Core Connections Algebra 2

Selected Answers 

for 

Core Connections Algebra 2

Lesson 1.1.1 

1-4. a: 1 2 b: 3 

1-5. a: h(x) then g(x) b: Yes, g(x) then h(x). 

1-6. See graph above right. 

# of Buses 

4 

3 

2 

1 

45 90 135 180 

# of Students 

1-7. a: y 

b: c: y 

d: 

y 

y 

x 

x 

x 

x 

1-8. a: Not linear. b: The exponent. c: A parabola. 

1-9. Answers will vary. 

Lesson 1.1.2 (Day 1) 

1-12. y = 2x + 10 

See graph and table at right. 

x 0 1 2 3 4 

y 10 12 14 16 18 

y 

1-13. a: x = –13 or 17 b: x = – 3 2 or 7 3 

c: x = 0 or 3 d: x = 0 or 5 

e: x = 7 or –5 f: x = 1 3 

or –5 

x 

1-14. a: 14, –4, 3x – 1 b: f(x) = 3x – 1 

1-15. a: y = 5x – 2 b: x = 2 5 

1-16. a: 21, 15, (0, 15) b: –3, 3, (0, 3) 

1-17. a: 16 b: 9 c: 478.38 

Temperature 

1-18. a: y depends on x; x is independent. Explanations will vary. 

b: Temperature is dependent; time is independent. 

c: See graph above right. 

Time 

2 © 2013 CPM Educational Program. All rights reserved. Core Connections Algebra 2


1-19. y = 30 – x 

Graph and table shown at right. 

Answers will vary. 

x 0 1 6 20 

y 30 29 24 10 

y 

1-20. See graph below. Possible inputs: x –4 –2 0 1 6 

all real numbers; possible outputs: y 8 2 0 0.5 18 

any number greater than or equal to zero. 

1-21. a: 1 b: x = 12 

c: 13 d: no solution 

y 

x 

e: x = ± 13 2 

≈ ± 2.55 f: x = ± 7 ≈ ± 2.65 

x 

1-22. Cube each input: f (x) = x 3 

1-23. a: The more gas you buy, the more money you spend. I: gallons, D: dollars 

b: People grow a lot in their early years and then their growing slows down. 

I: age, D: height 

c: As time goes by, the ozone concentration goes down, although the effect is slowing. 

I: year, D: ozone 

d: As the number of students grows, more classrooms are used and each classroom holds 

30 students. I: students, D: classrooms 

e: Possible inputs: x can be any number between and including 0 and 120, 

possible outputs: y = 1, 2, 3, 4 

1-24. They are similar by AA. 

a: n m b: m x 

1-25. Error in line 2: It should be –14, not +14; x = –37. 

Selected Answers © 2013 CPM Educational Program. All rights reserved. 3

Lesson 1.1.3 

1-34. a: The numbers between –2 and 4 inclusive. 

b: The numbers between –1 and 3 inclusive. 

c: No. He is missing all the values between those 

numbers. The curve is continuous, so the description 

needs to include all real numbers, not just integers. 

(–2,3) 

y 

x 

(4,–1) 

d: See graph at right. 

1-35. a: 70 b: 2 c: 43 d: undefined 

e: 3x 2 = x – 5 – 3 f: 3x 2 = x – 5 + 7 

g: all real numbers h: all real numbers greater than or equal to 5. 

i: They are different because the square root of a negative is undefined, whereas any real 

number can be squared. 

1-36. Chelita is correct about how to find the intercepts, but she makes an error with signs 

while factoring. The correct equation is (x − 7)(x − 3) = 0 and the x-intercepts are 

7 and 3. 

1-37. a: y = x–6 

3 

b: y = 

x+10 

5 

c: y = ± x 

d: y = ± x+4 

2 

e: y = ± x + 5 

1-38. a: –7 b: 3.5 c: The x- and y-intercepts. 

1-39. a: y = 3x + 24; Table and graph shown at right. 

1-40. a: x = 13 b: x = 8 

x 

0 

1 

2 

3 

4 

5 

y 

24 

27 

30 

33 

36 

39 

Height (in inches) 

Time (in weeks) 


Lesson 1.1.4 

1-46. (2, 1) 

1-47. a: 2 b: 10 c: 100 d: ≈ 142.86 

1-48. a: x = 5, 3 b: x ≈ 3.39, –0.89 or x = 

5± 73 

4 

1-49. a: 34 ≈ 5.83 units b: 3 5 

1-50. a: 1 52 

b: 

51 

52 

1-51. The error is in line 3. It should be: 0 = 5.4x + 23.7, x ≈ –4.39 

1-52. a: x ≈ –7.37 b: x = 2.8 


1-59. Table and graph shown below right. 

D: −∞ < x < ∞ ; R: −∞ < y < ∞ 

intercepts (0, –4) and 

3 

4, 0 

1-60. a: ≈ 5.18 b: ≈18.66 

( ) or (≈ 1.59, 0) 

x 

–3 

–2 

–1 

0 

1 

2 

3 

h(x) 

–31 

–12 

–5 

–4 

–3 

4 

23 

h(x) 

x 

c: ≈ 24.62º d: 180 ≈ 13.42 

1-61. a: A line, no variables are raised to a power. 

y 

b: y = 2 3 

x – 2 , graph shown at right. 

c: Substitute x = 0 and solve for y, substitute 

y = 0 and solve for x. (3, 0) and (0, –2) 

d: Answers will vary. 

x 

y 

e: The intercepts are (–9, 0) and (0, 6), 

graph shown at right. 

x 

1-62. a: D: x = –1, 1, 2 b: D: –1 ≤ x < 1 c: D: x ≥ –1 d: D: −∞ < x < ∞ 

R: y = –2, 1, 2 R: –1 ≤ y < 2 R: y ≥ –1 R: y ≥ –2 

1-63. There is an error in line 2. Both sides need to be multiplied by x: 5 = x 2 – 4x, 

0 = x 2 –4 –5 = (x –5)(x + 1), x = –1, 5. 

1-64. a: x = 3, –2 b: x = 3, –3 



1-65. a: 2 b: –4 c: 1 0 

is undefined d: Answers will vary. 

1-66. a: (0, 3) and ( – 

2 3 , 0 ) , see graph at right. 

b: These equations are equivalent, they just have different 

notation. 

1 

1-67. x ≈ 2.72 feet, y ≈ 1.27 feet 

1-68. a: D: –2, –1, 2 b: D: –1< x ≤ 1 c: D: x > –1 d: D: −∞ < x < ∞ 

R: –1, 0, 1 R: –1< y ≤ 2 R: y > –1 R: −∞ < y < ∞ 

1-69. l = 4w and l + w = 22 or w + 4w = 22 

The length is 17.6 cm, and the width is 4.4 cm. 

1-70. a: x = − 

17 1 ≈ −0.059 b: x = 

66 

13 

≈ 5.08 c: x = –1, 3 

1-71. a: (–1, 9) and (5, 21) b: x 2 +17 c: x 2 – 4x – 5 


1-72. a: x = 5(y–1) 

3 

b: x = –2y+6 

3 

c: x = ± y d: x = ± y +100 

1-73. y = π x 2 , table and graph shown at right. 

x 0 1 2 3 4 

1-74. a: 58 ≈ 7.62 

y 0 π 4π 9π 16π radius 

b: – 

7 

3 

1-75. Solve x 2 + 2x +1 = 1 ; 0 or –2. 

1-76. a: (0, 6) b: (0, 2) c: (0, 0) 

d: (0, –4) e: (0, 25) f: (0, 13) 

-3 

y 

3 

1 

-1 -1 

area 

x 



1-84. (1, 3) and (7, 81) 

1-85. a: x = –6 b: x = 38 

13 ≈ 2.92 

1-86. Graph shown at right. intercepts: (0, –2) and (4, 0), 

domain: x ≥ 0, range: y ≥ –2 

f(x) 

x 

1-87. x +(x + 18) or x + y = 84 and y = x +18; 33 and 51 meters long. 

1-88. a: Table and graph shown at right, y = 2x + 26. 

b: 37 weeks after his birthday. 

1-89. y = 0 

a: (–2, 0) b: (–10, 0) c: (0, 0) 

( ) e: (5, 0) f: 

3 

( 13, 0) 

d: ± 2, 0 

1-90. Graph shown at right. domain: −∞ < x < ∞ , range: y ≥ –8 

x 

0 

1 

2 

3 

4 

y 

26 

28 

30 

32 

34 

-2 

y 

13 

y 

x 

4 x 



1-91. a: x = y–b 

m b: r = ± A π 

c: W = V 

LH d: y = 1 

3–2x 

1-92. See table and graph at right. Answers will vary. 

1-93. a: Answers will vary. 

b: When the y-values are the same, they must be equal. 

c: 3x + 15 = 3 – 3x, x = –2 

d: y = 9 

y 

x 

x 

–3 

y 

–2 –1 

–1 –2 

–0.5 –4 

0 undef. 

0.5 4 

1 2 

2 1 

3 

e: They cross at the point (–2, 9). 

h(x) 

1-94. 7.5 feet 

1-95. ( ± 5, 0) ; Graph shown at right. 

x 

1-96. a: y 

b: y c: y-intercept (0, 3) for both, x-intercept 

x 

1-97. a: 4 b: 2 c: 3 d: 1 

x 

( – 

2 3 , 0 ) for (a) and none for (b). 

d: (0, 3) and (2, 7), solve 2x + 3 = x 2 + 3 to 

get x = 0 

or x = 2 


Lesson 1.2.3 

1-104. m = – 

4 3 , (4, 0), (0, 3), graph shown at right. 

5 

y 

1-105. y = 3 2 x – 3 

–4 

4 

x 

1-106. x = 

−3± 21 

7± 193 

2 

≈ –3.79, 0.79 b: x = 

6 

≈ 3.48, –1.15 

–5 

1-107. $12.00 

(Schools with an open campus) 

1-108. Sample graphs. 

# of People 

or 

# of People 

Time 

= 2 

3 

Time 

1-110. a: 1, 2, 3, 4, 5 or 6 b: 1 6 c: 4 6 

1-109. a: D: –3≤ x ≤ 3 b: D: x = 2 c: D: x ≥ −2 

R: y = –2, 1, 3 R: −∞ < y < ∞ R: −∞ < y < ∞ 


Lesson 1.2.4 

1-112. a: A portion of the trip at a specific speed. 

b: About 400 miles. It is the total distance on the graph. 

c: Graph shown below – a speed of approximately 30 mph for 1 hour, approximately 

80 mph for the next 3 hours, 0 mph for 2 hours, approximately 40 mph for 2 hours, 

and then approximately 20 mph for the last 2 hours. Note that the step graph assumes 

instantaneous change of speed, which is not technically possible. 

1-113. a: x = 2 b: x = 4 

1-114. mB = 39.8° , 244 ≈ 15.62 

1-115. 56 inches 

Speed (mph) 

Time (hours) 

1-116. The independent variable is the volume of 

water; the dependent variable is the height 

C 

of the liquid. The graph is 3 line segments 

A 

starting at the origin. C is the steepest, and 

B is the least steep. 

Volume of Water 

Added 

1-117. Diagrams vary; graph and table below, y = 3x. 

1-118. a: 

26 1 b: 25 1 x y 

1 3 

Height of Liquid 

2 6 

3 9 

B 


Lesson 2.1.1 

2-4. a: See graph at right. 

b: Yes, for every possible amount of water usage, 

there is only one possible cost. 

c: Domain: 0 to 1,000 cubic feet; range: discrete 

values including: $12.70, $16.60, $20.50, $24.40, 

$29.60, $34.80, $40, $45.20, $50.40, $55.60, 

$60.80 

Cost ($) 

Water Used (ft 3 ) 

2-5. Smallest: a: 2; b: 0; c: –3; d: none. 

Largest: a: none. b: none. c: none. d: 0. e: At the vertex. 

2-6. The negative coefficient causes parabolas to open 

downward, without changing the vertex. See graph at right. 

2-7. a: Parabola with vertex (3, 0), see graph at right. 

b: Shifted to the right three units. 

2-8. a: 4, 1, 0.25; t(n) = 256(0.25) n – 1 

b: They get smaller, but are never negative. 

c: See graph at right. They get very close to zero. 

d: The domain is n integers greater than or equal to zero. 

The domain of the function is all real numbers. 

y 

x 

t(n) 

2-9. a: y = − 2 3 x − 4 b: y = 2 

n 

c: x = 2 d: y = 2 3 x − 8 3 

2-10. n = 24; 650 = 5 26 


Lesson 2.1.2 


2-17. a: (0, –6) 

b: (–6, 0) and (1, 0) 

c: x-intercepts at (0, 0) and (–5, 0) and y-intercept at (0, 0); the graph of p(x) is 6 units 

lower than q(x) 

d: –6 

2-18. a: z = 1.5 b: z = – 18 5 

c: z = 8 d: z = –3, 2 

2-19. a: 3 b: 

1 

x 2 y 4 c: y 

x 

2-20. a: 3p + 3d = 22.50 and p + 3d + 3(8) = 37.5, so popcorn costs $4.50 and a soft drink costs 

$3.00. 

b: Answers will vary. 

2-21. a: 146 ≈ 12.1 b: 145 ≈ 12.0 c: 50 ≈ 7.1 d: 5 2 

2-22. Maximum profit is $25 million when n = 5 million. 

2-23. a: vertex at (–3, –8), opens up, vertically stretched. 

b: x-intercepts (–5, 0) and (–1, 0); y-intercept (0, 10) 

2-24. a, b, and c: Answers will vary. 

2-25. a: y = (x – 8) 2 – 5 b: y = 10(x + 6) 2 c: y = –0.6x(x + 7) 2 – 2 


2-27. a: 5 2 b: 6 2 c: 3 5 

2-28. a: x = 46.71 b: x = 8.19 

2-29. About $ 365.00. b: y = 300(1.04) x 


Lesson 2.1.3 

2-35. a: y = 0 or 6 b: n = 0 or –5 c: t = 0 or 7 d: x = 0 or –9 

2-36. a: (7, −16), y = (x − 7) 2 −16 b: (2, −16), y = (x − 2) 2 −16 

c: (7, −9), y = (x − 7) 2 − 9 d: (2, –1) 

2-37. a: (2, –1) 

b: When x = 2, (x – 2) 2 will equal zero and y = –1, the smallest possible value for y in 

the equation. So the y-value of the vertex is the minimum value in the range of the 

function. 

2-38. a: 9.015 gigatons 

b: C(x) = 8(1.01) (x+2) if x represents years since 2000 or 8.16(1.01) x . 

2-39. a: 2 b: 1 c: 1 d: 2 e: 2 f: 1 

h: If the factored version includes a perfect-square binomial factor, the parabola will 

touch at one point only. 

2-40. a: 4 b: 

1 

16x 4 y 10 

c: 6xy3 

2-41. a: 8 27 

b: 

12 

27 c: 6 

27 d: 1 

27 


Lesson 2.1.4 

2-50. a: f (x) = (x + 3) 2 + 6 (–3, 6), x = –3 

b: y = (x − 2) 2 + 5 , (2, 5) x = 2 

c: f (x) = (x − 4) 2 −16 , (4, –16), x = 4 

d: y = (x + 3.5) 2 −14.25 , (–3.5, –14.25), x = –13.5 

2-51. 

b 2 a 

2-52. The second graph is a reflection of the first 

across the x-axis. See graph at right. 

y 

x 

2-53. a: 45 = 3 5 ≈ 6.71; y = 1 2 

x + 5 b: 5; x = 3 

c: 725 ≈ 26.93; y = − 5 2 x + 5 2 

d: 4; y = –2 

2-54. After x is factored out, the other factor is a quadratic equation. After using the 

Quadratic Formula the solutions are x = 

−23± 561 

8 

or 0. 

2-55. a: x = 21 b: x = 10 5 ≈ 22.4 c: x = 50 

2-56. a: 1 4 b: 1 3 

2-57. B 

2-58. a: A cylinder b: 45π = 141.37 cubic units 

2-59. a and b: Answers will vary. c: A circle. 

2-60. (5, 14) 

2-61. a: 0.625 hours or about 37.5 minutes. 

b: 0.77 hours or about 46.2 minutes. 

c: About $22.99 per minute. 

2-62. a: 61 b: 30º c: tan −1 ( 4 5 ) d: 5 3 

2-63. a: Years; 1.06; 120,000; 120000(1.06) x 

b: Hours; 1.22; 180; 180(1.22) x 


Lesson 2.1.5 


2-70. See graph at right. 

a: It is the slope. 

b: No, because only lines have (constant) slopes. 

This 2 is the stretch factor. 

2-71. a and b: No. Answers will vary. 

2-72. a: y = 0.25 ⋅6 x b: y = 12 ⋅0.3 x 

y 

x 

2-73. a: x: (1, 0), ( – 5 2 , 0 ), y : (0, –5) b: x: (2, 0), y: none 

y 

2-74. See graphs at right. 

a: stretched parabola, vertex (0, 5) 

b: inverted parabola, vertex (3, –7) 

2-75. a: x = ± 5 b: x = ± 11 

y 

x 

x 



2-81. Possible equation: y = − 4 25 (x − 5)2 + 8 , standing at (0, 0) 

domain: 0 ≤ x ≤ 10; range: 4 ≤ y ≤ 8 

2-82. a: x : (− 1 2 , 0), (−1, 0); y : (0, 1) b: x = – 3 4 

( ) or (–0.75, –0.125) 

c: – 3 4 , – 1 8 

2-83. Move it up 0.125 units: y = 2x 2 + 3x +1.125 

2-84. a: 2 6 b: 3 2 c: 2 3 d: 5 3 

2-85. a: Years; 0.89; 12250; 12250(0.89) x b: Months; 1.005; 1000; 1000(1.005) x 

2-86. a: 32 b: x 2 y 2 x c: x2 y 

2-87. c + m = 18 and $4.89c + $5.43m = $92.07 

10.5 lbs. of Colombian and 7.5 lbs. of Mocha Java. 


