Chapter 2 y - Cengage · Copyright © Houghton Mifflin Company. All rights reserved. Precalculus with Limits, Answers to Section 2.1 4 (Continued) 75. (a) (b) (c) (d) (e) They are

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d.Precalculus with Limits, Answers to Section 2.1 1

Chapter 2Section 2.1 (page 134)

Vocabulary Check (page 134)1. nonnegative integer; real 2. quadratic; parabola3. axis 4. positive; minimum5. negative; maximum

1. g 2. c 3. b 4. h5. f 6. a 7. e 8. d

Vertical shrink Vertical shrink and reflection in the x-axis

Vertical stretch Vertical stretch and reflection in the x-axis

Vertical shift Vertical shift

Vertical shift Vertical shift

Horizontal shift Horizontal shrink andvertical shift

Horizontal stretch and Horizontal shiftvertical shift

Horizontal shift, Horizontal shift,vertical shrink, reflection vertical shrink, and in the x-axis, and vertical vertical shiftshift

Reflection in the axis, Horizontal shift,vertical shrink, horizontal vertical stretch, andshift, and vertical shift vertical shift

x-

y

x

2

1

−1

3

4

7

1−1−2−3−4 2 3 4

x−4

−4

−6

−8

−6−8

2

4

42 6

6

y

y

x−4−6−8 2 6 8

−4

−6

4

6

8

10

−6 −4 −2 2 6 8 10

4

6

8

x

y

−8 −6 −4 −2 2 4

2

8

10

x

−2

yy

x−6

−4

−2

2

4

6

8

−2 2 6

y

x−3 −2 −1

−11

3

4

5

2 3−2 −1 1 2 3 4−1

3

4

5

x

y

−6 –4 4 6

−4

4

6

8

x

y

−6 −4 −2 2 4 6−2

6

8

10

x

y

−3 −2 2 3

−2

1

2

3

4

x

y

−3 −2 −1 1 2 3−1

2

3

4

5

x

y

−6 −4 −2 2 4 6

4

2

6

x

y

−3 −2 −1 1 2 3−1

1

2

3

4

5

x

y

−6 −4 4 6

−6

−4

−2

2

4

6

x

y

−3 −2 −1 1 2 3−1

1

2

3

4

5

x

y9. (a)

10. (a) (b)

(d)(c)

(b)

(c) (d)

11. (a) (b)

(c) (d)

12. (a) (b)

(c) (d)

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d.

Precalculus with Limits, Answers to Section 2.1 2

(Continued)

13. 14.

Vertex: Vertex:Axis of symmetry: y-axis Axis of symmetry:x-intercepts: x-intercepts:

15. 16.

Vertex: Vertex:Axis of symmetry: y-axis Axis of symmetry:x-intercepts: x-intercepts:

17. 18.

Vertex: Vertex:Axis of symmetry: Axis of symmetry:x-intercepts: No x-intercept

19. 20.

Vertex: Vertex:Axis of symmetry: Axis of symmetry:x-intercept: x-intercept:

21. 22.

Vertex: Vertex:Axis of symmetry: Axis of symmetry:

No x-intercept x-intercepts:

23. 24.

Vertex: Vertex:Axis of symmetry: Axis of symmetry:x-intercepts: x-intercepts:

25. 26.

Vertex: Vertex:Axis of symmetry: Axis of symmetry:No x-intercept No x-intercept

27. 28.

Vertex: Vertex:Axis of symmetry: Axis of symmetry:x-intercepts: x-intercepts: �3, 0�, �6, 0��4, 0�, �12, 0�

x �92x � 4

�92, 34��4, �16�

x4 6−2

2

8 10−2

−4

−6

y

−12

−16

−20

−8

4

4 8 16x

y

x �14x �

12

�14, 78�� 1

2, 20�−1−2−3 1 2 3

1

3

4

5

6

x

y

−4−8 4 8

10

20

x

y

��2 ± �5, 0��1 ± �6, 0�x � �2x � 1

��2, 5��1, 6�

−6 −5 −3 −2 −1 1 2

−3

−2

1

2

4

5

x

y

−4 2 6

−4

−2

6

x

y

��32 ± �2, 0�

x � �32x �

12

��32, �2�� 1

2, 1�

−5 −4 −3 −2 −1 1 2

−3

−2

1

2

3

4

x

y

−2 −1 1 2 3

1

3

4

5

x

y

��1, 0��4, 0�x � �1x � 4

��1, 0��4, 0�−4 −3 −2 −1 1 2

1

2

3

4

5

6

x

y

−4 4 8 12 16

4

8

12

16

20

x

y

��5 ± �6, 0�x � 6x � �5

�6, 3��5, �6�−10−20 10 20 30

10

20

30

40

50

x

y

−20 −12 4 8

−8

12

16

20

x

y

�±8, 0��±2�2, 0�x � 0

�0, 16��0, �4�

x

12

3 6 9

18

−3

3

6

9

−3−6−9

y

−4 −3 −1 1 2 3 4

−5

−3

−2

1

2

3

x

y

�±5, 0��±�5, 0�x � 0

�0, 25��0, �5�

−10−20 10 20

30

x

y

−4 −3 −1 1 3 4

−6

−3

−2

1

2

x

y

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(Continued)

29. Vertex:Axis of symmetry:x-intercepts:





34. Vertex:Axis of symmetry:No x-intercept



37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54. 55.

56.

57. 58.

59. 60.

61. 62.

63. 64.

65. 66.

67. 68.

69. 70.

71. 55, 55 72. 73. 12, 6 74. 21, 7S

2,

S

2

g�x� � �2x2 � x � 10g�x� � �2x2 � 7x � 3

f �x� � 2x 2 � x � 10f �x� � 2x 2 � 7x � 3

g�x� � �x 2 � 12x � 32g�x� � �x2 � 10x

f �x� � x 2 � 12x � 32f �x� � x2 � 10x

g�x� � �x 2 � 25g�x� � �x2 � 2x � 3

f �x� � x2 � 25f �x� � x2 � 2x � 3

��15, 0�, �3, 0��7, 0�, ��1, 0�

−18 4

−60

10

−10

−6

14

10

��7, 0�, �34, 0��5

2, 0�, �6, 0�

−9 2

−70

10

−5

−40

10

10

��2, 0�, �10, 0��3, 0�, �6, 0�

−4 12

−40

10

−8

−4

16

12

�0, 0�, �5, 0��0, 0�, �4, 0�

−1 6

−6

14

−4 8

−4

4

��3, 0�, �12, 0�

�5, 0�, ��1, 0��3, 0��±4, 0�

f �x� � �450�x � 6�2 � 6f �x� � �163 �x �

52�2

f �x� �1981�x �

52�2

�34f �x� � �

2449�x �

14�2

�32

f �x� � 2�x � 2� 2 � 2f �x� �34�x � 5�2 � 12

f �x� � �14�x � 2�2 � 3f �x� � �

12�x � 3�2 � 4

f �x� � �x � 4� 2 � 1f �x� � �x � 2�2 � 5

y � 2�x � 2� 2y � �2�x � 2�2 � 2

y � �x � 2� 2 � 1y � ��x � 1�2 � 4

y � �x 2 � 1y � �x � 1� 2

��3 ± �14, 0�x � �3

��3, �425 �

−14 10

−10

6

��2 ± �6, 0�x � �2

��2, �3�

−8 4

4

−4

x � 3�3, �5�

0 6

−20

0

�4 ± 12�2, 0�x � 4

�4, �1�

−6 12

−12

48

��5 ± �11, 0�x � �5

��5, �11�−20 10

−15

5

��4 ± �5, 0�x � �4

��4, �5�

−18 12

−6

14

��6, 0�, �5, 0�x � �

12

��12, 121

4 �−10 10

−80

35

�1, 0�, ��3, 0�x � �1

��1, 4�

−8 7

−5

5

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d.


(Continued)

75. (a)

(b)

(c)

(d) (e) They are identical.

76. (a) (b)

(c)

77. 16 feet

78. (a) 1.5 feet (b)(c) feet

79. 20 fixtures 80. 1222 units 81. 350,000 units

82. $2000

83. (a) $14,000,000; $14,375,000; $13,500,000(b) 24; $14,400

84. (a) $408; $468; $432(b) $6.25 per pet; $468.75

Answers will vary.

85. (a)

(b) 4299; answers will vary.(c) 8879; 24

86. (a) and (c)

(b) (d) 1995(e) Answers will vary. (f)

87. (a) (b) 69.6 miles per hour

88. (a) and (c)

(b)(d) 45.5 miles per hour

89. True. The equation has no real solutions, so the graph hasno -intercepts.

90. True. The vertex of is and the vertex of is

91.

92. Conditions (a) and (d) are preferable because profitswould be increasing.

93. Yes. A graph of a quadratic equation whose vertex is onthe x-axis has only one x-intercept.

94. Answers will vary. 95.

96. 97.

98. 99. 27 100. 7

101. 102. 103. 109

104. 72 105. Answers will vary.

�43�

140849

y � �3x � 20

y �54 x � 3y �

32 x �

134

y � �13 x �

53

f �x� � a�x �b

2a�2

�4ac � b2

4a

��54, �71

4 �.g�x��5

4, 534 �f �x�

x

y � �0.0082x2 � 0.746x � 13.47

10 8020

31

0

−5

100

25

�1,381,000y � 4.30x2 � 49.9x � 886

4650

12

950

00

43

5000

�228.64

665764 feet � 104.02 feet

x � 50 meters, y �100�

metersA � x�200 � 2x

� �;

y �200 � 2x

�r �

1

2y; d � y�

A � �83 �x � 25�2 �

50003

x � 25 feet, y � 33 13 feet

00

60

2000

x � 25 feet, y � 33 13 feet

A �8x�50 � x�

3

x 5 10 15 20 25 30

A 600 1067 1400 1600 1667 1600

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Section 2.2 (page 148)

Vocabulary Check (page 148)1. continuous 2. Leading Coefficient Test3. ; 4. (a) solution; (b) (c) x-intercept5. touches; crosses 6. standard7. Intermediate Value

1. c 2. g 3. h 4. f5. a 6. e 7. d 8. b

−4 −3 −2 2 3 4

−4

−3

−2

−1

2

3

4

x

y

−4 −3 −2 2 3 4

1

2

3

4

x

y

−5 −4 −2 1 2 3

−4

x

y

−4 −3 −2 2 3 4

−4

−3

−2

−1

1

2

3

4

x

y

y

x−4 −3 −1 1

6

5

4

3

2

1

−13 4

y

x−4 −3 −2 −1

−1

−2

1

6

5

2 3 4

−4 −3 −2 −1 1 2 3 4

−2

x

y

−4 −3 −2 1 2 3 4

−2

1

2

3

5

6

x

−1

y

−4 −3 −2 2 3 4

−4

1

2

3

4

x

y

−5 −4 −3 −2 −1 1 2 3

−2

1

2

3

4

5

6

x

y

−5 −4 −3 −2 1 2 3

−4

−3

1

2

3

4

x

y

−4 −3 −2 2 3 4

−4

−3

2

3

4

x

y

−4 −3 −2 1 2 3 4

−4

−3

2

3

4

x

y

−4 −3 1 2 3 4

−4

−3

1

2

3

4

x

y

−3 −2 1 2 4 5

−5

−4

−3

−2

1

2

3

x

y

−4 −3 −2 2 3 4

−4

−3

−2

1

2

3

4

x

y

−4 −3 −2 2 3 4

−5

−4

1

2

3

x

y

−3 −2 2 3 4 5

−4

−3

−2

1

2

3

4

x

y

�x � a�;n � 1n

9. (a) (b)

(c) (d)

10. (a) (b)

(c) (d)

11. (a) (b)

(c) (d)

(e) (f)

12. (a) (b)

(c) (d)

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d.


