5144 demana te ans pp915-943...5144_demana_te_ans_pp915-943 1/24/06 8:24 am page 919 63. (a) h 16t2...

ADDITIONAL ANSWERS 915

SECTION 1.7 Exploration 11.

2. 3. Linear: r2 � 0.9758; power: r2 � 0.9903; quadratic: R2 � 1; cubic: R2 � 1; quartic: R2 � 1

5. Since the quadratic curve fits the points perfectly, there is nothing to be gained by adding a cubic term or a quartic term. The coefficients of these terms in the regressions are zero.

Exercises 1.711. Let C be the total cost and n be the number of items produced; C � 34,500 � 5.75n.12. Let C be the total cost and n be the number of items produced; C � (1.09)28,000 � 19.85n.13. Let R be the revenue and n be the number of items sold: R � 3.75n.14. Let P be the profit, and s be the amount of sales; then P � 200,000 � 0.12s.21. x � 4x � 620; x � 124; 4x � 496 22. x � 2x � 3x � 714, so x � 119; the second and third numbers are 238 and 357.24. 179.9 25. 182 � 52t, so t � 3.5 hr 26. 560 � 45t � 55(t � 2), so t � 4.5 hours on local highways.27. 0.60(33) � 19.8, 0.75(27) � 20.25; The $33 shirt is a better bargain, because the sale price is cheaper.31. (a) 0.10x � 0.45(100 � x) � 0.25(100) (b) Use x � 57.14 gallons of the 10% solution and about 42.86 gal of the 45% solution.32. 0.20x � 0.35(25 � x) � 0.26(25). Use x � 15 liters of the 20% solution and 10 liters of the 35% solution.34. 2x � 2(x � 16) � 136. Two pieces that are x � 26 ft long are needed, along with two 42 ft pieces.38. 900 � 0.07x � 0.085(12,000 � x); $8000 was invested at 7%; the other $4000 was invested at 8.5%.41. True; the correlation coefficient is close to 1 if there is a good fit. 42. False; quadratic regression is useful for modeling free-fall.47. (d) The point of intersection corresponds to the break-even point, where C � R.49. (e) (f) You should recommend stringing the rackets; fewer strung rackets need to

be sold to begin making a profit (since the intersection of y2 and y4 occurs for smaller x than the intersection of y1 and y3).

[0, 10,000] by [0, 500,000]

[3, 11] by [0, 40]

n = 10; d = 35n = 9; d = 27n = 8; d = 20n = 7; d = 14

n = 6; d = 9n = 5; d = 5n = 4; d = 2

n = 3; d = 0

5144_Demana_TE_Ans_pp915-943 1/24/06 8:24 AM Page 915

50. (a) 51. (a)

CHAPTER 1 REVIEW EXERCISES11. (a) All reals (b) All reals 12. (a) All reals (b) All reals 13. (a) All reals (b) [0, �) 14. (a) All reals (b) [5, �)15. (a) All reals (b) [8, �) 16. (a) [�2, 2] (b) [�2, 0] 17. (a) All reals except 0 and 2 (b) All reals except 0

18. (a) (�3, 3) (b) ��13�, �� 19. Continuous 20. Continuous 21. (a) Vertical asymptotes at x � 0 and x � 5 (b) y � 023. (a) none (b) y � 7 and y � �7 24. (a) x � �1 (b) y � 1 and y � �1 27. (��, �1), (�1, 1), (1, �)33. (a) none (b) �7, at x � �134. (a) 2, at x � �1 (b) �2, at x � 1 35. (a) �1, at x � 0 (b) none 36. (a) 1, at x � 2 (b) �1, at x � �245. 46. 47.

48. 49. 51.

52. f(x) � x � 3, x � �1; f(x) � x2 � 1, x � �1 53. ( f � g)(x) � �x2� �� 4�; (��, �2] � [2, �)

54. (g � f )(x) � x � 4; [0, �) 55. ( f � g)(x) � �x�(x2 � 4); [0, �) 56. ��gf��(x) � �g

f((xx))

� � ; [0, 2) � (2, �)

57. limx → �

�x� � � 58. limx → ��

�x2� �� 4� � �

65. (a) (b) The regression line is y � 61.133x � 725.333.

66. (a) (d)

[45, 110] by [–2, 18][45, 110] by [50, 70]

[4, 15] by [940, 1700][4, 15] by [940, 1700]

�x��x2 � 4

[–5, 5] by [–5, 5][–5, 5] by [–5, 5][–5, 5] by [–5, 5]

[–5, 5] by [–5, 5][–5, 5] by [–5, 5][–5, 5] by [–5, 5]

[0, 22] by [100, 200][–1, 15] by [9, 16]

916 ADDITIONAL ANSWERS



67. (a) (b) 2�r2�3� �� r�2� (c) [0, �3�] 68. (c)(d) (e) 12.57 in.3

Chapter 1 Project1. 4. The actual growth in the number of locations is

slowing while the model increases more rapidly.

5. y �

SECTION 2.1 Exploration 11. �$2000 per year 2. v(t) � �2000t � 50,000

Quick Review 2.11. y � 8x � 3.6 2. y � �1.8x � 2 3. y � �0.6x � 2.8 4. y � �

83

�x � �73

�

Exercises 2.11. Not a polynomial function because of the exponent �5 2. Polynomial of degree 1 with leading coefficient 23. Polynomial of degree 5 with leading coefficient 2 4. Polynomial of degree 0 with leading coefficient 135. Not a polynomial function because of the radical 6. Polynomial of degree 2 with leading coefficient �5

y

6

x5

(1, 5)

(�2, �3)

y

7

x5

(3, 1)(�2, 4)

4914.198��1 � 269.459e�0.468x

[–1, 13] by [–100, 2600]

[0, 13] by [0, 20] [0, 6] by [0, 180]3

3 3h2r r

3 – r2h = 2


7. f(x) � �57

�x � �178� 8. f(x) � ��

79

�x � �83

� 9. f(x) � ��43

�x � �23

� 10. f(x) � �54

�x � �34

�

11. f(x) � �x � 3 12. f(x) � �12

�x � 2

19. Translate the graph of y � x2 3 units right and the 20. Vertically shrink the graph of y � x2 by a factor of �14

�

result 2 units down. and translate the result down 1 unit.

21. Translate the graph of y � x2 2 units left, vertically 22. Vertically stretch the graph of y � x2 by a factor of 3, reflect

shrink the resulting graph by a factor of �12

�, and translate the result across the x-axis, and then translate up 2 units.

that graph 3 units down.

23. Vertex: (1, 5); axis: x � 1 24. Vertex: (�2, �1); axis: x � �2 25. Vertex: (1, �7); axis: x � 1

26. Vertex: (�3�, 4); axis: x � �3� 27. Vertex: ��56�, ��7132��; axis: x � ��56�; f(x) � 3�x � �

56

��2

� �7132�

28. Vertex: ��74�, �285��; axis: x � �74�; f(x) � �2�x � �

74

��2

� �285� 29. Vertex: (4, 19); axis: x � 4; f(x) � �(x � 4)2 � 19

30. Vertex: ��14�, �243��; axis: x � �14�; f(x) � 4�x � �

14

��2

� �243� 31. Vertex: ��35�, �

151��; axis: x � �35�; g(x) � 5�x � �

35

��2

� �151�

x

y

10

10

x

y

10

10

x

y

10

10

x

y

10

10

y

10

x10(–4, 0)

(0, 2)

y

5

x5

(0, 3)

(3, 0)

y

10

x10

(1, 2)

(5, 7)

y

10

x10

(–4, 6)

(–1, 2)

y

10

x10

(–3, 5)

(6, –2)

y

5

x3

(2, 4)

(–5, –1)



32. Vertex: ��74�, �187��; axis: x � ��74�; h(x) � �2�x � �

74

��2

� �187�

33. f(x) � (x � 2)2 � 2; Vertex: (2, 2); 34. g(x) � (x � 3)2 � 3; Vertex: (3, 3); 35. f(x) � �(x � 8)2 � 74; axis: x � 2; opens upward; axis: x � 3; opens upward; Vertex: (�8, 74); axis: x � �8; opens does not intersect x-axis does not intersect x-axis downward; intersects x-axis at about

�16.602 and 0.602, or (�8 � �74� )

36. h(x) � �(x � 1)2 � 9; 37. f(x) � 2�x � �32��2

� �52

�; 38. g(x) � 5�x � �52��2

� �747�;

