so, find a linear equation that models the data. x g(x) 1 4 3 9 7 11 19 12 Table 4.22 277. At 6 am, an online company has sold 120 items that day. If the company sells an average of 30 items per hour for the remainder of the day, write an expression to represent the number of items that were sold n after 6 am. For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular. 278. x β 9 y = 3 4 β4x β 3y = 8
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279. β2x + y = 3 3x + 3 2 y = 5 Find the x- and y-intercepts of 280. 2x + 7y = β 14. the equation 281. Given below are descriptions of two lines. Find the slopes of Line 1 and Line 2. Is the pair of lines parallel, perpendicular, or neither? Line 1: Passes through (β2, β6) and (3, 14) Line 2: Passes through (2, 6) and (4, 14) 282. Write an equation for a line perpendicular f (x) = 4x + 3 and passing through the point (8, 10). to 283. Sketch a line with a y-intercept of (0, 5) and slope β 5 2. Figure 4.53 274. Write an equation for line in Figure 4.54. Figure 4.54 275. Does Table 4.21 represent a linear function? If so, find a linear equation that models the data. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 4 Linear Functions 473 284. Graph of the linear function f (x) = β x + 6. 285. For the two linear functions, find the point of intersection: x = y + 2. 2x β 3y = β 1 286. A car rental company offers two plans for renting a car. Plan A: $25 per day and $0.10 per mile Plan B: $40 per day with free unlimited mileage How many miles would you need to drive for plan B to save you money? 287. Find the area of a triangle bounded by the y axis, the line f (x) = 12 β 4x, and the line perpendicular to f that passes through the origin. 288. A townβs population increases at a constant rate. In 2010 the population was 65,000. By 2012 the population had increased to 90,000. Assuming this trend continues, predict the population in 2018. 289. The number of people afflicted with the common cold in the winter months dropped steadily by 25 each year since 2002 until 2012. In 2002, 8,040 people were inflicted. Find the linear function that models the number of people afflicted with the common cold C as a function of the year, t. When will less than 6,000 people be afflicted? For the following exercises, use the graph in Figure 4.55, showing the profit, y, in
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thousands of dollars, of a company in a given year, x, where x represents years since 1980. Figure 4.55 290. Find the linear function y, where y depends on x, the number of years since 1980. 291. Find and interpret the y-intercept. 292. In 2004, a school population was 1250. By 2012 the population had dropped to 875. Assume the population is changing linearly. a. How much did the population drop between the year 2004 and 2012? b. What year? c. Find an equation for the population, P, of the school t years after 2004. is the average population decline per 293. Draw a scatter plot for the data provided in Table 4.23. Then determine whether the data appears to be linearly related. 0 2 4 6 8 10 β450 β200 10 265 500 755 Table 4.23 294. Draw a best-fit line for the plotted data. For the following exercises, use Table 4.24, which shows the percent of unemployed persons 25 years or older who are college graduates in a particular city, by year. Year 2000 2002 2005 2007 2010 8.5 8.0 7.2 6.7 6.4 Percent Graduates Table 4.24 295. Determine whether the trend appears linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places. 296. In what year will the percentage drop below 4%? 474 Chapter 4 Linear Functions 297. Based on the set of data given in Table 4.25, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient. Round to three decimal places of accuracy. x y 16 18 20 24 26 106 110 115 120 125 Table 4.25 For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years: (4, 500, 2000); (4, 700, 2001); (5, 200, 2003); (5, 800, 2006) 298. Use linear regression to determine a function y, where the year depends on the population. Round to three decimal places of accuracy. 299. Predict when the population will hit 20,000. 300. What is the correlation coefficient for this model? This content is available for free at https://
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cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 475 5 | POLYNOMIAL AND RATIONAL FUNCTIONS Figure 5.1 35-mm film, once the standard for capturing photographic images, has been made largely obsolete by digital photography. (credit βfilmβ: modification of work by Horia Varlan; credit βmemory cardsβ: modification of work by Paul Hudson) Chapter Outline 5.1 Quadratic Functions 5.2 Power Functions and Polynomial Functions 5.3 Graphs of Polynomial Functions 5.4 Dividing Polynomials 5.5 Zeros of Polynomial Functions 5.6 Rational Functions 5.7 Inverses and Radical Functions 5.8 Modeling Using Variation Introduction Digital photography has dramatically changed the nature of photography. No longer is an image etched in the emulsion on a roll of film. Instead, nearly every aspect of recording and manipulating images is now governed by mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When we open an image file, software on a camera or computer interprets the numbers and converts them to a visual image. Photo editing software uses complex polynomials to transform images, allowing us to manipulate the image in order to crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications. 476 Chapter 5 Polynomial and Rational Functions 5.1 | Quadratic Functions Learning Objectives In this section, you will: 5.1.1 Recognize characteristics of parabolas. 5.1.2 Understand how the graph of a parabola is related to its quadratic function. 5.1.3 Determine a quadratic functionβs minimum or maximum value. 5.1.4 Solve problems involving a quadratic functionβs minimum or maximum value. Figure 5.2 An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr) Curved antennas, such as the ones shown in Figure 5.2, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described