(Continued)

13. Falls to the left, rises to the right14. Rises to the left, rises to the right15. Falls to the left, falls to the right16. Falls to the left, falls to the right17. Rises to the left, falls to the right18. Falls to the left, rises to the right19. Rises to the left, falls to the right20. Rises to the left, rises to the right21. Falls to the left, falls to the right22. Rises to the left, falls to the right23. 24.

25. 26.

27. (a)(b) odd multiplicity; number of turning points: 1(c)


29. (a) 3(b) even multiplicity; number of turning points: 1(c)

30. (a)(b) even multiplicity; number of turning points: 1(c)


32. (a)

(b) odd multiplicity; number of turning points: 1(c)



−6

−16

6

16

0, 1 ±�2

−6

−24

6

8

0, 2 ± �3

−8

−5

4

3

�5 ± �37

2

−6

−4

6

4

�2, 1

−25

−5

15

25

�5

−18

−20

18

4

−30

−5

30

55

±7

−30

−30

30

10

±5

−6 6

−3

g

f

5

−8

−20

8

g

f

12

−6

gf

6

−9 9

−8

−4 4

8

fg

y

x−4 −3 −2

−2

−1−1 1 2 3 4

y

x

−4

2−2−6−8 6 8

(e) (f)

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(Continued)

35. (a) 0, 2(b) 0, odd multiplicity; 2, even multiplicity; number of

turning points: 2(c)

36. (a)(b) odd multiplicity; 0, even multiplicity; number

of turning points: 3(c)

37. (a)(b) 0, odd multiplicity; even multiplicity; number

of turning points: 4(c)


39. (a) No real zeros(b) number of turning points: 1(c)




43. (a)

(b) -intercepts: (c)(d) The answers in part (c) match the -intercepts.

44. (a)

(b) x-intercepts:(c)(d) The answers in part (c) match the x-intercepts.

45. (a)

(b) -intercepts:(c)(d) The answers in part (c) match the -intercepts.

46. (a)

(b) (c)(d) The answers in part (c) match the x-intercepts.

47. 48.

49. 50.

51. 52. f �x� � x3 � 7x 2 � 10xf �x� � x3 � 5x2 � 6x

f �x� � x 2 � x � 20f �x� � x2 � 4x � 12

f �x� � x2 � 3xf �x� � x2 � 10x

x � 0, 3, �3�0, 0�, �±3, 0�

−18 18

−12

12

xx � 0, 1, �1, 2, �2

�0, 0�, �±1, 0�, �±2, 0�x

−6 6

−4

4

x � �2, ��2, �1��2, 0�, ��2, 0�, ��1, 0�

−3

−11

3

2

xx � 0, 52�0, 0�, �5

2, 0�x

−2

−4

6

12

−9

−20

9

140

4, ±5

−8

−16

7

4

±2, �3

−6

−60

6

20

±�5

−4

−5

4

40

−9

−6

9

6

0, ±�2

−9

−6

9

6

±�3,0, ±�3

−6

−150

6

25

5, �4, 0,5, �4

−7

−5

8

5

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d.


(Continued)

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67. (a) Falls to the left, rises to the right(b) (c) Answers will vary.(d)

68. (a) Rises to the left, rises to the right(b) (c) Answers will vary.(d)

69. (a) Rises to the left, rises to the right(b) No zeros (c) Answers will vary.(d)

70. (a) Falls to the left, falls to the right

(b) 2, 8 (c) Answers will vary.

(d)

71. (a) Falls to the left, rises to the right

(b) 0, 3 (c) Answers will vary.

(d)

72. (a) Rises to the left, falls to the right

(b) 1 (c) Answers will vary.

(d)

73. (a) Falls to the left, rises to the right

(b) 0, 2, 3 (c) Answers will vary.

(d)

x−1

2

1

41

(2, 0) (3, 0)(0, 0)

−2

−15 6−2−3

3

4

5

6

7

y

−2 −1 2

−1

2

3

x(1, 0)

y

−1 1 2 4

−4

−3

1

x(0, 0) (3, 0)

y

4 6 10

2

4

6

8

10

x(2, 0) (8, 0)

y

−4 −2 2 4

2

6

8

t

y

x−1−3−4

2

1

1 43

4

−4

(−2, 0) (0, 0) (2, 0)

3

y

0, ±2

x

−4

−8

−4−8−12

12

4

4

(−3, 0)

8 12

(3, 0)

(0, 0)

y

0, ±3

f �x� � x5 � 10x4 � 14x3 � 88x2 � 183x � 90

f �x� � x5 � 16x4 � 96x3 � 256x2 � 256x

f �x� � x4 � 4x3 � 23x2 � 54x � 72

f �x� � x4 � x3 � 15x2 � 23x � 10

f �x� � x3 � 27x2 � 243x � 729

f �x� � x3 � 3x

f �x� � x3 � 9x2 � 6x � 56

f �x� � x3 � 2x2 � 3x

f �x� � x2 � 12x � 32

f �x� � x2 � 4x � 4

f �x� � x3 � 10x2 � 27x � 22

f �x� � x2 � 2x � 2

f �x� � x5 � 5x3 � 4x

f �x� � x 4 � 4x3 � 9x2 � 36x

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(Continued)

74. (a) Rises to the left, falls to the right(b) (c) Answers will vary.(d)

75. (a) Rises to the left, falls to the right(b) (c) Answers will vary.(d)

76. (a) Rises to the left, rises to the right(b) (c) Answers will vary.(d)

77. (a) Falls to the left, rises to the right(b) 0, 4 (c) Answers will vary.(d)

78. (a) Falls to the left, rises to the right(b) 0, 4 (c) Answers will vary.(d)

79. (a) Falls to the left, falls to the right(b) (c) Answers will vary.(d)

80. (a) Falls to the left, rises to the right(b) (c) Answers will vary.(d)

81. 82.

Zeros: Zeros:odd multiplicity odd multiplicity; 0,

even multiplicity83. 84.

Zeros: Zeros:even multiplicity; even multiplicity

odd multiplicity3, 92,

�2, 53,�1,

−12 12

−3

21

−12

−6

18

14

±2�2,0, ±2,

−9 9

−6

6

−9 9

−6

6

−6 −4 −2 4 6 8

2

4

6

x(−1, 0) (3, 0)

y

�1, 3

−3 −1 1 2 3

−6

−5

−2

−1

t(−2, 0) (2, 0)

y

±2

−4 −2 2 4 6 8 10 12

4

6

8

10

12

14

x(0, 0) (4, 0)

y

−4 −2 2 6 8

2

x(0, 0) (4, 0)

y

x(0, 0)

−300

2−2−6

100

(4, 0)

(−4, 0)

6

−200

y

0, ±4

x−10

(−5, 0) (0, 0)

−20

5 10−15

5

y

�5, 0

x41

(0, 0)

32−2−3−4

4

8

16

20

32

− , 0 (( 52, 0 ((

12

y

�32, 0, 52

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d.


(Continued)

85.

86.

87.

88.

89. (a)

(b) Domain:(c)

6 inches 24 inches 24 inches(d)

90. (a) Answers will vary. (b) Domain:(c)

91. (a)(b)(c) inches(d)

When the volume is maximum at dimensions of gutter are 3 inches 6 inches3 inches

(e)

The maximum value is the same.(f) No. Answers will vary.

92. (a) (b)(c)

(d) Radius feet; Length feet93.

The model is a good fit.94.

The model is a good fit.95. Region 1: 259,370 Region 2: 223,47096. Answers will vary.

Answers will vary.97. (a)

(b)(c) Vertex:(d) The results are approximately equal.

98.99. False. A fifth-degree polynomial can have at most four

turning points.100. True. has one repeated solution.101. True. The degree of the function is odd and its leading

coefficient is negative, so the graph rises to the left andfalls to the right.

102. (a) Degree: 3; Leading coefficient: positive(b) Degree: 2; Leading coefficient: positive(c) Degree: 4; Leading coefficient: positive(d) Degree: 5; Leading coefficient: positive

103.

−3 −2 −1 1 2 3−1

1

2

3

4

5

x

y

f �x� � �x � 1�6

x � 200

�15.22, 2.54�t � 15

−10 45

−5

60

7120

13

180

7140

13

200

� 7.72� 1.93

00

2

150

r ≥ 0V �163 �r 3

00

6

4000

��

V � 3456;x � 3,

0 inches < x < 6V � �384x2 � 2304xA � �2x2 � 12x

x � 2.51 2 3 4 5 6

120

240

360

480

600

720

x

V

0 < x < 6

x � 6

0 180

3600

��

0 < x < 18 � x�36 � 2x�2

� �36 � 2x��36 � 2x�x V � l � w � h

��4, �3, ��1, 0, �0, 1, �3, 4; � ±3.113, ±0.556

��2, �1, �0, 1; � �1.585, 0.779

�0, 1, �6, 7, �11, 12; � 0.845, 6.385, 11.588

��1, 0, �1, 2, �2, 3; � �0.879, 1.347, 2.532

x 1 2 3 4 5 6 7

V 1156 2048 2700 3136 3380 3456 3388

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(Continued)

(a) Vertical shift of two units; Even(b) Horizontal shift of two units; Neither even nor odd(c) Reflection in the y-axis; Even(d) Reflection in the x-axis; Even(e) Horizontal stretch; Even(f) Vertical shrink; Even(g) Neither odd nor even(h) Even

104. (a)

is decreasing. is increasing.(b) Either always increasing or always decreasing. The

behavior is determined by a.(c)

Because is not always increasing or alwaysdecreasing, cannot be written in the form

105. 106.107. 108.

109. 110. 111.

112. 113. 114.

115. 116.