Vertex: (1, 9); axis: x � 1; Vertex: ��32�, �52

��; axis: x � ��32�; Vertex: ��52

�, ��747��; axis: x � �52�; opens

opens downward; intersects opens upward; does not intersect upward; intersects x–axis at about

x-axis at �2 and 4 x-axis; vertically stretched by 2 0.538 and 4.462, or ��52� � �110��385��;

vertically stretched by 5

49. (a) (b) Strong positive 50. (a) (b) Strong negative

53. (a) y � 0.541x � 4.072. The slope tells us that hourly compensation for production workers increases about 54¢/yr. (b) About $25.7055. (a) [0, 100] by [0, 1000] is one possibility.58. (b) 59. (b) (c) 90 cents per can; $16,200

61. (a) About 215 ft above the field (c) About 117 ft/sec downward

[0, 15] by [10,000, 17,000][0, 25] by [200,000, 260,000]

[0, 90] by [0, 70][15, 45] by [20, 50]

[–5, 10] by [–20, 100]

[–3.7, 1] by [2, 5.1][–9, 11] by [–100, 10]

[–20, 5] by [–100, 100]

[–2, 8] by [0, 20] [–4, 6] by [0, 20]



63. (a) h � �16t2 � 80t � 10 64. 32�3� or about 55.426 ft/sec67. (a) (c) On average, the children gained

0.68 pounds per month.(d)

68. (a) y � 548.30x � 21027.56 (b) About $59,400

69. The Identity Function f(x) � x

Domain: (��, �); Range: (��, �); Continuous; Increasing for all x; Symmetric about the origin; Not bounded; No local extrema; No horizontal or vertical asymptotes; End behavior: lim

x → ��f(x) � ��, lim

x → �f(x) � �

70. The Squaring Function f(x) � x2

Domain: (��, �); Range: [0, �); Continuous; Increasing on [0, �), decreasing on (��, 0]; Symmetric about the y-axis; Bounded below, but not above; Local minimum of 0 at x � 0; No horizontal or vertical asymptotes; End behavior: lim

x → ��f(x) � lim

x → �f(x) � �

72. True. We can rewrite f in the form f(x) � �x � �14��2

� �34

�, so f � 0.

80. (a) (b) y � 0.115x � 8.245 (c) y � 0.556x � 6.093 (d) The median–median line appears to be the better fit, becauseit approximates more of the data val-ues more closely.

81. (a) The two solutions are��b � �

2ab2 � 4�ac�� and �

�b � �2a

b2 � 4�ac��; their sum is 2��2

ba�� ba�.

(b) The product of the two solutions given above is � �ac

�.

82. f(x) � (x � a)(x � b) � x2 � (a � b)x � ab; the axis is given by x � �[�(a � b)]/2, or x � (a � b)/2.

83. ��a �2b

�, ��(a �

4b)2

�� 84. x1 and x2 are given by the quadratic formula ��b � �2ab2 � 4�ac��; then x1 � x2 � ��

ba

�, and the line

of symmetry is x � ��2ba�, which is exactly equal to �

x1 �2

x2�.

b2 � (b2 � 4ac)��

4a2

[0, 15] by [0, 15][0, 17] by [2, 16]

[0, 17] by [2, 16]

[–4.7, 4.7] by [–1, 5]

[–4.7, 4.7] by [–3.1, 3.1]

[15, 45] by [20, 40]

[15, 45] by [20, 40][0, 5] by [–10, 100]



SECTION 2.2 Exploration 11. The pairs (0, 0), (1, 1) and (�1, �1) are

common to all three graphs.

2. The pairs (0, 0), (1, 1), and (�1, 1) are common toall three graphs.

Exercises 2.2

1. power � 5, constant � ��12

� 2. power � �53

�, constant � 9 3. not a power function 4. power � 0, constant � 13

5. power � 1, constant � c2 6. power � 5, constant � �2k

� 7. power � 2, constant � �g2

�

8. power � 3, constant � �43�� 9. power � �2, constant � k

10. power � 1, constant � m 11. degree � 0, coefficient � �4 12. not a monomial function; negative exponent13. degree � 7, coefficient � �6 14. not a monomial function: variable in exponent 15. degree � 2, coefficient � 4�16. degree � 1, Coefficient � l 23. The weight w of an object varies directly with its mass m, with the constant of variation g.24. The circumference C of a circle is proportional to its diameter D, with the constant of variation �.25. The refractive index n of a medium is inversely proportional to v, the velocity of light in the medium, with constant of variation c, the constant velocity of light in free space. 26. The distance d traveled by a free-falling object dropped from rest varies directly with the square

of its speed p, with the constant of variation �21g�.

27. power � 4, constant � 2; Domain: (��, �); 28. power � 3, constant � �3; Domain: (��, �); Range: [0, �); Continuous; Decreasing on (��, 0). Range: (��, �); Continuous; Decreasing for all x; Odd. Increasing on (0, �); Even. Symmetric with respect Symmetric with respect to origin; to y-axis; Bounded below, but not above; Not bounded above or below; Local minimum at x � 0; Asymptotes: none; No local extrema; Asymptotes: none; End Behavior: lim

x → ��2x4 � �, lim

x → �2x4 � � End Behavior: lim

x → ��3x3 � �, lim

x → ��3x3 � ��

[–5, 5] by [–20, 20][–5, 5] by [–1, 49]

[–1.5, 1.5] by [–0.5, 1.5] [–5, 5] by [–5, 25] [–15, 15] by [–50, 400]

[–5, 5] by [–15, 15] [–20, 20] by [–200, 200][–2.35, 2.35] by [–1.5, 1.5]



29. power � �14

�, constant � �12

�; Domain: [0, �); 30. power � �3, constant � �2; Domain: (��, 0) � (0, �);

Range: [0, �); Continuous; Increasing on [0, �); Range: (��, 0) � (0, �); Discontinuous at x � 0; Bounded below; Neither even nor odd; Increasing on (��, 0) and on (0, �).; Odd. Symmetric Local minimum at (0, 0); Asymptotes: none; with respect to origin; Not bounded above or below;

End Behavior: limx → �

�12

��4 x� � � No local extrema; Asymptotes at x � 0 and y � 0.; End Behavior: lim

x → ��2x�3 � 0, lim

x → ��2x�3 � 0.

31. shrink vertically by �23

�; f is even. 32. stretch vertically by 5; f is odd. 33. stretch vertically by 1.5 and reflect

over the x-axis; f is odd.

34. stretch vertically by 2 and reflect 35. shrink vertically by �14

�; f is even. 36. shrink vertically by �18

�; f is odd.over the x-axis; f is even.

43. k � 3, a � �14

�. f is increasing in Quadrant I. f is undefined for x 0. 44. k � �4, a � �23

�. f is decreasing in Quadrant IV.

f is even. 45. k � �2, a � �43

�. f is decreasing in Quadrant IV. f is even. 46. k � �25

�, a � �52

�. f is increasing in Quadrant I.

f is undefined for x 0. 47. k � �12

�, a � �3. f is decreasing in Quadrant I. f is odd.

48. k � �1, a � �4. f is increasing in Quadrant IV. f is even.

49. y � �x82�, power � �2, constant � 8 50. y � �2�x�, power � �

12

�, constant � �2 54.

[–5, 5] by [–19, 1][–5, 5] by [–50, 50][–5, 5] by [–1, 49]

[–5, 5] by [–20, 20]

[–5, 5] by [–20, 20][–5, 5] by [–1, 19]

[–5, 5] by [–5, 5][–1, 99] by [–1, 4]


Wind Speed (mph) Power (W)

10 0015

0120

40 0960

80 7680

20


55. (a) (c) (d) Approximately 37.67 beats/min, which is very close to Clark’s observed value

57. (a) (c) (d) Approximately 2.76 �mW

2�

and 0.697 �mW

2�, respectively

59. False, because f(�x) � (�x)1/3 � �x1/3 � �f(x). The graph of f is symmetric about the origin.65. (a) The graphs of f(x) � x�1 and h(x) � x�3

are similar and appear in the 1st and 3rdquadrants only. The graphs of g(x) � x�2 andk(x) � x�4 are similar and appear in the 1stand 2nd quadrants only. The pair (1, 1) iscommon to all four functions.