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by a quadratic function. In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior. Recognizing Characteristics of Parabolas The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry. These features are illustrated in Figure 5.3. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 477 Figure 5.3 The y-intercept is the point at which the parabola crosses the y-axis. The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of x at which y = 0. 478 Chapter 5 Polynomial and Rational Functions Example 5.1 Identifying the Characteristics of a Parabola Determine the vertex, axis of symmetry, zeros, and y- intercept of the parabola shown in Figure 5.4. Figure 5.4 Solution The vertex is the turning point of the graph. We can see that the vertex is at (3, 1). Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is x = 3. This parabola does not cross the x- axis, so it has no zeros. It crosses the y- axis at (0, 7) so this is the y-intercept. Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions The general form of a quadratic function presents the function in the form where a, b, and
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c are real numbers and a β 0. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry. f (x) = ax2 + bx + c (5.1) The axis of symmetry is defined by x = β 2a. If we use the quadratic formula, x = βb Β± b2 β 4ac ax2 + bx + c = 0 for the x- intercepts, or zeros, we find the value of x halfway between them is always x = β 2a b, to solve b 2a, the equation for the axis of symmetry. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 479 Figure 5.5 represents the graph of the quadratic function written in general form as y = x2 + 4x + 3. In this form, a = 1, b = 4, and c = 3. Because a > 0, the parabola opens upward. The axis of symmetry is x = β 4 2(1) = β2. This also makes sense because we can see from the graph that the vertical line x = β2 divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, (β2, β1). The x- intercepts, those points where the parabola crosses the x- axis, occur at (β3, 0) and (β1, 0). Figure 5.5 The standard form of a quadratic function presents the function in the form f (x) = a(x β h)2 + k (5.2) where (h, k) is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function. As with the general form, if a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, the parabola opens downward, and the vertex is a maximum. Figure 5.6 represents the graph of the quadratic function written in standard form as y = β3(x + 2)2 +
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4. Since x β h = x + 2 in this example, h = β2. In this form, a = β3, h = β2, and k = 4. Because a < 0, the parabola opens downward. The vertex is at (β2, 4). 480 Chapter 5 Polynomial and Rational Functions Figure 5.6 The standard form is useful for determining how the graph is transformed from the graph of y = x2. Figure 5.7 is the graph of this basic function. Figure 5.7 If k > 0, the graph shifts upward, whereas if k < 0, the graph shifts downward. In Figure 5.6, k > 0, so the graph is shifted 4 units upward. If h > 0, the graph shifts toward the right and if h < 0, the graph shifts to the left. In Figure This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 481 5.6, h < 0, so the graph is shifted 2 units to the left. The magnitude of a indicates the stretch of the graph. If |a| > 1, the point associated with a particular x- value shifts farther from the x-axis, so the graph appears to become narrower, and there is a vertical stretch. But if |a| < 1, the point associated with a particular x- value shifts closer to the x-axis, so the graph appears to become wider, but in fact there is a vertical compression. In Figure 5.6, |a| > 1, so the graph becomes narrower. The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form. a(x β h)2 + k = ax2 + bx + c ax2 β 2ahx + (ah2 + k) = ax2 + bx + c For the linear terms to be equal, the coefficients must be equal. β2ah = b, so h = β b 2a This is the axis of symmetry we defined earlier. Setting the constant terms equal: ah2 + k = c k = c β ah2 2 β β b 2a β β = c β a β = c β b2 4a In practice, though, it is usually easier to remember that k is the output value of