117. Horizontal translation four units to the left of

118. Reflection in the x-axis and vertical shift of three units upof

119. Horizontal translation one unit left and vertical translationfive units down of

120. Reflection in the x-axis, horizontal translation, and verti-cal translation of

121. Vertical stretch by a factor of 2 and vertical translationnine units up of

122. Vertical shrink, reflection in the x-axis, horizontal shift of 3 units to the left, and vertical shift of 10 units up of

y

x1−1 2

1

2

3

4

5

6

7

8

9

3 4 5 6 7 8 9

y � x�

y

x−6

−2

1

2

3

4

5

6

−3 −2 −1 21

y � x�

y

x3

−3

3

6

9

12

15

−3 6 9 12 15

y � �x

y

x−3 −2 −1

−5

−3

−2

−1

1

1 2 3

y � �x

y

x−4 −3 −1

−4

−3

−2

−1

1

2

4

1 3 4

y � x2

y

x−7 −6 −5 −4 −3 −2 −1

−1

1

2

3

4

5

6

7

1

y � x2

�2 ± �313

�5 ± �1854

4 ± �141 ± �22�12

�54, 13�

23, 8�

72, 4

�y � 6��y2 � 6y � 36�x2�4x � 5��x � 3�x�6x � 1��x � 10��5x � 8��x � 3�

H�x� � a�x � h�5 � k.H

H�x�

−9 9

−6

6

y2y1

−12 12

−8

y1y2

8

g�x� � x16;g�x� � x3;

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d.



Vocabulary Check (page 159)1. dividend; divisor; quotient; remainder2. improper; proper 3. synthetic division4. factor 5. remainder

1. Answers will vary. 2. Answers will vary.

3. 4.

5. 6. 7.

8. 9.

10. 11.

12. 13.

14. 15.

16. 17.

18. 19.

20. 21. 22.

23. 24.

25.

26.

27.

28.

29. 30.

31.

32.

33.

34. 35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45. (a) 1 (b) 4 (c) 4 (d) 195446. (a) 14 (b) 3122 (c) 434 (d) 247. (a) 97 (b) (c) 17 (d)48. (a) (b) 20 (c) 65.5 (d) 566849. Zeros:50. Zeros:51. Zeros:

52. Zeros:

53. Zeros:

54. Zeros:

55.Zeros:

56.Zeros:

57. (a) Answers will vary. (b)(c) (d)(e)


59. (a) Answers will vary. (b)(c)(d)(e)

60. (a) Answers will vary.

(b)(c) f �x� � �4x � 3��2x � 1��x � 2��x � 4�

�4x � 3�, �2x � 1�

−180

−6 6

20

1, 2, 5, �4f �x� � �x � 1��x � 2��x � 5��x � 4�

�x � 1�, �x � 2�

−4 3

−10

35

13, �3, 2f �x� � �3x � 1��x � 3��x � 2�

3x � 1−1

−6 6

7

12, �2, 1f �x� � �2x � 1��x � 2��x � 1�

2x � 12 � �5, 2 ��5, �3

�x � 2 � �5 ��x � 2 � �5 ��x � 3�;1, 1 ��3, 1 � �3

�x � 1��x � 1 � �3 ��x � 1 � �3 �;�2, ��2, �2�x � �2 ��x � �2 ��x � 2�;��3, �3, �2�x � �3 �� x � �3 ��x � 2�;

23, 34, 14�3x � 2��4x � 3��4x � 1�;

12, 5, 2�2x � 1��x � 5��x � 2�;

�4, �2, 6�x � 4��x � 2��x � 6�;2, �3, 1�x � 2��x � 3��x � 1�;

�2.5�199�

53

f �2 � �2 � � 0f �x� � �x � 2 ��2 ��3x2 � �2 � 3�2 �x � 8 � 4�2,f �1 � �3� � 0

f �x� � �x � 1 ��3��4x2 � �2 � 4�3�x � �2 � 2�3�,f ��5 � � 6

f �x� � �x � �5 ��x2 � �2 � �5 �x � 2�5 � 6,

f ��2 � � �8

f �x� � �x � �2 ��x2 � �3 � �2 �x � 3�2 � 8,

f �15� �

135f �x� � �x �

15��10x2 � 20x � 7� �

135 ,

f ��23� �

343f �x� � �x �

23��15x3 � 6x � 4� �

343 ,

f �x� � �x � 2��x2 � 7x � 3� � 2, f ��2� � 2

f (x) � �x � 4��x2 � 3x � 2� � 3, f �4� � 3

3x 2 �1

2x �

3

4�

49

8x � 12

4x2 � 14x � 30�x 2 � 3x � 6 �11

x � 1

�x3 � 6x2 � 36x � 36 �216

x � 6

�3x3 � 6x 2 � 12x � 24 �48

x � 2

�3x3 � 6x2 � 12x � 24 �48

x � 2

x2 � 9x � 81x2 � 8x � 64

x4 � 16x3 � 48x 2 � 144x � 312 �856

x � 3

10x3 � 10x2 � 60x � 360 �1360

x � 6

5x2 � 10x � 26 �44

x � 2

5x2 � 14x � 56 �232

x � 4

3x2 � 2x � 12�x2 � 10x � 25

9x 2 � 164x2 � 95x 2 � 3x � 2

3x2 � 2x � 52x �17x � 5

x2 � 2x � 1

x � 3 �6x2 � 8x � 3

�x � 1�3x2 �

x2 � 7

x3 � 1

x2 � 2x � 4 �2x � 11

x2 � 2x � 3x �

x � 9

x2 � 1

3x � 5 �2x � 3

2x2 � 14 �

9

2x � 1

7 �11

x � 2x2 � 7x � 18 �

42x � 3

x3 � 3x2 � 12x2 � 4x � 3

x2 � 3x � 15x � 32x � 4

−12 12

−8

8

−9 9

−6

6

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(Continued)

(d)

(e)





65. (a) Zeros are 2 and (b) (c)

66. (a) Zeros are 4 and (b) (c)

67. (a) Zeros are and (b)(c)

68. (a) Zeros are and (b)(c)

69. 70.

71. 72.73. (a) and (b)

(c)

Answers will vary.(d) 1614 thousand. No, because the model will approach

negative infinity quickly.74. (a) and (b)

(c) $49.3875. False. is a zero of f.76. True.

77. True. The degree of the numerator is greater than thedegree of the denominator.

78. (a)(b)

79. 80.81. The remainder is 0.82. Multiply the divisor and the quotient and add the remain-

der to obtain the dividend.83. 84.85. 0; is a factor of f.86. Because is in factored form, it is easier to evaluate

directly.

87. 88. 89. 90.

91. 92.

93.94.95.96. f �x� � x4 � 3x3 � 5x2 � 9x � 2

f �x� � x3 � x2 � 7x � 3f �x� � x2 � 5x � 6f �x� � x3 � 7x2 � 12x

�3 ± �212

�3 ± �32

54

, 32

�75

, 2±�21

4±

53

f �x�x � 3

c � 42c � �210

x2n � xn � 3x2n � 6xn � 9f �x� � ��x � 3�x2 � 1 � �x3 � 3x2 � 1f �x� � �x � 2�x2 � 5 � x3 � 2x2 � 5

f �x� � �2x � 1��x � 1��x � 2��x � 3��3x � 2��x � 4�

�47

R � 0.00260x3 � 0.0292x 2 � 1.558x � 15.63

20

12

40

M � �0.242x3 � 12.43x 2 � 173.4x � 2118

31200

13

1800

x 2 � 9x � 1, x � ±2x2 � 3x, x � �2, �1

x 2 � 7x � 8, x � �82x2 � x � 1, x �32

f �s� � �s � 6��s � �3 ��5 ��s � �3 ��5 �x � 6

�5.236.6, �0.764,h �t� � �t � 2��t � �2 � �3 � �t � �2 � �3 �x � �2

�3.732.�2, �0.268,g�x� � �x � 4��x � �2 ��x � �2 �x � 4�±1.414.

f �x� � �x � 2��x � �5 ��x � �5 �x � 2�±2.236.

−8 8

−240

60

±4�3, �3f �x� � �x � 4�3 ��x � 4�3 ��x � 3�

x � 4�3−6

−6 6

14

±�5, 12f �x� � �x � �5 ��x � �5 ��2x � 1��x � �5 �

−4 4

−80

100

3, �52, 35f �x� � �x � 3��2x � 5��5x � 3�

x � 3−40

−9 3

320

�7, �12, 23

f �x� � �x � 7��2x � 1��3x � 2�x � 7

−3 5

−380

40

�34, 12, �2, 4

t 3 4 5 6 7 8

1703 1608 1531 1473 1430 1402M(t)

t 9 10 11 12 13

1388 1385 1392 1409 1433M(t)

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d.



Vocabulary Check (page 167)1. (a) iii (b) i (c) ii 2.3. complex numbers; 4. principal square5. complex conjugates

1. 2.

3. 4. 5.

6. 7. 8.

9. 10. 11. 8

12. 45 13. 14.

15. 16. 17.

18. 19. 4 20.

21. 22. 4 23.

24. 25. 26.

27. 28. 29.

30. 31. 24 32. 18

33. 34. 35.

36. 37. 38.

39. 40.

41. 20 42.

43. 44.

45. 46. 47.

48. 49. 50.

51. 52. 53.

54. 55. 56.

57. 58. 59.

60. 61. 62.

63.

64.

65. 66. 67.

68. 69. 70.

71. 72. 73.

74. 75. 76.

77. 78. 79.

80. 81. 82.

83. (a)

(b)

84. (a) 8 (b) 8 (c) 8

85. (a) 16 (b) 16 (c) 16 (d) 16

86. (a) 1 (b) i (c) (d)

87. False. If the complex number is real, the number equals its conjugate.

88. True.

89. False.

90.

91–92. Proof

93.

94.

95.

96.

97. 98. 14

99. 100.

101.

102.

103. 1 liter

r ��m1m2F

F

a ��3V�b

2�b

�43

272

�31

4x2 � 20x � 25

3x2 �232 x � 2

x3 � x2 � 2x � 6

�x2 � 3x � 12

��6��6 � �6 i�6 i � 6i2 � �6

i44 � i150 � i74 � i109 � i61 � 1 � 1 � 1 � i � i � 1

56 � 56

36 � 6 � 14 �?

56

��i�6 �4� ��i�6 �2

� 14 �?