2-88. a: 15 ft 

b: Surface area of concrete: 793.14 sq. ft.; 528.76 cu. ft.; $1,263.74 

2-89. a: See graph at right. b: y = 3x +2 c: 2, 5, 8, 11 

d: One is continuous and one is discrete. They have 

the same slope so the “lines” are parallel, but they 

have different intercepts. 

2-90. a: 4.116 ⋅10 12 b: y = 1.665(10 12 )(1.0317) t 

c: Answers will vary. 

y 

x 

2-91. a: 6 x + 3 y b: 32 c: 5 d: 

3 

2 

y 

2-92. a: 6x 3 + 8x 4 y b: x 14 y 9 

2-93. See graph at right. line of symmetry x = 4 

x 

2-94. a: 4π + 4 3 π ≈ 16.755 m 3 b: No; r, r 2 , r 3 relationship; V = 80π 

3 ≈ 83.776 m 3 

c: V = 4 3 π r3 + 4π r 2 



2-95. a: y = 1 

x+2 b: y = x2 – 5 c: y = (x – 3) 3 d: y = 2 x – 3 

e: y = 3x – 6 f: y = (x + 2) 3 + 3 g: y = (x + 3) 2 – 6 h: y = –(x – 3) 2 + 6 

i: y = (x + 3) 3 – 2 

2-96. He should move it up 6 units or redraw the axes 6 units lower. 

2-97. a: 18 b: 3 2 c: 1 3 or 3 

3 

d: 11+ 6 2 

2-98. a: (2x – 3y)(2x + 3y) b: 2x 3 (2 + x 2 )(2 − x 2 ) 

c: (x 2 + 9y 2 )(x − 3y)(x + 3y) d: 2x 3 (4 + x 4 ) 

2-99. x = −by3 +c+7 

a 

2-100. a: t(n) = –6n + 26 b: t(n) = −1.5(4) n or − 6(4) n−1 

y 

2-101. a: See graph at right. b: 2 

c: –1 d: 

3 

–13 

e: no solution f: Three because the graphs cross three times. 

g: x 3 – x 2 – 2x 

x 



2-107. a: y = (x – 2) 2 + 3 b: y = (x – 2) 3 + 3 c: y = –2(x + 6) 2 

2-108. a: D: all real numbers, R: y ≥ 3 b: D and R: all real numbers 

c: D: all real numbers, R: y ≤ 0 

2-109. a: compresses or stretches b: shifts up or down 

c: shifts left or right d: shifts up or down 

2-110. a: y = 0.4 ⋅0.5 x b: y = 8 ⋅2 x 

2-111. a: 2 25 b: 3x2 y 3 

z 4 c: 54m 4 n d: y 5x 2 z 

3 

2-112. a: See table and graph at right. 

b: He had 28,900 miles in May. 

c: 5600 miles 

d: No, he will not be able to go in 

December, he will only have 

24,200 miles. 

Miles 

2-113. a: x = ± y 2 +17 b: x = (y + 7)3 − 5 

Month 

Month Miles 

1 15,000 

2 18,000 

3 22,900 

4 25,900 

5 28,900 

6 8,800 

7 11,800 

8 14,800 

9 19,700 

10 22,700 

11 25,700 

12 5,600 



2-114. a: (10, 48) b: 

29 

( 5 

, 9 5 ) 

2-115. a: 8 3 b: 3 x c: 12 d: 108 

2-116. a: g ( 1 2 = –4.75 ) b: g(h +1) = h 2 + 2h − 4 


a: y = 2x : (0, 0), y = – 1 2 

x + 6 : (0, 6), (12, 0) 

b: It should be a triangle with vertices 

(0, 0), (12, 0), and (2.4, 4.8). 

c: Domain 0 ≤ x < 12, Range 0 ≤ y ≤ 4.8 

d: A = 1 2 

(12)(4.8) = 28.8 square units 

y 

x 

2-118. y ≈ 2(x − 5) 2 + 2 and y ≈ − 1 2 (x − 5)2 + 2 


y = (x +1) 2 − 81; x-intercepts: (–10, 0), (8, 0), 

y-intercept: (0, –80); vertex: (–1, –81) 

2-120. Yes, when n = 117. 

y 

x 


Lesson 2.2.3 

2-125. a and : Neither c: Even 

2-126. a: b: c: Neither function is odd nor even. 

2-127. y = − 3 4 (x − 2)2 + 3 

2-128. a: x: (–1, 0), y: (0, 2) b: x: (0, 0), (2, 0), y: (0, 0) 

V: (–1, 0), y = 2(x +1) 2 V: (1, 1), y = –(x –1) 2 +1 

2-129. a: y = x b: 

1 

( 2 

, 1 3 ) c: 

1 

( 2 

, 1 3 ) 

d: The solution to the system is the point at which the lines intersect. 

2-130. a: t(n) = 40( 1 4 )n or 10( 1 4 )n−1 b: t(n) = −6n + 4 

2-131. a: x: (2, 0), (6, 0) y: (0, 2) vertex (4, –2), D: all real numbers; R: y ≥ –2 

b: x: (–4, 0), (2, 0) y: (0, 2) vertex (–1, 3), D: all real numbers; R: y ≤ 3 


Lesson 2.2.4 

2-139. y = (x + 3.5) 2 − 20.25 


b: Loudness depends on distance. 

2-141. See graph at right. The domain is all positive 

numbers (or d > 0). The range is all real 

numbers greater than or equal to 3 and that 

are multiples of 0.25. 

Cost ($) 

$5.00 

$4.00 

Loudness 

Distance 


$3.00 

2-143. The second graph shifts the first 5 units left and 

7 units up and stretches it by a factor of 4. 

2-144. a: x 2 –1 b: 2x 3 + 4x 2 + 2x 

c: x 3 − 2x 2 − x + 2 d: y: (0, 2), x: (1, 0), (–1, 0), (2, 0) 

2-145. a: (a, b) = ( 2, ± 1 2 ) b: (a, b) = ( 1 2 , ± 2 ) 

2-146. a: y = − 5 9 (x − 3)2 + 5 b: x = − 3 

25 (y − 5)2 + 3 

0.2 0.4 0.6 0.8 

Distance (miles) 


a: y = 2x 2 − 4x + 6 

b: There is no difference, but the explanations vary. 

c: y = x 2 

d: y = x 2 

y 

x 

2-148. a: The graph will be a circle with a center at (5, 8) and a radius of 7. 

b: See graph at right. 

2-149. a: –2 b: –2 c: 1 2 

d: –1 

e: The product of the slopes of any two perpendicular lines is –1. 


2-151. a: (0, 0), (–24, 0), and (0, 0) b: (6, 0), (10, 0), and (0, 60) 

2-152. (3, 2) 


Lesson 2.2.5 

2-162. x < 2, y = –(x – 2) 2 ; x ≥ 2, y = x + 2 

2-163. Any function for which f (x) = f (−x). On a graph, the function will have the y-axis as its 

line of symmetry. 

2-164. y = −2 x + 3 + 4 

2-165. a: (x + 2) 2 + (y − 3) 2 = 4 b: (x −12) 2 + (y +15) 2 = 81 

2-166. y = (x − 2.5) 2 + 0.75 , vertex (2.5, 0.75) 

2-167. He is incorrect. Answers will vary. 

2-168. f (x) = x 2 +1 

2-169. ± 11, ± 9, ±19 

Lesson 3.1.1 

3-5. a: 4x 2 –12x +14 b: 81y4 

x 4 

3-6. a: 3 b: 4 c: 1 d: 5 e: 2 

3-7. They are both correct: x12 y 3 

64 

. 

3-8. a: Horizontal line through (0, 3), domain: all real numbers, range: 3 

b: Vertical line through (–2, 0), domain: –2, range: all real numbers 

c: (–2, 3) 

3-9. m = 15, b = –3 

3-10. a: (4, 8, 4 3 ), (5, 10, 5 3 ) 

b: The long leg is n 3 units long, and the hypotenuse is 2n units long. 

3-11. a: 15, 21, 27, 33, t(n) = 6n –3 

b: 27, 81, 243, 729, t(n) = 3 n 

3-12. a: 1 5 b: 3 c: 27 d: 1 8 


Lesson 3.1.2 

3-23. a: not equivalent b: equivalent c: equivalent 

d: equivalent e: not equivalent f: not equivalent 

3-24. a: equal if x = 0 e: equal if x = 0 or x = 1 f: equal if a = 1 or a = 0 

3-25. a: Possibilities include x – 2 = 4 or 2x – 4 = 8. 

b: They have the solution x = 6. 

c: 3 – x = 7, x = –4 

3-26. a: t (n) = –3n + 17, points along a line with y-intercept (0, 17) and slope –3; 

b: t(n) = 50(0.8) n , points along a decreasing exponential curve with 

y-intercept (0, 50) 

3-27. a: 4 b: –30 c: 12 d: –2 1 4 e: x = –4, 1 3 

3-28. (0, 0) and (–6, 0) 

3-29. a: 2x 2 + 6x b: x 2 – 2x –15 c: 2x 2 – 5x – 3 d: x 2 + 6x + 9 

3-30. The first graph opens downward, is stretched, and has its vertex at (–1, –3). 

The second is the parent graph. 

3-31. a: (1, –4) b: (1, –4) c: (–2.5, –4.25) 

3-32. a: y8 

x 12 

d: Domain: −∞ < x < ∞ , Range: y ≥ –4.25 

b: −18x3 y + 6x 5 y 2 z 

3-33. a: odd b: even c: even 

3-34. a: $4.00 b: $4.00 

c: $4.00. $5.00 d: See graph above right. 

e: No, it is a step function. f: The graph will shift (translate) upward by $2.00. 

3-35. a: (x + 2) 2 + (y −13) 2 = 144 b: (x +1) 2 + (y + 4) 2 = 1 

c: (x − 3) 2 + (y + 8) 2 = 16 

-1 -1 1 2 3 4 5 6 7 8 

-2 

Hours 

3-36. a: 24 blocks per hr. b: 18 blocks per hr. c: 11.08 blocks per hr. 

Cost ($) 

7 

6 

5 

4 

3 

2 

1 


Lesson 3.1.3 

3-45. a: n = –2 b: x = –4, 1 

3-46. a: equivalent b: equivalent c: equivalent 

d: not equivalent e: not equivalent f: not equivalent 

3-47. d: equal if a = 0 or b = 0 e: equal if x = 1 f: equal if x = 5 and y = 2 

3-48. 10 = 15m + b and 106 = 63m + b; m = 2, b = –20, t (n) = 2n –20 

3-49. a: t(n) = 450000 (1.03) n 

b: They will make $154,762.37 or 34.39% profit. 

3-50. 5xy(x + 2)(x + 5) 

3-51. a: They both have the solution x = 2. 

b: She divided both sides of the equation by 150 and used the Distributive Property. 


3-52. a: –6, –14, –22, –30, t(n) = 18 – 8n 

b: 2 5 , 25 2 , 2 

125 , 2 

625 , t(n) = 50 ( 1 5 ) n 


3-53. a: 5 1/2 b: 9 1/3 or 3 2/3 c: 17 x/8 d: 7x 3/4 

3-54. a: x 2 + y 2 = 36 

b: (x − 2) 2 + (y + 3) 2 = 36 

c: (x − 4) 2 + (y + 5) 2 = 36 

3-55. 

741.8−25 

1800−0 

= 0.4 ºF/sec 

3-56. a: b: Shift the graph up $11. 

Price ($) 

100 

50 

Duration (days) 


Lesson 3.2.1 

3-63. odd numbers; 46 th term: 91; n th term: 2n – 1 

3-64. after 44 minutes 

3-65. a: 1.03 b: f (n) = 10.25(1.03) n c: $13.78 

3-66. (y –2)(y – 2)(y –2) 

3-67. a: x 1/5 b: x –3 3 

c: x 2 

d: x –1/2 

e: 

1 

xy 8 f: 1 

m 3 g: xy3 x h: 

1 

81x 6 y 12 

3-68. Yes, he can. a: x = 2 b: Divide both sides by 100. 

3-69. a: 5m 2 + 9m – 2 b: –x 2 + 4x +12 

c: 25x 2 –10xy + y 2 d: 6x 2 –15xy +12x 

Lesson 3.2.2 

3-78. a: x–4 

3x+2 b: 5 

x–3 

c: 2 

3-79. a: 1 b: none c: 2 d: 1 

3-80. a: x – 2 = 4 b: For each, x = 6. c: x + 3 = 8, x = 5 

3-81. a: x < 0 b: x ≤ –4 

3-82. a: 3 7 b: 5 4 

y 

3-83. Graph shown at right. 

a: y = x 3 ; The vertex has been shifted up 4 and left 2. 

b: It would not differ. 

x 


a: all real numbers 

b: See graph at far right. 

c: f(x) is a continuous function with 

range y > 0 while t(n) is a discrete 

sequence with positive integer inputs. 

y 

8 

4 

-2 

2 

x 

y 

15 

10 

5 

-2 

2 

x 


Lesson 3.2.3 

3-90. 

2x 

3(2x–1) = 2x 

6x–3 

3-91. a: x ≠ –4 or 2, x+4 

x–2 

b: 

x–4 

x+4 

b: x ≠ –2 or 3, 

2(x+2) 

(x–3) 2 


3-93. a: 

1 

( 3 

, –2) b: (4, –9) 

3-94. n ⋅ 3 15 = 72 million, n = 5; There were 5 bacteria at first. 

3-95. The function is even. A reflection across the y-axis results in the same graph. 

3-96. a: m = 6 b: x = 5.5 c: k = 4 d: x = 90 

Lesson 3.2.4 

3-102. a: Because if x = 4, then the denominator is zero. Since dividing by zero makes the 

expression undefined, x ≠ 4. 

b: a: x ≠ – 1 3 

and x ≠ 5; b: x ≠ 3 or –3 


3-103. a: 

8x+8 

(x–4)(x+2) b: 1 

x+2 

3-104. a: all real numbers b: –5 < x < 4 

c: no solution d: x = 1 3 

3-105. a: x – 4 b: 7m–1 

3m+2 

c: 

(4z–1)2 

z+2 

d: x–3 

x–2 

3-106. a: 1722 b: 1368 c: y = 1500(1.047) n+3 

3-107. a: 5(3x–1) 

2(4x+1) 

b: 1 c: 3 d: –m2 

3-108. See graph at right; x-intercept: (–2, 0), y-intercept: (0, –2); 

there is no value for f(1), which creates a break in the graph. 

3-109. a: –15 b: –4 c: 3 d: –m 2 


Lesson 3.2.5 

3-113. a: 

2 

3x+1 

b: 

x–7 

x–3 c: x–2 

2x+12 d: 1 

x+2 

3-114. a: 1 b: 4 c: 2 d: 5 

3-115. a: x = 3 b: 0 ≤ x ≤ 6 c: x = 1 or 5 d: x < 2 or x > 4 

3-116. Domain: all real numbers; Range y ≥ 0; g(–5) = 8 

g(a + 1) = 2a 2 +16a + 32 , x = 1 or x = 7, x = –3 

3-117. x = –3 or –11 

3-118. a: 1 b: 3 c: 2 

3-119. (–3, 8) and (1, –12) 

3-120. a: 

x+1 

x 2 −4 

b: 

x+6 

2(x+2) 2 c: 1 x d: – 1 2 

3-121. x = 62 

3-122. a: y = – 1 2 x +12 b: y = 2 3 x –15 

3-123. The width is 1.5 meters, and the outer dimensions are 8 m by 5 m. 

3-124. 80x + 0.5 = 100x, so x = 1 40 

of an hour or 1.5 minutes. 

3-125. 6 7 

3-126. a: (5x –1)(5x +1) b: 5x(x + 5)(x − 5) 

c: (x +9)(x –8) d: x(x –6)(x +3) 


Lesson 4.1.1 

4-7. See graph at right. x = 0 and x = 4 

y 

4-8. a: x = 5 or x = –3 b: m = 35 

c: no solution d: x = 7 

x 

4-9. a: y = 0 b: x = 0 

4-10. a: Combining the equations leads to an impossible result, 

so there is no solution. 


c: There can be no intersection because the lines are parallel. 

When assuming there is an intersection, students will find that their work results in a 

false statement. 

4-11. This is a scalene triangle, because the sides have lengths 29 , 17 and 20 . 

4-12. a: 63 b: 0 c: n 3 –1 d: Neither; answers will vary. 

y 

x 

4-13. a: (x−2)(x+6) 

(x+4)(2x+3) 

b: 

2x+1 

x−5 

c: 

9m+27 

m+3 

= 9 d: 

n+3 

n–1 

4-14. a: 0-2 times 

b: 0-4 times 

c: 0-4 times 

d: 1-3 times if you consider parabolas that open up or down. 0-4 times if you consider 

rotated parabolas. 


Lesson 4.1.2 

4-22. Graph y = (x − 3) 2 − 2 and y = x +1 and find the x-values of the points of intersection. 

Or, graph y = x 2 − 7x + 6 and find the x-intercepts. x = 1 or x = 6 

4-23. See graph at right. x = 3 or x = 6 

y 

4-24. a: x = 15 b: x = 7 3 

or x = –5 

x 

4-25. The lines intersect at the point (2, 6). 

Ted will solve the system algebraically by setting 18x – 30 = –22x + 50. 