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the function when the input is h, so f (h) = k. Forms of Quadratic Functions A quadratic function is a polynomial function of degree two. The graph of a quadratic function is a parabola. The general form of a quadratic function is f (x) = ax2 + bx + c where a, b, and c are real numbers and a β 0. The standard form of a quadratic function is f (x) = a(x β h)2 + k where a β 0. The vertex (h, k) is located at h = β b 2a, k = f (h) = f β β βb 2a β β Given a graph of a quadratic function, write the equation of the function in general form. 1. Identify the horizontal shift of the parabola; this value is h. Identify the vertical shift of the parabola; this value is k. 2. Substitute the values f (x) = a(x β h)2 + k. of the horizontal and vertical shift for h and k. in the function 3. Substitute the values of any point, other than the vertex, on the graph of the parabola for x and f (x). 4. Solve for the stretch factor, |a|. 5. Expand and simplify to write in general form. 482 Chapter 5 Polynomial and Rational Functions Example 5.2 Writing the Equation of a Quadratic Function from the Graph Write an equation for the quadratic function g in Figure 5.8 as a transformation of f (x) = x2, and then expand the formula, and simplify terms to write the equation in general form. Figure 5.8 Solution We can see the graph of g is the graph of f (x) = x2 shifted to the left 2 and down 3, giving a formula in the form g(x) = a(x β (β2))2 β 3 = a(x + 2)2 β 3. Substituting the coordinates of a point on the curve, such as (0, β1), we can solve for the stretch factor. β1 = a(0 + 2)2 β 3 2 = 4a a = 1 2 In standard form, the algebraic model for this graph is (g)x = 1 2 (x + 2)2
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β 3. To write this in general polynomial form, we can expand the formula and simplify terms. g(xx + 2)2 β 3 (x + 2)(x + 2) β 3 (x2 + 4x + 4) β 3 x2 + 2x + 2 β 3 x2 + 2x β 1 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 483 Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions. Analysis We can check our work using the table feature on a graphing utility. First enter Y1 = 1 2 (x + 2)2 β 3. Next, select TBLSET, then use TblStart = β 6 and ΞTbl = 2, and select TABLE. See Table 5.0. x y β6 β4 β2 0 5 β1 β3 β1 2 5 Table 5.0 The ordered pairs in the table correspond to points on the graph. A coordinate grid has been superimposed over the quadratic path of a basketball in Figure 5.9. Find an 5.1 equation for the path of the ball. Does the shooter make the basket? Figure 5.9 (credit: modification of work by Dan Meyer) Given a quadratic function in general form, find the vertex of the parabola. 1. Identify a, b, and c. 2. Find h, the x-coordinate of the vertex, by substituting a and b into h = β b 2a. 3. Find k, the y-coordinate of the vertex, by evaluating k = f (h) = f β ββ β β . b 2a 484 Chapter 5 Polynomial and Rational Functions Example 5.3 Finding the Vertex of a Quadratic Function Find the vertex of the quadratic function f (x) = 2x2 β 6x + 7. Rewrite the quadratic in standard form (vertex form). Solution The horizontal coordinate of the vertex will be at The vertical coordinate of the vertex will be at h = β b 2a = β6 2(2h Rewriting into standard form, the stretch factor will be the same as the a in the original quadratic
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. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the βaβ from the general form. f (x) = ax2 + bx + c f (x) = 2x2 β 6x + 7 The standard form of a quadratic function prior to writing the function then becomes the following: f (x) = 2 2 β βx β 3 2 β β + 5 2 Analysis One reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, k, and where it occurs, x. 5.2 Given the equation g(x) = 13 + x2 β 6x, write the equation in general form and then in standard form. Finding the Domain and Range of a Quadratic Function Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate at the turning point or less than or equal to the y-coordinate at the turning point, depending on whether the parabola opens up or down. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 485 Domain and Range of a Quadratic Function The domain of any quadratic function is all real numbers unless the context of the function presents some restrictions. The range of a quadratic function written in general form f (x) = ax2 + bx + c with a positive a value is f (x) β₯ f β ββ β β β ; the range of a quadratic function written in general form with a negative a β , β βββ, f β ββ value is f (x) β€ f β ββ β , or β‘ β£ f β b ββ 2a β , or β β b
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2a b 2a b 2a β€ β β¦. β β The range of a quadratic function written in standard form f (x) = a(x β h)2 + k with a positive a value is f (x) β₯ k; the range of a quadratic function written in standard form with a negative a value is f (x) β€ k. Given a quadratic function, find the domain and range. 1. Identify the domain of any quadratic function as all real numbers. 2. Determine whether a is positive or negative. If a is positive, the parabola has a minimum. If a is negative, the parabola has a maximum. 3. Determine the maximum or minimum value of the parabola, k. 4. If the parabola has a minimum, the range is given by f (x) β₯ k, or β‘ maximum, the range is given by f (x) β€ k, or (ββ, kβ€ β¦. β£k, β). If the parabola has a Example 5.4 Finding the Domain and Range of a Quadratic Function Find the domain and range of f (x) = β 5x2 + 9x β 1. Solution As with any quadratic function, the domain is all real numbers. Because a is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the x- value of the vertex. h = β b 2a = β 9 2(β5) The maximum value is given by f (h). = 9 10 2 β β + pβ β 9 10 β β β 1 f β β 9 10 9 10 β β β = 5 β = 61 20 The range is f (x) β€ 61 20, or β βββ, 61 20 β€ β¦. 486 5.3 Find the domain and range of f (x) = 2 2 β βx β 4 7 β β + 8 11. Chapter 5 Polynomial and Rational Functions Determining the Maximum and Minimum Values of Quadratic Functions The output of the quadratic function at the