56

x 4 � x2 � 14 � 56

�i�1

z �11,240

877�

4630877

i

z1 � 9 � 16i, z 2 � 20 � 10i

18

ii�8

�375�3i�i�5i

�4 � 2i�1 � 6i13

±�23

3i

57

±5�15

737

±�3414

i2 ± �2i

18

±�11

8i�

52

, �32

1

3± 2i

�2 ± 12i�3 ± i1 ± i

�2 � 4�6 i

�21 � 5�2� � �7�5 � 3�10�i�75�10�5�2

�2�3517 �

2017i62

949 �297949i

125 �

95i�

12 �

52i60

169 �25169i

�120

1681 �27

1681i8 � 4i�5 � 6i

4 � i45 �

35i5

2 �52i

841 �

1041i7i�5i

1 � �8, 9 � 4�2�8, 8

��15 i, 15�2�5i,

�3 � �2 i, 11�1 � �5 i, 6

7 � 12i, 1936 � 3i, 45�8i

�10�5 � 12i�9 � 40i

32 � 72i

12 � 30i6 � 22i5 � i

�4.2 � 7.5i16 �

76i17 � 18i

�14 � 20i3 � 3�2 i

�3 � 11i8 � 4i

11 � i0.02i0.3i

4 � 2i�1 � 6i

2i5�3 i

1 � 2�2 i2 � 3�3 i3 � 4i

4 � 3ia � 0, b � �52a � 6, b � 5

a � 13, b � 4a � �10, b � 6

a � bi

��1; �1

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Vocabulary Check (page 179)1. Fundamental Theorem of Algebra2. Linear Factorization Theorem 3. Rational Zero4. conjugate 5. irreducible over the reals6. Descartes’ Rule of Signs 7. lower; upper

1. 2. 0, 3. 4. 8

5. 6. 2, 7.

8.

9.

10. 11. 1, 2, 3 12.

13. 14. 1, 2, 6 15.

16. 3 17. 18. 19.

20. 21. 22.

23. 24.

25. (a)(b)

(c)

26. (a)

(b) (c)

27. (a)(b) (c)

28. (a)

(b) (c)

29. (a)(b) (c)

30. (a)(b) (c)

31. (a)

(b) (c)

32. (a)

(b) (c)

33. (a)

(b)

34. (a)

(b)

35. (a)

(b)

36. (a)

(b)

37. 38.

39. 40.

41.

42. x 4 � 8x 3 � 9x2 � 10x � 100

3x4 � 17x3 � 25x2 � 23x � 22

x3 � 10x 2 � 33x � 34x3 � 4x 2 � 31x � 174

x3 � 4x 2 � 9x � 36x3 � x2 � 25x � 25

g�x� � �x � 3��x � 3��2x � 3��3x � 1�±3, 1.5, 0.333

h�x� � x�x � 3��x � 4��x � �2 ��x � �2 �0, 3, 4, �±1.414

P�t) � �t � 2��t � 2��t � �3 ��t � �3 �±2, �±1.732

f �x� � �x � 1��x � 1��x � �2 ��x � �2 �±1, �±1.414

�2, 1

8±�145

8−8 8

−24

8

±14, ±3

4, ±94±1, ±2, ±3, ±6, ±9, ±18, ±1

2, ±32, ±9

2,

1, 34, �18

−1 3

−2

6

±1, ±3, ±12, ±3

2, ±14, ±3

4, ±18, ±3

8, ± 116, ± 3

16, ± 132, ± 3

32

±2, ±12

−8 8

−15

9

±1, ±2, ±4, ±12, ±1

4

�12, 1, 2, 4

−4 8

−8

16

±1, ±2, ±4, ±8, ±12

�1, 32, 52

−9 −6 −3 6 9 12

12

15

x

y

±1, ±3, ±5, ±15, ±12, ±3

2, ±52, ±15

2 , ±14, ±3

4, ±54, ±15

4

�14, 1, 3

−6 −4 −2 2 4 6 8 10

−6

−4

2

4

x

y

±1, ±3, ±12, ±3

2, ±14, ±3

4

23, 2, 4

−4 −2 6 8 10 12

−6

−4

2

4

6

8

10

x

y

±1, ±2, ±4, ±8, ±16, ±13, ±2

3, ±43, ±8

3, ±163

�2, �1, 2

−6 −4 4 6

−8

−6

−4

2

4

x

y

±1, ±2, ±4

�2, 0, 1�6, 12, 1

0, �1, �3, 4�1, 2±1, 5, 52

�2, 3, ±23

13, 31

2, �1

�1, �101, �1, 4

�2, �1, 3±1, ±2, ±12, ±1

4

±52, ±9

2, ±152 , ±45

2±1, ±3, ±5, ±9, ±15, ±45, ±12, ±3

2,

±1, ±2, ±4, ±8, ±16

±1, ±3±3i3,�6, ± i

�5,2, �4±1�3,0, 6

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d.


(Continued)

43. (a) (b)

(c)

44. (a)

(b)

(c)

45. (a)

(b)

(c)

46. (a)

(b)

(c)

47. 48. 49.

50. 51. 52.

53. 54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73. 74.

75. 76.

77. 78.79. No real zeros 80. Two or no positive zeros81. No real zeros 82. Two or no positive zeros83. One positive zero 84. One or three positive zeros

85. One or three positive zeros

86. One or three negative zeros

87–90. Answers will vary.

91. 92. 93. 94.

95. 96. 97. 98.

99. d 100. a 101. b 102. c

103. (a)

(b)Domain:

(c)

(d) 8 is not in the domain of V.

104. (a) Answers will vary.(b)

(c) the value is

physically impossible because x is negative.

105. or $384,000

106. or $315,000

107. (a)

(b) 4 feet by 5 feet by 6 feet

108. (a) feet squared

(b) feet

feet

(c) the corral hasdimensions 313.2 feet by 191.6 feet.

109. or 4000 units

110. No. Setting and solving the resulting equationyields imaginary roots.

111. No. Setting and solving the resultingequation yields imaginary roots.

p � 9,000,000

h � 64

x � 40,

A � �160 � x��250 � 2x� � 60,000;

250 ��410 � �248,100

2� 294.05

160 ��410 � �248,100

2� 204.05

A � �160 � x��250 � x� � 60,000

V � x3 � 9x2 � 26x � 24 � 120

x � 31.5,

x � 38.4,

x �15 � 15�5

215,

15 ± 15�5

2;

20 inches � 20 inches � 40 inches

0 300

18,000

12, 72, 8;

1.82 centimeters � 5.36 centimeters � 11.36 centimeters

Length of sides ofsquares removed

V

x1 3 42 5

25

50

75

100

125

Vol

ume

of b

ox

0 < x < 92

V � x�9 � 2x��15 � 2x�

15

9

x

x

x9 − 2x 15 − 2x

�2, �13, 12±1, 14�3, 12, 4±2, ±3

2

23�

34�

32, 13, 321, �1

2

1 ± �3 i, 2�2, �12, ± i

13 ± 2

3i, 1�34, 1 ± 1

2i

1 ± 2i, 12�10, �7 ± 5i

�x � 2i��x � 2i��x � 5i��x � 5i�±2i, ±5i;

�x � i��x � i ��x � 3i��x � 3i�± i, ±3i;

�x � 3�2�x � i��x � i��3, ± i;

�x � 2�2�x � 2i��x � 2i�2, ±2i;

�3x � 2��x � 1 � �3 i��x � 1 � �3 i�1 ± �3i, �23;

�5x � 1��x � 1 � �5 i��x � 1 � �5 i��15, 1 ± �5 i;

�x � 5��x � 2 � �3i��x � 2 � �3 i��2 ± �3i, �5;

�x � 2��x � 1 � �2 i��x � 1 � �2 i��2, 1 ± �2 i;

�x � 4��x � 3 � 2i��x � 3 � 2i�3 ± 2i, �4;

�x � 2��x � 2 � i ��x � 2 � i �2, 2 ± i;

�x � 1��x � 1 � i��x � 1 � i�1 ± i, 1;

�z � 1 � i��z � 1 � i�1 ± i;

�y � 5��y � 5��y � 5i��y � 5i�±5, ±5i;

�x � 3��x � 3��x � 3i��x � 3i�±3, ±3i;

�x � 5 � �2 ��x � 5 � �2 ��5 ± �2;

�x � 2 � �3 ��x � 2 � �3 �2 ± �3;

�x �1 � �223i

2 ��x �1 � �223i

2 �1 ± �223 i

2;

�x � 5i ��x � 5i �±5i;

�2, �1 ± 3i2, �3 ± �2 i, 1

�23, 1 ± �3 i�3 ± i , 14�3, 5 ± 2i

±2i, 1, �12�1, ±3i�

32, ±5i

�x �3 � �29

2 ��x �3 � �29

2 ��x � 2i��x � 2i�

�x 2 � 4��x �3 � �29

2 ��x �3 � �29

2 ��x 2 � 4��x 2 � 3x � 5�

�x � 1 � �2 i��x � 1 � �3 ��x � 1 � �3 ��x � 1 � �2 i��x � 1 � �3 ��x � 1 � �3 ��x2 � 2x � 3��x2 � 2x � 2��x2 � 2x � 3�

�x � 1 � �2 i��x � �6��x � �6��x � 1 � �2 i��x � �6 ��x � �6 ��x 2 � 2x � 3��x 2 � 6��x 2 � 2x � 3��x � 3i ��x � 3i ��x � �3 ��x � �3 �

�x2 � 9��x � �3 ��x � �3 ��x2 � 9��x2 � 3�

333202CB02_AN.qxd 1/1/70 09:37 AM Page 16

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(Continued)

112. (a)

(b) (c) 2000(d) 2001(e) Yes, the attendance

will continue to grow.

113. False. The most complex zeros it can have is two, and theLinear Factorization Theorem guarantees that there arethree linear factors, so one zero must be real.

114. False. f does not have real coefficients.

115. 116.

117. 118.

119. The zeros cannot be determined.

120.

121. (a) (b) (c) (d)

122. (a) No (b) No123. Answers will vary. There are infinitely many possible

functions for f. Sample equation and graph:

124. Answers will vary. Sample graph:

125. Answers will vary.

126. (a)

(b) The graph touches the axis at

(c) The least possible degree of the function is 4, becausethere are at least four real zeros (1 is repeated) and afunction can have at most the number of real zerosequal to the degree of the function. The degree cannotbe odd by the definition of multiplicity.

(d) Positive. From the information in the table, it can be concluded that the graph will eventually rise to theleft and rise to the right.

(e)

(f)

127. (a) (b)

128. (a) Not correct because f has as an intercept.

(b) Not correct because the function must be at least afourth-degree polynomial.

(c) Correct function

(d) Not correct because has as an intercept.

129. 130.

131. 132. 106

133. 134.

135. 136.