[–2, 2] by [–2, 2][0, 3] by [0, 3][0, 1] by [0, 5]

[0.8, 3.2] by [�0.3, 9.2][0.8, 3.2] by [–0.3, 9.2]

[–2, 71] by [50, 450][–2, 71] by [50, 450]


f g h k

Domain x 0 x 0 x 0 x 0

Range y 0 y � 0 y 0 y � 0

Continuous yes yes yes yes

Increasing (��, 0) (��, 0)

Decreasing (��, 0), (0, �) (0, �) (��, 0), (0, �) (0, �)

Symmetry w.r.t. origin w.r.t. y-axis w.r.t. origin w.r.t. y-axis

Bounded not below not below

Extrema none none none none

Asymptotes x-axis, y-axis x-axis, y-axis x-axis, y-axis x-axis, y-axis

End Behavior limx → ��

f(x) � 0 limx → ��

g(x) � 0 limx → ��

h(x) � 0 limx → ��

k(x) � 0


(b) The graphs of f(x) � x1/2 and h(x) � x1/4

are similar and appear in the 1st quadrantonly. The graphs of g(x) � x1/3 and k(x) � x1/5 are similar and appear in the 1stand 3rd quadrants only. The pairs (0, 0),(1, 1) are common to all four functions.

SECTION 2.3Quick Review 2.33. (3x � 2)(x � 3) 5. x(3x � 2)(x � 1)8. x � 0, x � �2, x � 5

Exercises 2.31. Shift y � x3 to the right by 3 units, 2. Shift y � x3 to the left by 5 units 3. Shift y � x3 to the left by 1 unit,

stretch vertically by 2. and then reflect over the x-axis. vertically shrink by �12

�, reflect over the y-intercept: (0, �54) y-intercept: (0, �125)

x-axis, and then vertically shift up

2 units. y-intercept: �0, �32��

x

y

5

5

x

y

15

200

x

y

10

10

[–3, 3] by [–2, 2][0, 3] by [0, 2][0, 1] by [0, 1]


f g h k

Domain [0, �) (��, �) [0, �) (��, �)

Range y � 0 (��, �) y � 0 (��, �)

Continuous yes yes yes yes

Increasing [0, �) (��, �) [0, �) (��, �)

Decreasing

Symmetry none w.r.t. origin none w.r.t. origin

Bounded below not below not

Extrema min at (0, 0) none min at (0, 0) none

Asymptotes none none none none

End behavior

limx → �

f(x) � � limx → �

g(x) � �

limx → ��

g(x) � ��

limx → �

h(x) � � limx → �

k(x) � �

limx → ��

k(x) � ��


4. Shift y � x3 to the right by 3 units, 5. Shift y � x4 to the left 2 units, 6. Shift y � x4 to the right 1 unit,

vertically shrink by �23

�, and vertically vertically stretch by 2, reflect over vertically stretch by 3, and vertically

shift up 1 unit. y-intercept: (0, �17)the x-axis, and vertically shift down shift down 2 units. y-intercept: (0, 1)3 units. y-intercept: (0, �35)

7. local maximum: � (0.79, 1.19), zeros: x � 0 and x � 1.26.8. local maximum: (0, 0), local minima: � (1.12, �3.13) and (�1.12, �3.13), zeros: x � 0, x � 1.58, x � �1.58.13. One possibility: 14. One possibility: 15. One possibility:

16. One possibility: 17. 18.

limx → �

f(x) � �; limx → ��

f(x) � �� limx → �

f(x) � ��; limx → ��

f(x) � �

19. 20. 21.

limx → �

f(x) � ��; limx → ��

f(x) � � limx → �

f(x) � �; limx → ��

f(x) � �� limx → �

f(x) � �; limx → ��

f(x) � �

22. 23. 24.

limx → �

f(x) � �; limx → ��

f(x) � � limx → �

f(x) � �; limx → ��

f(x) � � limx → �

f(x) � ��; limx → ��

f(x) � ��

[–4, 3] by [–20, 90][–3, 5] by [–50, 50][–2, 6] by [–100, 25]

[–5, 5] by [–14, 6][–10, 10] by [–100, 130][–8, 10] by [–120, 100]

[–100, 100] by [–2000, 2000]

[–5, 5] by [–15, 15][–5, 3] by [–8, 3]

[–50, 50] by [–1000, 1000][–50, 50] by [–1000, 1000][–100, 100] by [–1000, 1000]

x

y

5

40

x

y

5

5

x

y

10

20



29. (a) There are 3 zeros: they are �2.5, 1, and 1.1. 30. (b) There are 3 zeros: they are 0.4, approximately 0.429 �actually �37��,and 3. 31. (c) There are 3 zeros: approximately �0.273 �actually ��1

31��, �0.25, and 1. 32. (d) There are 3 zeros: �2, 0.5,

and 3. 33. �4 and 2 35. �23

� and ��13

� 37. 0, ��23

�, and 1

39. Degree 3; zeros: x � 0 (mult. 1, graph crosses x-axis), 40. Degree 4; zeros: x � 0 (mult. 3, graph crosses x-axis),x � 3 (mult. 2, graph is tangent) x � 2 (mult. 1, graph crosses x-axis)

41. Degree 5; zeros: x � 1 (mult. 3, graph crosses x-axis), 42. Degree 6; zeros: x � 3 (mult. 2, graph is tangent),x � �2 (mult. 2, graph is tangent) x � �5 (mult. 4, graph is tangent)

43. 44. 45.

�2.43, �0.74, 1.67 �1.73, 0.26, 4.47 �2.47, �1.46, 1.94

46. 47. 48.

�4.53, 2 �4.90, �0.45, 1, 1.35 �1.98, �0.16, 1.25, 2.77, 3.62

53. f(x) � x3 � 5x2 � 18x � 72 55. f(x) � x3 � 4x2 � 3x � 1256. f(x) � x3 � 3x2 � x � 1 61. It follows from the Intermediate Value Theorem. 62. It follows from the Intermediate Value Theorem.

[–3, 4] by [–100, 100][–6, 4] by [–100, 20][–5, 3] by [–20, 90]

[–3, 3] by [–10, 10][–2, 5] by [–8, 22][–3, 2] by [–10, 10]

x

y

10

100,000

y

10

–10

x–5 5

x

y

4

5

x

y

6

10



63. (a) (c)

65. (a) 66. (a) The height of the box will be x, the width will be 15 � 2x, and the length 60 � 2x.(b) Any value of x between approximately 0.550 and 6.786 inches.

67. 0 x � 0.929 or 3.644 � x 569. True. Because f is continuous and f(1) � �2 and f(2) � 2, the Intermediate Value Theorem assures us

that the graph of f crosses the x-axis between x � 1 and x � 2.70. False. If a � 0, the graph of g is obtained by translating the graph of f a units to the left.77. The exact behavior near x � 1 is hard to see. A zoomed-in view around the point (1, 0) suggests that

the graph just touches the x-axis at 0 without actually crossing it — that is, (1, 0) is a local maximum.One possible window is [0.9999, 1.0001] by [�1 � 10�7, 1 � 10�7].

78. This also has a maximum near x � 1 — but this time a window such as [0.6, 1.4] by [�0.1, 0.1] reveals that the graph actually rises above the x-axis and has a maximum at (0.999, 0.025). 79. A maximum and minimum are not visible in the standard window, but canbe seen on the window [0.2, 0.4] by [5.29, 5.3]. 80. A maximum and minimum are not visible in the standard window, but can be seen on thewindow [0.95, 1.05] by [�6.0005, �5.9995].81. The graph of y � 3(x3 � x) increases, then decreases, then increases; the graph of y � x3 only increases. Therefore, this graph can not beobtained from the graph of y � x3 by the transformations studied in Chapter 1 (translations, reflections, and stretching/shrinking). Since the rightside includes only these transformations, there can be no solution. 82. The graph of y � x4 has a “flat bottom,” while the graph of y � x4 �3x3 � 2x � 3 is “bumpy.” Therefore, this graph cannot be obtained from the graph of y � x4 through the transformations of Chapter 1. Since theright side includes only these transformations, there can be no solution.

(b)

84. (a) Note that f(a) � an and f(�a) � �an; (b) y � an/(n � 1) � an � 1(x � a1/(n � 1))

m � �yx

2

2

�

�

yx

1

1� � �

�

�

aa

n �

�

aa

n

� � ��

�

22aa

n

� � an � 1 (c) y � 9x � 6�3�, y � x3

SECTION 2.4 Quick Review 2.4

7. 4(x � 5)(x � 3) 8. x(3x � 2)(5x � 4)

Exercises 2.4

1. f(x) � (x � 1)2 � 2; �xf�

(x)1

� � x � 1 � �x �

21

� 2. f(x) � (x2 � x � 1)(x � 1) � 2; �xf�

(x)1

� � x2 � x � 1 � �x �

21

�

[–5, 5] by [–30, 30]

(c) The line L also crosses the graph off(x) at (�2, �13).

