Which of the following represents the zeros of f(x) = 5x3 − 6x2 − 59x + 12?

5

343

Polynomial and Rational Functions

Figure 1 35-mm film, once the standard for capturing photographic images, has been made largely obsolete by digital photography. (credit “film”: modification of work by Horia Varlan; credit “memory cards”: 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.

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SECTION 5.1 sectioN exercises 357

5.1 SeCTIOn exeRCISeS

veRbAl

1. Explain the advantage of writing a quadratic function in standard form.

2. How can the vertex of a parabola be used in solving real-world problems?

3. Explain why the condition of a ≠ 0 is imposed in the definition of the quadratic function.

4. What is another name for the standard form of a quadratic function?

5. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

AlgebRAIC

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. 6. f (x) = x 2 − 12x + 32 7. g(x) = x 2 + 2x − 3 8. f (x) = x 2 − x

9. f (x) = x 2 + 5x − 2 10. h(x) = 2x 2 + 8x − 10 11. k(x) = 3x 2 − 6x − 9

12. f (x) = 2x 2 − 6x 13. f (x) = 3x 2 − 5x − 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.

14. y(x) = 2x 2 + 10x + 12 15. f(x) = 2x 2 − 10x + 4 16. f(x) = −x 2 + 4x + 3

17. f(x) = 4x 2 + x − 1 18. h(t) = −4t 2 + 6t − 1 19. f(x) = 1 __ 2 x 2 + 3x + 1

20. f(x) = −  1 __ 3 x 2 − 2x + 3

For the following exercises, determine the domain and range of the quadratic function.

21. f(x) = (x − 3)2 + 2 22. f(x) = −2(x + 3)2 − 6 23. f(x) = x 2 + 6x + 4

24. f(x) = 2x 2 − 4x + 2 25. k(x) = 3x 2 − 6x − 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) = (−2, −1), (x, y) = (−4, 3) 28. (h, k) = (0, 1), (x, y) = (2, 5)

29. (h, k) = (2, 3), (x, y) = (5, 12) 30. (h, k) = (−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. f(x) = x 2 − 2x 35. f(x) = x 2 − 6x − 1 36. f(x) = x 2 − 5x − 6

37. f(x) = x 2 − 7x + 3 38. f(x) = −2x 2 + 5x − 8 39. f(x) = 4x 2 − 12x − 3

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358 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

For the following exercises, write the equation for the graphed function. 40.

–4 –3 –2 –1 1 2 3 4 5 6

–5 –4 –3 –2 –1

1 2 3 4 5

x

y 41.

–6 –5 –4 –3 –2 –1 1 2 3 4

–2 –1

1 2 3 4 5 6 7 8

x

y 42.

–3 –2 –1 1 2 3 4 5 6 7

–3 –2 –1

1 2 3 4 5 6 7

x

y

43.

–6 –5 –4 –3 –2 –1 1 2 3 4

–7 –6 –5 –4 –3 –2 –1

1 2 3

x

y 44. 45.

–7 –6 –5 –4 –3 –2 –1 1 2 3

–5 –4 –3 –2 –1

1 2 3 4 5

x

y

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. 47. 48.

49. 50.

TeChnOlOgy For the following exercises, use a calculator to find the answer.

51. Graph on the same set of axes the functions f (x) = x2, f(x) = 2x 2, and f(x) = 1 __ 3 x

2. What appears to be the effect of changing the coefficient?

52. Graph on the same set of axes f(x) = x 2, f(x) = x 2 + 2 and f(x) = x 2, f(x) = x 2 + 5 and f(x) = x 2 − 3. What appears to be the effect of adding a constant?

53. Graph on the same set of axes f(x) = x 2, f(x) = (x − 2)2, f(x − 3)2, and f(x) = (x + 4)2. What appears to be the effect of adding or subtracting those numbers?

54. The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function h(x) = −32 ____ (80)2 x

2 + 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.

55. A suspension bridge can be modeled by the quadratic function h(x) = 0.0001x 2 with −2000 ≤ x ≤ 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.

–2 –1 1 2 3 4 5 6 7 8

–2 –1

1 2 3 4 5 6 7 8

x

y

x −2 −1 0 1 2

y 5 2 1 2 5 x −2 −1 0 1 2

y 1 0 1 4 9

x −2 −1 0 1 2

y −2 1 2 1 −2

x −2 −1 0 1 2

y −8 −3 0 1 0

x −2 −1 0 1 2

y 8 2 0 2 8

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SECTION 5.1 sectioN exercises 359

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, −2), opens up. 57. Vertex (−1, 2) opens down.

