Name class date Precalculus Unit 1 Practice

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

LeSSon 1-1 1. Marena makes 16 bracelets per week for charity. She has already made 112 bracelets for charity.

a. Complete the following table to indicate the total number of bracelets made as the weeks progress.

Total number of Bracelets

Week, n 1 2 3 4 5

Total number of Bracelets, Bn

b. List the next five terms of the sequence {Bn}.

c. Model with mathematics. Plot the sequence {Bn} for n 5 1, 2, 3, … 10.

n

Bn

100125150175200

Tota

l Num

ber

of B

race

lets

225250275300325

1 2 3 4 5

Week 6 7 8 9 10

Total Number of Bracelets

d. Write an algebraic equation in terms of n for Bn, the total number of bracelets made.

2. How many weeks will it take for Marena to make a total of 496 bracelets?

A. 29 weeks

B. 30 weeks

C. 31 weeks

D. 32 weeks

3. Consider the sequence an 5 {2, 6, 10, …}.

a. Write an algebraic equation for the nth term an in terms of n.

b. Model with mathematics. Plot the sequence {an} for n 5 1, 2, 3, … 10.

n

an

5101520253035404550

1 2 3 4 5 6 7 8 9 10

4. Consider the sequence An11 5 An 1 an11, where an 5 {2, 6, 10, …}.

a. List the first ten terms of the sequence {An11}.

b. Model with mathematics. Plot the sequence {An} for n = 1, 2, … 5.

5101520253035404550

1 2 3 4 5 6 7 8 9 10n

An

Precalculus Unit 1 Practice

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

5. Construct viable arguments. How does the graph for {an} compare to the graph for {An11}? Explain.

LeSSon 1-2 6. Model with mathematics. A retirement plan

provides $22,500 per year with an annual 5% cost of living increase. What is the total amount of retirement collected at the end of the fifth year? (Hint: You will have a total of six installments.)

7. Critique the reasoning of others. Pamela started with 20 pencils and added 5 pencils each week. She then used sigma notation to represent the partial sums:

∑15k 1

7

5

1 5k 5 20 1 25 1 30 1 35 1 40 1 45 1 50

5 245. Pamela states that she has 245 pencils at the end of the seventh week. Is her reasoning sound? Explain.

8. Which of the following represents the associated series of the first 15 terms of the arithmetic sequence with a1 5

35

and d 5 213

?

A. ∑

k

35

13k 1

15

25

B. ∑

k

13

35k 1

15

25

C. ∑

k

415

13k 1

15

25

D. ∑

k

415

13k 1

15

25

9. Evaluate ∑( )k 1k

3

1

10

25

.

10. Find the values of n for which ∑k

n

15

(6 2 4k) . 250.

LeSSon 1-3 11. The daily high temperature in a city, starting on

June 1, increases 1 degree each day. Will this pattern continue? Explain using mathematical induction.

12. Consider the formula f (n) 5 n12 2

n1

.

a. Find the values of f(1), f (2), f(3), and f(4). Try other values of n . 4. Is f(n) a negative number for your choices of n?

b. Construct viable arguments. Based on your work in part a, do you think that f(n) is a negative number for all positive integers n? Explain.

13. Which of the following expressions is equivalent

to ∑

k

12k

n

15

?

A. n n( 1)

21

B. n n( 1)

41

C. n n( 1)1

D. n n4 ( 1)1

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14. Verify ∑ k( 1)k

n

12

5

5 n n( 1)

21

using mathematical

induction.

15. Does finding three different instances of a statement true mean that it is true for all instances? Explain.

LeSSon 2-1 16. Which of the following equations represents a

geometric sequence with a1 5 0.875 and r 5 0.2?

A. an 5 0.2(0.875)n

B. an 5 0.875(n)0.2

C. an 5 15

78

n

D. an 5 78

15

n

17. a. Model with mathematics. Plot the sequence an 5 1.5, 4.5, 13.5, 40.5, 121.5.

102030405060708090

100110120130

1 2 3 4 5 6 7 8 9 10n

an

b. Critique the reasoning of others. Carter states that this sequence is arithmetic. Is he correct? Explain.