4-26. a = 18.5, b = 5.5 

4-27. x = 13, x = 5 is extraneous 

4-28. x = 36 b: x = 20 2 or x ≈ 28.28 

4-29. See graph at right. x = 1 and x = 3; No. 

4-30. a: 1 2 (x − 2)3 +1 = 2x 2 − 6x − 3, x = 0 or x = 4 

y 

5 

b: x = 6 is also a solution. 

c: 1 2 (x − 2)3 +1 = 0 , x ≈ 0.74 

5 

x 

d: Domain and range of f(x): all real numbers, domain of g(x): all real numbers, 

range of g(x): y ≥ –7.5 

4-31. a: x = –3 b: x = 1 or x = 3 c: x = –8 or x = 13 d: x = 1.2 

4-32. a: y = 5 3 x – 4 b: m 2 = Fr2 

Gm 1 

c: m = 2E 

v 2 d: y = ± 10 − (x − 4)2 +1 

4-33. (a + b) 2 = a 2 + 2ab + b 2 , substitute numbers, etc. 

4-34. a: See graph at top right. 

b: See graph at bottom right. 

c: Graph (b) is similar to graph (a), but is rotated 

90º clockwise. 

d: (a) D: all real numbers, R: y ≥ 0; (b) D: x ≥ 0, R: all real numbers 

y 

y 

x 

x 

4-35. a: 21.00 b: 117.58 


Lesson 4.1.3 

4-40. a: (–2, –11); The lines intersect at one point. 

b: infinite solutions; The equations are equivalent. 

c: (2, 45), and(–1, 3); The line and parabola intersect twice. 

d: (3, 6); The line is tangent to the parabola. 

4-41. a: y = 3 or y = –5 b: x = – 99 4 

c: y = 1 d: x = –13 

4-42. a: E t(n) = −2 + 3n ; R t(0) = −2, t(n +1) = t(n) + 3 

b: E t(n) = 6( 1 2 )n ; R t(0) = 6, t(n +1) = 1 2 t(n) 

c: t(n) = 10 – 7 d: t(n) = 5(1.2) n e: t(n) = 1620 

4-43. 19.79 feet 

4-44. a: m = – 6 5 , b = (0, –7) b: m = 2 3 , b = (0, –5) c: m = 2, b = (0, –12) 

4-45. a: not a function; D: –3 ≤ x ≤ 3; R: –3 ≤ y ≤ 3 

b: a function; D: –2 ≤ x ≤ 3; R: –2 ≤ y ≤ 2 

4-46. (–7, 11) 

Lesson 4.1.4 

4-51. 4c + 5p = 32, c + 8p = 35, cylinders weigh 3 oz. and prisms weigh 4 oz. 

4-52. Yes. No. Any value of x such that –3 ≤ x ≤ 2 is a solution. 

4-53. a: x = 4 b: x = 6 c: x = 6 d: x = 3 2 

4-54. a: (4, –6) b: (4, –6) c: 

3 

( 2 

, – 9 4 ) 

y 

4-55. a: b: 

y 

x 

x 

4-56. B 

y 

4-57. a: See graph at right. b: x ≈ 0.71 

x 


Lesson 4.2.1 

4-65. a: boundary point x = –4 b: boundary points x = 4, – 3 2 

4-66. a: –4 < x < 1 b: x ≤ –4 or x ≥ 1 c: –1 < x < 4 

d: x ≤ –1 or x ≥ 4 e: –1 < x < 4 f: x ≤ –1 or x ≥ 4 

4-67. a: y = –3x + 8 b: y = –x – 1 2 

4-68. a: No real solutions b: y = 7, y = 13 3 

is extraneous 

4-69. a: 

3x 2 +x−3 

2x 3 +9x 2 −5x 

b: 3x−5 

2x+3 

4-70. x = −6 + 4 6 or x = –6 – 4 6 

c: 

x+4 

4 x−3 

d: 

m+5 

m+4 

4-71. a: x(b + a) b: x(1 + a) c: 

a 

x+1 

d: x–b 

a 


a: Rectangle; perpendicular lines or slopes. 

x 

b: (1, 4), (–3, –3), (–5, 1), (3, 0) 

4-78. a: y = 1 2 x – 2 b: y = 2x + 2 

4-73. a: –5 < x < 13 b: x ≥ 250 or x ≤ –70 c: 

2 3 ≤ x ≤ 2 

7 

4-74. a: C = 800 + 60m b: C = 1200 + 40m c: 20 months d: 5 years 

4-75. a: input x, output x 

b: Replace x with c in first function machine resulting in c – 5 , then substitute this 

expression for x in the second function machine, yielding 6(c−5)+8 

2 

= 3c −11 . 

Substitute this a third time in the final machine, giving (3c−11)+11 

3 

= c . 

4-76. a: 

x–3 

3x–14 

b: 

2x–1 

x+1 

4-77. (1, 12) and (–5, 42) 

y 


Lesson 4.2.2 

4-83. x = –2, y = 3, z = –5; Solve the system to two equations with x and y, then substitute these 

values into the third equation to find z. 

4-84. a: x ≤ 4 b: x < –6 or x > 6 

4-85. red = 10 cm, blue = 14 cm 

–5 0 5 –6 

0 6 

x 

4-86. The points on the line y = 2x – 2 are excluded from the solution region of y < 2x –2. 

4-87. a: y = 1 3 x – 4 b: y = 6 5 x – 1 5 c: y = (x + 1) 2 + 4 d: y = x 2 + 4x 

4-88. y = 0, x = 0 

4-89. 2.11 feet 

Lesson 4.2.3 

4-92. There is no solution, so the lines are parallel. 


a: A square, justifications will vary. 

b: (0, –3), (4, 1), (–4, 1), (0, 5) 

c: 32 square units. 

y 

x 

4-94. a: x < 13 b: 

5− 57 5+ 57 

4 

≤ x ≤ 

4 

or −0.637 ≤ x ≤ 3.137 

13 –1 0 3 

4-95. a: no solution b: y ≈ 4.3 or 10.7 

4-96. (25, –3) 

a: x 2 + 3y = 16 and x 2 − 2y = 31 

b: The solutions to the new system are (5, –3) and (–5, –3). 

4-97. a: See graph at right; y = –2x +8 

b: 63.43º or 116.57º 

y 

x 


Lesson 4.2.4 

4-99. a: y = 1 2 (x + 3)2 − 2 b: y = x +5 c: x = 1 or x = –5 

d: (1, 6) and (–5, 0) e: x < –5 and x > 1 f: x = –1 or x = –5 

g: x = –1 h: Answers will vary. 

4-100. y ≤ 3x + 3, y ≥ 0.5x − 2 , y ≤ −0.75x + 3 

4-101. a: x ≤ 1 or x ≥ 7 3 b: x < 3 

0 1 2 x 1 2 3 x 

4-102. a: y b: y c: 

y 

x 

x 

x 

4-103. 60º 

4-104. a: y > 3x –3 b: y < 3 c: y ≥ 3 2 x – 3 d: y ≥ x2 – 9 

4-105. a: w = 0 or w = –4 b: w = 0 or w = 2 5 

c: w = 0 or w = 6 

Lesson 5.1.1 

5-8. a: y = 2(x +3) b: Yes, y = x. See graph at right. 

y 

5-9. a: 9 b: 4 c: x ≈ 1.89 

x 

5-10. x = sin −1 (0.75) ≈ 48.59° ; to check: sin(48.59 ) ≈ 0.75 

5-11. x must equal y. 

5-12. a: x = 12 5 b: x = 5 2 c: x = 8 d: x = 80 3 

5-13. The area between an upward parabola with vertex (0, –5) 

and the downward parabola with vertex (1, 7). See graph at right. 

5-14. a and b: See graph at right. 

c: Possible equation: y = 10x – 5 

d: For this equation, approximately $495. 

5-15. ≈ 17.74 feet 

x 

x 

Weight (ounces) 

Selected Answers © 2013 CPM Educational Program. All rights reserved. 33 

Cost (dollars) 

y 

y

Lesson 5.1.2 


y 

5-27. a: y = 1 3 

(x + 8) b: y = 2(x – 6) c: y = 2x – 6 

x 

5-28. x ≈ 0.53 

5-29. a: x 2 – 5x –14 b: 6m 2 +11m – 7 c: x 2 – 6x + 9 d: 4y 2 – 9 

5-30. (x + 3) 2 + (y − 5) 2 = 9 . See graph at right. 

y 

5-31. a: 

x−3 

x(x−4) b: 4 

x–2 

c: 2 d: x–1 

x+1 

x 

5-32. a: f (x) ≈ 1.5(1.048) x 

b: ≈ $425.04 

y 

4 


For f(x), domain: –2≤ x ≤ 5, range: –3 ≤ y ≤ 3 

For f –1 (x), domain: –3 ≤ x ≤ 3, range: –2≤ y ≤ 5 

-4 

-4 

4 

x 

5-34. a: L(x) = x 2 −1, R(x) = 3(x + 2) 

b: 30 

c: Order does matter – show by substituting numbers; output is 224 if x = 3 for L(R(x)). 

5-35. a: The system has no solution. 

b: The graphs do not intersect, they are parallel lines. 

5-36. If she adds nothing else to the account and it just sits there making interest, she will have 

$440.13 on her eighteenth birthday. 

5-37. a: x 2 –10x – 56 b: 4m 2 + 8m – 5 

c: x 2 – 81 d: 9y 2 +12y + 4 

5-38. a: (2, 0), (–1, 0) b: (–5, 0), (–3, 0) 

5-39. x = 2.5 


Lesson 5.1.3 

5-48. 121 b: 17 

5-49. a: 2x 3 + 2x 2 − 3x − 3 b: x 3 − x 2 + x + 3 

c: 2x 2 +12x +18 d: 4x 3 − 8x 2 − 3x + 9 

5-50. a: x = – 10 7 b: x = 1 3 or x = 1 

c: x = 115 d: x = 0 or x = 4 

5-51. a: y = ± x − 3 b: 

3 

y = 4 ( x − 6) 

c: y = x2 +6 

5 

5-52. (x − 2) 2 + y 2 = 20 ; circle, x 2 + y 2 = r 2 , center (2, 0) 

and radius ≈ 4.5; See graph at right. 

5-53. 70 

5-54. a: 3 b: y – 4 c: 1 3x d: x 

x–2 

y 

x 


Lesson 5.2.1 

5-60. Domain: x > 0; Range: −∞ < y < ∞ ; x-intercept: (1, 0) no y-intercept; 

asymptote at x = 0 

5-61. a: undefined b: x ≠ 7 c: g(3) = 11 d: f (g(3)) = − 1 2 

5-62. a: e(x) = (x −1) 2 − 5 

b: One machine undoes the other so e( f (−4)) = −4. 

c: They would be reflections of each other across the line y = x. 


y 

x 


a: Domain: all real numbers, range: y > –3 

b: No 

c: (0, –2), (1.585, 0) 

d: Sample: y +a = 2 x , where a ≤ 0. 

5-64. a: x ≈ 36.78 b: x ≈ 31.43 

5-65. a: B = 0.07(0.3x) or B = 0.021x 

b: S = 0.09(0.7x) or S = 0.063x 

c: 0.084x = 5000; $59,523.81 

y 

x 

5-66. a: (x + 7)(x – 7) b: 6x(x + 8) 

c: (x – 9)(x + 8) d: 2x(x + 2)(x – 2) 

y 

5-67. The region between the two parabolas, see graph at right. 

-5 

x 

-5 


Lesson 5.2.2 

5-73. x = 2 y , no, yes, yes; They have the same graph or give the same table of (x, y) values, 

or one is just a rewritten equation of the other. 

5-74. a: x = log 5 (y) b: x = 7 y c: x = log 8 (y) 

d: K = log A (C) e: C = A K f: K = ( 1 2 ) N 

5-75. a: $1.90, 1.38, 0.96, 0.94, 0.90, 0.88 

b: decrease 

c: Smaller size. Note: Sketching a graph of rate with respect to 

bag size like the one at right may help here. 

Rate 

5-76. Answers will vary. Possible answers: 

5-77. x = –4 

a: Factor and use the Zero b: Take the square root (undo) 

Product Property (rewrite), (–8, 0) and (1, 0) 

c: Quadratic Formula d: Complete the square (rewrite) 

Bag size (pounds) 

5-78. a: x = 17 3 ≈ 29.44 b: x = 4 2 ≈ 5.66 

y 

5-79. See graph at right. domain: x ≥ 0, range: y ≥ 0, x- and y-intercept: (0, 0), 

no asymptotes, half of parabola: y = πx 2 

5-80. a: A good sketch would be a parabola opening upwards with a locator 

point at (–6, –7). 

b: Shift the graph up 9 units. 

c: The graph is the same except the region below the x-axis is reflected across the axis so 

that the graph is entirely above the x-axis. 


y 

x 

e: y = x + 7 − 6 

x 


Lesson 5.2.3 

5-84. Possible answer: y = 2 x +15 5-85. y = log 7 x 

5-86. n ≈ 3.66 5-87. (x + 2) 2 + (y − 3) 2 = 4r 2 

5-88. $0.66 


a: The second is just the first shifted up ten units. 

b: y = km x + b 

5-90. a: x = 10 or x = –8 b: x = 2 or x –4 

c: –2 < x < 4 d: x ≥ 3 or x ≤ –13 

5-91. a: x(x + 8) b: (xy + 9z)(xy – 9z) 

c: 2(x +8)(x –1) d: (3x + 1)(x – 4) 

y 

x 

5-92. a: 2 b: 

1 

x+2 c: x−4 

(x−2)(x−1) 

d: 

4 x+16 

x(x+2) 

Lesson 5.2.4 

5-96. a: 12 because 12 .926628408 = 10 b: Answers will vary 

5-97. a: x = 25 b: x = 2 c: x = 343 d: x = 3 e: x = 3 f: x = 4 

5-98. Less than one; Answers will vary. 

5-99. x ≈ 17.973 

5-100. a: (2x + 1)(2x – 1) b: (2x +1) 2 c: (2y + 1)(y + 2) d: (3m + 1)(m – 2) 

5-101. a: –1 < x < 3 b: x ≤ 1 or x ≥ 2 

5-102. No; log 3 2 < 1and log 2 3 > 1 

5-103. a: a = y 

b x 

b: b is the x th root of y a , or b = x y a 

. 


38 © 2013 CPM Educational Program. All rights reserved. Core Connections Algebra 2 

y 

x 

y 

x

Lesson 5.2.5 

5-112. f (g(x)) = g( f (x)) = x ; They are inverses. 

5-113. No. For f (x) = mx + b, f (a) + f (b) = ma + b + mb + b = m(a + b) + 2b and 

f (a + b) = m(a + b) + b . 

5-114. x ≈ 1.585 

5-115. a: t(n) is arithmetic, h(n) is geometric, q(n) is neither 

b: No, because h(n) is increasing much faster than the other two. 

c: h(1) = q(1) = 12 and t(2) = h(2) = 36 ; continuous graphs for t(n) and q(n) intersect but 

not for an integer n. h(n) is increasing much faster than q(n). 

5-116. s(n) = (50 + 7n) 2 − 6(50 − 7n) +17 , neither, it is quadratic and there is no common 

difference or multiplier. 

5-117. a: 1 

10 


b: 10x+m 

y 

2 

5-119. (–3, 0, 5) 

5-120. m ≈ 2.19 

-5 5 

-2 

y 

x 

5-121. a and b: g( f (x)) = log x or f (g(x)) = log x , 

see graphs at right. 

c: The log of an absolute value is very different 

from the absolute value of a log. 

-5 

2 

-2 

5 

x 

d: See graph at right. Note that x = 0 is an asymptote 

y 

5-122. 1 2 

no matter where X is placed. 

5 

5-123. x ≈ 1.68 

5 

x 

5-124. a: 

6x−21 

(x−4)(x+1) 

b: 

5+6x 

2(x−5) c: 1 

x+1 

d: 

5 

x 2 −9 

y 

5-125. a: b + a b: 3d + 2c 2 c: x – 1 d: xy 

x 


Lesson 6.1.1 

6-8. a: Their y- and z-coordinates are zero. b: Answers will vary. 

6-9. x = –2, y = 5 

6-10. a: 9 b: 4N – 3, arithmetic 

6-11. a: x ≈ 1.204 b: x ≈ 1.613 c: x = 6 d: x ≈ 2.004 

6-12. a: 

25 1 b: x 

y 2 c: 1 

x 2 y 2 d: b10 

a 

6-13. a: x b: 

6 

x 2 −3x+2 

6-14. a: 1 2 

b: –2 

c: The product of the slopes is –1, or they are negative reciprocals of each other. 

6-15. Heather is correct, because a 4% decrease does not “undo” a 4% increase. 


Lesson 6.1.2 

6-21. a: (0, 10, 0), (0, 0, 4) b: (8, 0, 0), (0, 6, 0), (0, 0, 12) 

c: (0, 0, 4), (0, 0, –4), (2, 0, 0) d: (0, 0, 6) 

6-22. A line b: They do not intersect. c: They do not intersect. 

6-23. a: y = −2(x + 4) 2 + 2 b: y = 1 

x−2 c: y = −x3 + 3 

6-24. It is not the parent. The second equation does not have a vertical asymptote, and it has a 

maximum value, while y = 1 x 

does not. 

6-25. a: x = b 3 b: x = b 

5a 

c: x = b 

1+a 

6-26. a: No, input equals output only if x ≥ 0. 

b: The output is the absolute value of the input value. 

c: n + 2, n 2 − 4 , n 

d: Because x 2 = x . 

6-27. It is the log 5 (x) graph shifted 2 units to the right. 

See graph at right. 

6-28. a: 254,000 people/year b: 1,574,000 people/year c: 1960 to 2010 

6-29. a: –7 b: –102 c: –102 d: –132 


Lesson 6.1.3 


6-36. Yes. 