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vertex is the maximum or minimum value of the function, depending on the orientation of the parabola. We can see the maximum and minimum values in Figure 5.10. Figure 5.10 There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue. Example 5.5 Finding the Maximum Value of a Quadratic Function A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side. a. Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length L. b. What dimensions should she make her garden to maximize the enclosed area? Solution This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 487 Letβs use a diagram such as Figure 5.11 to record the given information. It is also helpful to introduce a temporary variable, W, to represent the width of the garden and the length of the fence section parallel to the backyard fence. Figure 5.11 a. We know we have only 80 feet of fence available, and L + W + L = 80, or more simply, 2L + W = 80. This allows us to represent the width, W, in terms of L. W = 80 β 2L Now we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so A = LW = L(80 β 2L) A(L) = 80L β 2L2 This formula represents the area of the fence in terms of the variable length L. The function, written in general form, is A(L) = β2L2 + 80L. b. The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. This is why we rewrote the function in general form above. Since a is the coefficient of the squared term, a = β2, b = 80, and c = 0. To find the vertex: h = β b 2
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a = β 80 2(β2) = 20 k = A(20) and = 80(20) β 2(20)2 = 800 The maximum value of the function is an area of 800 square feet, which occurs when L = 20 feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet. Analysis This problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function in Figure 5.12. 488 Chapter 5 Polynomial and Rational Functions Figure 5.12 Given an application involving revenue, use a quadratic equation to find the maximum. 1. Write a quadratic equation for a revenue function. 2. Find the vertex of the quadratic equation. 3. Determine the y-value of the vertex. Example 5.6 Finding Maximum Revenue The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue? Solution Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. We can introduce variables, p for price per subscription and Q for quantity, giving us the equation Revenue = pQ. Because the number of subscribers changes with the price, we need to find a relationship between the variables. We know that currently p = 30 and Q = 84,000. We also know that if the price rises to $32, the newspaper would lose 5,000 subscribers, giving a second pair of values, p = 32 and Q = 79,000. From this we can find a linear equation relating the two quantities. The slope will be This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 489 m = 79,000 β 84,
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000 32 β 30 = β5,000 2 = β2,500 This tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the y-intercept. Q = β2500p + b 84,000 = β2500(30) + b b = 159,000 Substitute in the pointQ = 84,000 and p = 30 Solve forb This gives us the linear equation Q = β2,500p + 159,000 relating cost and subscribers. We now return to our revenue equation. Revenue = pQ Revenue = p(β2,500p + 159,000) Revenue = β2,500p2 + 159,000p We now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maximize revenue for the newspaper, we can find the vertex. h = β 159,000 2(β2,500) = 31.8 The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function. maximum revenue = β2,500(31.8)2 + 159,000(31.8) = 2,528,100 Analysis This could also be solved by graphing the quadratic as in Figure 5.13. We can see the maximum revenue on a graph of the quadratic function. Figure 5.13 Finding the x- and y-Intercepts of a Quadratic Function Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the y- intercept of a quadratic by evaluating the function at an input of zero, and we find 490 Chapter 5 Polynomial and Rational Functions the x- intercepts at locations where the output is zero. Notice in Figure 5.14 that the number of x- intercepts can vary depending upon the location of the graph. Figure 5.14 Number of x-intercepts of a parabola Given a quadratic function f(x), find the y- and x-intercepts. 1. Evaluate f (0) to find the y-intercept. 2. Solve the quadratic equation f (x) = 0 to find the x-intercepts. Example 5.7 Finding the y- and x-Intercepts
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of a Parabola Find the y- and x-intercepts of the quadratic f (x) = 3x2 + 5x β 2. Solution We find the y-intercept by evaluating f (0). f (0) = 3(0)2 + 5(0) β 2 = β2 So the y-intercept is at (0, β2). For the x-intercepts, we find all solutions of f (x) = 0. In this case, the quadratic can be factored easily, providing the simplest method for solution. 0 = 3x2 + 5x β 2 h = β b 2a = β 4 2(2) = β1 0 = (3x β 1)(x + 2) k = f (β1) = 2(β1)2 + 4(β1) β 4 = β6 So the x-intercepts are at β β β β and (β2, 0)., 0 1 3 Analysis This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 491 By graphing the function, we can confirm that the graph crosses the y-axis at (0, β2). We can also confirm that the graph crosses the x-axis at β β β β and (β2, 0). See Figure 5.15, 0 1 3 Figure 5.15 Rewriting Quadratics in Standard Form In Example 5.7, the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form. Given a quadratic function, find the x- intercepts by rewriting in standard form. 1. Substitute a and b into h = β b 2a. 2. Substitute x = h into the general form of the quadratic function to find k. 3. Rewrite the quadratic in standard form using h and k. 4. Solve for when the output of the function will be zero to find the x- intercepts. Example 5.8 Finding the x-Intercepts of a Parabola Find the x- intercepts of the quadratic function f (x) = 2x2 + 4x β 4. Solution We begin
