answers to precalculus unit 1 practicepshs.psd202.org/documents/jfinley/1506287062.pdfa4 springboard...

A1© 2015 College Board. All rights reserved. SpringBoard Precalculus, Unit 1 Practice

LeSSon 1-1 1. a. Total number of Bracelets

Week n 1 2 3 4 5

Total number of Bracelets Made Bn

128 144 160 176 192

Answers to Precalculus Unit 1 Practice

b. 208, 224, 240, 256, 272

c.

n

Bn

100125150175200

Tota

l Num

ber

of B

race

lets

225250275300

1 2 3 4 5

Week 6 7 8 9 10

Total Number of BraceletsMade

d. Bn 5 112 1 16n or Bn 5 128 1 16(n 2 1)

2. C

3. a. an 5 22 1 4n or an 5 2 1 4(n 2 1)

b.

n

an

5101520253035404550

1 2 3 4 5 6 7 8 9 10

4. a. 2, 8, 18, 32, 50, 72, 98, 128, 162, 200

b.

n

An

20406080

100120140160180

220

1 2 3 4 5 6 7 8 9 10

200

5. The graph for {An 1 1} increases more quickly than for the same n-values of {an} because {An 1 1} is a sequence that adds the values of the terms in {an}.

LeSSon 1-2 6. $153,043.03

7. Sample answer: She will not have 245 pencils because the values for the sequence are the total number of pencils each week.

8. D

9. 3015

10. n # 6

LeSSon 1-3 11. no, because logically the temperature cannot

increase infinitely

12. a. 0, 214

, 229

, 2316

; sample answer: Yes, the next

three numbers are 2425

, 2536

, and 2649

,

which are all negative.


b. Sample answer: Yes, the formula f (n) 5 n12 2

n1

always subtracts a larger number from a smaller number, which will always be negative.

13. A

14. For n 5 1: 1 2 1 5 1(1 1)

22

, so Step 1 is verified

for n 5 1. Assume that k 2 1 5 k k( 1)

22

is true.

So k k( 1)( 1 1)

21 1 2

5 k k( 1)

21

.

15. Sample answer: No, concluding that a statement is true comes from logical induction of the conditions initially set.

LeSSon 2-1 16. D

17. a.

102030405060708090

100110120130

1 2 3 4 5 6 7 8 9 10n

an

b. Sample answer: No, he is not correct because the points are not linear.

18. Sample answer: No, it is not true because in 30 years the investment will be worth less than $1 million. It will take about 100 years to be worth $1 million.

19. n 5 3

20. 803

LeSSon 2-2

21. S4 5 858 and S5 5

34132

22. A

23. 68.2448

24. ∑65(0.2)kk 1

5

5

25. 111.111

LeSSon 2-3 26. a. convergent to 0

b. divergent

c. divergent

d. convergent to 0

27. A

28. 310

3100

31000

...1 1 1

29. 6.85810

30. 79

LeSSon 3-1 31. a. The population is increasing each year.

b. 7.8 billion

c. u0 5 7.3; un 5 1.0114un21

d. 62 years

32. C 33. u0 5 7; un 5 un21 1 5

34. 11,264

35. 64,766

LeSSon 3-2 36. an 5 7.3(1.0114)n

37. Sample answer: The explicit expression has the variable as an exponent.

38. an 5 23.5 1 11.5n

39. a55 5 20.2105

40. C

LeSSon 4-1 41. C

42. Year 2232


43. a. M(t) 5 1000(1.1)t

b. Sample answer: No, it is not true because in 50 years the investment will be worth about $117,391. It will actually take 73 years for the mutual fund to be worth more than $1 million.

44. about 28.5

45. a. neither

b. both

c. both

d. neither

e. both

LeSSon 4-2 46. a. D

b. A2(t) 5 35,572(1.1169)t

47. $36,327; $980,702

48. 4739 years

49. Sample answer: Neither will reach the goal of $2 million. Instead of investing, I would spend the money on college since a bachelor’s degree holder will earn at least a third more than a high school diploma holder. [Source: http://education-portal.com/articles/How_Much_More_Do_College_Graduates_Earn_Than_Non-College_Graduates.html]

50. about 37 years

LeSSon 4-3 51. a. 3.47%

b. 3.5%

52. Sample answer: Investing in a home is more lucrative than investing in a savings account, but it is less rewarding than investing in the stock market. One other factor to consider is risk; a savings account is a lot less risky than the stock market.

53. A

54. $63.80

55. 20.00866

LeSSon 5-1 56. B 57. a.

x

y

023 22 21

21

22

23

3

2

1

1 2 3

b. domain: (0, ∞); range: (2∞, ∞); x-intercept: 1; asymptote: x 5 0; end behavior: as x → 0, y → 2∞ and x → ∞, y → ∞.

58. 428

59. 2.4

60. 0.0000316

LeSSon 5-2 61. 1.269

62. Sample explanation: The equation 75(1.05)t 5 150 can be used to determine the time t in years for the number of users to reach 150 million. Divide both sides of the equation by 75: 1.05t 5 2. Write the logarithmic form of the equation: t 5 log1.05 2. Apply

the Change of Base Formula: t 5 log 2

log 1.05 ≈ 14. So,

it will take about 14 years for the number of users to reach 150 million.