[1.8, 2.2] by [6, 8]

83. (a) Substituting x � 2, y � 7, we find that7 � 5(2 � 2) � 7, so Q is on line L,and also f(2) � �8 � 8 � 18 � 11 � 7,so Q is on the graph of f(x).

[0, 0.8] by [0, 1.20]

[0, 60] by [–10, 210][0, 60] by [–10, 210]



3. f(x) � (x2 � x � 4)(x � 3) � 21; �xf�

(x)3

� � x2 � x � 4 � �x

2�

13

� 4. f(x) � �2x2 � 5x � �72��(2x � 1) � �92

�;

�2x

f(�

x)1

� � 2x2 � 5x � �72

� � �2x

9�

/21

� 5. f(x) � (x2 � 4x � 12)(x2 � 2x � 1) � 32x � 18;

�x2 �

f(2xx)� 1

� � x2 � 4x � 12 � �x�2

3�

2x2x

�

�

181

� 6. f(x) � (x2 � 3x � 5)(x2 � 1); �x2

f(�

x)1

� � x2 � 3x � 5

33. ��1, �2

�

,1�3, �6�; 1 34. �

�1, ��

21,,�

�

73, �14

�; �73

� 35. ��1

�

,1�

,3�

,2�9

�; �32

� 36. ; ��43

� and �32

�

49. Rational zero: �32

�; irrational zeros: ��2�

53. Rational: �1 and 4; irrational: ��2� 54. Rational: �1 and 2; irrational: ��5� 55. Rational: ��12

� and 4; irrational: none

56. Rational: �23

�; irrational: about �0.6823 61. (c) (x � 2)(x3 � 4x2 � 3x � 19) (d) One irrational zero is x � 2.04.(e) f(x) � (x � 2)(x � 2.04)(x2 � 6.04x � 9.3116)

62. (a) D � 0.0669t3 � 0.7420t2 � 2.1759t � 0.8250

63. False. (x � 2) is a factor if and only if f(�2) � 0. 64. True because the remainder is f(1) which is equal to 3.69. (d) x � 0.6527 m 71. (a) Shown is one possible view, on the window [0, 600] by [0, 500].

(c) P � 0 when t � 523.22 — about 523 days after release.

SECTION 2.5 Exploration 11. f(2i) � (2i)2�i(2i) � 2 � �4 � 2 � 2 � 0; f (�i) � (�i)2 � i(�i) � 2 � �1 � 1 � 2 � 0; no.2. g(i) � i2 � i � (1 � i) � �1 � i � 1 � i � 0; g(1 � i) � (1 � i)2 � (1 � i) � (l � i) � � 2i � 2i � 0; no.3. The Complex Conjugate Zeros Theorem does not necessarily hold true for a polynomial function with complex coefficients.

Quick Review 2.5

7. �52

� � ��

219��i 8. ��

34

� � ��

447��i

Exercises 2.52. x3 � 2x2 � 3x � 6; zeros: �2, ��3�i; x-intercept: x � �23. x4 � 2x3 � 5x2 � 8x � 4; zeros: 1 (mult. 2), �2i; x-intercept: x � 1 4. x4 � 3x3 � 4x2 � 2x; zeros: 0, 1, 1 � i; x-intercepts: x � 0, x � 1 7. x3 � x2 � 9x � 9 8. x3 � 2x2 � 6x � 8 9. x4 � 5x3 � 7x2 � 5x � 610. x4 � 3x3 � 2x2 � 2x � 4 11. x3 � 11x2 � 43x � 65 13. x5 � 4x4 � x3 � 10x2 � 4x � 815. x4 � 10x3 � 38x2 � 64x � 40 16. x4 � 6x3 � 14x2 � 14x � 5

[0, 600] by [0, 500]

[–1, 8.25] by [0, 5]

�1, �2, �3, �4, �6, �12��

�1, �2, �3, �6



27. Zeros: x � 1, x � ��12

� � ��

219��i; f(x) � �

14

�(x � 1)(2x � 1 � �19�i)(2x � 1 � �19�i)

28. Zeros: x � 3, x � �72

� � ��

243��i; f(x) � �

14

�(x � 3)(2x � 7 � �43�i); (2x � 7 � �43�i)

29. Zeros: x � �1, x � ��12

� � ��

223��i; f(x) � �

14

�(x � 1)(x � 1)(2x � 1 � �23�i)(2x � 1 � �23�i)

30. Zeros: x � �2, x � �13

�, x � ��12

� � ��

23��i; f(x) � �

14

�(x � 2)(3x � 1)(2x � 1 � �3�i)(2x � 1 � �3�i)

31. Zeros: x � ��73

�, x � �32

�, x � 1 � 2i; f(x) � (3x � 7)(2x � 3)(x � 1 � 2i)(x � 1 � 2i)

32. Zeros: x � ��35

�, x � 5, x � �32

� � i; f(x) � (5x � 3)(x � 5)(2x � 3 � 2i)(2x � 3 � 2i)

33. Zeros: x � ��3�, x � 1 � i; f(x) � (x � �3� )(x � �3� )(x � 1 � i)(x � 1 � i)34. Zeros: x � ��3�, x � �4i; f(x) � (x � �3� )(x � �3� )(x � 4i)(x � 4i)35. Zeros: x � ��2�, x � 3 � 2i; f(x) � (x � �2�)(x � �2�)(x � 3 � 2i)(x � 3 � 2i)36. Zeros: x � ��5�, x � 1 � 3i; f(x) � (x � �5�)(x � �5� )(x � 1 � 3i)(x � 1 � 3i)49. f(x) � �2x4 � 12x3 � 20x2 � 4x � 30 50. f(x) � 2x4 � 8x3 � 22x2 � 28x � 20

51. (a) D � �0.0820t3 � 0.9162t2 � 2.5126t � 3.3779

(b) Sally walks toward the detector, turns and walks away (or walks backward), then walks toward the detector again.

(c) t � 1.81 sec (D � 1.35 m) and t � 5.64 sec (D � 3.65 m).

52. (a) D � 0.2434t2 � 1.7159t � 4.4241

(b) Jacob walks toward the detector, then turns and walks away (or walks backward).

(c) The model “changes direction” at t � 3.52(when D � 1.40 m).

53. False. If 1 � 2i is a zero, then 1 � 2i must also be a zero.54. False. The polynomial f(x) � x(x � 1)(x � 2) � x3 � 3x2 � 2x has degree 3, real coefficients, and no non-real zeros.

[–1, 9] by [0, 6]

[–1, 9] by [0, 5]



SECTION 2.6 Exploration 1

1. g(x) � �x �

12

� 2. h(x) � ��x �

15

� 3. k(x) � �x �

34

� � 2

Exercises 2.61. Domain: all x �3; lim

x → �3�f(x) � ��, lim

x → �3�f(x) � � 2. Domain: all x 1; lim

x → 1�f(x) � �; lim

x → 1�f(x) � ��

3. Domain: all x �2, 2; limx → �2�

f(x) � ��, limx → �2�

f(x) � �, limx → 2�

f(x) � �, limx → 2�

f(x) � ��

4. Domain: all x �1, 1; limx → �1�

f(x) � �, limx → �1�

f(x) � ��, limx → 1�

f(x) � ��, limx → 1�

f(x) � �

5. Translate right 3 units. 6. Translate left 5 units, vertically 7. Translate left 3 units, reflect across x-axis,Asymptotes: x � 3, y � 0 stretch by 2, reflect across x-axis. vertically stretch by 7, translate up 2 units.