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by solving for when the output will be zero. 0 = 2x2 + 4x β 4 492 Chapter 5 Polynomial and Rational Functions Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form. We know that a = 2. Then we solve for h and k. f (x) = a(x β h)2 + k h = β b 2a = β 4 2(2) = β1 k = f (β1) = 2(β1)2 + 4(β1) β 4 = β6 So now we can rewrite in standard form. We can now solve for when the output will be zero. f (x) = 2(x + 1)2 β 6 0 = 2(x + 1)2 β 6 6 = 2(x + 1)2 3 = (x + 1) The graph has x-intercepts at (β1 β 3, 0) and (β1 + 3, 0). We can check our work by graphing the given function on a graphing utility and observing the x- intercepts. See Figure 5.16. Figure 5.16 Analysis We could have achieved the same results using the quadratic formula. Identify a = 2, b = 4 and c = β4. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 493 x = βb Β± b2 β 4ac 2a = β4 Β± 42 β 4(2)(β4) 2(2) = β4 Β± 48 4 = β4 Β± 3(16) 4 = β1 Β± 3 So the x-intercepts occur at β ββ1 β 3, 0β β and β ββ1 + 3, 0β β . 5.4 In a Try It, we found the standard and general form for the function g(x) = 13 + x2 β 6x. Now find the y- and x-intercepts (if any). Example 5.9 Applying the Vertex and x-Intercepts of a Parabola A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ballβs height above ground can be modeled by the
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equation H(t) = β 16t 2 + 80t + 40. a. When does the ball reach the maximum height? b. What is the maximum height of the ball? c. When does the ball hit the ground? Solution a. The ball reaches the maximum height at the vertex of the parabola. h = β 80 2(β16) = 80 32 = 5 2 = 2.5 The ball reaches a maximum height after 2.5 seconds. b. To find the maximum height, find the y- coordinate of the vertex of the parabola. k = Hβ ββ b 2a = H(2.5) = β16(2.5)2 + 80(2.5) + 40 = 140 The ball reaches a maximum height of 140 feet. β β c. To find when the ball hits the ground, we need to determine when the height is zero, H(t) = 0. We use the quadratic formula. 494 Chapter 5 Polynomial and Rational Functions t = β80 Β± 802 β 4(β16)(40) 2(β16) = β80 Β± 8960 β32 Because the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions. t = β80 β 8960 β32 β 5.458 or t = β80 + 8960 β32 β β 0.458 The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds. See Figure 5.17. Figure 5.17 Note that the graph does not represent the physical path of the ball upward and downward. Keep the quantities on each axis in mind while interpreting the graph. A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet 5.5 per second. The rockβs height above ocean can be modeled by the equation H(t) = β16t 2 + 96t + 112. a. When does the rock reach the maximum height? b. What is the maximum height of the rock? c. When does the rock hit the ocean? Access these online resources for additional instruction and practice with quadratic equations. β’ Graphing Quadratic Functions in General Form (http://openstaxcollege.org/l/ graphquadgen) β’ Graphing Quadratic Functions in Standard
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Form (http://openstaxcollege.org/l/ graphquadstan) β’ Quadratic Function Review (http://openstaxcollege.org/l/quadfuncrev) β’ Characteristics of a Quadratic Function (http://openstaxcollege.org/l/characterquad) This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 495 5.1 EXERCISES Verbal Explain the advantage of writing a quadratic function in 1. standard form. How can the vertex of a parabola be used in solving real- 2. world problems? 3. Explain why the condition of a β 0 is imposed in the definition of the quadratic function. What 4. quadratic function? is another name for the standard form of a What two algebraic methods can be used to find the 5. horizontal intercepts of a quadratic function? Algebraic For the following exercises, rewrite the quadratic functions in standard form and give the vertex. 6. 7. 8. 9. 10. 11. 12. 13. f (x) = x2 β 12x + 32 g(x) = x2 + 2x β 3 f (x) = x2 β x f (x) = x2 + 5x β 2 h(x) = 2x2 + 8x β 10 k(x) = 3x2 β 6x β 9 f (x) = 2x2 β 6x f (x) = 3x2 β 5x β 1 For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. y(x) = 2x2 + 10x + 12 f (x) = 2x2 β 10x + 4 f (x) = β x2 + 4x + 3 f (x) = 4x2 + x β 1 h(t) = β4t 2 + 6t β 1 14. 15. 16. 17. 18. 19. f (x) = 1 2 x2 + 3x + 1 20. f (x) = β 1 3 x2 β 2x + 3 For the following exercises, determine the domain and range of the quadratic function. 21. 22. 23. 24. 25. f (x) = (x β
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3)2 + 2 f (x) = β2(x + 3)2 β 6 f (x) = x2 + 6x + 4 f (x) = 2x2 β 4x + 2 k(x) = 3x2 β 6x β 9 For the following exercises, use the vertex (h, k) and a point on the graph (x, y) to find the general form of the equation of the quadratic function. 26. (h, k) = (2, 0), (x, y) = (4, 4) 27. (h, k) = (β2, β1), (x, y) = (β4, 3) 28. (h, k) = (0, 1), (x, y) = (2, 5) 29. (h, k) = (2, 3), (x, y) = (5, 12) 30. (h, k) = ( β 5, 3), (x, y) = (2, 9) 31. (h, k) = (3, 2), (x, y) = (10, 1) 32. (h, k) = (0, 1), (x, y) = (1, 0) 33. (h, k) = (1, 0), (x, y) = (0, 1) Graphical For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts. 34. 35. 36. 37. f (x) = x2 β 2x f (x) = x2 β 6x β 1 f (x) = x2 β 5x β 6 f (x) = x2 β 7x + 3 Chapter 5 Polynomial and Rational Functions 43. 44. 45. 496 38. 39. f (x) = β2x2 + 5x β 8 f (x) = 4x2 β 12x β 3 For the following exercises, write the equation for the graphed quadratic function. 40. 41. 42. Numeric For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function. 46. This content is available for free at https://cnx.org/content/col