63. log x 1

216

3 2

64. y log x 1 log z

65. C

LeSSon 5-3 66. 2.46


67. a. 10,000 5 22(22t)

b. t ≈ 4.41 hours

c. It will take about 4.4 hours for the population to reach 10,000.

d. In 4 hours, the population will have doubled 8 times, so the population will be 22(28) 5 5632. In 4.5 hours, the population will have doubled 9 times, so the population will be 22(29) 5 11,264. So, it is reasonable that the population will be 10,000 in close to 4.4 hours.

68. x 5 1.61

69. x ≈ 2.95

70. x 5 7 3 52

6

LeSSon 6-1 71. a. g (x) 5 3|x 2 2| 1 1

b.

x

g(x)

21

21

1

2

3

1 2 3 4 5

72. a. g(x) 5 216t2 1 20t 1 5

b.

x

g(x)

22

22

5

10

5

73. a. g(x) 5 ln 2x 1 1

b.

x

g(x)

21

22

23

25 24 23

1

2

3

22 121

74. f (x) is neither odd nor even since f (2x) fi f (x) and f (2x) fi f (x). There is symmetry about the y-axis between x 5 20.5 and x 5 0.5.

75. a. T(t) 5 216t2 1 17t 1 3.5

b. 3.5 feet; 17 feet per second

LeSSon 6-2

76. a. f (x) 5

x xx xx x

0.2 31,8660.4 6373 31866 150,0000.45 53,627 150,000

,

1 < <

1 .

b. f (x) 5

x xx xx x

0.22 31,8660.42 7011 31866 150,0000.47 56,628 150,000

,

1 < <

1 .

77. a. g (x) 5 x x

x x

2 if 0if 0

3

2

2 >

,

b.

x

g(x)


78. g (x) 5 x x

x x

2 if 0if 0

3

2

1 <

2 .

x

g(x)

79. ( f 1 g)(x) 5 3x 2 1 2x 2 1; D :

( f 2 g)(x) 5 2x 2 1 2x 1 1; D :

( f g)(x) 5 2x 4 1 4x 3 2 x 2 2 2x; D :

fg

(x) 5 xx

22 2

2

21

2; D : x x 1

2≠

| 6

80. ( f 1 g)(x) 5 3x 1 2 1 32x; D :

( f 2 g)(x) 5 3x 1 2 2 32x; D :

( f g)(x) 5 3x 1 2; D :

fg

(x) 5 32x 1 2; D :

LeSSon 7-1

81.

x

y

200400600800

1000

Acc

iden

ts 12001400160018002000

10 20 30 40 50

Speed Limit (mph)60 70 80

Hwy. 123

82. a. Sample answer: The scatterplot shows that the data points do not appear to lie on or very close to a single line. In addition, a linear regression of the data has a correlation coefficient of 0.543, which does not indicate a strong linear correlation.

b. Take the logarithm to the base 10 of the speed limit data and the logarithm to the base 10 of the accident data.

c. y 5 3.085x 2 3.004

d. r 5 0.921; Sample answer: The correlation coefficient for the transformed data is much closer to 1 than the correlation coefficient for the original data. This result indicates that a power function is a better model of the original data than a linear function.

83. D

84. about 1,060 accidents

85. about 41 mph

LeSSon 7-2 86.

x

y

5

10

15

Acc

iden

ts

20

5 10

Speed Limit (mph)

15 20 25

Hwy. 456

87. C

88. Sample answer: f (25) ≈ 400 and f (50) ≈ 6,750 makes sense since the higher the speed of vehicles on a road, the more likely it is that accidents will happen. All of the other choices go against this basic principle.

89. domain: ; range: ; symmetry: origin; max/min: neither; end behavior: y → ∞ as x → ∞, y → 2∞ as x → 2∞; increasing

90. domain: ; range: y # 0; symmetry: y-axis; max/min: maximum at (0, 0); end behavior: y → 2∞ as x → ∞, y → 2∞ as x → 2∞; increasing then decreasing


LeSSon 8-1 91. f ( g(x)) 5

x4164

2; {x | x fi 22, 2, 6i 4}

g ( f (x)) 5

x

4 4

216; {x | x fi 0}

92. f ( g(x)) 5 x4 22 ; {x | 22 # x # 2}

g ( f (x)) 5 6 2 x 2 16; {x ∈ }

93. C

94. g (x) 5 80x3; f (x) 5 0.67x; f (g(x)) 5 53.6x3; {x | x $ 0}; f (g (x)) represents the value in dollars of a cube of loam with a side length of x feet.

95. f (g(10)) 5 $53,600

LeSSon 8-2 96. B

97. f 21(x) 5 x3 33 2 1 1; D:

98. yes

f ( g(x)) 5

x

13

3 1

13

1

2 5 x

g ( f (x)) 5

x

3

3 1 13

1

2 1

5 x

99. No, g(x) includes only the negative side of the function.

100. a. f 21(x) 5 x3

53

2 . The domain of f is restricted

to {x | x $ 22}. The domain of f 21 is {x | x $ 21} and the range of f 21 is {y | y $ 22}.

b.

x

y

22

24

22 2

2

4

4 6 8 10

answers to precalculus unit 1 practicepshs.psd202.org/documents/jfinley/1506287062.pdfa4 springboard...

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