Asymptotes: x � �5, y � 0 Asymptotes: x � �3, y � 2

8. Translate right 1 unit, translate up 9. Translate left 4 units, vertically 10. Translate right 5 units, vertically 3 units. Asymptotes: x � 1, y � 3 stretch by 13, translate down 2 units. stretch by 11, reflect across x-axis,

Asymptotes: x � �4, y � �2 translate down 3 units. Asymptotes: x � 5, y � �3

19. Vertical asymptote: none; Horizontal asymptote: y � 2; limx → ��

f(x) � limx → �

f(x) � 2

20. Vertical asymptote: none; Horizontal asymptote: y � 3; limx → ��


f(x) � 3

21. Vertical asymptotes: x � 0, x � 1; Horizontal asymptote: y � 0; limx → 0�

f(x) � �, limx → 0�

f(x) � ��, limx → 1�

f(x) � ��,lim

x → 1�f(x) � �, lim

x → ��f(x) � lim

x → �f(x) � 0

22. Vertical asymptotes: x � �3, x � 0; Horizontal asymptote: y � 0; limx → �3�

f(x) � ��, limx → �3�

f(x) � �, limx → 0�

f(x) � �,lim

x → 0�f(x) � ��, lim

x → ��f(x) � lim

x → �f(x) � 0

x

y

10

20

x

y

6

8

x

y

6

6

x

y

6

10

x

y

4

5

x

y

5

5

[–8, 2] by [–5, 5][–1, 9] by [–5, 5][–3, 7] by [–5, 5]



23. Intercepts: �0, �23�� and (2, 0) 24. Intercepts: �0, ��23

�� and (�2, 0) 25. No intercepts Asymptotes: x � �1, x � 3, Asymptotes: x � �3, x � 1, Asymptotes: x � �1, x � 0,and y � 0 and y � 0 x � 1, and y � 0

26. No intercepts 27. Intercepts: (0, 2), (�1.28, 0), and 28. Intercepts: (0, �3), (�1.84, 0), and Asymptotes: x � �2, x � 0, (0.78, 0); Asymptotes: x � 1, (2.17, 0); Asymptotes: x � �2,x � 2, and y � 0. x � �1, and y � 2 x � 2, and y � �3

29. Intercept: �0, �32�� 30. Intercepts: �0, ��73��, (�1.54, 0), and (4.54, 0) Asymptotes: x � �2, y � x � 4 Asymptotes: x � �3, y � x � 6

31. (d); Xmin � �2, Xmax � 8, Xscl � 1, and Ymin � �3, Ymax � 3, Yscl � 132. (b); Xmin � �6, Xmax � 2, Xscl � 1, and Ymin � �3, Ymax � 3, Yscl � 133. (a); Xmin � �3, Xmax � 5, Xscl � 1, and Ymin � �5, Ymax � 10, Yscl � 134. (f); Xmin � �6, Xmax � 2, Xscl � 1, and Ymin � �5, Ymax � 5, Yscl � 135. (e); Xmin � �2, Xmax � 8, Xscl � 1, and Ymin � �3, Ymax � 3, Yscl � 136. (c); Xmin � �3, Xmax � 5, Xscl � 1, and Ymin � �3, Ymax � 8, Yscl � 1

37. Intercept: �0, ��23��; asymptotes: x � �1, x � �32

�, y � 0; limx → �1�

f(x) � �, limx → �1�

f(x) � ��, limx → (3/2)�

f(x) � ��,

limx → (3/2)�

f(x) � �; Domain: x �1, �32

�; Range: ��, ��1265�� (0, �); Continuity: all x �1, �32�;

Increasing: (��, �1), ��1, �14��, Decreasing: ��14

�, �32

��, ��32�, ��; Unbounded; Local Maximum at ��14�, ��

1265��; Horizontal asymptote: y � 0;

Vertical asymptotes: x � �1, x � �32

�; End behavior: limx → ��


f(x) � 0[–4.7, 4.7] by [–3.1, 3.1]

[–30, 30] by [–40, 20][–20, 20] by [–20, 20]

[–5, 5] by [–8, 2][–5, 5] by [–4, 6][–4, 4] by [–5, 5]

[–4.7, 4.7] by [–10, 10][–6, 4] by [–5, 5][–4, 6] by [–5, 5]



38. Intercept: �0, �23��; asymptotes: x � �3, x � �1, y � 0; limx → �3� g(x) � �, limx → �3� g(x) � ��, limx → �1� g(x) � ��,lim

x → �1�g(x) � � Domain: x �3, �1; Range: (��, �2] � (0, �);

Continuity: all x �3, �1; Increasing: (��, �3), (�3, �2]; Decreasing: [�2, �1), (�1, �); No symmetry; Unbounded; Local maximum at (�2, �2); Horizontal asymptote: y � 0; Vertical asymptotes: x � �3, x � �1; End behavior: lim

x → ��g(x) � lim

x → �g(x) � 0

39. Intercepts: �0, �112��, (1, 0); asymptotes: x � �3, x � 4, y � 0; limx → �3� h(x) � ��, limx → �3� h(x) � �, limx → 4� h(x) � ��,

limx → 4�

h(x) � � Domain: x �3, 4; Range: (��, �); Continuity: all x �3, 4; Decreasing: (��, �3), (�3, 4), (4, �); No symmetry; Unbounded; No extrema; Horizontal asymptote: y � 0; Vertical asymptotes: x � �3, x � 4; End behavior: lim

x → ��h(x) � lim

x → �h(x) � 0

40. Intercepts: (�1, 0), (0, �0.1); asymptotes: x � �2, x � 5, y � 0; limx → �2�

k(x) � ��, limx → �2�

k(x) � �, limx → 5�

k(x) � ��,

limx → 5�

k(x) � � Domain: x �2, 5; Range: (��, �); Continuity: all x �2, 5; Decreasing: (��, �2), (�2, 5), (5, �); No symmetry; Unbounded; No extrema; Horizontal asymptote: y � 0; Vertical asymptotes: x � �2, x � 5; End behavior: lim

x → ��k(x) � lim

x → �k(x) � 0

41. Intercepts: (�2, 0), (1, 0), �0, �29��; asymptotes: x � �3, x � 3, y � 1; limx → �3� f(x) � �, limx → �3� f(x) � ��, limx → 3� f(x) � ��,lim

x → 3�f(x) � � Domain: x �3, 3; Range: (��, 0.260] � (1, �); Continuity: all x �3, 3;

Increasing: (��, �3), (�3, �0.675); Decreasing: (�0.675, 3), (3, �); No symmetry; Unbounded; Local maximum at (�0.675, 0.260); Horizontal asymptote: y � 1; Vertical asymptotes: x � �3, x � 3; End behavior: lim

x → ��f(x) � lim

x → �f(x) � 1

42. Intercepts: (�1, 0), (2, 0), �0, �14��; asymptotes: x � �2, x � 4, y � 1; limx → �2� g(x) � �, limx → �2� g(x) � ��, limx → 4� f(x) � ��,lim

x → 4�g(x) � � Domain: x �2, 4; Range: (��, 0.260] � (1, �);

Continuity: all x �2, 4; Increasing (��, �2), (�2, 0.324]; Decreasing: [0.324, 4), (4, �); No symmetry; Unbounded; Local maximum at (0.324, 0.260); Horizontal asymptote: y � 1; Vertical asymptotes: x � �2, x � 4; End behavior: lim

x → ��g(x) � lim

x → �g(x) � 1

[–9.4, 9.4] by [–3, 3]

[–9.4, 9.4] by [–3, 3]

[–9.4, 9.4] by [–1, 1]

[–5.875, 5.875] by [–3.1, 3.1]

[–6.7, 2.7] by [–5, 5]



43. Intercepts: (�3, 0), (1, 0), �0, ��32��; asymptotes: x � �2, y � x; limx → �2� h(x) � �, limx → �2� h(x) � ��Domain: x �2, Range: (��, �); Continuity: all x �2; Increasing: (��, �2), (�2, �); No symmetry; Unbounded; No extrema; Horizontal asymptote: none; Vertical asymptote: x � �2; Slant asymptote: y � x; End behavior: lim

x → ��h(x) � ��, lim

x → �h(x) � �

44. Intercepts: (�1, 0), (2, 0), �0, �23��; asymptotes: x � 3, y � x � 2; limx → 3� k(x) � ��, limx → 3� k(x) � �Domain: x 3; Range: (��, 1] � [9, �); Continuity: all x 3; Increasing: (��, 1], [5, �); Decreasing: [1, 3), (3, 5]; No symmetry; Unbounded; Local max at (1, 1), local min at (5, 9); Horizontal asymptote: none; Vertical asymptote: x � 3; Slant asymptote: y � x � 2; End behavior: lim

x → ��k(x) � ��, lim

x → �k(x) � �

45. y � x � 3 46. y � 2x � 4 47. y � x2 � 3x � 6(a) (a) (a)