6-38. y ≤ −x + 4, y ≥ 1 3 x 

6-39. a: x+3 

2x−1 b: 1 

(x−3) 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

x 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

z 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

• • • • • • • • • • • • • • • • 

y 

6-40. a: Most solving strategies will yield x = 8 or x = 1. 

b: x = 1 does not check, so it is extraneous. 

6-41. a: x = –4 or x = 5 2 

b: x = –4, 2, or 3 

6-42. a: Neither b: Even 

6-43. x = 3, y = 1, z = 3 

Lesson 6.1.4 

6-51. (1, –2, 4) 

6-52. a: ≈ $140,809.30 b: ≈ 24.2 years c: ≈ $164,706.25 

6-53. x = 7 

6-54. a: They both equal 16, but this is a special case (for example, 5 3 ≠ 3 5 ). 

6-55. a: x = 6.5 b: x = –3.75 or x = 5 

6-56. a: y = 1 3 x + 5 b: y = 2x + 5 c: y = − 1 2 x + 15 2 

d: y = 2x 

6-57. a: y = −x 2 + 4x b: y = 5 ± x − 3 

6-58. a: See graph right. 

b: See graph far right. 

6-59. 384 feet 

y 

x 

y 

5 

-5 5 

-5 

x 


Lesson 6.1.5 

6-71. x = –1, y = 3, z = 5 

6-72. y = 3x 2 − 5x + 7 

6-73. a: x+3 

x−4 b: 1 

x(x+2) 

6-74. a: y + x 2 b: 2b + 4a2 c: 6x – 1 d: xy 

6-75. a: x = 12 y b: y x = 17 c: 2x = log 1.75 y d: 7 = log x 3y 

6-76. x = 14 

6-77. a: ≈ 0.0488 grams 

b: Roughly between 4600 and 6700 depending on how the base is rounded. 

c: Never 


b: x > −2; y = x + 2 and x ≤ −2; y = (x + 2) 3 

6-79. a: 2 4 b: 2 –3 c: 2 1/2 d: 2 2/3 

6-80. x = –1, y = 3, z = 6 

6-81. y = 2x 2 − 3x + 5 

6-82. a: 24 = b a b: 7 = (2y) 3x c: 5x = log 2 3y d: 6 = log 2q 4 p 

6-83. a: 

3 

x+1 

b: 

x−4 

x 2 −3x+2 

6-84. Yes, Hannah is correct; 4(x − 3) 2 − 29 = 4x 2 − 24x + 7 and 4(x − 3) 2 − 2 = 4x 2 − 24 + 34 

6-85. a: y = 2(x − 2) 2 −1, vertex (2, –1), axis of symmetry x = 2 

b: y = 5(x −1) 2 −12 , vertex (1, –12), axis of symmetry x = 1 

6-86. See graph at right. y = log(x − 6) + 3 

6-87. a: 2a 2 − 4 b: 18a 2 − 4 c: 2a 2 + 4ab + 2b 2 − 4 

d: 2x 2 + 28x + 94 e: 50x 2 + 60x +14 f: 10x 2 −17 


Lesson 6.2.1 

6-95. y = 3 x 

6-96. In 2 = 1.04 x the variable is the exponent, but in 56 = x 8 the exponent is known so 

you can take the 8 th root. 

6-97. x > 100, because 10 2 = 100 


6-99. a: 1 8 b: 1 x 

c: m ≈ 1.586 d: n ≈ 2.587 

e: Answers will vary. x = b 1/a 

6-100. 2 1/2 = 2 and 2 −1 = 1 2 

6-101. a: –3 < x < 3 b: –2 < x < 1 c: x ≤ –2 or x ≥ 1 

6-102. a: Yes 

b: See graph at right, (it is not a function). 

c: Not necessarily. 

d: Functions that have inverse functions have no 

repeated outputs; a horizontal line can intersect 

the graph in no more than one place. 

e: Yes; for example, a sleeping parabola is not a function, 

but its inverse is a function. 

6-103. a: x = –3, y = 5, z = 10 

b: There are infinitely many solutions. 

c: The planes intersect in a line. 

y 

x 


Lesson 6.2.2 

6-113. a: 5.717 b: 11.228 

6-114. a: x2 

x−1 

b: b+a 

a−a 2 b 

6-115. log 5 7 

log5 2 

y 

6-116. It is the log 3 (x) graph shifted 4 units to the left. See graph at right. 

6-117. 16.5 months; 99.2 months 

x 

z 

6-118. They are correct. Vertex: (2.5, –23.75), line of symmetry: x = 2.5. 

6-119. a: f (x) = 4(x −1.5) 2 − 3, vertex (1.5, –3), line of symmetry x = 1.5 

b: g(x) = 2(x + 3.5) 2 − 20.5 , vertex (–3.5, –20.5), line of symmetry x = –3.5 

6-120. a: Consider only x ≥ –2 or x ≤ –2. 

b: Depending on the original domain restriction, y = x+7 

3 

− 2 or y = − x+7 

3 

− 2 . 

c: x ≥ –7 and y ≥ –2 or x ≥ –7 and y ≤ –2. 

6-121. a: 

6x−21 

x 2 −3x−4 

b: 

5 

x 2 −9 

6-122. a: 20, 100, 500 b: n = 7 

c: No, because there are no terms between the 6 th term (62,500) and the 7 th term 

(312,500). 


Lesson 6.2.3 

6-127. a: y = 40(1.5) x 

b: When x = –9, or 9 days before the last day of October (October 22). 

6-128. Possible answer: 4 (x+1) = 6 


6-130. The graph should show a decreasing exponential function which will have an asymptote 

at room temperature. 

6-131. y = x 2 − 6x + 8 

6-132. a: x ≥ 1 2 and y ≥ 3 b: g(x) = (x−3)2 +1 

2 

c: x ≥ 3 and y ≥ 1 2 

d: x e: x (They are the same, because f and g are inverses.) 

6-133. a: x ≈ 6.24 b: x = 5 

6-134. a: (x −1) 2 + y 2 = 9 b: (x + 3) 2 + (y − 4) 2 = 4 

6-135. a: x + 5 b: a + 5 c: x – y d: x2 +1 

x 2 −1 

6-136. a: p −1 (x) = 3 ( x 3 − 6) b: k −1 (x) = 3 ( x−6 

3 ) 

c: h −1 (x) = x+1 

x−1 

d: j−1 (x) = 3x−2 

x 

= − 2 x + 3 


Lesson 6.2.4 

6-138. a: Decreasing by 20% means you multiply by 0.8 each time, and the presence of a 

multiplier implies exponential. 

b: y = 23500(0.8 x ) c: $9625.60 

d: ≈ 6.12 years e: $42,926.44 

6-139. a: x = 1 2 

b: x > 0 c: x = 1023 

6-140. a: x = 2.236 b: x = 4.230 c: x = 0.316 

d: x = 2.021 e: x = 3.673 

6-141. a: 16 b: 12 c: 12 4 = 20736 d: 54 

e: No, they are not inverses (if they were, then the answers to parts (c) and (d) would 

have to be 2). 

6-143. c(x) = x 2 − 5 

6-144. x = 17 

6-145. a: 2(x+1) 

x+3 b: 3x2 −5x−3 

(2x+1) 2 

6-146. a: 30° b: 22.6° 

6-147. y ≤ − 

4 3 x + 3, y ≥ − 4 3 x − 3, x ≤ 3, x ≥ −3 


Lesson 7.1.1 

7-3. a: The shape would be stretched vertically. In other words, there would be a larger 

distance between the lowest and highest points of each cycle. 

b: Each cycle would be longer horizontally. Fewer cycles would fit on a page of the 

same length. 

y 

7-4. See graph at right. domain: x ≠ 3, range: y ≠ 0, 

asymptotes at x = 3 and y = 0 f −1 (x) = 2 x + 3 

x 

7-5. a: ≈ 27.04 feet b: ≈ 176.88 cm c: ≈ 28.94 meters 

7-6. 30º - 60º: 1 2 , 3 

2 ; 45º - 45º: 1 , 1 or 2 

2 2 2 , 2 

2 

7-7. y = 6x − x 2 

7-8. x = 5, x ≈ 19.69 does not check. 

y 

7-9. a: y = ( x + 5 2 ) 2 + 

4 3 , vertex ( − 5 2 , 4 

3 ) b: (0, 7) 

c: (–5, 7); See graph at right. 

7-10. No x-intercepts, y-intercept: (0, 88) 

7-11. (x −1) 2 + y 2 = 30 ; See graph at right. 

center: (1, 0), intercepts: (± 30 +1, 0) and (0, ± 29) 

y 

x 

x 



7-15. a: 30 – 60: hypotenuse: 2, leg: 3 ; isosceles: hypotenuse: 2 , leg: 1 

b: See diagram at right. 

7-16. ≈ 17.46° 

60° 

60° 60° 

7-17. x = 2 , − 5 2 , y = –10 

7-18. a: 0 b: 3 c: 4 d: 64 

7-19. y : 3; 4; 5; undefined; 7; 8 

a: See graph at right. It is linear. The data does not all 

connect because f (1) is undefined. 

b: y = x + 5, f (0.9) = 5.9, f (1.1) = 6.1, no asymptote. 

c: The complete graph is the line y = x +5 with a hole at (1, 6). 

7-20. a: An exponential function b: y = 60000 +12000(0.93) t 

y 

x 

7-21. If he drives down the center of the road, the height of the tunnel at the edge of the house 

is only approximately 23.56 feet. The house will not fit. 

7-22. a: x ≈ 33.752 b: x ≈ 9.663 

7-23. x = 18, y = 13, z = 9 



7-24. −∞ < θ < ∞ 

7-25. ≈ 40.5º or 139.5º 

7-26. She should have subtracted 3⋅16 = 48 to 

account for the factor of three. The vertex is (4, 7). 

7-27. 

1 

7 

7-28. See graph above right. 

y 

x 

y 

7-29. a: b: c: d: 

y 

y 

y 

x 

x 

x 

x 

7-30. x = 3 2 or − 1 4 , y = –3 

W -1 (x) 


1 

b: No; when the points are interchanged, the 

input x = 0 has two outputs. 

–1 

–1 

1 

x 

7-32. R + B + G = 40, R = B + 5, R = 2G ; 18 red, 13 blue and 9 green 


Lesson 7.1.3 


7-37. (a): Above ground just past the highest point. 

(b): Just below ground. 

(c): Back to the starting point. 

See diagram at right. 

7-38. ≈ 82.4 feet 

B 

A 

C 

y 

1 

(a) 

-1 

(c) (c) (c) 

(c) 

90 270 450 630 

(b) 

(b) 

x 

7-39. a: log 6 = log 3+ log 2 ≈ 0.7781 b: log15 = log 3+ log 5 ≈ 1.1761 

c: log 9 = 2 log 3 ≈ 0.9542 d: log 50 = 2 log 5 + log 2 ≈ 1.6990 

7-40. x = −3± 6 

3 

, y = 1 

7-41. y = 3(x +1) 2 − 2 ; See graph at right. 

y 

7-42. x ≤ –5 

x 

7-43. No real solution. 

7-44. C + W + P = 40 , W = C − 5 , C = 2P ; 18 from California, 13 from Washington, and 

9 from Pennsylvania 



7-53. 

15 

( 4 , 1 4 ) 

15 

or ( − 

4 , 1 4 ) 

7-54. P: (cos 50º , sin 50º ) or (~0.643, ~0.766); Q: (cos110º , sin110º ) or (~ –0.342, ~0.940) 

( 2 ) 

7-55. a: 300° b: 1 2 , 3 

2 c: 1 

2 

, − 3 

7-56. a: 30° b: 60 ° c: 67° d: 23 ° 

y 

7-57. x = 11 

5 

x 

7-58. a: downward parabola, vertex (2, 3), see graph above right. 

b: cubic, point of inflection (1, 3), see graph below right. 

y 

7-59. Solving graphically: x ≈ –3.2 

x 

7-60. a: y = 25d + 0.50m and y = 0.03(2) m−1 

b: R vs. T: $55 vs. $15.36, $60 vs. $15,728.64, $100 vs. ~ $1.901×10 28 

7-61. All of these problems could be solved using the same system of equations. 



7-62. 58º, 122º, 238º, or 302º 

7-63. a: An angle in the 4 th quadrant. b: 270 ° or –90° 

c: An angle in the 3 rd quadrant. d: Approximately 160° 

e: No, an angle with sine equal to 0.9 has cosine equal to ±0.4359, so the point (0.8, 0.9) 

is not on the unit circle. 

7-64. a: (0.3420, 0.9397) b: (cos70° , sin70 ° ) 

c: (cos 70°) 2 + (sin 70°) 2 = 0.1170 + 0.8830 = 1 

7-65. Graph 2 is sine, while graph 1 is cosine. Answers will vary. 

7-66. a: All yes. 

b: Answers will vary. 

c: x = (−180° + 360°n) for all integers n 

7-67. y-intercept: (0, –17), x-intercepts: (−2 + 21, 0) and (−2 − 21, 0) 

7-68. a: x = –4 b: x = 

7-69. 7.07 ' 

5± 57 

4 

c: no solution 

d: If a = 

x+2 3 , then a + 5 ≠ a. Or, solving yields x = –2, but when substituted, –2 gives a 

zero denominator. 

7-70. Tess is correct: A sequence has no more than one output for each input. A sequence is a 

function with domain limited to positive integers. 


Lesson 7.1.5 

7-77. a: Same; π 3 

and 60° are measures of the same angle. 

b: 45º, 135º, 405º, etc. 

7-78. a: 

2 

2 ≈ 0.707 b: 3 

2 ≈ 0.866 

7-79. a: Set each factor equal to zero to get x = 0, 1 2 

, or 3. 

b: Factor to get x(x – 1)(2x + 3) = 0. x = 0, 1, – 3 2 

7-80. a: x ≈ 2.657 b: x ≈ 0.936 c: x ≈ –0.711 

7-81. He should have subtracted 2 ⋅ 9 4 = 9 2 to account for the factor of 2. The vertex is 3 

( 2 

, − 5 2 ) 

. 

7-82. a: y = 3(x − 3) 2 −1, vertex: (3, –1), axis of symmetry x = 3 

b: y = 3( x − 2 3 ) 2 − 37 3 , vertex: 2 

( 3 

, − 37 3 ) , axis of symmetry: x = 2 3 

7-83. a: x = 2.5121 b: x = 5 57y 


a: No 

b: –10 ≤ x ≤ 10, –10 ≤ y ≤ 10 

c: 200π 

3 

≈ 209.44 sq. units 

y 

x 

y 

7-85. f −1 (x) = (x −1) 2 + 3 for x ≥ 1; See graph at right. 

x 


Lesson 7.1.6 

7-90. a: –0.76 b: – 3 

2 

7-91. 

π 

6 

, 5π 6 

7-92. 

π 

4 

, π 3 , π 2 , 2π 3 , 3π 4 , 5π 6 , π, 7π 6 , 5π 4 , 4π 3 , 3π 2 , 5π 3 , 7π 4 , 11π 

6 , 2π 

y 

7-93. See diagram at right. 

a: A little less than 360° (almost 344° ). 

b: sin6 ≈ –0.3 

x 

7-94. a: 1 b: 1 2 

c: undefined d: 9 

7-95. ~ 4.73% annual interest 

7-96. 

sin A 

cos A = 3 

10 

91 

100 

≈ −0.3145 

7-97. a: f −1 (x) = x3 +1 

4 

b: g −1 (x) = 7 x 

7-98. a: x = 4 or x = –2 b: x ≈ 2.81 


Lesson 7.1.7 

7-104. 420° 

a: π 3 ± 2π n 

b: See diagram at right. 

c: 

3 

2 , 1 2 , 3 

7-105. a: 0 b: 0 c: –1 

d: 0.5 e: 0 f: undefined 

y 

–1 

1 

0 1 x 

–1 

7-106. Some may set up a proportion, others may use 

π 

180 

. 

7-107. a: 210° b: 300 ° c: π 4 radians 

d: 5π 9 radians e: 9π 2 

radians f: 630 ° 


y 

f(x) 

7-109. f (x) = 2(x − 4) 2 + 2 

f -1 (x) 

x 

7-110. a: – 5 

13 b: 5 

12 

7-111. a: a + b b: 2c c: a + 2b d: 3a + c 


b: Yes, the pizza will never get below room temperature. 

temperature 

time 


Lesson 7.2.1 


y 

b: y = 1+ sin x 

c: y : (0, 1), x : ( − 

2 π , 0 3π 

), ( 2 

, 0 

7π 

), ( 2 

, 0), ... 

x 

d: Yes, there are infinitely many, at intervals of 2π . 

7-117. a: π b: y = sin(x + π ) 

7-118. a: This may go up and down, but the cycles are probably of differing length. 

b: This may or may not be periodic. 

c: This is probably approximately periodic. 

7-119. y = 100 sin ( x + 

2 

π ) − 50 or y = 100 cos x − 50 

7-120. Only one needs to be a parent, since y = sin(x + 90°) is the same as y = cos x. 

7-121. a: y = 3⋅6 x b: y = −2(0.5) x 

7-122. a: x = ± 3 5 = ± 15 

5 

b: x = 4, –1 c: x = 4 

7-123. a: – 3 b: 

3 

3 

7-124. a = − 

3125 3 = −0.00096 , possible equation: y = − 

3 

3125 (x −125)2 +15 


Lesson 7.2.2 

7-129. a: y = sin x − π 4 

( ) + 2 

( ) + 0.5 

( ) + 2 or y = sin ( x + 5π 6 ) + 2 

( ) −1 or y = −3sin ( x + 

3 

π ) −1 

b: y = 1.5 sin x + π 2 

c: y = − sin x − π 6 

d: y = 3sin x − 2π 3 

7-130. 360° is the period of y = cosθ , so shifting it 360° left lines up the cycles perfectly. 