18. Mutual Fund XYZ claims that they can turn $100 into $1 million in 30 years. If the stock market has an average return of 10%, is the investment true? Explain.

19. Find n where an 5 1.953125 for a geometric sequence with a1 5 125, r 5 0.25.

20. Determine the first term of a geometric sequence

with r 5 23 and a3 5

64081

.

LeSSon 2-2

Use the geometric sequence an 5 32

14

n

for Items 21 and 22.

21. Make use of structure. Find the partial sums S2, S3, and S4 of an.

22. Which of the following represents the general formula for the partial sum Sn for an?

A. Sn 5 8(1 0.25 )0.75

n2

B. Sn 5 32(1 0.25 )

0.25

n1

C. Sn 5

132 1 14

14

n

D. Sn 5

8 1 1

434

n

2

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23. Find the sum of the first five terms of the geometric sequence whose first term is 65 and ratio is 0.2.

24. Express the sum in Item 23 using sigma notation.

25. Attend to precision. Evaluate ∑1000(0.1)kk 1

6

5.

LeSSon 2-3 26. Indicate whether the given sequences are divergent

or convergent. If convergent, determine the value to which the terms are approaching.

a. 0.6, 0.36, 0.216, 0.1296, …

b. 130

, 110

, 310

, 910 , …

c. an 5 12

(20.4)n

d. an 5

57

n

27. Which of the following represents the sum 10.5 1 7.35 1 5.145 1 … using summation notation.

A. k 1∑∞

5

15(0.7)k

B. k 1∑∞

5

0.7(15)k

C. k 1∑∞

5

15(0.7)k21

D. k 1∑∞

5

0.7(15)k21

28. Make use of structure. Express 0.3333… as an infinite series.

29. Attend to precision. Find the sum of an infinite series with a common ratio of 0.5327 and a1 = 3.20479. Round to the nearest hundred-thousandth.

30. Solve for r: ∑∞

r7 k

k 15

5 3.

LeSSon 3-1 31. Model with mathematics. The population of

Earth is increasing at a rate of 1.14%. The world population in 2014 was approximately 7.3 billion people. [Source: www.worldometers.info/world-population/]

a. Describe the growth of the population.

b. According to the model, about how many people will live on Earth in 2020?

c. Write expressions u0 and un for this situation.

d. Use appropriate tools strategically. Use a spreadsheet or calculator to determine how long it will take the world population to double from 2014.

32. Which of the following represents the recursive sequence 4, 3.7, 2.885, 1.5293, …?

A. u0 5 4; 0.925un 2 1

B. u0 5 4; 0.7797un 2 1

C. u0 5 4; 1.05un 2 1 2 0.5n

D. u0 5 4; 0.05un 2 1 2 1.5n

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33. An arithmetic sequence has a first term of 7 and a constant difference of 5. Write expressions for u0 and un to represent this sequence.

34. A geometric sequence has a first term of 11 and a ratio of 2. Determine u10.

35. Find u100 given u0 5 1.7 and un 5 1.1un–1 1 0.3. Round to the nearest whole number.

LeSSon 3-2 36. Model with mathematics. The population of

Earth is increasing at a rate of 1.14%. The world population in 2014 was approximately 7.3 billion people. [Source: www.worldometers.info/world-population/] Write an explicit expression for an.

37. Make sense of problems. How does the explicit expression for an in Item 36 compare to its recursive expression?

38. Write the explicit expression for u0 5 8 and un 5 3un 21 1 7.

39. Determine a55 using the explicit form of u0 5 2.17 and un 5 0.525un 21 2 0.1. Round to the nearest ten-thousandth.

40. Jamal is saving pennies with a goal of earning 5 pennies each day for doing chores. For each day he does his chores, his mother pledges to double the number of pennies he had from the day before in addition to the five pennies earned per day. Assuming Jamal does his chores and he starts off with 5 pennies, use the explicit form to find a30 or the approximate amount of money he will have at the end of 30 days of saving.