(b) (b) (b)

48. y � x2 � x � 1 49. y � x3 � 2x2 � 4x � 6(a) (a)

(b) (b)

[–20, 20] by [–5000, 5000][–50, 50] by [–1500, 2500]

[–5, 5] by [–100, 200][–8, 8] by [–20, 40]

[–50, 50] by [–1500, 2500][–40, 40] by [–40, 40][–40, 40] by [–40, 40]

[–10, 10] by [–30, 60][–15, 10] by [–30, 20][–10, 20] by [–10, 30]

[–9.4, 9.4] by [–10, 20]

[–9.4, 9.4] by [–15, 15]



50. y � x3 � x(a) There are no vertical asymptotes.(b)

51. Intercept: �0, �45��; 52. Intercepts: ��12

�, 0�, (0, 0); Domain: (��, �); Range: [0.773, 14.227]; Domain: (��, �); Range: [�0.028, 9.028]; Continuity: (��, �); Increasing: [�0.245, 2.445]; Continuity: (��, �); Increasing: [�0.235, 3.790]; Decreasing: (��, �0.245], [2.445, �); Decreasing: (��, �0.235], [3.790, �); No symmetry; Bounded; No symmetry; Bounded; Local max at (2.445, 14.227), local min at (�0.245, 0.773); Local max at (3.790, 9.028), local min at (�0.235, �0.028); Horizontal asymptote: y � 3; Vertical asymptote: none; Horizontal asymptote: y � 4; Vertical asymptote: none; End behavior: lim

x → ��f(x) � lim

x → �f(x) � 3 End behavior: lim

x → ��g(x) � lim

x → �g(x) � 4

53. Intercepts: (1, 0), �0, �12��; 54. Intercepts: (�3 2�, 0), (0, �1); Domain: x 2; Range: (��, �); Continuity: x 2; Domain: x �2; Range: (��, �); Continuity: x �2; Increasing: [�0.384, 0.442], [2.942, �); Increasing: [�3.104, �2], (�2, �); Decreasing: (��, �0.384], [0.442, 2), (2, 2.942]; Decreasing: (��, �3.104]; No symmetry; Not bounded; No symmetry; Not bounded; Local max at (0.442, 0.586), local min at (�0.384, 0.443) Local min at (�3.104, 28.901); and (2.942, 25.970); Horizontal asymptote: none; Horizontal asymptote: none; Vertical asymptote: x � 2; Vertical asymptotes: x � �2; End behavior: lim

x → ��h(x) � lim

x → �h(x) � �; End behavior: lim

x → ��k(x) � lim

x → �k(x) � �;

End-behavior asymptote: y � x2 � 2x � 4 End-behavior asymptote: y � x2 � 2x � 4

[–10, 10] by [–20, 50][–10, 10] by [–20, 50]

[–10, 15] by [–5, 10][–15, 15] by [–5, 15]

[–20, 20] by [–5000, 5000]



55. Intercepts: (1.755, 0), (0, 1); 56. Intercepts: (�1.189, 0), (0, �2.5);

Domain: x �12

�; Range: (��, �); Continuity: x �12

�; Domain: x 2; Range: (��, �); Continuity: x 2;

Increasing: [�0.184, �12�), ��12�, ��; Increasing: [2.899, �); Decreasing: (��, �0.184]; Decreasing: (��, 2), (2, 2.899]; No symmetry; Not bounded; No symmetry; Not bounded; Local min at (�0.184, 0.920); Local min at (2.899, 37.842); Horizontal asymptote: none; Vertical asymptote: x � �

12

�; Horizontal asymptote: none; Vertical asymptote: x � 2;

End behavior: limx → ��


f(x) � �; End behavior: limx → ��

g(x) � limx → �

g(x) � �;

End-behavior asymptote: y � �12

�x2 � �34

�x � �18

� End-behavior asymptote: y � 2x2 � 2x � 3

57. Intercept: (0, 1); 58. Intercepts: (�1.108, 0), (0, �1); 59. Intercepts: (1, 0), �0, ��12��; Asymptote: x � �1; Asymptotes: x � �1; Asymptote: x � �2; End-behavior asymptote: End-behavior asymptote: End-behavior asymptote:y � x3 � x2 � x � 1 y � 2x3 � 2x � 1 y � x4 � 2x3 � 4x2 � 8x � 16

60. Intercepts: (�1, 0), (0, �1); 61. Intercepts: (�1.476, 0), (0, �2); 62. Intercepts: (1, 0), (0, �4); Asymptote: x � 1; Asymptote: x � 1; Asymptote: x � �1; End-behavior asymptote: End-behavior asymptote: y � 2 End-behavior asymptote: y � 3y � x4 � x3 � x2 � x � 1

63. False. �x2 �

11

� is a rational function and has no vertical asymptotes.

64. False. A rational function is the quotient of two polynomials, and �x2 � 4� is not a polynomial.69. (a) No: the domain of f is (��, 3) � (3, �); the domain of g is all real numbers.(b) No: while it is not defined at 3, it does not tend toward �� on either side.(c) Most grapher viewing windows do not reveal that f is undefined at 3. (d) Almost—but not quite, they are equal for all x 3.70. (a) The functions are identical at all points except x � 1 (b) The functions are identical except at x � �1.(c) The functions are identical except at x � �1. (d) The functions are identical except at x � 1.

[–5, 5] by [–25, 50]

[–5, 5] by [–10, 10][–5, 5] by [–5, 5]

[–10, 10] by [–200, 400][–3, 3] by [–20, 40][–5, 5] by [–30, 30]

[–10, 10] by [–20, 60][–5, 5] by [–10, 10]



71. (b) If f(x) � kxa, where a is a negative integer, 72. (a) (c)then the power function f is also a rational function.

73. Horizontal asymptotes: y � �2 and y � 2; 74. Horizontal asymptotes: y � �3;

Intercepts: �0, ��32��, ��32

�, 0�; Intercepts: �0, �53��, ��53

�, 0�;

h(x) �� h(x) ��

75. Horizontal asymptotes: y � �3; 76. Horizontal asymptotes: y � �2;

Intercepts: �0, �54��, ��53

�, 0�; Intercepts: (0, 2), (1, 0);

f(x) �� f(x) ��

SECTION 2.7 Quick Review 2.71. 2x2 � 8x 2. x2 � 2x � 1 3. LCD: 36; ��

316� 4. LCD: x(x � 1); �

2xx2 �

�

x1

�

6. LCD: (x � 2)(x � 3)(x � 2);�(x �

�22x)(

�

x �8

2)� if x 3 7. �

3 �4�17�� 8. �

5 �4�33�� 9. �

�1 �3

�7�� 10. �

3 �23�5��

Exercises 2.71. x � �1 4. x � �

11 �8�73�� 2.443 or x � �

11 �8�73�� 0.307

[–7, 13] by [–3, 3][–10, 10] by [–5, 5]

�2x�

�

21x

� x � 0

2 x 0

�5x�

�

34x

� x � 0

��

5x�

�

3x4

� x 0

[–5, 5] by [–5, 5][–5, 5] by [–5, 5]

�3xx�

�

35

� x � 0

��

3xx

�

�

53

� x 0

�2xx�

�

23

� x � 0

��

2xx

�

�

32

� x 0

[0.5, 3.5] by [0, 7][0.5, 3.5] by [0, 7]



5. x � �4 or x � 3, the latter is extraneous. 6. x � �13 �

1�6

105�� 1.453 or x � �

13 �1�6

105�� 0.172

11. x � �12

� or x � �1, the latter is extraneous.

12. x � �32

� or x � �4, the latter is extraneous. 13. x � ��13

� or x � 2, the latter is extraneous.

14. x � ��34

� or x � 1, the latter is extraneous.

23. x � 3 � �2� � 4.414 or x � 3 � �2� � 1.586 24. x � ��3 �

2�31�� 1.284 or x � �

�3 �2�31�� 4.284

26. x � �1 �

6�13�� 0.768 or x � �

1 �6�13�� 0.434

31. (a) The total amount of solution is (125 � x) mL; of this, the amount of acid is x plus 60% of the original amount, or x � 0.6(125).

(c) C(x) � �xx�

�

17255

� � 0.83; x � 169.12 mL 32. (a) C(x) � �x �

x0�

.3150(1000)

� � �xx�

�

13050

�

33. (a) C(x) � �3000 �

x2.12x� (c) 6350 hats per week

34. (c) limt→�

P(t) � limt→� �500 � �t9�00200�� 500, so the bear population will never exceed 500.