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11758/1.5 Chapter 5 Polynomial and Rational Functions 497 x y β2 β2 β1 1 0 β2 β1 β2 1 β2 β1 β8 β3 β2 β2 2 0 2 8 47. 48. 49. 50. Technology For the following exercises, use a calculator to find the answer. What appears to be the effect of adding or subtracting those numbers? The path of an object projected at a 45 degree angle 54. with initial velocity of 80 feet per second is given by the function h(x) = β32 (80)2 x2 + x where x is the horizontal distance traveled and h(x) is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally. A suspension bridge can be modeled by the quadratic 55. function h(x) =.0001x2 with β2000 β€ x β€ 2000 where |x| is the number of feet from the center and h(x) is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet. Extensions For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. 56. Vertex (1, β2), opens up. 57. Vertex (β1, 2) opens down. 58. Vertex (β5, 11), opens down. 59. Vertex (β100, 100), opens up. For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. 60. Contains (1, 1) and has shape of f (x) = 2x2. Vertex is on the y- axis. 61. Contains (β1, 4) and has the shape of f (x) = 2x2. Vertex is on the y- axis. Graph on the same set of axes x2. 51. f (x) = x2, f (x) = 2x2, and f (x) = 1 3 the functions 62. Contains (2, 3) and has the shape of f (x) = 3x2. Vertex is on the y- axis. What appears to be the effect of changing the coefficient? on the same Graph 52. f (x) = x
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2, f (x) = x2 + 2 f (x) = x2, f (x) = x2 + 5 and f (x) = x2 β 3. appears to be the effect of adding a constant? set of axes and What Graph axes 53. f (x) = x2, f (x) = (x β 2)2, f (x β 3)2, and f (x) = (x + 4)2. same the set on of 63. Contains (1, β3) and has the shape of f (x) = β x2. Vertex is on the y- axis. 64. Contains (4, 3) and has the shape of f (x) = 5x2. Vertex is on the y- axis. 65. 498 Chapter 5 Polynomial and Rational Functions Contains (1, β6) has the shape of f (x) = 3x2. Vertex has x-coordinate of β1. Real-World Applications Find the dimensions of corral 66. producing the greatest enclosed area given 200 feet of fencing. rectangular the Find the dimensions of the rectangular corral split into 67. 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing. Find the dimensions of corral 68. producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing. rectangular the Among all of the pairs of numbers whose sum is 6, find 69. the pair with the largest product. What is the product? Among all of the pairs of numbers whose difference is 70. 12, find the pair with the smallest product. What is the product? Suppose that the price per unit in dollars of a cell phone 71. production is modeled by p = $45 β 0.0125x, where x is in thousands of phones produced, and the revenue represented by thousands of dollars is R = x β
p. Find the production level that will maximize revenue. A rocket is launched in the air. Its height, in meters 72. above sea level, as a function of time, in seconds, is given by h(t) = β4.9t 2 + 229t + 234. Find the maximum height the rocket attains. A ball is thrown in the air from the top of a building. Its 73. height, in meters above ground, as a function of time, in seconds, is given by h(t) = β
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4.9t 2 + 24t + 8. How long does it take to reach maximum height? A soccer stadium holds 62,000 spectators. With a ticket 74. price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue? A farmer finds that if she plants 75 trees per acre, each 75. tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest? This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 499 5.2 | Power Functions and Polynomial Functions Learning Objectives In this section, you will: 5.2.1 Identify power functions. 5.2.2 Identify end behavior of power functions. 5.2.3 Identify polynomial functions. 5.2.4 Identify the degree and leading coefficient of polynomial functions. Figure 5.18 (credit: Jason Bay, Flickr) Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in Table 5.1. Year 2009 2010 2011 2012 2013 Bird Population 800 897 992 1, 083 1, 169 Table 5.1 The population can be estimated using the function P(t) = β 0.3t 3 + 97t + 800, where P(t) represents the bird population on the island t years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes. Identifying Power Functions Before we can understand the bird problem, it will be helpful to understand a different type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. As an example, consider functions for area or volume. The function for the area of a circle with radius r is and the function for the volume of a sphere with radius r is A(r) =