7-131. Graphing form: y = 2(x −1) 2 + 3 ; vertex (1, 3); 


7-132. a: x = (0, 0), 

5±3 3 

( 2 

, 0) and y = (0, 0) 

y 

7-133. 17.67 years 

x 

7-134. a: y = −2 ( x + 1 4 ) 2 + 105 

8 , x = all real numbers, y = −∞ < y < 25 8 

; Yes it is a function. 

b: y = −3(x +1) 2 +15 , domain: all real numbers, range: −∞ < y < 15 ; Yes it is a function. 

7-135. 64.16 ° , unsafe 

7-136. a: 5,000,000 bytes b: ≈ 12.3 minutes 

c: According to the equation, technically never, but for all practical purposes, after 

23 minutes. 


a: The vertex of the graph is at (6, –4) with 

two rays emanating at slopes of ±1. 

b: See graph at right. Flip all parts of the graph 

that are below the x-axis above the x-axis. 

y 

y 

x 

x 


Lesson 7.2.3 

7-144. a: Amplitude 3, period 4π 


c: The differences are the period and 

amplitude, and therefore some of the 

x-intercepts. They have the same basic shape. 

y 

x 

7-145. 1, 2π 

2π 

= 1 or 2π (1) = 2π 

7-146. Colleen’s calculator was in radian mode, while Jolleen’s calculator was in degree mode. 

Colleen’s calculation is wrong. 

7-147. y = sin 2(x −1) is correct. To shift the graph one unit to the right, subtract 1 from x 

before multiplying by anything. 

7-148. They are both wrong. The equation needs to be set equal to zero before the Zero Product 

Property can be applied. 2x 2 + 5x − 3 = 4 is equivalent to (2x + 7)(x −1) = 0 . 

x = 1 or x = − 7 2 

7-149. a: 3 b: 1.5 c: 2 d: 12 

7-150. a: y = 20 ( 1 2 ) x + 5 b: w = 5.078 

7-151. a: Answers vary, if g(x) is linear, tangent lines only. 

b: Any line y = b such that b < –8. 


Lesson 7.2.4 

7-158. a: Yes b: y = cos ( x + 

2 

π ) c: y = –sin x 

7-159. 6 cycles, period: π 3 


7-161. a: 180º b: 540º c: π 6 radians 

d: 45º e: 5π 4 

radians f: 270º 

7-162. a: − 2 

2 

b: 3 c: – 1 2 d: 2 

2 

e: 1 f: − 1 3 

g: π 4 or 5π 4 h: 3π 4 or 7π 4 

7-163. ( –1, 1 2 , 2 ) 

7-164. a: x = 0, x = – 1 2 , or x = 5 3 

b: x = 6 or x = –1 


7-165. a, b, and c: Answers will vary. 

7-166. a: About $564,240 b: In 2025 c: About $36,585 



8-8. See graphs and tables below. Parent functions: 

a: y = x 3 b: y = x 4 c: y = x 3 

x y 

–2 –9 

–1 0 

0 1 

1 0 

2 3 

x y 

–2 9 

–1 0 

0 1 

1 0 

2 9 

x y 

–2 0 

–1 3 

0 0 

1 –3 

2 0 

y 

y 

y 

x 

x 

x 

8-9. Functions in parts (a), (b), and (e) are polynomial functions; explanations vary. 

8-10. Graphs will vary. 

a: 0, 1, or ∞ b: 0. 1, or 2 c: 0, 1, 2, 3 

d: 0, 1, 2, 3, or 4 (1 and 3 require the parabola to be tangent to the circle.) 

8-11. (–2, –1) and (3, 4) 

8-12. a: adds 2; multiplies by 3; ; subtracts 1 

b: f −1 (x) = ( x−3 

2 )2 +1, g −1 (x) = 3(x + 2) −1 

8-13. The second graph is shifted up 5 from the first. 

y 

8-14. a: b: c: 

y 

y 

x 

x 

x 

8-15. a: 4n– 27 b: At least 2507 times. 

8-16. a: 60º, 300º b: 135º, 315º c: 60º, 120º d: 150º, 210º 



8-17. The functions in parts (a), (b), (d), (e), (h), (i), and (j) are polynomial functions. 

8-18. They are not equivalent. Explanations vary. Students may substitute numbers to check. 

Also, the second equation can be written y = – x + 12, which is a line, not a circle. 

8-19. a: x = 2 or x = 4 b: x = 3 c: x = –2, x = 0, or x = 2 


y 

a: 2 b: x = 7, – 7 

8-21. x = –1± 6 

a: 2 b: At x ≈ 1.45 and x ≈ –3.45 

x 


y 

8-23. x = –1 or 5 

x 

8-24. a: y = ( 3 x ) – 4 b: y = 3 (x–7) 

8-25. y: –21.2' ; 0'; 21.2'; 30'; 21.2'; 0'; …; –30' 

a: Repeat the pattern for several cycles. 

b: 30' 

c: y = 30 sin x 


Lesson 8.1.2 

8-36. At (−3 ± 5, 0) . 

8-37. At (74, 0), a double root, and at (–29, 0). 

8-38. Possible Answers: a: y = x 2 + x – 6 b: y = 2x 2 + 5x – 3 

8-39. a: 2 b: 5 c: 3 d: 6 

8-40. Lines, parabolas (vertically oriented), and cubics are polynomial functions because they 

can be written in the form y = ax n . Exponentials are not polynomial functions because 

“x” is the exponent. Circles are not functions. 

8-41. a: y 

b: 

5 

y 

-5 5 

-5 

x 

x 

8-42. a: (x − 2) 2 + (y − 6) 2 = 4 b: (x − 3) 2 + (y − 9) 2 = 9 


y 

8-44. a: 30º, 150º b: 60º, 240º 

c: 30º, 330º d: 225º, 315º 

x 


Lesson 8.1.3 

8-54. Stretch factor is –2. f (x) = −2(x + 2) 2 (x −1) 

8-55. a: degree 4, a 4 = 6 , a 3 = –3 , a 2 = 5 , a 1 = 1, a 0 = 8 

b: degree 3, a 3 = −5 , a 2 = 10 , a 1 = 0 , a 0 = 8 , 

c: degree 2, a 2 = –1, a 1 = 1, a 0 = 0 

d: degree 3, a 3 = 1, a 2 = –8 , a 1 = 15 , a 0 = 0 

e: degree 1, a 1 = 1 

f: degree 0, a 0 = 10 

8-56. Possible equation: p(x) = 2.5(x + 4)(x −1)(x − 3) 

8-57. a: y = 4x 2 + 5x − 6 b: y = x 2 − 5 

8-58. There is no real solution, because a radical cannot be equal to a negative value. 

8-59. a: C: (3, 7), r: 5 b: C: (0, –5), r: 4 

c: x = 4 d: x = 7 

8-60. a: x = log17 

log 2 

b: x = 242 c: x = 4 d: x = 7 

8-61. a: –3 < x < 2 b: x ≤ –1 or x ≥ 7 3 

8-62. y = 2 + 4sinx 


Lesson 8.2.1 

8-70. a: –18 – 5i b: 1± 2i c: 5 + i 6 

8-71. i 3 = i 2 i = −1i = −i ; 1 

8-72. a: –21 b: –10 + 7i c: –22 + i 

8-73. Yes, substitute it into the equation to check. 

8-74. x = –8 

8-75. Yes; both are equivalent to x 2 –10x + 25 . 

8-76. a: 7i b: 2i or i 2 c: –16 d: –27i 

8-77. a: x+3 

2 b: x − 2 + 3 

8-78. a: x ≈ 2.24 b: x ≈ ± 2.25 


Lesson 8.2.2 

8-87. Possible Functions: 

a: f (x) = x 2 + 6x +10 b: g(x) = x 2 −10x + 22 

c: h(x) = x 3 + 2x 2 − 7x −14 d: p(x) = x 3 + 2x 2 −14x − 40 

8-88. a: b 2 − 4ac = −7 , complex b: b 2 − 4ac = 49 , real 

8-89. See graph at right. Area = 25 sq. units 

y 

8-90. a: Repeat 1, i, –1, –i, etc. b: 1, i, –i, 1 

c: 1 d: i, –1, –i 

e: 1, i, –1, –i 

x 

8-91. a: 1 b: i c: –1 

8-92. If n is a multiple of 4, the value is 1; if it is 1 more than a multiple of 4, the value is i; if it 

is 2 more than a multiple of 4, the value is –1; if it 3 more than a multiple of 4, the value 

is –i. 

8-93. a: x = 

log 17 

log 3 

b: 

3 

x = 17 

8-94. a: 2 b: 4 c: 5 d: 3 e: 1 

8-95. a: Standard form for y-intercept at (0, 400) and graphing form for vertex at (0.5, 404). 

b: 400 ft; 404 ft 

8-96. a: y = log x b: x = 2 c: y = log 2 (x − 2) is one possibility. 


Lesson 8.2.3 

8-104. a: three real linear factors (one repeated), therefore two real (one single, one double) and 

zero complex (non-real) roots 

b: one linear and one quadratic factor, therefore one real and two complex (non-real) 

roots 

c: four linear factors, therefore four real and zero complex (non-real) roots 

d: two linear and one quadratic factor, therefore two real and two complex (non-real) 

roots 

y 

8-105. a: b: c: d: 

y 

y 

y 

x 

x 

x 

x 

8-106. a: (3, 0), (0, 0), and (–3, 0) b: See graph at right. 

y 

8-107. See graphs below. 

a: x-intercepts ( − 5 2 , 0 ), (0, 0) , and 

7 

( 2 

, 0), y-intercept (0, 0) 

b: x-intercepts (−3, 0) and 

15 

2 

, 0 

a: b: 

y 

( ) (double root), y-intercept (0, 675) 

y 

x 

x 

x 

y 

8-108. See graph at far right. 

x 

8-109. a: Platform is 11.27 meters off the ground. h = −4.9(t − 5) 2 +133.77 ; 

Therefore, the maximum height is 133.77 meters. Time when h = 0 is 10.22 sec. 

b: h ≈ −4.9(t −10.22)(t + 0.22) ; Factored form reveals the intercepts, or how long it took 

the firework to reach the ground.) 

8-110. b ≥ 20 or b ≤ –20 

8-111. a: (i − 3) 2 = i 2 − 6i + 9 = −1− 6i + 9 = 8 − 6i 

b: (2i −1)(3i +1) = 6i 2 − 3i + 2i −1 = −6 − i −1 = −7 − i 

c: (3− 2i)(2i + 3) = 6i − 4i 2 − 6i + 9 = 4 + 9 = 13 

8-112. ( ±6, 1 2 ) 


Lesson 8.3.1 

8-120. a: –7 c: (x + 7) d: (x 2 − 2x − 2) f: −7, 1± 3 

8-122. Part (c), because (–2)(3)(–5) = 30 and (x)(x)(x) = x 3 not 2x 3 . 

8-123. Part (b), because 5 is a factor of the last term, but 2 and 3 are not. 

8-124. (x − 5)(x 2 − 4x −1); zeros: 5, 2 ± 5 

8-125. a: (x − 2)(5x + 3) b: − 

5 3 , 2 c: Explanations will vary. 

d: 3 and 2 are factors of 6, while 5 is a factor of the lead coefficient. 

8-126. a: See the combination histogram boxplot at right. 

The five number summary (for the box plot) is 

0, 2.75, 8, 15.7, 36.5. 

b: The distribution has a right skew and an outlier at 

36.5 pounds so the center is best described by the 

median of 8.0 pounds and the spread by the IQR 

of 12.95 pounds. 

c: The median is better in this case because it is not 

affected by skewing and outliers. 

d: The IQR is better in this case because it is less affected by skewing and outliers than 

the standard deviation. 

e: If you remove the outlier from the data the mean drops to 8.7 pounds which is below 

the profitable minimum. You could suggest running the test a few more weeks 

because perhaps as people get used to the composting program they will participate 

even more. 

10 

5 

0 6 12 18 24 32 36 42 

8-127. a: b: 

y 

8-128. a: See graph below. 

b: Yes, it is a solution to the equation. 

z 

x 

x 

y 



8-138. a: It shows that (x – 3) is a double factor and 3 is a double root. 

b: p(x) = (x − 3) 2 (x 2 + 2x −1) ; x = 3, −1± 2 

8-139. a: x 2 − 6x + 25 = 0 b: x 2 − 6x + 25 = 0 c: Answers will vary. 

8-140. a: 3+2i 

−4+7i ⋅ −4−7i 

−4−7i = 2−29i 

65 

b: 2 

65 − 29 

65 i 

8-141. a: 12 5 − 1 5 i b: − 3 

13 + 11 

13 i 

8-142. See graph at right; x + 3 −1; x ≥ −3, y ≥ −1 

y 

8-143. a: See graph below left. −1, 1± 3 i 

2 

b: See graph below right. 2, −1± 3 i 

a: b: 

y 

y 

x 

x 

x 

8-144. (3, 4, –1) 

8-145. x = 1 2 

8-146. a: See graph below left, locator( − 

2 π , 0 ) , period = 2π, amplitude: 3 

b: See graph below right, locator (0, 0), period: 

2 π , amplitude: 2, inverted 

y 

y 

x 

x 



8-147. p(x) = x 3 + 5x 2 + 33x + 29 

8-148. a: p(2) = 0 b: (x – 2) c: (x 2 − 4x −1) d: 2, 2 ± 5 

8-149. a: 

17 8 + 15 

17 

i b: 2 + 5i 

8-150. a: Regular: (361, 367, 369 373, 380 grams); Diet (349, 354, 356.5, 361, 366 grams) 

b: See histograms at right. 

Regular 

c: Regular: The mean is 369.6 grams, which falls at the middle 

of the distribution on the histogram. The shape is singlepeaked 

and symmetric, so the mean should be a good 

measure of the center. There are no outliers, so the standard 

deviation of 4.34 grams could be used to describe spread. 

Diet: The mean is 357.5 grams; this mean also falls at the 

center of the data on the histogram. The data is doublepeaked 

but still fairly symmetric so the mean could be used 

to represent the center. There are no outliers so the standard 

deviation of 5.12 grams could be used to describe spread. 

d: The regular cola cans are noticeably heavier (or had more mass) than the diet cans. 

The lightest regular can is at the third quartile of the diet sample and the median of 

the regular cans is heavier than the most massive diet can. The spread of each 

distribution is similar and they are both reasonably symmetric but the diet cans have a 

double peaked distribution. 

e: Answers will vary. 

348 360 372 384 

Diet 

348 360 372 384 

y 

8-151. See graph at right. a: 4 b: (±4, 3) and (±3, −4) 

x 

8-152. a: x = 4 ( 1 is extraneous) b: x = 1 4 

8-153. a: b: 

y 

x 

y 

c: d: 

y 

x 

x 

8-154. a: x = 2, (x – 2) b: x = 2, −3 ± 2i ; (x − 2) ( x − (−3+ 2i) )( x − (−3− 2i) ) 

8-155. a: x = 2π 3 , 4π 3 b: x = π 6 , 7π 6 c: x = 0, π d: x = π 4 , 7π 4 



8-156. a: − 1 5 − 5 7 i b: 1 – 2i 

8-157. At 6 years, it will be worth $23,803.11. At 7 years it will be worth $25,707.36. 

8-158. a: x = 5 9 

b: x = 3 c: x = 48 d: x ≈ 1.46 

8-159. Students should show the substitution of the coordinates of the point into both equations 

to verify. 

8-160. x = 2 or x ≈ 1.1187 

8-161. a: x ≈ 781.36 b: x = 6 c: x = 1, 1 5 

d: x = 0, 1, 2 

8-162. When you find the complement of the angle, the x and y values reverse. 

8-163. a: −7 ⋅ −7 = i 7 ⋅i 7 = i 2 49 = −7 

b: She multiplied −7 ⋅ −7 to get 49 = 7 . 

c: −7 is undefined in relation to real numbers, and is only defined as the imaginary 

number 7i , so it must be written in its imaginary form before operations such as 

addition or multiplication can be performed. 

d: a and b must be non-negative real numbers. 

8-164. a: π 3 

b: 

5π 

12 c: 7π 6 d: 5π 4 


Lesson 8.3.3 

8-169. (0, 0), (3, 0), and (−0.5, 0) 

y 


8-171. a: (x + 10)(x − 10) b: x − 

3+ 37 

( 2 ) 

3− 37 

( x − 

2 ) 

c: (x + 2i)(x − 2i) d: (x − (1+ i))(x − (1− i)) 

x 

8-172. a: real b: complex c: complex 

d: real e: real f: complex 

8-173. It is not; 16 + 8 ≠ 32 − 40 

8-174. a: x = 5 or 1 b: x = 4 or 0 c: x = 7 d: x = 1 

8-175. a: 24 

b: (x 3 − 3x 2 − 7x + 9) ÷ (x − 5) = (x 2 + 2x + 3) with a remainder of 24. 

8-176. a: y = x 2 +1 b: y = x 2 – 2x –1 

8-177. a: b: 

y 

y 

x 

x 


Lesson 9.1.1 

9-7. a: Population: U.S. employees; the population is too large to conveniently measure so 

sampling should be used. 

b: Population: students in the class. A census can be taken. 

c: Population: All carrots. To measure the Vitamin A in a carrot, it must be destroyed, so 

even if it were possible to measure all carrots, it would not be wise. Sampling must be 

used. 

d: Population: The public (the media does not generally make this population very clear). 