35. (a) P(x) � 2x � �36x4

� (b) x � 13.49 (a square); P � 53.96

36. (a) A(x) � (x � 1.75)��4x0� � 2.5� (b) x � 5.29, so the dimensions are about 7.04 in. � 10.06 in.; A � 70.8325 in2

37. (a) S � �2�x3 �

x1000� (b) Either x � 1.12 cm and h � 126.88 cm or x � 11.37 cm and h � 1.23 cm

38. (a) A(x) � (x � 4)��10x00� � 4� (b) x � �1000� � 31.62, so the dimensions are about 35.62 ft � 35.62 ft; A � 1268.98 ft2

40. (a) P(x) � 2x � �40

x0

� (b) 7.1922 m � 27.8078 m

41. (a) D(t) � �4.

47.575

�

tt

� 42. (a) T � �1x7� � �

x �53

43�

43. (a) 44. (a) 45. False. An extraneous solution is a solution of the equation cleared of fractions that is not a solution of the original equation.

46. True. For a fraction to be equal to zero,the numerator must be zero and 1 is not zero.

(b) About 2102 wineries

51. (a) f(x) � �xx

2

2�

�

22xx

� (c) f(x) � {(d) The graph appears to be the horizontal line

y � 1 with holes at x � �2 and x � 0.

[–4.7, 4.7] by [–3.1, 3.1]

x �2, 0x � �2 or x � 0

1,undefined,

[0, 35] by [0, 3000][0, 15] by [0, 120]



SECTION 2.8 Exploration 1

1. (a) 2. (a)

(b) (b)

3. (a) (b)

Quick Review 2.8

7. �2xx

2

2�

�

75xx�

�

23

� 9. (a) �1, ��12

�, �3, ��32

� (b) (x � 1)(2x � 3)(x � 1)

10. (a) �1, ��13

�, �2, ��23

�, �4, ��43

�, �8, ��83

� (b) (x � 2)(x � 1)(3x � 4)

Exercises 2.81. (a) x � �2, �1, 5 (b) �2 x �1 or x � 5 (c) x �2 or �1 x 5 2. (a) x � 7, ��

13

�, �4

(b) �4 x ��13

� or x � 7 (c) x �4 or ��13

� x 7 3. (a) x � �7, �4, 6 (b) x �7 or �4 x 6 or x � 6

(c) �7 x �4 4. (a) x � ��35

�, 1 (b) x ��35

� or x � 1 (c) ��35

� x 1 5. (a) x � 8, �1 (b) �1 x 8 or x � 8

(c) x �1 6. (a) x � �2, 9 (b) �2 x 9 or x � 9 (c) x �2 21. (a) (��, �) (b) (��, �)(c) There are no solutions. (d) There are no solutions. 22. (a) There are no solutions. (b) There are no solutions. (c) (��, �)

(d) (��, �) 23. (a) x �43

� (b) (��, �) (c) There are no solutions. (d) x � �43

� 24. (a) x �32

� (b) (��, �)

(c) There are no solutions. (d) x � �32

� 25. (a) x � 1 (b) x � ��32

�, 4 (c) ��32

� x 1 or x � 4 (d) x ��32

�, or 1 x 4

26. (a) x � �72

�, �1 (b) x � �5 (c) �5 x �1 or x � �72

� (d) x �5 or �1 x �72

� 27. (a) x � 0, �3 (b) x �3

(c) x � 0 (d) �3 x 0 28. (a) x � 0, ��92

� (b) None (c) x ��92

�, 0 (d) None 29. (a) x � �5

(b) x � ��12

�, x � 1, x �5 (c) �5 x ��12

� or x � 1 (d) ��12

� x 1 30. (a) x � 1 (b) x � 4, x � �2

(c) �2 x 1 or x � 4 (d) 1 x 4 31. (a) x � 3 (b) x � 4, x 3 (c) 3 x 4 or x � 4 (d) f(x) is never negative.

32. (a) None (b) x � 5 (c) 5 x � (d) None 33. (��, �2) � (1, 2) 37. (��, �4) � (3, �) 39. [�1, 0] � [1, �)41. (0, 2) � (2, �)

43. ��4, �12�� 47. (��, 0) � (�3 2�, �)49. (��, �1) � [1, 3) 50. (��, �5) � (�2, 1) 51. [�3, �)

[–5, 5] by [–3000, 2000]

Positive Negative Negative(+)(+) (–)(–) (+)(+) (+)(–) (+)(+) (+)(–)

2–4x

[–3, 1] by [–30, 20][–5, 5] by [–250, 50]

Positive Positive Negative(+)(–) (–)(+) (+)(–) (–)(+) (+)(–) (+)(+)

–1–2xNegative Negative Positive

(+)(–)(+) (+)(–)(+) (+)(+)(+)

2–3

x



59. 0 in. � x � 0.69 in. or 4.20 in. � x � 6 in.61. (b) 1.12 cm � x � 11.37 cm, 1.23 cm � h � 126.88 cm 62. (a) R � �

x2�

.32x.3

�

63. (a) y � 993.870x � 19025.768 64. (a) y � 7.883x3 � 214.972x2 � 6479.797x � 62862.27865. False, because the factor x4 does not change sign at x � 0. 66. True, because the factor (x � 2) changes sign at x � �2.

71. Vertical asymptotes: x � �1, x � 3; x-intercepts: (�2, 0), (1, 0); y-intercept: �0, �43��

By hand: Grapher

72. Vertical asymptotes: x � �4, x � 0; x-intercept: (3, 0); y-intercept: none

Sketch: Graph:

73. (a) x � 3 13 ⇒ 3x � 9 1 ⇒ 3x � 5 � 4 1 ⇒ f(x) � 4 1.(b) If x stays within the dashed vertical lines, f(x) will stay within the dashed horizontal lines.(c) x � 3 0.01 ⇒ 3x � 9 0.03 ⇒ 3x � 5 � 4 0.03 ⇒ f(x) � 4 0.03. The dashed lines would be closer when x � 3 and

y � 4.74. When x2 � 4 � 0, y � 1, and when x2 � 4 � 0, y � 0.

[–10, 10] by [–10, 10] [–20, 0] by [–1000, 1000]x

y

10

600

Positive Negative Positive Positive30–4

0unde

fine

d

unde

fine

d

x

(–)4

(–)(–)

(–)4

(–)(+)

(–)4

(+)(+)

(+)4

(+)(+)

[–5, 5] by [–5, 5] [0, 10] by [–40, 40]

x

y

105

30

–10

-30

Negative3–1 1–2

0 unde

fine

d

0 unde

fine

dx

(–)(–)2

(–)(–)

Negative

(–)(+)2

(–)(–)

Positive

(–)(+)2

(–)(+)

Negative

(+)(+)2

(–)(+)

Positive

(+)(+)2

(+)(+)


support:


CHAPTER 2 REVIEW EXERCISES1. y � �x � 5 2. y � �2x

3. Starting from y � x2, translate 4. Starting from y � x2, translate 5. Vertex: (�3, 5); axis: x � �3right 2 units and vertically stretch left 3 units and reflect across the 6. Vertex: (5, �7); axis: x � 5by 3 (either order), then translate x-axis (either order), then translate 7. Vertex: (�4, 1); axis: x � �4up 4 units. up 1 unit. 8. Vertex: (1, �1); axis: x � 1

11. y � �12

�(x � 3)2 � 2

12. y � ��12

�(x � 4)2 � 5

13. 14. 15. 16.

19. The force F needed varies directly with the distance x from its resting position, with constant of variation k.

20. The area of a circle A varies directly with the square of its radius. 21. k � 4, a � �13

�, f is increasing in the first quadrant, f is odd.

22. k � �2, a � �34

�, f is decreasing in the fourth quadrant, f is not defined for x 0. 23. k � �2, a � �3, f is

increasing in the fourth quadrant, f is odd. 24. k � �23

�, a � �4, f is decreasing in the first quadrant, f is even.

25. 2x2 � x � 1 � �x �

23

� 26. x3 � x2 � x � 1 � �x �

52

� 27. 2x2 � 3x � 1 � ��

x22x�

�

43

� 28. x3 � 2x2 � 1 � �3x

�

�

71

�

37. �1, �2, �3, �6, ��12

�, ��32

� ; ��32

� and 2 are zeros.