58. Vertex (−5, 11), opens down. 59. Vertex (−100, 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) = 2x 2. Vertex is on the y-axis.

61. Contains (−1, 4) and has the shape of f(x) = 2x 2. Vertex is on the y-axis.

62. Contains (2, 3) and has the shape of f(x) = 3x 2. Vertex is on the y-axis.

63. Contains (1, −3) and has the shape of f(x) = −x 2. Vertex is on the y-axis.

64. Contains (4, 3) and has the shape of f(x) = 5x 2. Vertex is on the y-axis.

65. Contains (1, −6) has the shape of f(x) = 3x 2. Vertex has x-coordinate of −1.

ReAl-WORld APPlICATIOnS

66. Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.

67. Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.

68. Find the dimensions of the rectangular corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.

69. Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?

70. Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?

71. Suppose that the price per unit in dollars of a cell phone production is modeled by p = $45 − 0.0125x, where x is in thousands of phones produced, and the revenue represented by thousands of dollars is R = x ⋅ p. Find the production level that will maximize revenue.

72. A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by h(t) = −4.9t 2 + 229t + 234. Find the maximum height the rocket attains.

73. A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by h(t) = −4.9t2 + 24t + 8. How long does it take to reach maximum height?

74. A soccer stadium holds 62,000 spectators. With a ticket 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?

75. A farmer finds that if she plants 75 trees per acre, each 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?

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372 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

5.2 SeCTIOn exeRCISeS

veRbAl

1. Explain the difference between the coefficient of a power function and its degree.

2. If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

3. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

4. What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

5. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x → −∞, f (x) → −∞ and as x → ∞, f (x) → −∞.

AlgebRAIC

For the following exercises, identify the function as a power function, a polynomial function, or neither.

6. f (x) = x5 7. f (x) = (x2)3 8. f (x) = x − x4

9. f (x) = x 2 _____ x2 − 1

10. f (x) = 2x(x + 2)(x − 1)2 11. f (x) = 3x + 1

For the following exercises, find the degree and leading coefficient for the given polynomial.

12. −3x 13. 7 − 2x2 14. −2x2 − 3x5 + x − 6

15. x(4 − x2)(2x + 1) 16. x 2 (2x − 3)2

For the following exercises, determine the end behavior of the functions.

17. f (x) = x4 18. f (x) = x3 19. f (x) = −x4

20. f (x) = −x9 21. f (x) = −2x4 − 3x2 + x − 1 22. f (x) = 3x2 + x − 2

23. f (x) = x2(2x3 − x + 1) 24. f (x) = (2 − x)7

For the following exercises, find the intercepts of the functions.

25. f (t) = 2(t − 1)(t + 2)(t − 3) 26. g(n) = −2(3n − 1)(2n + 1) 27. f (x) = x4 − 16

28. f (x) = x3 + 27 29. f (x) = x(x2 − 2x − 8) 30. f (x) = (x + 3)(4x2 − 1)

gRAPhICAl

For the following exercises, determine the least possible degree of the polynomial function shown.

31.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

32.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

33.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

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SECTION 5.2 sectioN exercises 373

34.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

35.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

36.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

37.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5 –6

–5

1

3

3

21 4

4

5

5 6

38.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5 6

For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

39.

x

y 40.

x

y 41.

x

y 42.

x

y

43.

x

y 44.

x

y 45.

x

y

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374 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

nUmeRIC

For the following exercises, make a table to confirm the end behavior of the function.

46. f (x) = −x3 47. f (x) = x4 − 5x2 48. f (x) = x2(1 − x)2

49. f (x) = (x − 1)(x − 2)(3 − x) 50. f (x) = x 5 __ 10 − x

4

TeChnOlOgy

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

51. f (x) = x3(x − 2) 52. f (x) = x(x − 3)(x + 3) 53. f (x) = x(14 − 2x)(10 − 2x)

54. f (x) = x(14 − 2x)(10 − 2x)2 55. f (x) = x3 − 16x 56. f (x) = x3 − 27

57. f (x) = x4 − 81 58. f (x) = −x3 + x2 + 2x 59. f (x) = x3 − 2x2 − 15x

60. f (x) = x3 − 0.01x

exTenSIOnS

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or −1. There may be more than one correct answer.