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Οr 2 V(r) = 4 3 Οr 3 500 Chapter 5 Polynomial and Rational Functions Both of these are examples of power functions because they consist of a coefficient, Ο or 4 3 Ο, multiplied by a variable r raised to a power. Power Function A power function is a function that can be represented in the form f (x) = kx p where k and p are real numbers, and k is known as the coefficient. Is f(x) = 2 x a power function? No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function. Example 5.10 Identifying Power Functions Which of the following functions are power functions? f (x) = 1 f (x) = x f (x) = x2 f (x) = x3 f (x) = 1 x f (x) = 1 x2 f (x) = x f (x) = x3 Solution Constant function Identify function Quadratic function Cubic function Reciprocal function Reciprocal squared function Square root function Cube root function All of the listed functions are power functions. The constant and identity functions are power functions because they can be written as f (x) = x0 and f (x) = x1 respectively. The quadratic and cubic functions are power functions with whole number powers f (x) = x2 and f (x) = x3. The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as f (x) = xβ1 and f (x) = xβ2. The square and cube root functions are power functions with fractional powers because they can be written as f (x) = x 1 2 or f (x) = x 1 3. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 501 5.6 Which functions are power functions? f (x) = 2x β
4x3 g(x) = βx5 + 5x3 h(x) = 2x5 β 1 3x2 + 4 Identifying End Behavior of Power Functions Figure 5.19 shows the graphs of f (x) = x2, g(x) = x4 and h(x) = x6,
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which are all power functions with even, wholenumber powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin. Figure 5.19 Even-power functions To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol β for positive infinity and ββ for negative infinity. When we say that β x approaches infinity,β which can be symbolically written as x β β, we are describing a behavior; we are saying that x is increasing without bound. With the positive even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as x approaches positive or negative infinity, the f (x) values increase without bound. In symbolic form, we could write as x β Β± β, f (x) β β Figure 5.20 shows the graphs of f (x) = x3, g(x) = x5, and h(x) = x7, which are all power functions with odd, wholenumber powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin. 502 Chapter 5 Polynomial and Rational Functions Figure 5.20 Odd-power functions These examples illustrate that functions of the form f (x) = xn reveal symmetry of one kind or another. First, in Figure 5.19 we see that even functions of the form f (x) = xn, n even, are symmetric about the y- axis. In Figure 5.20 we see that odd functions of the form f (x) = xn, n odd, are symmetric about the origin. For these odd power functions, as x approaches negative infinity, f (x) decreases without bound. As x approaches positive infinity, f (x) increases without bound. In symbolic form we write as x β β β, as x β β, f (x) β β f (x) β β β The behavior of the graph of a function as the input values get very small ( x β ββ ) and get very large ( x β β ) is
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referred to as the end behavior of the function. We can use words or symbols to describe end behavior. Figure 5.21 shows the end behavior of power functions in the form f (x) = kxn where n is a non-negative integer depending on the power and the constant. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 503 Figure 5.21 Given a power function f(x) = kxn where n is a non-negative integer, identify the end behavior. 1. Determine whether the power is even or odd. 2. Determine whether the constant is positive or negative. 3. Use Figure 5.21 to identify the end behavior. Example 5.11 Identifying the End Behavior of a Power Function Describe the end behavior of the graph of f (x) = x8. Solution The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As x approaches infinity, the output (value of f (x) ) increases without bound. We write as x β β, f (x) β β. As x approaches negative infinity, the output increases without bound. In symbolic form, as x β ββ, f (x) β β. We can graphically represent the function as shown in Figure 5.22. 504 Chapter 5 Polynomial and Rational Functions Figure 5.22 Example 5.12 Identifying the End Behavior of a Power Function. Describe the end behavior of the graph of f (x) = β x9. Solution The exponent of the power function is 9 (an odd number). Because the coefficient is β1 (negative), the graph is the reflection about the x- axis of the graph of f (x) = x9. Figure 5.23 shows that as x approaches infinity, the output decreases without bound. As x approaches negative infinity, the output increases without bound. In symbolic form, we would write as x β ββ, as x β β, f (x) β β f (x) β ββ This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 505 Figure 5.23 Analysis We can check our work by using the table feature on a graphing utility. x f(x) β10 1,000,000,