137. 138.

x42 6 8

6

4

8

10

−2(−4, 0)

(4, 2)(0, 2)

(8, 4)

yy

xx−1−2 1 2

4

3

(−1, 0)

(0, 2)(1, 2)

(2, 4)

y

x1−1−2−3−4

1

2

3

4

−1

−2

(2, 0)

(0, 2)

(−2, 2)

(−4, 4)

y

x42 6 8

6

8

10

−2(−2, 0)

(2, 4)

(0, 4)

(4, 8)

y

x2 31−1−2 4

2

3

−2

−3

(2, 0)(0, 0)

(4, 2)

y

(−2, −2)

x2 3 4 5 6

1

−1

−2

2

3

4

(0, 0)

(4, 2)

(2, 2)

(6, 4)

y

20 � 40i

12 � 11i�11 � 9i

��1, 0�k

�0, 0�

x2 � 2ax � a2 � b2x2 � b

x−1−3 2

(−2, 0)(1, 0) (4, 0)

−4−6−8

2

3 5

−10

y

f �x� � x4 � 4x3 � 3x2 � 14x � 8

x � 1.x-

�2, 1, 4

x4 5

50

10

(−1, 0)

(3, 0)

(1, 0) (4, 0)

y

x−4

8

4

4 8 12−8

(−2, 0)

12

, 0 ( ((3, 0)

y

f �x� � �2x3 � 3x2 � 11x � 6

k > 4k < 0k � 40 < k < 4

�r1, �r2, �r3

r1

2,

r2

2,

r3

25 � r1, 5 � r2, 5 � r3

r1, r2, r3r1, r2, r3

70

13

12

A � 0.0167x3 � 0.508x2 � 5.60x � 13.4

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d.



Vocabulary Check (page 193)1. rational functions 2. vertical asymptote3. horizontal asymptote 4. slant asymptote

1. (a)

(b) Vertical asymptote:Horizontal asymptote:

(c) Domain: all real numbers x except

2. (a)

(b) Vertical asymptote:Horizontal asymptote:


3. (a)

(b) Vertical asymptotes:Horizontal asymptote:


4. (a)

(b) Vertical asymptotes:Horizontal asymptote:

(c) Domain: all real numbers x except 5. Domain: all real numbers x except

Vertical asymptote:Horizontal asymptote:

6. Domain: all real numbers x except Vertical asymptote:Horizontal asymptote:



9. Domain: all real numbers x except Vertical asymptotes:

10. Domain: all real numbers x except Vertical asymptote:Horizontal asymptote: none

11. Domain: all real numbers xHorizontal asymptote:

12. Domain: all real numbers xHorizontal asymptote:No vertical asymptote

13. d 14. a 15. c 16. b17. 1 18. None 19. 6 20. 221. Domain: all real numbers x except

Vertical asymptote: horizontal asymptote:22. Domain: all real numbers except

Vertical asymptote: horizontal asymptote:23. Domain: all real numbers x except 3;

Vertical asymptote: horizontal asymptote:24. Domain: all real numbers x except 2;

Vertical asymptote: horizontal asymptote:25. Domain: all real numbers x except

Vertical asymptote: horizontal asymptote:26. Domain: all real numbers x except

Vertical asymptote: horizontal asymptote:27. (a) Domain: all real numbers x except

(b) y-intercept:(c) Vertical asymptote:

Horizontal asymptote:(d)

−3 −1

−2

−1

1

2

x

( (0, 12

y

y � 0x � �2

�0, 12�x � �2

y � 1x � �13;

32;x � �

13,

y �12x �

12;

x � �1, 12;y � 1x � 1;

x � 1,y � 1x � 3;

x � �1,y � 0x � 3;

x � ±3;xy � 0x � �4;

x � ±4;

y � 3

y � 3

x � �1x � �1

x � ±1x � ±1

y � �52

x � �12

x � �12

y � �1x � 2

x � 2y � 0

x � 2x � 2

y � 0x � 0

x � 0x � ±1

y � 0x � ±1

x � ±1y � 3

x � ±1

x � 1y � 5

x � 1

x � 1y � 0

x � 1

x

0.5

0.9

0.99

0.999 �1000

�100

�10

�2

f �x� x

1.5 2

1.1 10

1.01 100

1.001 1000

f �x� x

5 0.25

10

100

1000 0.001

0.01

0.1

f �x�

0.5

0.9

0.99

0.999 �4995

�495

�45

�5

f �x�x

1.5 15

1.1 55

1.01 505

1.001 5005

f �x�x

5 6.25

10

100

1000 5.005

5.05

5.55

f �x�x

x

0.5

0.9

0.99

0.999 �1498

�147.8

�12.79

�1

f �x� x

5 3.125

10

100

1000 3

3.0003

3.03

f �x�x

1.5 5.4

1.1 17.29

1.01 152.3

1.001 1502

f �x�

0.5

0.9

0.99

0.999 �1999

�199

�18.95

�2.66

f �x�x

1.5 4.8

1.1 20.95

1.01 201

1.001 2001

f �x�x

5

10

100 0.04

1000 0.004

0.40

0.833

f �x�x

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(Continued)

28. (a) Domain: all real numbers x except (b) y-intercept:(c) Vertical asymptote:






31. (a) Domain: all real numbers x except (b) x-intercept:

y-intercept:(c) Vertical asymptote:

Horizontal asymptote:

(d)




33. (a) Domain: all real numbers x(b) Intercept:(c) Horizontal asymptote:(d)

34. (a) Domain: all real numbers t except (b) t-intercept: (c) Vertical asymptote:


−2 −1 1 2

−3

−1

t

( ), 012

y

y � �2

t � 0�12, 0�

t � 0

−2 −1 1 2

−1

2

3

x(0, 0)

y

y � 1�0, 0�

−2 −1 2 3 4

4

5

6

x

(0, 1) , 013( )

y

y � 3x � 1

�0, 1��1

3, 0�x � 1

( (

−6 −4 2 4x

6

52

− , 0

(0, 5)

y

y � 2x � �1

�0, 5��5

2, 0�x � �1

1 2 4

−3

−2

−1

1

2

3

x

( (0, 13

y

y � 0x � 3

�0, 13�x � 3

−4 −3 −1

−2

−1

1

2

x( (0, − 1

2

y

y � 0x � �2

�0, �12�

x � �2

2 4 5 6

−3

−2

−1

1

2

3

x

( (0, 13−

y

y � 0x � 3

�0, �13�

x � 3

333202CB02_AN.qxd 1/1/70 09:37 AM Page 19

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d.


(Continued)

35. (a) Domain: all real numbers s

(b) Intercept:

(c) Horizontal asymptote:

(d)

36. (a) Domain: all real numbers x except

(b) y-intercept:

(c) Vertical asymptote:


(d)


(b) x-intercepts: and

y-intercept:

(c) Vertical asymptotes:


(d)


(b) x-intercepts:

y-intercept:



(d)


(b) x-intercept:

y-intercept:



(d)

40. (a) Domain: all real numbers x except 1, 3

(b) x-intercepts:

y-intercept:



(d)


(b) Intercept:



(d) y

x−2−4−6 4 6

−4

−6

2

4

6

(0, 0)

y � 1

x � 2

�0, 0�x � 2, �3

x

1

2

3

2 4 5

−2

−3

−4

−5

4

13

−0,( (

( 1, 0)−

(2, 0)

y

y � 0

x � 3x � 1,x � �2,

�0, �13�

��1, 0�, �2, 0�x � �2,

y

x

0, −32( (

, 012( (−

(3, 0)

−4 −3 43

3

6

9

y � 0

x � ±1x � 2,

�0, �32�

�3, 0�, ��12, 0�

x � ±1, 2

y

x−4−6 2 4 6

−2

−4

−6

2

4

6

(−2, 0)

(4, 0)

(0, 0.88)

y � 1

x � ±3

�0, 0.88�

��2, 0�, �4, 0�

x � ±3

y

x−4−6 6

2

4

6

(1, 0)

(4, 0)

y � 1

x � ±2

�0, �1�

�4, 0��1, 0�

x � ±2

1 3

−4

−3

−2

−1

x( (0, − 1

4

y

y � 0

x � 2

�0, �14�

x � 2

1 2

−2

−1

1

2

s(0, 0)

y

y � 0

�0, 0�

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(Continued)


(b) y-intercept:



(d)


(b) x-intercept:

y-intercept:



(d)

44. (a) Domain: all real numbers x except 2

(b) x-intercept:

y-intercept:



(d)

45. (a) Domain: all real numbers t except

(b) t-intercept:

y-intercept:

(c) Vertical asymptote: None

Horizontal asymptote: None

(d)


y-intercept:(c) Vertical asymptote: None

Horizontal asymptote: None(d)

47. (a) Domain of f : all real numbers x except Domain of g: all real numbers x

(b) Vertical asymptotes: none(c)

(d)

(e) Because there are only a finite number of pixels, thegraphing utility may not attempt to evaluate the function where it does not exist.

48. (a) Domain of f : all real numbers x except Domain of g: all real numbers x

(b) x; Vertical asymptotes: none(c)

x � 0, 2

−4

−3

2

1

x � 1;

x � �1

y

x

(0, 4)

(−4, 0)

−2−6 2 4 6−2

2

4

6

8

10

�0, 4��4, 0�

x � 4

y

t−1−2−3 1 2 3

−1

−2

−3

1

2

3

(1, 0)

(0, −1)

�0, �1��1, 0�

t � �1

y

x−1−2−3−4 3 4

1

(0, −2)

, 023( (

y �32

x � �0.5

�0, �2��2

3, 0�x � �0.5,

y

x−2−3−4−5 2 3

2

3

4

, 0 12 ))

0, − 13 ))

1

y � 1

x � �32

�0, 13��1

2, 0�x � �

32, 2

y

x

(0, −1.66)2 4 6 8

−4

−6

2

4

6

y � 0

x � 3

�0, �1.66�

�4x � 3,

0 1

Undef. 0

0�1�1.5�2�2.5�3�4g�x�

�1�1.5�2.5�3�4f �x�

�0.5�1�1.5�2�3x

0 1 1.5 2 2.5 3

Undef. 1 1.5 Undef. 2.5 3

0 1 1.5 2 2.5 3�1g�x�

�1f �x�

�1x

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d.


(Continued)

(d)


49. (a) Domain of f: all real numbers x except Domain of g: all real numbers x except

(b) Vertical asymptote:

(c)

(d)

(e) Because there are only a finite number of pixels, thegraphing utility may not attempt to evaluate thefunction where it does not exist.