It could be all voting adults, all adults, or all people in the state. In any case, the 

population is too large, so sampling must be used. 

e: Population: Elevator cables. To find this, elevator cables must hold greater and greater 

weight until they break. If all elevator cables were tested, there would be none left to 

use for elevators. Sampling must be used. 

f: Population: Your friends. A census can be taken. 

9-8. a: The five-number summary is (1, 19.5, 29, 40.5, 76 cups of coffee per hour). 

b: The typical number of cups sold in an hour is 29 as determined by the median. 

Looking at the shape of the distribution the median is a satisfactory representation of 

the distribution. The distribution has a skew. There is a gap between 60 and 70 cups. 

The IQR is 21 cups. Seventy-six cups of coffee in one hour is an apparent outlier. 

9-9. a: (x ±1), (x ± 7) 

9-10. a ≤ 25 

24 

b: Neither are factors. Use substitution to determine whether x = –1 and x = 1 are zeros. 

Or you could use the Remainder Theorem and divide to see that neither are factors 

because there is a remainder. 

9-11. a: 30º or 150º b: 120º or 240º c: 45º or 225º 

d: 35.26º, 144.74º, 215.26º, or 324.76º 

9-12. f (g(x)) = g( f (x)) = x 

9-13. a: 10 times stronger b: 10 0.8 = 6.3 times stronger 

c: log(0.5) ≈ –0.3, so 6.2 – 0.3 ≈ 5.9 on the Richter scale 

9-14. a: (x + 2 − 3)(x + 2 + 3) = x 2 + 4x +1 

b: (x + 2 − i)(x + 2 + i) = x 2 + 4x + 5 

9-15. posts: $3, boards: $2, piers: $10 


Lesson 9.1.2 

9-22. a: The question implies that the questioner holds this opinion, thus biasing results. 

b: The question assumes that the respondent believes that the climate is changing and 

will think that one of the given factors is important, and that it is important to slow 

global climate change, biasing results. 

c: The question implies that teacher salaries should be raised. 

9-23. Sample questions given: 

a: Are a majority of Americans in favor of replacing the Electoral College with a 

popular vote? 

b: Do consumers prefer the taste of the new “improved” cookie recipe? 

c: What was the class average on the semester final exam? 

d: What was the average for high school students taking the state math proficiency 

examination last year? 

Hush Puppy 

9-24. a: See plots at right. Hush Puppy: min = 19.7, 

Q1 = 44.5, med = 58.3, Q3 = 70.1, max = 79.5; 

Quiet Down: min = 14.2, Q1 = 37.4, med = 54.85, 

Q3 = 63.3, max = 102.1 

b: Hush Puppy: The distribution is left skewed so its 

center and spread are best described by the median 

of 58.3 dB and IQR of 25.6 dB there are no apparent 

outliers. Quiet Down: Has some potential outliers 

over 100 dB or is perhaps dual-peaked. The main 

body of data has a left skew. The center and spread 

are best described by the median of 54.85 dB and IQR 

of 25.9 dB. 


d: The Hush Puppy looks better now because those three 

high readings from the Quiet Down model are a lot 

more significant. Or perhaps the Quiet Down could 

be redesigned to eliminate those high readings. 

12 24 36 48 60 72 84 96 10 

8 

Quiet Down 

12 24 36 48 60 72 84 96 10 

8 

9-25. y = 2(x + 2) 2 − 3 , (–2, –3). See graph at right. 

y 

9-26. y = 6, z = 2 

9-27. a: x = 4 b: x = 200 

x 

9-28. a: π 6 b: π 12 

c: – 

5π 

12 d: 7π 2 

9-29. y = − 1 4 

(x − 2)(x + 2)2 

9-30. a: 160π 

3 

, about 167.6 cubic feet b: 6 feet 

c: It is not changing, the angle is ≈ 68.20° 


Lesson 9.1.3 

9-36. a: This information could be found on the web for all American League players. 

It would be a census and the answer would be a parameter. 

b: An experiment would need to be conducted on a sample of eggs. The findings would 

be a statistic. 

c: Random high school students could be surveyed, possibly from different high schools 

in different parts of the country. Surveying every high school student would be almost 

impossible, so this survey would be a sample and the answer would be a statistic. 

9-37. a: closed b: open c: open d: closed 


9-39. 

2 

9 

; k = 7, 8 are factorable. 

9-40. blue block: 8 grams, red block: 16 grams 

9-41. a: The more rabbits you have, the more new ones you get, a linear model would grow by 

the same number each year. A sine function would be better if the population rises 

and falls, but more data would be needed to apply this model. 

9-42. 

3 

x+5 

9-43. 

x 

x+2 

b: R = 80, 000(5.4772...) t 

c: ≈ 394 million 

d: 1859, it seems okay that they grew to 80,000 in 7 years, if they are growing 

exponentially. 

e: No, since it would predict a huge number of rabbits now. The population probably 

leveled off at some point or dropped drastically and rebuilt periodically. 

9-44. a: 4π 5 

b: 

15π 

9 

c: 100º d: 255º e: 1710º f: 

11π 

9 


Lesson 9.2.1 

9-50. a: When asked to choose between an “m” and a “q,” most people prefer “m,” regardless 

of the Cola taste. 

b: The “to protect” interpretation is not part of the Bill of Rights; it will bias the results. 

c: The statistics chosen for the lead-in will bias the results. 

9-51. a: Only certain types of people typically respond. 

b-d: Answers will vary. 

e: Students should observe some clear preferences for some numbers, letters, and colors. 

This would provide evidence that people cannot behave or choose randomly. 

9-52. Mean: 7.6 g, mean distance-squared: 2.56+0.16+0.16+1.96+0.36 

5−1 

= 1.3 , 

sample standard deviation≈ 1.14 g (and not ≈ 1.02). 

9-53. 

1 

( 2 

, 6, –3) 

9-54. x 2 + 25 

9-55. 2, ±5i 

9-56. ± 15 

17 

9-57. a: 3 + 2i b: 1 + 4i c: 5 + i d: − 1 2 + 5 2 i 

9-58. a: x = 32 b: x = 1 6 


Lesson 9.2.2 

9-62. a: Not likely; this samples the population of people with phone numbers listed online that 

are home midday. In each of these, we could also notice that we only get responses 

from those who agree to participate in our polling activities—already a very 

unrepresentative sample. 

b: Not likely; this samples the population of people who shop at this particular grocery 

store. 

c: Not likely; this samples the population of people who attend movies. 

d: Not likely: this samples the population of people who go downtown at the time you are 

there. 

e: This sample is likely to be more representative, as it is closer to random. 

9-63. Sample diagram: 

30 get 

classroom 

SAT prep 

90 student 

volunteers 

take SAT 

random assignment 

30 get 

on-line 

SAT prep 

30 get no 

additional 

instruction 

All Students retake 

the SAT. Compare 

average change in 

scores between the 

groups 

9-64. Children would need to be randomly assigned to treatment groups, one that gets spanked 

and another where there is no spanking. After a period of years the IQ of both groups can 

be tested and compared. Any variable that has the suggestion or potential to lower the IQ 

of a human does not belong in a clinical experiment. Who would decide who, when, and 

how the spankings would be administered? Would you spank kids randomly? 

9-65. Mean: 52 g, mean distance-squared: 64+64+4+144+4 

5−1 

≈ 8.37g (and not ≈ 7.48g). 

9-66. one point of intersection: (2, 2) 

= 70 , sample standard deviation 

y 

2 

9-67. See graph at right; a sphere, V = 32π 

3 

cubic units 

-2 

2 x 

9-68. x 3 + 8 = (x + 2)(x 2 − 2x + 4) 

y 

9-69. a: b: 

y 

-2 

x 

x 


Lesson 9.3.1 

9-75. a: They will not show the people who named other stations or people saw the station 

logo and knew what station the interviewer was from. 

b: Surveying outside the gym does not give you a random sample. 

c: No, more people drive during the day. You should look at the probability of being in 

an accident. 

d: About half of all power plants are below average. It does not mean that it is unsafe. 


9-77. a: 3 2 = 9 b: 3 0 = 1 c: 3 3 + 3 1 = 27 + 3 = 30 d: – 9 2 

9-78. a: n = 2; 1 7 b: n = 0, 1; 2 7 c: n = 3, 4, 5, 6; 4 7 

9-79. – 1 5 

9-80. a: x = 4 b: x = 9 

9-81. ≈ 278 months or 23 years 

9-82. a: y 

b: 

y 

x 

x 

9-83. a: No 

b: No, no number of trials will assure there are no red ones. 

c: Not possible. 


Lesson 9.3.2 

9-88. a: The typical number of tardy students (center) is the median, 3 students, because 50% 

of the students fall below 3. The shape is single-peaked and symmetric with no gaps 

or clusters. The IQR (spread) is 1 student, because Q1 is 2 and Q3 is 3. There are no 

apparent outliers. 

b: (0.43 + 0.13 + 0.07)(30) = 19 days 

c: ≈ 30% 

9-89. a: Answers will vary. 

b: Yes, assuming a sufficient sample size, a controlled randomized experiment can show 

cause and effect because of the random assignment of subjects to the groups. 

9-90. Experiments can be very expensive, time consuming, and in some cases involving 

humans or animals, unethical. 

9-91. 6 sq. units; see graph at right. 

y 

9-92. a: f −1 3 

(x) = x +1 − 3 

b: See graph below right. 

c: Yes, each x is paired with no more than one y. 

9-93. a: double roots at –1, 2, 5 b: Same as the previous except 

reflected over the x-axis 

y 

y 

x 

x 

y 

x 

x 

9-94. a: (– 4, 0), (–2, 0) and (0, –16) b: domain: all real numbers, range: y ≤ 2 

9-95. a: y = 1 4 (x +1)2 + 

8 3 , vertex = ( −1, 8 

3 ) , x = –1 

b: y = 1 4 (x +10)2 +16 , vertex = (–10, 16), x = –10 

9-96. a: x + 3 b: (x + 3) 2 + (y − 2) 2 


Lesson 9.3.3 

9-103. a: 667.87 lunches; sample standard deviation is 56.17 lunches. 

b: (576, 624, 665, 700.5, 785) 

c: See histogram at right. 

d: Since the shape is fairly symmetric, we’ll use 

mean as the measure of center; the typical number 

of lunches served is 668. The shape is single 

peaked and fairly symmetric with no gaps or 

clusters, the standard deviation is about 

56 lunches, and there are no apparent outliers. 

e: 10.8% 

f: Use half of the 680 – 720 bin. 0.216 + 0.351 + (0.5)(0.108) ≈ 62.1% 


b: 2.7333 tardy students, 1.1427 

c: See graph middle right. 

d: See graph middle right. 11.0% 

e: See graph far right. 

normalcdf(4, 10^99, 2.7333, 1.1427) = 0.1338. 0.1338(180) ≈ 24 days 


b: 50 % 

c: See graph at right. 

normalcdf(11.5, 12.5, 12, 0.33) = 87% 

9-106. a: log 3 (5m) b: log 6 ( p m 

) c: not possible d: log(10) = 1 

0.108 

0.216 

0.351 

0.135 

0.108 

0.081 

9-107. a: 

2x 2 −2x−7 

(x−3)(x+1)(x−2) 

b: −x2 +2x−4 

c: 

x(x−2) x+2 

x−5 d: x(x2 − 2x + 4) 

z 

z 

9-108. a: x = ± 26 b: x = 

9-109. See graphs at far right. 

−2± 10 

3 

–3 

x 

2 

–1 

y 

x 

y 

9-110. a: 1224 ≈ 34.99 b: (x +1) 2 + (y +1) 2 

y 

9-111. a: f −1 (x) = 1 3 ( x−5 

2 )2 +1 = 1 

12 (x − 5)2 +1 for x ≥ 5 



x

Lesson 10.1.1 

10-7. a: $165 b: t(n) = 50 + 5(n −1) c: $930 

10-8. a: 4050 b: 300 + 550 + 800 + … + 4050; t(n) = 300 + 250(n −1) 

10-9. a: t(n) = 3+ 7(n −1) b: t(n) = 20 − 9(n −1) 

10-10. –2 

10-11. a: x-intercepts (2.71, 0) and (5.29, 0), y-intercept (0, 43) 

b: x-intercepts (–1, 0) and (2.5, 0), y-intercept (0, –5) 

10-12. a: normcdf(70, 79, 74, 5) = 0.629, About 63% would be considered average. 

b: normcdf(–10^99, 66, 74, 5) = 0.055, Between 5 and 6% of would be in excellent 

shape. 

c: normcdf(–10^99, 66, 70, 5) – 0.0548 from part (b) = 0.157; There would be a nearly 

16% increase in young women classified as being in excellent shape. 

10-13. a: 233 units b: (x − 5) 2 + (y − 2) 2 units 

10-14. $20.14 

10-15. 34,800 people 

10-16. Yes, because the sum of the diameters is 830 mm. 

10-17. 235 

10-18. (−6) + (−3) + 0 + 3+ 6 + 9 +12 +15 +18 

10-19. It is the 55 th term. 

10-20. 220 

10-21. a: 10 P 5 = 30, 240 b: 10 ⋅9 4 = 65, 610 

10-22. n = ±6 2 

10-23. a: 2 b: 3 4 

10-24. a: It looks like an endless wave repeating the original cycle over and over again. 

b: A polynomial of degree n has at most n roots, but f(x) = sin(x) has infinitely many 

roots. Also, every polynomial eventually heads away from the x-axis. 


Lesson 10.1.2 

10-34. a: odds: t(n) = 1+ 2(n −1) , evens: t(n) = 2 + 2(n −1) 

b: odds: 5625, evens: 5700 

10-35. a: t(n) = 21− 4(n −1) 

b: 31; You can solve the equation 25 − 4n = −99 . 

c: –1209 

10-36. 15 

y 

10-37. 6 P 4 = 360 

10-38. Degree 4; Graph shown at right. 

10-39. f (x) = 1 4 

(x − 3)(x − 2)(x +1) 

10-40. a: f (x) = (x + 3) 2 − 2 , vertex (–3, –2) b: f (x) = (x − 5) 2 − 25 , vertex (5, –25) 

y 

x 

10-41. For David: normcdf(122, 10^99,149,13.6) = 0.976 

For Regina: normcdf(130, 10^99,145,8.2) = 0.966 

For now David is relatively faster. 

10-42. a: π 4 

b: 

5π 

12 c: − π 12 

x 

y 

x 

d: 5π 2 


Lesson 10.1.3 

10-49. a: 16,200 b: 16,040 c: 564 

10-50. 11+ 22 + 33+ ...+ 99 = 495 

10-51. a: Sample response: The terms decrease by two, then add seven, then decrease by two, 

and then add seven continually. 

b: It is not arithmetic because the difference from one term to the next is not constant. 

c: Find the sum of each “unzipped” series and then add these sums together. 

The sum is 32,240. 

10-52. a: 12 C 10 = 66 b: 9 C 7 = 36 

10-53. Some may substitute for x, others may set x equal to 3+ i 2 and work back to the 

equation, others may write the two factors and multiply to get the original equation, and 

others may solve by completing the square. 

10-54. The graphs of y = 2 x and y = 5 − x intersect at only one point. 

10-55. h = $2.50, m = $1.75 

10-56. a: 17 b: 5 

10-57. 

y 

x 


Lesson 10.1.4 

11 

10-62. a: ∑ (60 −13k) = −198 b: ∑ 

k=1 

10-63. 495,550 

n 

(3+ 7(k −1)) = 

n[3+(3+7(n−1))] 

k=1 

10-64. It works for the integers from 1 through 39. 

10-65. a: 7 ⋅ 3 n b: 10(0.6) n−1 

10-66. 9!= 362,880 

10-67. a: normalcdf(–10^99, 59, 63.8, 2.7) = 0.0377; 3.77% 

b: (0.0377)(324)(half girls) = 6 girls 

2 

= n(−1+7n) 

2 

c: normalcdf(72,10^99, 63.8, 2.7) = 0.00119. (0.00119)(324)(half) = 0.19 girls. 

We would not expect to see any girls over 6ft tall. 

z 

10-68. a: b: 

z 

x 

y 

x 

y 

10-69. cos( 2π 3 ) = – 1 2 , cos( 4π 3 ) = – 1 2 , cos( 5π 3 ) = 1 2 



10-87. a: 3+ 30 + 300 + 3,000 + 30,000 + 300,000 = 333,333 

b: Write the series 3+ 30 + ...+ 300,000 = S(6) twice. Multiply one of them by 10. 

Subtract 10S(6) − S(6) = 2,999,997 = 9S(6) . Divide by 9 to get 333,333. 

n 

c: ∑ 3⋅10 n−1 = 3⋅10n −3 

9 

i=1 

10-88. a: A sequence would represent the list of the class sizes of the graduating classes as the 

number of years since the school opened increased. The corresponding series would 

represent the growing number of alumni. 

b: t(10) = 150 ; total = 960 

c: n(36 + 6n) = 36n + 6n 2 

10-89. a: 15 b: –615 

10-90. 210, arithmetic 

10-91. a: 23 P 3 = 10, 626 b: 23 C 3 = 1771 c: 1⋅22 ⋅22 = 484 d: 4 ⋅22 ⋅22 = 1848 


y 

10-93. a: x−2 

x+2 

b: 

2x+1 

x−3 

c: Use the Distributive Property to factor and the multiplicative property of 1 to reduce. 

10-94. a and b: no amplitude, period = π , LP = (0, 0). 


10-95. a: x = 125 

2 b: x = − 4 5 

c: x = 0.04 d: y = 9 4 

x 

f(x) 

x 



10-96. Calculate the sums of two geometric series, the first with 25 terms, the second with 15. 