38. �1, �7, ��12

�, ��72

�, ��13

�, ��73

�, ��16

�, ��76

�; �73

� is a zero.

49. Rational: 0. Irrational: 5 � �2�. No nonreal zeros. 50. Rational: �2. Irrational: ��3�. No nonreal zeros.51. Rational: none. Irrational: approximately �2.34, 0.57, 3.77. No nonreal zeros.

52. Rational: none. Irrational: approximately �3.97, �0.19. Two nonreal zeros. 53. ��32

�, 3 � i;

f(x) � (2x � 3)(x � 3 � i)(x � 3 � i) 54. �45

�, 2 � �7�; f(x) � (5x � 4)(x � 2 � �7�)(x � 2 � �7�)

55. 1, �1, �23

�, and ��52

�; f(x) � (3x � 2)(2x � 5)(x � 1)(x � 1)

56. 3 � i, 1 � 2i; f(x) � (x � 1 � 2i)(x � 1 � 2i)(x � 3 � i)(x � 3 � i)

[–6, 7] by [–50, 30][–4, 3] by [–30, 30][–2, 4] by [–50, 10][–10, 7] by [–50, 10]

x

y

10

10

x

y

6

10

[–5, 5] by [–5, 5][–15, 5] by [–15, 5]



59. f(x) � (2x � 3)(x � 1)(x2 � 2x � 5)60. f(x) � (3x � 2)(x � 1)(x2 � 4x � 5) 63. 6x4 � 5x3 � 38x2 � 5x � 6 65. x4 � 4x3 � 12x2 � 32x � 6467. Translate right 5 units and vertically stretch by 2 (either order), then translate down 1 unit.; Horizontal asymptote: y � �1; vertical asymptote: x � 5. 68. Translate left 2 units and reflect across x-axis (either order),then translate up 3 units.; Horizontal asymptote: y � 3; vertical asymptote: x � �2.

69. Asymptotes: y � 1, x � �1, and 70. Asymptotes: y � 2, x � �3, and 71. End behavior asymptote: y � x � 7; x � 1. Intercept: (0, �1). x � 2. Intercept: �0, ��76��. Vertical asymptote: x � �3.

Intercept: �0, �53��.

72. End behavior asymptote: y � x � 6; 73. y-intercept: �0, �52��, x-intercept: (�2.55, 0); Vertical asymptote: x � �3. Domain: x �2; Range: (��, �); Continuity: all x �2;

Intercepts: approx. (�1.54, 0), (4.54, 0) and �0, ��73��. Decreasing: (��, �2), (�2, 0.82]; Increasing: [0.82, �); Unbounded; Local minimum: (0.82, 1.63); Vertical asymptote: x � �2; End-behavior asymptote: y � x2 � x;

limx → ��


f(x) � �

74. y-intercept: (0, �1), x-intercepts: (�1.27, 0), (1.27, 0); 79. [�3, �2) � (2, �)Domain: x 1; Range: (��, �); 80. (��, �2) � (1, 3)Continuity: all x 1; Decreasing: (��, 1), (1, �); Unbounded; No extrema; Vertical asymptote: x � 1; 81. x � �3, x � �

12

�

End-behavior asymptote: y � �x3 � x2; 82. (1, 4) � (4, �)lim

x → ��f(x) � �; lim

x → � f(x) � ��

[–4.7, 4.7] by [–10, 10]

[–10, 10] by [–10, 20]

[–15, 10] by [–30, 10]

[–7, 3] by [–50, 30]

[–10, 10] by [–10, 10][–5, 5] by [–5, 5]



84. (a) h � �16t2 � 170t � 6 (b) When t � 5.3125, 85. (a) V � x(30 � 2x)(70 � 2x) in3

h � 457.5625. (b) Either x � 4.57 or x � 8.63 in.86. (a) & (b)

86. (d) The two pilings are made of different materials, for example.

87. (b) (c) The largest volume occurs when x � 70 (so it is actually a sphere).

This volume is �43

��(70)3 � 1,436,755 ft3.

88. (a) y � 18.694x2 � 88.144x � 2393.0222 (b) y � �0.291x4 � 7.100x3 � 35.865x2 � 48.971x � 2336.634

(c) Using quadratic regression: $5768; Using quartic regression: $3949

89. (a) y � 1.401x � 4.331 (b) y � 0.188x2 � 1.411x � 13.331 (c) Using linear regression: In 2008;Using quadratic regression: In 2003

92. (a) R2 � �x1�

.21x.2

� (b) 2 ohms 93. (a) C(x) � �50

5�

0x

� (b) about 33.33 ounces of distilled water (c) x � �1030

� � 33.33

94. (a) S � 2�x2 � �20x00� (b) Either x � 2.31 cm and h � 59.75 cm, or x � 10.65 cm and h � 2.81 cm.

94. (c) Approximately 2.31 x 10.65 (graphically) and 2.81 h 59.75.

95. (a) S � x2 � �40x00� (b) 20 ft by 20 ft by 2.5 ft or x � 7.32, giving approximate dimensions 7.32 by 7.32 by 18.66.

94. (c) 7.32 x 20 (lower bound approximate), so y must be between 2.5 and about 18.66.

[0, 15] by [0, 30][0, 15] by [0, 30]

[0, 15] by [0, 4500][0, 15] by [0, 4500]

[0, 70] by [0, 1,500,000]

[0, 255] by [0, 2.5]

[0, 11] by [0, 500]


[0, 50] by [0, 1]


Chapter 2 ProjectAnswers are based on the sample data shown in the table.

1. 2. y � �4.962(x � 1.097)2 � 0.8303. The sign of a affects the direction the parabola opens.

The magnitude of a affects the vertical stretch of the graph.Changes to h cause horizontal shifts to the graph, while changes to k cause vertical shifts.

4. y � � 4.962x2 � 10.887x � 5.1415. y � � 4.968x2 � 10.913x � 5.1606. y � � 4.968 (x � 1.098)2 � 0.833

SECTION 3.1 Exploration 11. (0, 1) is in common; Domain: (��, �); Range: (0, �); Continuous; Always increasing; Not symmetric; No local extrema; Bounded below by

y � 0, which is also the only asymptote; limx → �

f(x) � �. limx → ��

f(x) � 0

2. (0, 1) is in common; Domain: (��, �); Range: (0, �); Continuous; Always decreasing; Not symmetric; No local extrema; Bounded below by y� 0, which is also the only asymptote; lim

x → �g(x) � 0, lim

x → ��g(x) � �

Exploration 21.

Exercises 3.118. Reflect f(x) � 2x over the y-axis and then shift by 5 units to the right. 19. Vertically stretch f(x) � 0.5x by a factor of 3 and then shift 4units up. 20. Vertically stretch f(x) � 0.6x by a factor of 2 and then horizontally shrink by a factor of 3. 21. Reflect f(x) � ex across they-axis and horizontally shrink by a factor of 2. 22. Reflect f(x) � ex across the x-axis and y-axis. Then, horizontally shrink by a factor of 3.23. Reflect f(x) � ex across the y-axis, horizontally shrink by a factor of 3, translate 1 unit to the right and vertically stretch by a factor of 2.24. Horizontally shrink f(x) � ex by a factor of 2, vertically stretch by a factor of 3, and shift down one unit. 25. Graph (a) is the only graphshaped and positioned like the graph of y � bx, b � 1. 29. Graph (b) is the graph of y � 3�x translated down 2 units. 30. Graph (f) is thegraph of y � 1.5x translated down 2 units. 31. Exponential decay; lim

x → �f(x) � 0, lim

x → ��f(x) � �

40. y2 � y3 since 2 � 23x � 2 � 21 � 23x � 2 � 21 � 3x � 2 � 23x � 1

41. 42. 43. 44.

y-intercept: (0, 4) y-intercept: (0, 3) y-intercept: (0, 4) y-intercept: (0, 3) Horizontal asymptotes: Horizontal asymptotes: Horizontal asymptotes: Horizontal asymptotes:y � 0, y � 12 y � 0, y � 18 y � 0, y � 16 y � 0, y � 9

[–5, 10] by [–5, 10][–5, 10] by [–5, 20][–5, 10] by [–5, 20][–10, 20] by [–5, 15]

[–4, 4] by [–2, 8]

[0, 1.6] by [–0.1, 1]



5144 demana te ans pp915-943...5144_demana_te_ans_pp915-943 1/24/06 8:24 am page 919 63. (a) h 16t2...

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