50. (a) Domain of f : all real numbers x except Domain of g: all real numbers x except


(c)

(d)


51. (a) Domain: all real numbers x except (b) x-intercepts:(c) Vertical asymptote:

Slant asymptote:

(d)

52. (a) Domain: all real numbers x except (b) No intercepts(c) Vertical asymptote:

Slant asymptote:(d)


Slant asymptote:(d)

54. (a) Domain: all real numbers x except (b) x-intercepts:(c) Vertical asymptote:

Slant asymptote:(d)


Slant asymptote: y � xx � 0

x � 0

−8 −6 −4 −2 4 6 8

−8

−6

−4

2

4

6

8

x

y = −x

(−1, 0) (1, 0)

y

y � �xx � 0

��1, 0�, �1, 0�x � 0

y = 2x

−6 −4 −2 2 4 6

−6

2

4

6

x

y

y � 2xx � 0

x � 0

y

x−4 −2−6 2 4 6

−2

−4

2

4

6

y = x

y � xx � 0

x � 0

y

x−4−6 6

−2

−4

−6

2(−2, 0)

(2, 0)

y = x

y � xx � 0

�2, 0�, ��2, 0�x � 0

−1

−3

8

3

x � 42

x � 4;

x � 4x � 3, 4

−3

−2

3

2

x � 01x;

x � 0x � 0, 2

−2

−2

4

2

0 0.5 1 1.5 2 3

Undef. 2 1 Undef.

Undef. 2 1 13

12

23�2g�x�

13

23�2f �x�

�0.5x

x 0 1 2 3 4 5 6

Undef. Undef. 2 1

Undef. 2 1�2�1�23�

12g�x�

�1�23�

12f �x�

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(Continued)

(d)

56. (a) Domain: all real numbers x except (b) Intercept:(c) Vertical asymptote:

Slant asymptote:(d)

57. (a) Domain: all real numbers t except (b) intercept:(c) Vertical asymptote:

Slant asymptote:(d)

58. (a) Domain: all real numbers x except (b) Intercept:(c) Vertical asymptote:

Slant asymptote:(d)

59. (a) Domain: all real numbers x except (b) Intercept:(c) Vertical asymptotes:

Slant asymptote:(d)


Slant asymptote:(d)


Slant asymptote:(d)


Slant asymptote:(d)

−9 −6 −3 3 6 9 12 15

−9

3

6

9

12

15

x

( (0, − 52

y = 2x − 1

y

y � 2x � 1x � 2

�0, �52�

x � 2

−4 −2 2 4 6 8

−4

2

4

6

8

x(0, −1)

y = x

y

y � xx � 1

�0, �1�x � 1

−8 −6 −4 4 6 8

4

6

8

x

y = x

(0, 0)

12

y

y �12x

x � ±2�0, 0�

x � ±2

−6 −4 −2 2 4 6x

2(0, 0)

y = x

y

y � xx � ±1

�0, 0�x � ±1

y

x

13

123

23

43

13

1−1

y = x − 13

19 (0, 0)

y �13x �

19

x � �13

�0, 0�x � �

13

y

t

y = 5 − t

(0, −0.2)

−10−15−20 10

5

15

20

25

−5

y � �t � 5t � �5

�0, �0.2�y-t � �5

−4 2 4 6 8

−4

−2

2

4

6

8

x(0, 0)

y = x + 1

y

y � x � 1x � 1

�0, 0�x � 1

−6 −4 −2 2 4 6

−6

2

4

6

x

y = x

y

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d.


(Continued)


(b) y-intercept:

x-intercepts:


Slant asymptote:

(d)

64. (a) Domain: all real numbers x except 2

(b) y-intercept:

x-intercepts:


Slant asymptote:

(d)

65.

Domain: all real numbers x except

Vertical asymptote:

Slant asymptote:

66.


Vertical asymptote:

Slant asymptote:

67.

Domain: all real numbers x except Vertical asymptote:Slant asymptote:

68.


Vertical asymptote:

Slant asymptotes:

69. (a) (b)

70. (a) (b) 0

71. (a) (b)

72. (a) (b) 1, 2

73. (a)

(b) $28.33 million; $170 million; $765 million(c) No. The function is undefined at

74. (a)

(b) $4411.76; $25,000; $225,000(c) No. The function is undefined at

75. (a) 333 deer, 500 deer, 800 deer (b) 1500 deer

76. (a) Answers will vary. (b)(c)

(d) Increases more slowly; 75%

200 400 600 800 1000

0.2

0.4

0.6

0.8

1.0

x

C

�0, 950

p � 100.

00

100

300,000

p � 100.

00

100

2,000

�1, 0�, �2, 0�±1�1, 0�, ��1, 0�

�0, 0��1��1, 0�

y � �12 x � 1

y � �12 x � 1

x � �4

x � �4

−16

−6

8

10

y � �x � 3y � �x � 3

x � 0x � 0

−12

−4

12

12

y � 2x � 1

y � 2x � 1

x � �1

x � �1

−12

−10

12

6

y � x � 2

y � x � 2

x � �3

x � �3

−14

−8

10

8

y

x2−2

12

18

24

30

−6 4 6

(−2, 0)

(0, −2)

y = 2x + 7

, 0− 12( (

y � 2x � 7

x � 1

��2, 0�, ��12, 0�

�0, �2�x � 1,

y

x−1−3−4−5−6 3

−12

−18

−24

−30

−36

6

12

18

(0.5, 0)

(0, 0.5)(1, 0)

y = 2x − 7

y � 2x � 7

x � �2

�0.5, 0�, �1, 0��0, 0.5�

x � �1, �2

333202CB02_AN.qxd 1/1/70 09:37 AM Page 24


(Continued)

77. (a) Answers will vary. (b)

(c)

78.

79. (a) Answers will vary.



(c)

(d)

(e) Yes. You would expect the average speed for the roundtrip to be the average of the average speeds for the twoparts of the trip.

(f) No. At 20 miles per hour you would use more time inone direction than is required for the round trip at anaverage speed of 50 miles per hour.

80. (a)

The model is a good fit for the data.

(b) $763.8 million in sales

(c) No, horizontal asymptote at

81. False. Polynomials do not have vertical asymptotes.

82. False. The graph of crosses which

is a horizontal asymptote.

83.

84.

85. 86.

87.

88.

89.

90.

91.

92.

93. Answers will vary.

−

−2−4−6−8

x

0

22713

42

x ≥ 72x ≤ �

132 ,

20−2−4

−3x

4 6

7

8

�3 < x < 7

0

x

1−1−2−3 2 3

x < 0

0

x

1 2 3 4 5 6

103

x ≥ 103

�x � 6��x � �2��x � �2�

�x � 5��x � 2i��x � 2i�

�3x � 4��x � 9��x � 7��x � 8�

f �x� �x3

�x � 2��x � 1�

f �x� �2x2

x2 � 1

y � 0,f �x� �x

x2 � 1

y � 1454.

80

13

600

025 65

200

y � 25

x � 25

12.8 inches � 8.5 inches

11.75 inches � 5.87 inches

40

40

200

�4, �

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30 35 40 45 50 55 60

150 87.5 66.7 56.3 50 45.8 42.9y

x

333202CB02_AN.qxd 1/1/70 09:37 AM Page 25


Vocabulary Check (page 204)1. critical; test intervals 2. zeros; undefined values3.

1. (a) No (b) Yes (c) Yes (d) No

2. (a) Yes (b) No (c) Yes (d) Yes

3. (a) Yes (b) No (c) No (d) Yes

4. (a) No (b) Yes (c) Yes (d) No

5. 6. 7. 8.

9. 10.

11. 12.

13. 14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25. 26. All real numbers

27. 28.29. 30.31. 32.33. 34.

(a) (a)

(b) (b)35. 36.

(a) (a)

(b) (b)37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

−1−2−3 0 1 2 3

x

−1−2−3 0 1 2 3

x

��3, 0� � �2, ��3, �2 � �0, 3�

−1−2−3−4 10

x

−2−4

−33

4 6 820

4x

��, �3� � �0, ��34, 3� � �6, �

−14

−5

−2

−10−15 0 5

6

10

x2

−1−2−3−4−5

−

0

3

x

��14, �2� � �6, ��5, �32� � ��1, �

−2 −1

−

0 21

12

x3 6 9

5

12 15 18

x

��, �12� � �1, ��5, 15�

−1−2 0 1 2 3

x

−2 −1 10 2 3 4 5

x

��2, 3��, �1� � �4, �10

14

−1

x−1−2 210

x

��, 0� � � 14, ��, �1� � �0, 1�

x � �2, 5 ≤ x < x ≤ 41 ≤ x ≤ 42 ≤ x <

� < x ≤ �4,�2 ≤ x ≤ 0,

−12 12

−24

48

−12 12

−8

8

x ≤ �2, x ≥ 60 ≤ x ≤ 2

2 � �2 ≤ x ≤ 2 � �2x ≤ �1, x ≥ 3

−10

−4

14

12

−5 7

−2

6

��, 3��2, ��, 0 � �2, ��2, 0 � �2, ��3, �(�, 0� � �0, 32�

1 2 30−3 −2 −1

x

210−1

x

−2

12

x �12

4

x

−2−4−6−8 20

132

−

��132 , �2 � �2, �

x

−4 −3 −2 −1 0 1 2 3 4 5

��3, 2 � �3, �

0 1 2 3 4

x

��, 2

0 1 2 43

x

−1

��1, 1� � �3, �

x

543210−1−2

32 2

39− 32 2

39+

��, 32

��39

2 � � 32

��39

2, �

20

x

−8 −6 −4 −2−10

−4 + 21−4 − 21

��, �4 � �21 � ��4 � �21, �0−2−4 2 4 6 8

x2 − 5 2 + 5

��, 2 ��5 � � �2 � �5, �10−1−2−3

x

��3, 1�

−1−2−3−4 0 1 2

x

��, �3� � �1, �

0−1−2−3 1 2

x

��3, 2�−2

−1

2 40 6

7

8

xx

210−1−2−3−4−5−6

��1, 7��, �5 � �1, �

31 2 54

x

−8 −4−6

−7

0 2

3

4

x

−2

��, 2 � �4, ��7, 3�

x

−8 −6 −4 −2 0 2 4 6 8−1−2−3 0 1 2 3

x

��6, 6��3, 3�2, �1, 1, 47

2, 50, 2592, �3

2

P � R � C


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333202CB02_AN.qxd 1/1/70 09:37 AM Page 26


(Continued)

49.

50.

51. 52.

(a) (a)

(b) (b)

53. 54.

(a) (a)

(b) (b)55. 56.57. 58.59. 60.61. 62.63. 64.65. 66.67. (a) seconds (b)68. (a) seconds

(b) 0 seconds seconds and

69. 13.8 meters 36.2 meters70. 45.97 feet 174.03 feet71.72. 90,000 units 100,000 units; 73. (a)

(b)

2011

(c)(d)

2016 to 2021(e)(f) Answers will vary.

74. (a)

(b) 3.83 inches

75. ohms

76. (a) 1995 (b) (c) 2006 (d)

77. True. The test intervals are and

78. True. The -values are greater than zero for all values of

79. 80.

81.

82.