Retirement at age 55: $1,093,777; at age 65: $1,115,934 

10-97. $20,000 at 8% and $30,000 at 6.5% 

10-98. a: 8!= 40320 b: 1⋅ 7!= 5040 c: 1⋅ 7! + 7!⋅1 = 10080 

10-99. a: 272 = 4 17 units b: (x + 3) 2 + (y + 5) 2 units 

10-100. a: x = 5 2 b: y =10 c: x = –3, 2 d: y = − 15 4 

10-101. a: (2, 8) and (4, 4) b: (3+ i,6 − 2i) and (3− i,6 + 2i) 

c: In system (a), the solutions are the points of intersection. In system (b), the solutions 

show that they do not intersect. 

10-102. tan(160°) = –0.3640 , tan(200°) = 0.3640 , tan(340°) = –0.3640 

10-103. a: x = 4 b: x = 48 c: x = 3 d: x = 6 

10-104. 26 + i 


Lesson 10.2.2 

10-117. 500 miles 

10-118. a: 7 C 2 = 21 b: 7 C 3 = 35 c: 7 C 4 = 35 

d: Choosing three points to form a triangle is the same as choosing four points to not be 

part of the triangle. Those four points form a quadrilateral, 7 C 3 = 

4!3! 7! = 

3!4! 7! = 7 C 4 . 

10-119. a: 10t + u − (10u + t) = 27 b: x = –13 

10-120. $1157 

10-121. a, d 

10-122. a: (x − 3)(x 2 − 2x + 5) b: 3, 1± 2i 

10-123. 

3 

11 

10-124. (−1+ i, 3), (−1− i, 3) 

10-125. When r ≥ 1, r n increases in size as n increases, so the expression 1− r n does not get 

close to 1, and being able to replace that expression with 1 is a key part of the 

derivation of the formula. 

10-126. 

121 

27 

10-127. a: 45 b: 792 c: 7 

10-128. a: x = –3, 4 b: x = –1.5, 3 c: x = 

d: Never e: x = 0, –2, 4 3 f: x = 0 

10-129. a: −16x 5/2 y 4 z 2 b: 3 1/2 x 3/2 y 8/3 

−1 ± 57 

4 

10-130. a: 0, 5 seconds b: 0 ≤ t ≤ 5 c: 5 seconds d: 1 < t < 4 

10-131. Yes; use the Quadratic Formula or direct substitution. 

10-132. a: x = –1, 4 b: x ≤ –1 or x ≥ 4 c: –1 ≤ x ≤ 4 



10-145. x 7 + 7x 6 y + 21x 5 y 2 + 35x 4 y 3 + 35x 3 y 4 + 21x 2 y 5 + 7xy 6 + y 7 

10-146. −640w 2 z 3 

10-147. 1365 

10-148. a: 16 b: Not possible. r > 1, and the terms keep increasing. 

10-149. 4 C 0 = 1, 4 C 1 = 4, 4 C 2 = 6, 4 C 3 = 4, 4 C 4 = 1 

a: The number of possibilities are the elements of the 4 th row of the triangle. 

b: 1, 6, 15, 20, 15, 6, 1; Use the 6 th row of the triangle. 

10-150. a: 9 C 3 = 84 b: 9 C 2 = 36 c: 10 C 3 = 9 C 3 + 9 C 2 

d: The tenth row entry of Pascal’s Triangle is the sum of the two ninth row entries 

above it and these numbers correspond to the total number of combinations when 

one more choice is added. 

y 


10-152. a: See graph below right. 

b: Answers will vary. See graph below right. 


normalcdf (6.71, 13.29, 10, 2) = 0.9000 

x 

10-153. (−2, 3, − 1 2 ) 

10-154. a: y = x 2 − 4x + 5 = (x − 2) 2 +1 

b: (2, 1) 



10-155. 42x 3 

10-156. 728 

10-157. 81x 4 +108x 3 + 54x 2 +12x +1 

10-158. a: f (x) = (x + 2) 2 + 2 b: (x + 3) 2 + (y − 4) 2 = 25 

y 

y 

10-159. 32.9 mm 

x 

x 

10-160. y = −2x + 34 

y 

10-161. a: See graph at right. b: f −1 (x) = [2(x +1)] 2 – 4 

c: x ≥ –1, y ≥ –4 d: 5 

x 

10-162. a: x = 7 b: x = 1.5 c: x ≈ 1.75 d: x ≈ 1.87 

10-163. a: (0, –5), (4, 3), (8, 3) b: See graph at right. 

c: y = − 1 4 x2 + 3x − 5 d: 10 seconds 

height 

e: 0 ≤ x ≤ 10 f: 0 ≤ x < 2 

time 


Lesson 10.3.2 

10-169. Robin: $11,887.58; Tyrell: $11,808.38; difference: $79.20 

10-170. a: $10,304.55; it rounds off to the same amount. 

b: $10,832,870, 680 and $10,832,775,720, a difference of $94,960. 

c: Maybe billionaires, or other investors of large amounts. 

10-171. a: 1+ 3 n + 3 

n 2 + 1 

n 3 

b: 1+ 5 n + 10 

n 2 + 10 

n 3 + 5 

n 4 + 1 

n 5 

10-172. a: 14.7 lbs./sq. in. b: ≈ 12.55 lbs./sq. in. c: ≈ 14.83 lbs./sq. in. 

10-173. a: 2 = (1.015) 4t , 2 = e 0.06t 

b: Quarterly, 11.64 years; Continuously, 11.55 years 

c: The difference is about one month, so probably not. 

10-174. (−5, 0), ( 2 3 , 0), (− 1 4 , 0) 

10-175. 8x 3 − 36x 2 + 54x − 27 

10-176. a: log 2 (5x) b: log 2 (5x 2 ) c: x = 17 

d: x = − 

20 9 = −0.45 e: x = 15 f: x = 4 

10-177. a: 10t + u 

b: 10u + t 

c: t + u = 11 and 10t + u − (10u + t) = 27 

d: 74 and 47 


Lesson 11.1.1 

11-5. a: preface b: biased wording c: desire to please d: fair question 

11-6. a: Using the normal model here is not a good idea because the data is not symmetric, 

single-peaked, and bell-shaped. A different model would represent the data better. 

b: 9/48 ≈ 19 th percentile; In 19% of the hours over the two day period the coffee shop 

was not profitable. 

c: 41/48 = 85 th percentile. In 85% of the hours over the two-day period, the coffee shop 

would not have been profitable. If this data represents a typical 48-hour period, now 

would not be the time to expand. 

11-7. If the nickel is tossed and the die rolled there are 12 equally likely outcomes. He can 

make a list of each of the possible outcomes, assign one player to each outcome 

(H1-Juan, H2-Rolf, … T6-Jordan), and then toss the coin and roll the die to select. 

11-8. a: No, by observation, a curved regression line may be 

better. See graph at right. 

b: Exponential growth. 

4 5 

c: m = 8.187 ⋅1.338 d , where m is the percentage of mold, 

and d is the number of days. Hannah predicted the 

5 

1 2 3 

mold covered 20% of a sandwich on Wednesday. 

Day 

Hannah measured to the nearest percent. 

b: 0.60 

0.80 = 75% y 

11-9. a: f −1 (x) = ln x 

b: f(x): domain is all real numbers, range is y > 0, y-intercept is (0, 1), 

asymptote is y = 0. f –1 (x): domain is x > 0, range is all real numbers, 

x-intercept is (1, 0), asymptote is x = 0. See graphs at right. 

c: 2 and 3 

d: Above log base 2 for x > 1 and below it for 0 < x < 1, below 

y 

log base 3 for x > 1 and above for 0 < x < 1. 

11-10. a: 9.00646832 for both b: 3.10628372 for both 

c: It will take about 9 years to double. 

d: After about three years and one month, the car 

will be worth less than half of the original price. 

11-11. 765 

Garage? 

yes no 

0.80 0.20 

11-12. a: 2i b: –2 + 2i 

yes 

0.60 0.15 

0.75 

11-13. a: 5%. See the table at right. 

no 

0.20 0.05 

0.25 

Selected Answers © 2013 CPM Educational Program. All rights reserved. 91 

% Mold 

40 

20 

Large 

Backyard? 

x 

x

Lesson 11.1.2 

11-18. Answers should be close to P(sum 6 or less) = 15 

36 

≈ 0.42 , so 0.42(7) = 2.94 or about 

3 days per week. 

11-19. Theoretical probabilities: P(sum 6 or less) = 15 

36 ≈ 0.42 , 

P(sum 7 or more) = 

36 21 ≈ 0.58 , so about3 days a week. 

11-20. a: F 2%, D 16 – 2 = 14%, C 84 – 16 = 68%, B 98 – 84 = 14%, A 100 – 98 = 2% 

b: F 0, D 76.5 – (2)17.4 = 41.7, C 75.5 – 17.4 = 58.1, A 76.5 – (2)17.4 = 111.3 

c: The normal model is not a good idea because the distribution of scores is strongly 

skewed left. Also, the minimum grade required for an A would be 111.3, which is 

probably not possible. 

11-21. Answers vary. 

11-22. a: 

13 

( 5 

, 11 5 ) b: ( −5, 1 2 ) 

11-23. a: 12 b: 1 2 

11-24. Both equal 3 8 . 

11-25. The base of the natural logs is e; e is between 2 and 3, and ln e = 1; x = e 

11-26. a: 1.79175 b: 2.4849 c: 2.7726 d: –1.0986 


Lesson 11.1.3 

11-28. Theoretically, the series is expected to last for 5.8125 games. 

11-29. Theoretically, a streak of 4 or more has probability of about 48%, a streak of five or more 

about 25%, a streak of six or more 12%, and a streak of seven or more about 6%. 

11-30. The survey is not random. They will get a lot more representation from adults. People 

may be influenced by the responses of others. An individual may have several favorites 

but only states one. 

11-31. a: x = 1 4 y2 −1, parabola b: x 2 + y 2 = 49 , circle 

11-32. a: 51 b: 64.77 

11-33. They are equivalent and simplify to x 2 . 

11-34. a: 1 

12 

b: 580 c: (–9, 1) d: y − 2 = 

1 

12 

(x − 3) 

11-35. a: ≈ 266.67 b: 37.5% c: 27% d: 135 

11-36. a: See diagram at right. 

b: In the RR rectangle, 18 

38 ⋅ 18 

38 = 1444 324 ≈ 22.44% . 

c: Add the RR, BR, and GR rectangles, 

324 

1444 + 1444 324 + 1444 36 ≈ 47.37%. 

d: Considering only the column in which the 

second spin is red (RR, BR, GR), the probability 

the first spin is red (RR) 

is 

324 

1444 

324 

1444 + 1444 324 + 1444 

36 

≈ 47.37%. 

R 

B 

G 

R 

B 

G 

e: They are the same. 


Lesson 11.2.1 

11-41. Theoretically, there will be 7.39 streaks of three or more in 60 days. 

11-42. a: 7.83% 

b: 5% and 11% 

c: About 7.83% ± 3% def. flashlights 

11-43. Each used a convenience sample. Each sample came from a distinct population with 

people who have many things in common, including attitudes and beliefs about the 

subject matter of murals. 

11-44. a: –344 b: –6740 

11-45. a: x = 10p b: x = 3q – 2p 

11-46. y = 5, z = 3 

11-47. See table at right. 0.6 ⋅0.5 = 0.3 = 30% 

11-48. (–1, –3) and (3, 5) 

11-49. a: 10 log 2 ≈ 3 

b: 20 ⋅10 6 

c: Two sounds have equivalent pressures, or one sound has a pressure of 20 micropascals. 

d: 100 

Second Jar 

yellow 

0.30 

orange 

0.50 

white 

0.20 

black 

0.60 

First Jar 

purple 

0.40 

0.18 0.12 

0.30 0.20 

0.12 0.08 


Lesson 11.2.2 

11-52. a: 0.56 and 0.74 b: ≈ 65% ± 9% 

11-53. 3% ± 0.85%. Answers vary, but should be 3% plus/minus a number smaller than 1.7% 

11-54. The energy usage will change by –1.3% to 1.7%. Indiana could save up to $11.7 million, 

but could end up spending $15.3 million. 

11-55. a: Answers will vary. Like a tournament, the students could be paired and the coin 

tossed for each pair. The “winners” are paired and the coin tossed again, repeating 

until there is just one student. 

b: 3 tosses. Tossing a coin 3 times has 8 equally likely outcomes: HHH, HHT, HTH, HTT, 

THH, THT, TTH, TTT. Assign each student an outcome and toss the coin 3 times. 

c: Yes. Use the method outlined in part b repeatedly until a baseball player is not 

selected. 

11-56. a: y = 2(x + 

4 7 )2 − 105 

8 

, graph shown at right, 

vertex (− 

4 7 , − 105 

8 ) , axis of symmetry x = − 4 

7 

b: y = 3(x − 1 6 )2 − 

12 97 , graph shown at far right, 

vertex( 1 6 , − 12 97 ) , axis of symmetry x = 1 6 

y 

11-57. a: b: 

y 

–5 

y 

–5 

x 

y 

y 

–3 

3 

x 

x 

x 

x 

c: For part (a), the parts above the x-axis stay the same, the parts below the x-axis are 

reflected upward across the axis. For part (b), the part of the graph to the right of the 

y-axis remains the same, and the part to the left of the y-axis is replaced by a reflection 

of the part on the right of the axis. 

11-58. a: Some may predict the amount due will be far too much for a state to pay. 

b: ≈ 1.126 ⋅ 10 15 dollars 

c: A ≈ $2.791⋅10 16 , or about 2.68 ⋅10 16 dollars more. 

11-59. a: x = log(3) 

log(2) 

c: x = log(12) 

log(7) 

log(8) 

b: x = 

log(5) 

log(b) 

d: x = 

log(a) 

11-60. Both 31.5%. Neither 16.5%. See table at right. 

Dimples 

yes 

0.45 

Widow’s Peak 

yes no 

0.70 0.30 

0.315 

no 

0.55 0.165 


Lesson 11.2.3 

11-64. Upper bound: 25145 lower bound: 24563; Students should report that 90% of the time we 

can expect that the copy machine will need maintenance after 24854 ± 291 copies. The 

interval of confidence is from 24563 to 25145. 25000 copies is within the interval of 

confidence, so the research company may support the copy machine company’s claim. 

They may also state that the copy machine company is pushing the limits with a claim of 

“at least 25000 copies” since the range within the margin of error is from 24563 to 25145. 

11-65. a: 0.12 

b: A difference of zero is not within the margin of error, so it is not a plausible result. A 

difference of zero means that there is no difference in the percentage of food removed 

with detergent compared with the percentage of food cleaned off with plain water. 

c: Yes. Because a difference of zero is not within the margin of error, a difference of 

zero is not a plausible result for the population of all food cleaned. We are convinced 

there is between 3.5% and 20.5% (12% ± 8.5%) more food removed with detergent 

than with plain water. 

11-66. a: 1/8 or 12.5%, yes 

b: Her study provides no reliable evidence of her conclusion. She used a convenience 

sample of only 4 people. Her question introduced bias with the preface about violence 

and crime. There may also have been a desire to please the interviewer. She used a 

closed question forcing romantic comedies as the only alternative to action movies. 

Even if there is no real preference among moviegoers, it is plausible that 4 people will 

have the same opinion between any two types of movies. 

11-67. a: The number of cards on the field is 768 ×1029 = 790, 272 cards. The probability is 

or 0.000 001265 . 

b: 

1 

790272 

790,272 cards 

52 cards 

1 pack 

= 15,198 packs of cards. 

The maximum loss is if the first player chooses a card and wins: 

−$1, 000, 000 prize − ($0.99)(15,198) cost of packs + $5 from the player 

= −$1, 015, 041.02 . (If nobody plays, then the million dollars is not paid out, and the 

boosters do not have the maximum possible loss.) 

c: If all of the chances were purchased, 

−1, 000, 000 prize − ($0.99)(15,198) cost of packs + ($5)(790, 272) from players 

= $2, 936, 313.98 

d: On average half the cards would be sold before there was a winner, 

−1, 000, 000 prize − ($0.99)(15,198) cost of packs + ($5)(395,136) from players 

= $960, 633.98 . 

e: 176,000,000 

790,272 

≈ 223 football fields would have to be covered to give the same odds as 

winning the state lottery! 


11-68. (0, 0) 

11-69. a: 5 + i b: –1 + 9i c: 26 + 7i d: –0.56 + 0.92i 

11-70. a: 4 ≤ y ≤ 10 b: m > 1 or m < –2 

11-71. a: x = –3 or x = 2 b: x < –3 or x > 2 


Lesson 11.2.4 

11-75. Yes, 20% is within the margin of error of 13% ± 10%. 

11-76. a: 0.10 

b: i: 25% of 40 + 15% of 40 = 16 putts went into the hole 

ii: 1 to 16 will represent a putt that went into the hole; 17 to 80 will represent a putt 

that missed. 

c: A difference of zero means that there is no difference between the proportion of putts 

that went in the hole with the new club and the proportion of putts that went in with 

the old club. A difference of zero is within the margin of error, so it is a plausible 

result. 

d: No. A difference of zero is within the margin of error, and a difference of zero is a 

plausible result for the population of all putts. We are not convinced there is a true 

difference in the number of putts that go in with the new club compared to the old 

club. 

11-77. Yes. As little as 11.75 ounces is still within specifications. 

11-78. The first process is wildly out of control; systems wildly out of control are often caused 

by inexperienced operators. The second process is fully in control. The third process is 

technically out of control at only one point, but the cyclical nature of the process is 

disconcerting. Any explanation that is cyclical over 20 hours is acceptable. 