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000 β5 1,953,125 0 5 0 β1,953,125 10 β1,000,000,000 Table 5.1 We can see from Table 5.1 that, when we substitute very small values for x, the output is very large, and when we substitute very large values for x, the output is very small (meaning that it is a very large negative value). 506 Chapter 5 Polynomial and Rational Functions 5.7 Describe in words and symbols the end behavior of f (x) = β 5x4. Identifying Polynomial Functions An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius r of the spill depends on the number of weeks w that have passed. This relationship is linear. We can combine this with the formula for the area A of a circle. r(w) = 24 + 8w A(r) = Οr 2 Composing these functions gives a formula for the area in terms of weeks. Multiplying gives the formula. A(w) = A(r(w)) = A(24 + 8w) = Ο(24 + 8w)2 A(w) = 576Ο + 384Οw + 64Οw2 This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power. Polynomial Functions Let n be a non-negative integer. A polynomial function is a function that can be written in the form f (x) = an xn +...a1 x + a2 x2 + a1 x + a0 (5.3) This is called the general form of a polynomial function. Each ai is a coefficient and can be any real number other than zero. Each expression ai xi is a term of a polynomial function. Example 5.13 Identifying Polynomial Functions Which of the following are polynomial functions? f (x) = 2x3 β
3x + 4 g(x) = βx(
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x2 β 4) h(x) = 5 x + 2 Solution This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 507 The first two functions are examples of polynomial functions because they can be written in the form f (x) = an xn +... + a2 x2 + a1 x + a0, where the powers are non-negative integers and the coefficients are real numbers. β’ β’ β’ f (x) can be written as f (x) = 6x4 + 4. g(x) can be written as g(x) = β x3 + 4x. h(x) cannot be written in this form and is therefore not a polynomial function. Identifying the Degree and Leading Coefficient of a Polynomial Function Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term. Terminology of Polynomial Functions We often rearrange polynomials so that the powers are descending. When a polynomial is written in this way, we say that it is in general form. Given a polynomial function, identify the degree and leading coefficient. 1. Find the highest power of x to determine the degree function. 2. 3. Identify the term containing the highest power of x to find the leading term. Identify the coefficient of the leading term. Example 5.14 Identifying the Degree and Leading Coefficient of a Polynomial Function Identify the degree, leading term, and leading coefficient of the following polynomial functions. f (x) = 3 + 2x2 β 4x3 g(t) = 5t 2 β 2t 3 + 7t h(p) = 6p β p3 β 2 508 Chapter 5 Polynomial and Rational Functions Solution For the function f (x),
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the highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, β4x3. The leading coefficient is the coefficient of that term, β4. For the function g(t), the highest power of t is 5, so the degree is 5. The leading term is the term containing that degree, 5t 5. The leading coefficient is the coefficient of that term, 5. For the function h(p), the highest power of p is 3, so the degree is 3. The leading term is the term containing that degree, βp3. The leading coefficient is the coefficient of that term, β1. 5.8 Identify the degree, leading term, and leading coefficient of the polynomial f (x) = 4x2 β x6 + 2x β 6. Identifying End Behavior of Polynomial Functions Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term. See Table 5.2. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 509 Polynomial Function Leading Term Graph of Polynomial Function f (x) = 5x4 + 2x3 β x β 4 5x4 β2x6 f (x) = β 2x6 β x5 + 3x4 + x3 Table 5.2 510 Chapter 5 Polynomial and Rational Functions Polynomial Function Leading Term Graph of Polynomial Function f (x) = 3x5 β 4x4 + 2x2 + 1 f (x) = β 6x3 + 7x2 + 3x + 1 3x5 β6x3 Table 5.2 Example 5.15 Identifying End Behavior and Degree of a Polynomial Function This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 5 Polynomial and Rational Functions 511 Describe the end behavior and determine a possible degree of
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the polynomial function in Figure 5.24. Figure 5.24 Solution As the input values x get very large, the output values f (x) increase without bound. As the input values x get very small, the output values f (x) decrease without bound. We can describe the end behavior symbolically by writing as x β ββ, as x β β, f (x) β ββ f (x) β β In words, we could say that as x values approach infinity, the function values approach infinity, and as x values approach negative infinity, the function values approach negative infinity. We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive. 512 Chapter 5 Polynomial and Rational Functions 5.9 Describe the end behavior, and determine a possible degree of the polynomial function in Figure 5.25. Figure 5.25 Example 5.16 Identifying End Behavior and Degree of a Polynomial Function Given the function f (x) = β 3x2(x β 1)(x + 4), express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function. Solution Obtain the general form by expanding the given expression for f (x). f (x) = β3x2(x β 1)(x + 4) β = β3x2 β βx2 + 3x β 4 β = β3x4 β 9x3 + 12x2 The general form is f (x) = β3x4 β 9x3 + 12x2. The leading term is β3x4; polynomial is 4. The degree is even (4) and the leading coefficient is negative (β3), so the end behavior is therefore, the degree of the as x β β β, as x β β, f (x) β β β f (x) β β β 5.10 Given the function f (x) = 0.2(x β 2)(x + 1)(x β 5), express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function. Identifying Local Behavior of Polynomial Functions In addition to the end behavior of polyn
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