83. (a) If and can be any real number. If and

(b) 0

84. (a)

(b)

(c) The real zeros of the polynomial

85. 86.

87. 88.

89. 90. 32b2 � b2x2 � x

2x�x � 3��x2 � 3x � 9��x � 3��x � 2��x � 2�

�x � 7��x � 1��2x � 5�2

ax

b

+ − +

− − +

− + +

x � a, x � b

c > 0, b < �2�ac or b > 2�ac.a > 0c ≤ 0, ba > 0

��, �2�10 � �2�10, ��, �2�30 � �2�30, �

��, ��, �4 � �4, �

x.y

�4, �.�1, 4�,��3, 1�,��, �3�,

t � 16.25t � 5.1

R1 ≥ 2

Depth of the beam

Max

imum

saf

e lo

add

L

4 6 8 10 12

5,000

10,000

15,000

20,000

25,000

37 ≤ t ≤ 41

t � 31

0 230

80

30.00 ≤ p ≤ 32.00≤ x ≤40,000 ≤ x ≤ 50,000; 50.00 ≤ p ≤ 55.00

≤ L ≤≤ L ≤

4 � 2�2 seconds < t ≤ 8 seconds≤ t < 4 � 2�2

t � 84 seconds < t < 6 secondst � 10�1.19, 1.30��2.26, 2.39��4.42, 0.42��0.13, 25.13��1.13, 1.13��3.51, 3.51��3, 0 � �3, ��5, 0 � �7, ��4, 4��, 3 � �4, ��, �2 � �2, ��2, 2

� < x ≤ 0� < x <

1 ≤ x ≤ 4�x� ≥ 2

−6 6

−4

4

−6 6

−2

6

�2 ≤ x < �12 < x ≤ 4

�1 < x ≤ 20 ≤ x < 2

−15 15

−6

14

−6 12

8

−4

0 2 64

x

−2−4

1

��, �4� � ��2, 1� � �6, �

x

−1

−

43210

23

��, �1� � ��23, 1� � �3, �

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t 24 26 28 30 32 34

C 70.5 71.6 72.9 74.6 76.8 79.6

t 36 37 38 39

C 83.2 85.4 87.8 90.5

t 40 41 42 43

C 93.5 96.8 100.4 104.4

d 4 6 8 10 12

Load 2223.9 5593.9 10,312 16,378 23,792

333202CB02_AN.qxd 1/1/70 09:37 AM Page 27

Review Exercises (page 208)

1. (a) (b)

Vertical stretch Vertical stretch and reflection in the x-axis

(c) (d)

Vertical shift Horizontal shift

2. (a) (b)

Vertical shift Reflection in the x-axis andvertical shift

(c) (d)

Horizontal shift Vertical shrink and verticalshift

3. 4.

Vertex: Vertex:Axis of symmetry: Axis of symmetry:x-intercepts: x-intercepts:

5. 6.


x-intercepts: x-intercepts:

7. 8.


t-intercepts:x-intercepts: �2, 0�, �6, 0��1 ±

�62

, 0�x � 4t � 1

�4, �4��1, 3�

4 8−2

2

6

4

8

10−2

−4

y

1 2−1−2−3 3 4 5 6

2

4

5

6

3

1

y

f �x� � �x � 4�2 � 4f �t� � �2�t � 1�2 � 3

�2 ± �7, 0��4 ±�6, 0�x � 2x � �4

�2, 7��4, �6�

x4−2

2

4

8

6

10

2 106 8

y

x2

2

−8 −4

−2

−4

−6

y

h�x� � ��x � 2�2 � 7f �x� � �x � 4�2 � 6

�0, 0�, �6, 0��0, 0�, �2, 0�x � 3x � 1

�3, 9��1, �1�

x4

−28−2

2

4

8

6

10

2 10

y

x1−1

−2

−1−2−3

3

4

5

6

7

2 3 4 5 6

y

f �x� � ��x � 3�2 � 9g�x� � �x � 1�2 � 1

y

x−4 −3 −2

−4

−3

−2

1

2

3

4

2 3 4

y

x−3 −2 −1

−3

−2

−1

1

2

3

4

5

21 3 4 5

y

x−4 −3 −1

−3

−2

−1

1

2

3

5

1 3 4

y

x−4 −3 −1

−5

−2

−1

2

3

1 3 4

y

x−4 −3 −2

−4

−3

−2

−1

1

4

−1 1 2 3 4

y

x−4 −3 −2

−4

−3

−2

−1

1

3

4

−1 1 2 3 4

y

x−4 −3 −2

−4

−3

2

1

3

4

−1 1 2 3 4

y

x−4 −3 −2

−4

−3

−2

−1

2

3

4

−1 1 2 3 4

Precalculus with Limits, Answers to Review Exercises 28

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333202CB02_AN.qxd 1/1/70 09:37 AM Page 28


(Continued)

9. 10.

Vertex: Vertex:

Axis of symmetry: Axis of symmetry:No x-intercept x-intercepts:

11. 12.

Vertex: Vertex:

Axis of symmetry: Axis of symmetry:

x-intercepts: No x-intercept

13. 14.

Vertex: Vertex:

Axis of symmetry: Axis of symmetry:

x-intercepts: x-intercepts:

15. 16.

17. 18.19. (a) (b)

(c)

20. (a) $12,000; $13,750; $15,000(b) Maximum revenue at $40; $16,000. Any price greater

or less than $40 per unit will not yield as muchrevenue.

21. 1091 units

22. 24 years old

23. 24.

25. 26.

27. 28.

29. Falls to the left, falls to the right30. Falls to the left, rises to the right31. Rises to the left, rises to the right32. Rises to the left, falls to the right33. odd multiplicity; turning point: 134. 0, odd multiplicity; even multiplicity; turning points: 235. odd multiplicity; turning points: 236. 8, odd multiplicity; 0, even multiplicity; turning points: 237. 0, even multiplicity; odd multiplicity; turning points: 238. odd multiplicity; 0, even multiplicity;

turning points: 3�1, 2,

53,

0, ±�3,�3,

�7, 32,

x6−2−4−6 2 4

4

8

6

y

x

2

1

1−2

5

−5

3

4

3 4 5 6 7

y

x−1−2−3 1 2 3 4 5 6

2

−2

−3

4

5

6

3

1

y

x

1

1 2 3−2−3

3

−1

−2

−3

y

x−1−2−3

2

1

1 2 3

3

−1

−2

−3

y

x

2

1

1 2 7−2

5

−3

−4

3

4

3 4 6

y

y

x20 21 22 23 24 25

22

23

24

25

26

27

Age of bride

Age

of

groo

m

x � 50, y � 50A � 100x � x2

y � 100 � x

x

y

f �x� �13�x � 2�2 � 3f �x� � �x � 1�2 � 4

f �x� �14�x � 2�2 � 2f �x� � �

12�x � 4�2 � 1

�2 ±�33

, 0��±�41 � 52

, 0�x � 2x � �

52

�2, �1��52, �41

12�

–6 –4 –2 4 6 8 10

2

4

6

8

10

12

14

x

y

x

4

−4

−2 2−4−6−8

−6

2

y

f �x� � 3�x � 2�2 � 1f �x� �13�x �

52�2

�4112

�±�41 � 52

, 0�x � �

12x � �

52

��12, 4��5

2, �414 �

x

8

10

12

2

−2−2

2 4 6−4−6−8

y

x

−2

−2 2−4−6−8

−4

−10

y

f �x� � 4�x �12�2

� 4h�x� � �x �52�2

�414

�3 ± 2�2, 0�x � 3x � �

12

�3, �8��12, 12�

x4 8

2

102−2

−2

−4

−6

−8

y

x−1−2−3 1 2 3

5

10

15

20

y

f �x� � �x � 3�2 � 8h�x� � 4�x �12�2

� 12

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333202CB02_AN.qxd 1/1/70 09:37 AM Page 29

(Continued)

39. (a) Rises to the left, falls to the right (b)(c) Answers will vary.(d)

40. (a) Rises to the right, falls to the left(b) (c) Answers will vary.(d)

41. (a) Rises to the right, rises to the left (b)(c) Answers will vary.(d)

42. (a) Falls to the left, falls to the right(b) (c) Answers will vary.(d)

43. (a) (b)44. (a) (b)45. (a) (b)46. (a) (b)

47. 48.

49. 50.

51.

52.

53.

54.

55. 56.57. (a) Yes (b) Yes (c) Yes (d) No58. (a) Yes (b) No (c) Yes (d) No59. (a) (b) 60. (a) (b) 061. (a) Answers will vary.

(b)(c)(d) 4(e)


63. (a) Answers will vary. (b)(c)(d) 3, 4(e)


65. 66. 67.68. 69. 70.71. 72. 73. �4 � 46i17 � 28i40 � 65i

��2 i3 � 7i�1 � 5i�1 � 3i3 � 5i6 � 2i

−6 12

−8

4

1, 2, 3, 5f �x� � �x � 1��x � 3��x � 2��x � 5��x � 1�, �x � 3�

−3

−10

5

40

�1,�2,f �x� � �x � 1��x � 4��x � 2��x � 3�

�x � 4��x � 1�,

−7 5

−100

50

�52, 3, �6f �x� � �2x � 5��x � 3��x � 6�

�2x � 5�, �x � 3�

−8 5

−60

80

�1,�7,f �x� � �x � 7��x � 1��x � 4�

�x � 1��x � 7�,

�3276�9�421

3x2 � 11x � 42x2 � 11x � 6

0.1x2 � 0.8x � 4 �19.5

x � 5

6x3 � 8x2 � 11x � 4 �8

x � 2

3x2 � 5x � 8 �10

2x2 � 1

x2 � 3x � 2 �1

x2 � 2

3x2 � 3 �3

x2 � 15x � 2

4

3�

29

3�3x � 2�8x � 5 �

2

3x � 2

� �0.509, � �1.211��2, �1, ��1, 0� �0.200, � 1.772��1, 0, �1, 2��4.479��5, �4��0.900��1, 0

y

x−4 −3 −1

−4

−3

−2

−1

2

3

4

1 3 4

( (3, 0 ( (3, 0−

(0, 0)

0, ±�3

y

x−4 −1−2

3

21 3 4

−15

−18

−21

(−3, 0)

(0, 0)

(1, 0)

�3, 0, 1

y

x−4 −3 −1

−4

−3

−2

−1

2

3

4

21 3 4

(−2, 0) (0, 0)

�2, 0

y

x−4 −3 −2

−4

−3

1

2

3

4

1 2 3 4

(−1, 0)

�1


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(Continued)

74. 75. 76. 77.

78. 79. 80. 81.

82. 83. 84. 4 85.

86. 0, 87. 88. 5, 8,89.

90.

91. 92. 6 93.94. 95. 96. 2,97.98.99. 100. 101.