11-79. a: y 

b: 

y 

x 

x 

11-80. a: x = a+b 

c 

b: x = ab + ac c: x = a, b 

d: x = 0, c e: x = a+b 

c 

f: x = 

b−a 

1 

11-81. a: x = −26 b: x = 10 or x = 3 

11-82. a: x = 1 b: x = ±2 

parents 

1 

3 

niece 

1 

6 

boyfriend 

1 

2 

11-83. a: See diagram at right. 

b: 1 4 / 1 2 = 0.5 

c: 1 9 + 1 6 + 1 6 + 1 4 = 25 

36 ≈ 69% 

d: 1 6 / 25 

36 = 6 25 = 24% 

parents 

1 

3 

niece 

1 

6 

boyfriend 

1 

2 

1 1 

9 18 

1 

18 

1 

6 

1 

36 

1 

12 

1 

6 

1 

12 

1 

4 


Lesson 11.3.1 

11-87. This is a “Gambler’s Ruin” problem. The player with more coins always has a better 

chance of winning in the long run. P(Jill) = 4 6 ≈ 67%, P(Jack) = 2 6 ≈ 33% . 

11-88. a: See possible diagrams at right and answers below. 

Cell A is the proportion of times the system correctly 

activated the alarm. 

Cell B is the proportion of times the alarm was 

correctly not activated. 

Cell C is the proportion of times A happened and the 

alarm was incorrectly not activated. 

Cell D is the proportion of times A did not happen and 

the alarm was activated. 

b: 0.03966/(0.03966 + 0.00096) = 97.6% 

c: Yes, it is independent because the accuracy of 

alarm is the same regardless of whether event A 

occurs or not. 

Detection System 

Event A 

Event A 

Yes No 

Correct 

Incorrect 

A B 

C D 

correct 

0.00096 

incorrect 

0.00004 

11-89. y = 1 5 x + 27 5 

11-90. (± 7, 3), (0, −4) 

Not Event A 

11-91. a: 10 + 11i b: 13 c: 29 d: a 2 + b 2 

correct 

incorrect 

0.95904 

0.03996 

y 

11-92. a: b: 

y 

x 

x 

c: d: 

y 

y 

x 

x 

11-93. a: –35 b: 123 

11-94. a: x = ±2 3 b: x = 2 c: x = 2 9 

d: x = 

−1± 13 

6 

or x ≈ 0.434 or –0.768 

11-95. a: any polynomial with 5 x-intercepts; b: a polynomial graph with 3 x-intercepts and 

another ‘bend’; c: no x-intercepts, could have two ‘humps’; d: 2 x-intercepts and up to 

two ‘humps.’ 


Lesson 12.1.1 

12-5. a: always b: never 

c: always d: True for x = π 4 + 2π n and x = 5π 4 + 2π n 

12-6. a: (x − 2)(x + 2) b: (y − 9)(y + 9) 

c: (1− x)(1+ x) d: (1− sin x)(1+ sin x) 

12-7. a: ≈ 80.86 b: ≈ 24.05 c: ≈ 15.50º 

trigonometric ratios, Law of Sines, Law of Cosines, Pythagorean theorem 

12-8. The graphs of y = sin(2x) and y = 2sin(x) intersect at integer multiples of π. Solving the 

equation algebraically yields x = 0 + πn. 

12-9. a: x ≥ 2, f (x) ≥ 2 b: f −1 (x) = (x−2)2 

2 

+ 2 c: x ≥ 2, f −1 (x) ≥ 2 

12-10. a: Stretched (amplitude = 3), shifted left 

2 π , and shifted down 4 


y 

x 

12-11. The first process is fully in control. The second process is wildly 

out of control. The third process is out of control; beginning at the 

9 th hour, there are 12 consecutive points above the centerline. 

12-12. a: 90 b: 190 c: 35 d: 405,150 

12-13. a: 12 P 5 = 95, 040 b: 12 C 5 = 792 

12-14. Sample answers: h = π 2 , 5π 2 , − 3π 2 

12-15. a: negative b: negative c: positive d: negative 

12-16. a ≤ 25 

24 


12-18. 

5 

x+4 

12-19. a: x = 9 b: x = –9 

12-20. a: 3π 5 

12-21. a: 5 C 2 i 4 C 1 

12 C 3 

d: 5 C 3 + 4 C 3 + 3 C 3 

12 C 3 

b: 

16π 

9 

c: 140º d: 285º e: 1530º f: 

13π 

9 

= 

11 2 b: 4 C 3 

12 C = 1 

3 55 c: 5 C 1 i 4 C 1 i 3 C 1 

12 C = 3 

3 11 

= 

44 3 e: 5 C 1 i 4 C 2 

12 C = 3 

3 22 

12-22. a: a 3 + 3a 2 b + 3ab 2 + b 3 b: 8m 3 + 60m 2 +150m +125 

f: 1− 

3 

( ) = 29 

11 + 44 

3 


44

Lesson 12.1.2 

12-29. a: x = π 6 , 5π 6 b: x = 5π 6 , 7π 6 c: x = π 4 , 3π 4 d: x = 0 

12-30. No, 52º and 308º have the same value for the cosine, while 128º 

has the exact opposite cosine value. See diagram at right. 

128º 

52º 

y 

x 


–x 

12-37. a: 24 

91 

b: 

20 

91 

12-32. f −1 (x) = −x + 6 

x 

12-33. 

x 

x+2 

12-34. a: yes b: x 4 + x 3 + x 2 + x +1; yes c: x n + x n−1 + x n−2 +…+ x +1 

Recipient 


Cell A is the proportion of people correctly 

identified as drug users. 

Correct 

Cell B is the proportion correctly identified as 

not drug users. 

Cell C is the proportion the test failed to identify 

Incorrect 

as drug users but who are. 

Cell D is the proportion identified as drug users 

who are not. 

This can also be modeled as a tree diagram. 

b: 0.02 are actually using; 0.01 are told they are 

User 

A 

C 

correct 

Not Using 

B 

D 

0.02277 

using, but are actually not. 

Drug User 

c: 0.00977/(0.00977 + 0.02277) ≈ 30% 

incorrect 

0.00023 

d: From part (b), about 1 out of 100 people 

receiving assistance will lose their assistance 

correct 0.98703 

because they have been falsely accused of 

Not Drug User 

using drugs. That seems high considering that 

only 2 out of 100 are actually using drugs. 

From part (c), 30% of the people identified as 

using drugs will be falsely accused and unfairly 

lose their money. 

incorrect 0.00977 

e: Yes, they are independent because the accuracy of the test stays the same whether or 

not a person uses drugs. To test, check whether P(A)⋅ P(B) = P(A and B) , for 

example, P(drug user)⋅ P(test correct) = P(drug user and test correct) . 

12-36. a: 0.0253 b: 26 C5 

52 C 5 

c: 0.000495 d: 13 C5 

52 C 5 

e: 0.0019808 

Drug Test 

360º – 52º 

=308º 



12-43. a: 30º, 50º or 

6 π , 5π 6 b: 120º, 140º or 2π 3 , 4π 3 

c: 45º, 225º or 

4 π , 5π 4 

d: ≈ 35.26º, 144.74º, 215.26º, 324.74º or 0.62, 2.53, 3.76, 5.67 

12-44. a: domain: –3 ≤ x ≤ 3, range: –3 ≤ y ≤ 3, not a function 

b: domain: –3 ≤ x ≤ 4, range: –2 ≤ y ≤ 4, not a function 

c: domain: x ≤ 3, range: y ≤ 4, yes a function 

d: domain: −∞ < x < ∞ , range: y ≥ –2, yes a function 

12-45. a: x = 2 b: no solution 

z 

12-46. a: b: 

z 

3 

–2 –2 

x 

y 

x 

y 

12-47. a: (4, 8) b: (0, –2, 3) 

Suspect 

Person 

Not Suspect 


Cell A is the proportion correctly identified as 

suspects. 

Cell B is the proportion correctly identified as not 

being suspects. 

Cell C is the proportion the software failed to 

identify but who actually are suspects. 

Cell D is the proportion the software identified as 

suspects who are not. 

This can also be modeled as a tree diagram. 

b: 0.000999995/(0.000004995 + 0.000999995) 

≈ 99.5% 

12-49. 12 C 5 + 12 C 4 + 12 C 3 + 12 C 2 + 12 C 1 + 12 C 0 = 1586 

12-50. 2x 4 − 2x + −1 

x−3 

Suspect 

Facial ID Software 

Not a Suspect 

Correct 

Incorrect 

correct 

incorrect 

correct 

incorrect 

A 

C 

B 

D 

0.000004995 

0.000000005 

0.99899501 

0.000999995 


b: The number of defects seems to be staying at or within 

control limits, but there is a cycle apparent every eight 

hours. Perhaps as the inspectors work through their 

shift, they get tired and catch fewer errors. 



12-52. The restrictions are needed so that the inverses will be functions. The domain of the sine 

function is restricted to − 

2 π ≤ x ≤ 2 π , the domain of the cosine function is restricted to 

0 ≤ x ≤ π , and the domain of the tangent function is restricted to − 

2 π < x < 2 π . 

12-53. a: x = 7π 6 

+ 2π n, 

11π 

6 + 2π n b: x = 6 π + 2π n, 

11π 

6 + 2π n 

c: x = 3π 4 + π n d: x = π n 

12-54. a: shifted up 1 unit b: shifted left 

4 

π 

c: reflected over the x-axis d: vertically stretched by a factor of 4 

12-55. a: y 

b: 

y 

x 

x 

c: g(x) = − f (x) 

12-56. a: 5 6 

b: 

3x+8 

2x 2 

c: x2 +2x+3 

(x+1)(x−1) 

12-57. f (x) = 2(x −1) 2 −1; domain: all real numbers; range: f (x) ≥ −1; 

vertex: (1, –1); line of symmetry: x = 1; See graph at right. 

d: sin2 θ+cosθ 

sinθ cosθ 

y 

12-58. a: x ≈ 1.356 b: x ≈ 2.112 c: x ≈ 1.792 

12-59. a: a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 

b: 81m 4 − 216m 3 + 216m 2 − 96m +16 

x 

12-60. a: 8 C 3 + 8 C 4 = 56 + 70 = 126 

b: If mushrooms are a known topping, then choose the rest of the toppings from only 

7 remaining toppings so 7 C3+7C2 

126 

= 

126 56 = 4 9 


Lesson 12.1.4 

12-67. a: a c b: c b c: c b d: b a 

12-68. a and b: ± π 6 , ± 5π 6 , ± 7π 6 , ± 11π 

6 

12-69. The solution is equivalent to the solutions of cos(x) = 1 2 

300°; or 0, π, 

3 π , 5π 3 

and sin(x) = 0 . 0°, 180°, 60°, 

12-70. 9.10 

12-71. 1.083, 8.3% 

12-72. a: a 3 + b 3 b: x 3 − 8 c: y 2 +125 d: x 3 − y 3 

e: They consist only of two terms; they are sums or differences of cubes. 

12-73. a: (x + y)(x 2 − xy + y 2 ) b: (x − 3)(x 2 + 3x + 9) 

c: (2x − y)(4x 2 + 2xy + y 2 ) d: (x +1)(x 2 − x +1) 

12-74. y = 10 

216 (x + 6)3 −10 

12-75. a: 3C 1 i 10 C 3 

13C4 

= 360 

715 ≈ 0.503 b: 11 

13C4 = 1 65 = 0.015 

12-76. a: c a b: a b c: b a d: c a 

12-77. The solution is equivalent to the solutions of sin(x) = − 1 2 

and sin(x) = 0 . 90° + 180°n, 

210° + 360°n, 330° + 360°n; or 

2 π + π n, 7π 6 

+ 2π n, 

11π 

6 + 2π n 

12-78. 2x 2 + 4x −1 

12-79. a: x 2 (x + 2y)(x 2 − 2xy + 4y 2 ) 

b: (2y 2 − 5x)(4y 2 +10xy 2 + 25x 2 ) 

c: (x + y)(x 2 − xy + y 2 )(x − y)(x 2 + xy + y 2 ) 

12-80. Possible equation: y = 1 8 (x − 3)3 + 3 Inverse: y = 3 8(x − 3) + 3 

12-81. y = 4(0.4) x + 5 

12-82. a: 126a 5 b 4 b: 1120x 4 y 4 

12-83. x 2 − 6x + 34 = 0 

12-84. P(3 or 4 or 5) ≈ 0.99 


Lesson 12.2.1 

12-90. a: 1 b: cos(4w) c: tan(θ) 

12-91. ≈ 75.52°, 75.52°, and 28.96° 

12-92. a: x = 30º + 360ºn or x = 150º + 360ºn b: no solution 

12-93. 3x 2 − x + 2 

12-94. x 3 − 2x 2 − 3x + 9 

12-95. 

1 

10! 

12-96. (5 – 2)! because 3! > 2! 

12-97. a: 18 b: –12 c: –1 + 7i d: –14.5 e: x = 0, –7 

12-98. a: p = 44, a = 20 b: y = 20 cos 

π 

( 22 

(x −15) ) + 3 

12-99. Possibilities include: sin 2 x = 1− cos 2 x , sin x = ± 1− cos 2 x , 

sin 2 x = (1− cos x)(1+ cos x) 

12-100. − 4 5 

12-101. a: sinθ 

cosθ b: 1 

sinθ 

c: 

cosθ 

sinθ d: 1 

cosθ 

12-102. a: y − 9 = 315 

2 

(x − 2) or y = 

315 

2 

12-103. They intersect at 

1 

( 2 

, 0) and (3, 10). 

12-104. a: 

1 

2(x−1) b: 4x 

3x 2 +10x+3 

12-105. a: 41.41° b: 28.30° 

x − 306 b: y = 0.25(6)x 

12-106. roots: −0.4 ± 0.8i 6 ; vertex: (–0.4, 19.2); f (x) = 5(x + 0.4) 2 +19.2 

12-107. a: (0.9) 5 ≈ 0.59 

b: 10(0.9) 2 (0.1) 3 + 5(0.9)(0.1) 4 + (0.1) 5 ≈ 0.00856 



12-109. 90.21 feet or 4.71 feet 

12-110. a: 2x 4 − x 2 + 3x + 5 = (x −1)(2x 3 + 2x 2 + x + 4) + 9 

b: x 5 − 2x 3 +1 = (x − 3)(x 4 + 3x 3 + 7x 2 + 21x + 63) +190 

12-111. a: x = 3π 2 b: x = π 3 , 5π 3 c: x = π 4 , 5π 4 d: x = 7π 6 , 11π 

6 

12-112. a: x ≈ 69.34 b: x ≈ 5.35 

12-113. a: 10 P 5 = 30, 240 b: 10 i 9 4 = 65, 610 

12-114. ( ± 

5 3 , 4 5 ) 

y 

12-115. (x + 4) 2 + (y − 6) 2 = 64 ; circle, center: (–4, 6), r = 8 


12-116. a: See graph at right below. 

b: The process has a lot of variability for the first 14 hours. 

There are two out-of-control points (one upper and one 

lower). Apparently an adjustment was made at hour 14 

because the process is much less variable, but now there are 

nine consecutive points above the centerline of 0.075. 

Apparently another adjustment was made at hour 22, but 

this apparently swung the process to the very low end. 

x 

12-117. a: 2x+1 

3x−2 

b: x2 +2x+4 

x(4x+5) 



12-118. a: (sinθ + cosθ) 2 = 

sin 2 θ + 2 sinθ cosθ + cos 2 θ = 

1+ 2 sinθ cosθ 

( ) = 

c: (tanθ cosθ) sin 2 θ + 1 

sec 2 θ 

( ) (sin 2 θ + cos 2 θ) = 

sinθ 

cosθ cosθ 

(sinθ)(1) = sinθ 

b: tanθ + cotθ = 

sinθ 

cosθ + cosθ 

sinθ = 

sin 2 θ+cos 2 θ 

sinθ cosθ 

= 

1 

sinθ cosθ 

= cscθ secθ 

12-119. See unit circle at right. θ = π 6 , 5π 6 , 7π 6 , 11π 

6 

12-120. m∠B = 86.17º or 1.5 radians 

12-121. The equation of the parabola is y = 1 8 (x − 3)2 + 3 . 

It is a function; every input has only one output. 

12-122. a: x ≈ 1.839 b: x ≈ –1.839 c: x ≈ 1.839 

12-123. a: The two lines intersect at (8, 17). 

b: No solution; the lines are parallel. 

12-124. a: There is one way to choose all five. 

5! 

5!0! 

= 1 . In order to have the formula give a 

reasonable result for all situations, it is necessary to define 0! as equal to 1. 

b: There is one way to choose nothing. 

5! 

0!5! = 1 

12-125. AC = 10 inches 

12-126. Possible answer: f (x) = x 3 − 5x 2 + 8x − 6 


Lesson 12.2.3 

12-130. sin π 12 = sin π 3 − π 4 

12-131. a: sin π 3 + π 4 

( ) = 6− 2 

( ) = 6+ 2 

4 

( ) = cos ( 7π 6 − 4 

π ) = − 6+ 2 

4 

b: cos 3π 4 + π 6 

4 

; cos π 12 = cos π 3 − π 4 

12-132. sin(x + x) = sin x cos x + cos x sin x = 2 sin x cos x 

( ) = 2+ 6 

cos(x + x) = cos x cos x − sin x sin x = cos 2 x − sin 2 x 


b: Possible answer: f (x) = − sin x 

c: cos x + π 2 

12-134. sin x + cos x 

( ) = cos x cos π 2 − sin x sin π 2 = − sin x 

12-135. Divide by x – 3, then solve the resulting quadratic; x = 1 ± i. 

12-136. a: 2 b: a – 2 

12-137. 3 C 1 ( 1 4 ) 2 3 

( 4 ) = 9 

64 ≈ 0.141 x 

4 

y

Core Connections Algebra 2

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