C H A P T E R 2 Polynomial and Rational Functions

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260 Chapter 2 Polynomial and Rational Functions 7. 9. 10. f x x kqx r 8. (a) (a) Cubic, passes through 2, 5, rises to the right a bia bi a 2 abi abi b 2 i 2 a 2 b 2 Since a and b are real numbers, a is also a real number. 2 b2 fx Vertical asymptote: Horizontal asymptote: (i) (ii) (iii) (iv) One possibility: f x x 2x 2 5 x 3 2x 2 5 (b) Cubic, passes through 3, 1, falls to the right One possibility: f x x 3x 2 1 x 3 3x 2 1 ax b cx d x d c y a c a > 0, b < 0, c > 0, d < 0 Both the vertical asymptote and the horizontal asymptote are positive. Matches graph (d). a > 0, b > 0, c < 0, d < 0 Both the vertical asymptote and the horizontal asymptote are negative. Matches graph (b). a < 0, b > 0, c > 0, d < 0 The vertical asymptote is positive and the horizontal asymptote is negative. Matches graph (a). a > 0, b < 0, c > 0, d > 0 The vertical asymptote is negative and the horizontal asymptote is positive. Matches graph (c). (b) (c) z m 1 z 1 1 1 i 1 i 1 i 1 i z m 1 z z m 1 z 1 i 2 1 1 3 i 3 i 3 i 3 i 3 i 10 1 2 8i 1 2 8i 2 8i 2 8i 2 8i 68 1 1 2 2 i 3 1 10 10 i 1 2 34 17 i 11. (a) is a vertical asymptote. a causes a vertical stretch if and a vertical shrink if For a > 1, the graph becomes wider as a increases. When a is negative the graph is reflected about the x-axis. 0 < fx b 0 ⇒ x b a > 1 a < 1. ax x b2 (b) a 0. Varying the value of b varies the vertical asymptote of the graph of f. For b > 0, the graph is translated to the right. For b < 0, the graph is reflected in the x-axis and is translated to the left.
Problem Solving for Chapter 2 261 12. (a) (b) 50 1 y 0.007x 0.44 0 0 70 1 y 0.0313x 0.007x 0.44 y 2 Age, x Near point, y 50 (c) Age, x Near point, y Quadratic Rational 16 3.0 Model Model 32 44 4.7 9.8 0 0 70 16 32 3.0 4.7 3.66 2.32 3.05 4.63 50 19.7 44 9.8 11.83 7.58 60 39.4 50 19.7 19.97 11.11 60 39.4 38.54 50.00 1.586x 21.02 The models are fairly good fits to the data. The quadratic model seems to be a better fit for older ages and the rational model a better fit for younger ages. (d) For x 25, the quadratic model yields y 0.9325 inches and the rational model yields y 3.774 inches. (e) The reciprocal model cannot be used to predict the near point for a person who is 70 years old because it results in a negative value y 20. The quadratic model yields y 63.37 inches.
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CHAPTER 2 Polynomial and Rational F
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9. (a) (c) 10. (a) (c) y 1 2 x2 (b
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13. f x x 2 5 14. Vertex: Axis of
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25. 27. 29. hx 4x 2 4x 21 Vertex
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41. 2, 2 is the vertex. 42. 2, 0 is
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61. fx 2x 10 2 7x 30 62. x-inter
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75. —CONTINUED— (d) A 8 x50 x
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86. (a) and (c) (b) (d) 1996 (e) Ve
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Section 2.2 Polynomial Functions of
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11. (a) (b) fx x4 fx x 3 3 4 y
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Section 2.2 Polynomial Functions of
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36. (a) 0 with multiplicity 2, 4 an
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49. 51. 53. 55. 57. 59. 61. fx x
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70. 72. 74. gx x 2 10x 16 x 2x
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81. 83. 85. Zeros: 0, 2, 2 all of m
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92. (a) 93. (c) 200 7 140 V 4 3r 3
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111. 113. 115. 117. 12x 2 11x 5
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2x 4 4. and y2 x 3 x (a) and (b
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20. 22. 24. 26. 28. 29. 31. 33. 3 5
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45. fx 4x (a) 1 4 0 13 4 4 4 4 9 3
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58. f x 3x Factors: x 3, x 2 3
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67. 69. 71. ht t 3 2t 2 7t 2 (a
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79. 83. x n 3 ) x 3n 9x 2n 27x n
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1. 4. 7. a 10 b 6 Section 2.4 Comp
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52. 53. 55. 57. 59. 8 16i 2i 3i 4
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74. 4.5x2 3x 12 0; a 4.5, b 3,
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Section 2.5 Zeros of Polynomial Fun
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18. 19. f x 3x 3 19x 2 33x 9 Po
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−24 (c) Real zeros: 2, 1 145 ± 8
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47. 48. fx 2x 3 3x 2 50x 75 Sin
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54. f x x Since 1 3i is a zero, s
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69. 71. 73. gx x 4 4x 3 8x 2 16
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79. Let fx x4 gx 5x 2. 5 10x 5
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99. fx x Rational zeros: 1 x 1 Ir
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111. 9,000,000 0.0001x 2 60x 150,
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129. 3 6i 8 3i 3 6i 8 3i 11
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2 x x 2 7. fx 2 x x 2 Domain:
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Page 79 and 80: 43. 45. f x 2x2 5x 2 2x 2 x 6
Page 81 and 82: 51. 53. 55. hx (a) Domain: all rea
Page 83 and 84: 60. gx (a) Domain: all real number
Page 85 and 86: 72. (a) x-intercepts: (b) 0 x 3
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Page 89 and 90: 4. 5. 8. 10. 3x2 x2 < 1 4 (a) x 2
Page 91 and 92: 17. 19. 20. x 2 2x 3 < 0 18. x 3
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Page 103 and 104: Review Exercises for Chapter 2 1. (
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Page 123 and 124: 142. 144. 146. 148. 12x 3 20x 2 <
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Page 129: 19. Use synthetic division to show
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C H A P T E R 2 Polynomial and Rational Functions

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