102.103.104.105.106.

107. Two or no positive zeros, one negative zero108. One or three positive real zeros, two or no negative real zeros109–110. Answers will vary.111. Domain: all real numbers x except 112. Domain: all real numbers x except 113. Domain: all real numbers x except 114. Domain: all real numbers x115. Vertical asymptote:

Horizontal asymptote:116. Horizontal asymptote:117. Vertical asymptote:

Horizontal asymptote:118. Vertical asymptotes:

Horizontal asymptotes: None





(d)


(b) x-intercept:

y-intercept:



(d)


(b) x-intercept:

y-intercept:



(d)

123. (a) Domain: all real numbers (b) Intercept:(c) Horizontal asymptote:

(d)

124. (a) Domain: all real numbers (b) Intercept:(c) Horizontal asymptote: y � 0

�0, 0�x

(0, 0)−3 −2 −1 2 3

−2

2

3

4

x

y

y � 1�0, 0�

x

−2 −1 1 4 5 6

−3

−2

3

4

5

x

(3, 0)

( (0, 32

y

y � 1

x � 2

�0, 32��3, 0�

x � 2

−8

−6

−4

−2

4

6

x2(−2, 0)

(0, 2)

y

y � �1

x � 1

�0, 2��2, 0�

x � 1

−3 −2 −1 1 2 3 4

−3

−2

1

2

3

4

x

y

y � 0x � 0

x � 0

−1 1 2

−3

−2

1

x

y

y � 0x � 0

x � 0

x � �2, �1y � 0

x � �3y � 2y � 0

x � �3

x � 6, 4x � �

13

x � �12

f �x� � �x � 3��x � 3��x � 4 � i ��x � 4 � i �±3, �4 ± i;

�x � 2 � 3i��4, 2 ± 3i; g�x� � �x � 4�2�x � 2 � 3i��2, 3, 6; g�x� � �x � 2��x � 3��x � 6�0, 1, �5; f (x� � x �x � 1��x � 5�0, 34, 1 ± i

�3, 12, 2 ± i2, ±4i4, ± ix 4 � x3 � 3x2 � 17x � 303x4 � 14x3 � 17x2 � 42x � 24

±25�3,�4, 3�2�5,

1, 853,�1,�1, �3, 6

±1, ±2, ±4, ±8, ±13, ±2

3, ±43, ±8

3

±1, ±3, ±5, ±15, ±12, ±3

2, ±52, ±15

2 , ±14, ±3

4, ±54, ±15

4

3 ± i�4, 6, ±2i±�6 i

8, 1�9,0, 2�14

±�71

4i

1 ± 3i±12

i±�33

i985

�8385

i

2113 �

113i17

26 �726 i23

17 �1017i9 � 20i

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333202CB02_AN.qxd 1/1/70 09:37 AM Page 31

(Continued)

(d)







(d)

129. (a) Domain: all real numbers x except (b) x-intercept:(c) Vertical asymptote:


130. (a) Domain: all real numbers x except (b) x-intercept:(c) Vertical asymptote:


131. (a) Domain: all real numbers x(b) Intercept: (c) Slant asymptote:(d)


Slant asymptote: y � x � 1x � �1

�0, 1�x � �1

1

x

−2

2

3

3

−3

1 2−1−2−3

y

(0, 0)

y � 2x�0, 0�

y

x

, 023( (

−1−2−3 2 3

2

y � 1.5x � �

12

�23, 0�

x � ±12

y

x−2−4−6−8 4 6 8

−2

−4

−6

−8

, 032( (

2

y � 2x � 0

�1.5, 0�

13x � 0,

−6 −4 4 6

4

6

x(0, 0)

y

y � 2x � ±2

�0, 0�x � ±2

x−2−4−6

2

6

−8

4

2 4

(0, 0)

y

y � �6�0, 0�

−3 −2 −1 2 3 4 5

1

3

5

6

7

x

(0, 4)

y

y � 0x � 1

�0, 4�x � 1

1 2

−2

−1

1

2

x(0, 0)

y

y � 0�0, 0�

−3

−2

−1

1

2

3

x1 2 3

(0, 0)

y


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333202CB02_AN.qxd 1/1/70 09:37 AM Page 32


(Continued)

(d)

133. (a) Domain: all real numbers x except (b) y-intercept:

x-intercepts:


Slant asymptote:(d)

134. (a) Domain: all real numbers x except (b) x-intercepts:


Slant asymptote:(d)

135. $0.50 is the horizontal asymptote of the function.

136. (a)

(b) $176 million; $528 million; $1584 million (or $1.584billion)

(c) No

137. (a)

(b)

(c)(d)

9.48 inches 9.48 inches138.

80.3 milligrams per square decimeter per hour

139. 140.

141. 142.

143. 144.145. 146.147. 4.9% 148. 9 days149. False. A fourth-degree polynomial can have at most four

zeros, and complex zeros occur in conjugate pairs.

150. False. The domain of is the set of all realnumbers.

151. Find the vertex of the quadratic function and write thefunction in standard form. If the leading coefficient is positive, the vertex is a minimum. If the leading coeffi-cient is negative, the vertex is a maximum.

152. Answers will vary. Sample answer:A polynomial of degree with real coefficients canbe written as the product of linear and quadratic factorswith real coefficients, where the quadratic factors have noreal zeros.Setting the factors equal to zero and solving for the variable can find the zeros of a polynomial function.To solve an equation is to find all the values of the variable for which the equation is true.

153. An asymptote of a graph is a line to which the graphbecomes arbitrarily close as increases or decreaseswithout bound.

x

n > 0

f �x� �1

x2 � 1

��, 0� � �2, ��4, �3 � �0, ��, 3� � �5, ��5, �1� � �1, ��, 0� � �0, 53��4, 0 � �4, ��, �3 � �5

2, ��43, 12�

00

100

90

�

04 32

200

4 < x <

�2x�2x � 7�

x � 4

Area � x�4x � 14x � 4 �

y �4x � 14x � 4

�x � 4�� y � 4� � 30

y

x

2 in.

2 in.

2 in. 2 in.

00

100

4000

y

x

, 043( (

−2−4−6 4 6−2

−6

2

4

(2, 0)

(0, −8)

y � x � 3

x �13

�0, �8��2, 0�, �4

3, 0�x �

13, �2

y

x(1, 0)

, 0

−1−2 2 3 4

−2

1

2

3

4

23( (

0, − 12( (

y � x �13

x �43

�23, 0�, �1, 0�

�0, �0.5�x �

43

x−2−4−6 62

4

4

y

(0, 1)

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333202CB02_AN.qxd 1/1/70 09:37 AM Page 33

Chapter Test (page 212)1. (a) Reflection in the x-axis followed by a vertical

translation(b) Horizontal translation

2.

3. (a) 50 feet

(b) 5. Yes, changing the constant term results in a verticaltranslation of the graph and therefore changes themaximum height.

4. Rises to the left, falls to the right

5. 6.

7.

Solutions:

8. (a) (b) 7 9.

10.

11.

12. 13.

14. -intercepts:

No -intercept

Vertical asymptote:


15. -intercept:-intercept:

Vertical asymptote:Horizontal asymptote:

16. No -intercept-intercept:

Vertical asymptote:Slant asymptote:

17. 18.

−2−4−6−8 0 2 4 6

x

210−1−2−3−4−5 3x

32

x < �6 or 0 < x < 4x < �4 or x > 32

−8 −6 −4 2 4 6 8

−6

−4

2

4

6

8

10

x

(0, −2)

y

y � x � 1x � 1

�0, �2�yx

y

x−4−6−8 2 4

−2

−4

2

4

6

8

(−1.5, 0)

(0, 0.75)

y � 2x � �4

�0, 0.75�y��1.5, 0�x

−2 −1 1 2

−2

1

2

3

4

x(2, 0)(−2, 0)

y

y � �1

x � 0

y

��2, 0�, �2, 0�x

�2, 4, �1 ± �2 i�2, ±�5i

f �x� � x4 � 6x3 � 16x2 � 24x � 16

f �x� � x4 � 9x3 � 28x2 � 30x

2 � i�3 � 5i

14, ±�3

�4x � 1��x � �3 ��x � �3 �;

2x3 � 4x 2 � 3x � 6 �9

x � 23x �

x � 1

x2 � 1

t−1

−2

−3

−4

−5

−2−3−4 2

5

3 4 5

3

4

y

y � �x � 3� 2 � 6

Precalculus with Limits, Answers to Chapter Test 34

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333202CB02_AN.qxd 1/1/70 09:37 AM Page 34

Precalculus with Limits, Answers to Problem Solving 35

Problem Solving (page 215)1. Answers will vary.2. (a)

(b) (c) (d) (e)(f) (g)

3. 2 inches 2 inches 5 inches

4. False, the statement would be true if the 2 were replaced by

5. (a) and (b)

6. (a) less than (b) greater than

(c) less than (d)

(e) The values are the same.

(f)

7. (a)

(b)

8. (a) (b) (c)

9.

10. (i) d (ii) b (iii) a (iv) c

11. (a) As increases, the graph stretches vertically. Forthe graph is reflected in the x-axis.

(b) As increases, the vertical asymptote is translated.For the graph is translated to the right. For

the graph is reflected in the x-axis and is trans-lated to the left.

12. (a)

(b)

(c) The models are a good fit for the original data.(d)

The rational model is the better fit for the original data.(e) The reciprocal model should not be used to predict the

near point for a person who is 70 years old because anegative value is obtained. The quadratic model is abetter fit.

y2 � 3.861y1 � 1.125;

00

70

50

y2 ��134.82x � 59.93

00

70

50

y1 � 0.031x2 � 1.57x � 21.0

b < 0,b > 0,

�b�a < 0,

�a�

� a2 � b2

�a � bi ��a � bi� � a2 � abi � abi � b2i2�

134 �

217 i3

10 �110i1

2 �12 i

f �x� � ��x � 3�x 2 � 1 � �x3 � 3x 2 � 1

f �x� � �x � 2�x 2 � 5 � x3 � 2x 2 � 5

m tan � 4 since h � 0.

mh � 3, 5, 4.1;

mh � h � 4m3 � 4.1;

m2 � 3;m1 � 5;

y � �x2 � 5x � 4

f ��1�.

��

x � 3x � 6x � 10x � 3x � 6x � 6

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y 1 2 3 4 5

2 12 36 80 150y3 � y2

y 6 7 8 9 10

252 392 576 810 1100y3 � y2

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333202CB02_AN.qxd 1/1/70 09:37 AM Page 36

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Chapter 2 y - Cengage · Copyright © Houghton Mifflin Company. All rights reserved. Precalculus with Limits, Answers to Section 2.1 4 (Continued) 75. (a) (b) (c) (d) (e) They are