61. The y-intercept is (0, −4). The x-intercepts are (−2, 0), (2, 0). Degree is 2. End behavior: as x → −∞, f (x) → ∞, as x → ∞, f (x) → ∞.

62. The y-intercept is (0, 9). The x-intercepts are (−3, 0), (3, 0). Degree is 2. End behavior: as x → −∞, f (x) → −∞, as x → ∞, f (x) → −∞.

63. The y-intercept is (0, 0). The x-intercepts are (0, 0), (2, 0). Degree is 3. End behavior: as x → −∞, f (x) → −∞, as x → ∞, f (x) → ∞.

64. The y-intercept is (0, 1). The x-intercept is (1, 0). Degree is 3. End behavior: as x → −∞, f (x) → ∞, as x → ∞, f (x) → −∞.

65. The y-intercept is (0, 1). There is no x-intercept. Degree is 4. End behavior: as x → −∞, f (x) → ∞, as x → ∞, f (x) → ∞.

ReAl-WORld APPlICATIOnS

For the following exercises, use the written statements to construct a polynomial function that represents the required information.

66. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of d, the number of days elapsed.

67. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of m, the number of minutes elapsed.

68. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by x inches and the width increased by twice that amount, express the area of the rectangle as a function of x.

69. An open box is to be constructed by cutting out square corners of x-inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of x.

70. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width (x).

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390 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

5.3 SeCTIOn exeRCISeS

veRbAl

1. What is the difference between an x-intercept and a zero of a polynomial function f ?

2. If a polynomial function of degree n has n distinct zeros, what do you know about the graph of the function?

3. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

4. Explain how the factored form of the polynomial helps us in graphing it.

5. If the graph of a polynomial just touches the x-axis and then changes direction, what can we conclude about the factored form of the polynomial?

AlgebRAIC

For the following exercises, find the x- or t-intercepts of the polynomial functions.

6. C(t) = 2(t − 4)(t + 1)(t − 6) 7. C(t) = 3(t + 2)(t − 3)(t + 5) 8. C(t) = 4t(t − 2)2(t + 1)

9. C(t) = 2t(t − 3)(t + 1)2 10. C(t) = 2t4 − 8t3 + 6t2 11. C(t) = 4t4 + 12t3 − 40t2

12. f (x) = x4 − x2 13. f (x) = x3 + x2 − 20x 14. f (x) = x3 + 6x2 − 7x

15. f (x) = x3 + x2 − 4x − 4 16. f (x) = x3 + 2x2 − 9x − 18 17. f (x) = 2x3 − x2 − 8x + 4

18. f (x) = x6 − 7x3 − 8 19. f (x) = 2x4 + 6x2 − 8 20. f (x) = x3 − 3x2 − x + 3

21. f (x) = x6 − 2x4 − 3x2 22. f (x) = x6 − 3x4 − 4x2 23. f (x) = x5 − 5x3 + 4x

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

24. f (x) = x3 − 9x, between x = −4 and x = −2. 25. f (x) = x3 − 9x, between x = 2 and x = 4.

26. f (x) = x5 − 2x, between x = 1 and x = 2. 27. f (x) = −x4 + 4, between x = 1 and x = 3.

28. f (x) = −2x3 − x, between x = −1 and x = 1. 29. f (x) = x3 − 100x + 2, between x = 0.01 and x = 0.1

For the following exercises, find the zeros and give the multiplicity of each.

30. f (x) = (x + 2)3(x − 3)2 31. f (x) = x2(2x + 3)5(x − 4)2

32. f (x) = x3 (x − 1)3(x + 2) 33. f (x) = x2(x2 + 4x + 4)

34. f (x) = (2x + 1)3(9x2 − 6x + 1) 35. f (x) = (3x + 2)5(x2 − 10x + 25)

36. f (x) = x(4x2 − 12x + 9)(x2 + 8x + 16) 37. f (x) = x6 − x5 − 2x4

38. f (x) = 3x4 + 6x3 + 3x2 39. f (x) = 4x5 − 12x4 + 9x3

40. f (x) = 2x4(x3 − 4x2 + 4x) 41. f (x) = 4x4(9x4 − 12x3 + 4x2)

gRAPhICAl

For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity, and end behavior.

42. f (x) = (x + 3)2(x − 2) 43. g(x) = (x + 4)(x − 1)2 44. h(x) = (x − 1)3(x + 3)2

45. k(x) = (x − 3)3(x − 2)2 46. m(x) = −2x(x − 1)(x + 3) 47. n(x) = −3x(x + 2)(x − 4)