A. $1.55

B. $15

C. $110 million

D. $53.7 million

LeSSon 4-1 41. In 2014 the population of Japan was approximately

127 million with an annual growth rate of 20.11%. [Source: www.worldometers.info/world-population/] Which of the following could be used to model the population of Japan from 2014 if the growth rate remains constant?

A. J(t) 5 12720.0011t

B. J(t) 5 12721.0011t

C. J(t) 5 127(1.0011)2t

D. J(t) 5 127(21.0011)t

42. Use appropriate tools strategically. Use the model from Item 41 and a spreadsheet or calculator to predict the year in which the population of Japan will fall below 100 million.

43. Model with mathematics. Mutual Fund ABC claims that it can turn $1,000 into $1 million in 50 years.

a. Assume the stock market has an average return of 10%. Write an exponential model, M(t), for the investment where t represents the number of years.

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b. Is Mutual Fund ABC making a true statement? If not, how long will it take for the mutual fund to be worth $1 million?

44. Any principal amount invested at 5% annual interest takes 15 years to double. How many years does it take for the principal amount to quadruple?

45. Graph the functions f(x) 5 a(b)x and g(x) 5 a(b)2x on a graphing calculator. Choose various positive values for a and negative values for b, where b fi 21. Then determine if the descriptions below are true for the functions by writing f(x), g(x), both, or neither next to each.

a. y-intercept: (0, a)

b. horizontal asymptote: y 5 0

c. domain: all real numbers

d. range: y . 0

e. range: y , 0

LeSSon 4-2Use this information for Items 46-50.In 2014, the average savings account interest rate was 0.07% compounded quarterly. [Source: http://www.nerdwallet.com/blog/banking/studies/savings-rates/] The historical return on the stock market is 11.69% compounded annually. [Source: http://www.daveramsey.com/article/the-12-reality/lifeandmoney_investing/] Suppose you invested the average cost of 4 years of college ($35,572) in either investment for 30 years. [Source: http://www.collegedata.com/cs/content/content_payarticle_tmpl.jhtml?articleId=10064]

46. Model with mathematics.

a. Which of the following models the growth of the college savings in a savings account?

A. A1(t) 5 35, 572(1.07)t

B. A1(t) 5 35, 572(1.0007)t

C. A1(t) 5 35, 572(1.000175)t

D. A1(t) 5 35, 572(1.000175)4t

b. Write a function A2(t) that models the growth of the college savings in the stock market.

47. Attend to precision. How much will each investment be worth in 30 years? Round to the nearest whole dollar.

48. Approximately how long, in years, would it take for the savings account to equal the value of the mutual fund?

49. If you needed $2 million to retire and live comfortably in 30 years, would either investment be a sound choice? If not, what should you do with the cost of college instead?

50. How long would it take for the mutual fund to be worth $2 million?

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LeSSon 4-3 51. Modeling with mathematics. There are many

choices for investing for retirement. Generally, the investment in a home should double every 20 years. In 2014, the average savings account interest rate was 0.07% compounded quarterly. The historical return on the stock market is 11.69% compounded annually.

[Source: http://www.zillow.com/research/zillow-home-value-appreciation-5235/]

[Source: http://www.nerdwallet.com/blog/banking/studies/savings-rates/]

[Source: http://www.daveramsey.com/article/the-12-reality/lifeandmoney_investing/]

a. What approximate interest rate would you need to have in a certificate of deposit that compounds monthly in order to double your money in 20 years?

b. What approximate rate of return would you receive from a home value that compounded continuously and doubled your investment in 20 years?

52. Construct viable arguments. Is a home investment better than investing in a savings account? How does it compare to investing in the stock market? What other factor(s) should be considered when deciding on an investment?

53. Modeling with mathematics. A new RV is purchased for $14,500. It depreciates continuously at a rate of 15%. Which of the following exponential functions represents the value of the RV after t years of ownership?

A. A(t) 5 14,500e20.15t

B. A(t) 5 214,500e0.15t

C. A(t) 5 14,500e215t

D. A(t) 5 214,500e15t

54. Joey, a freshman in high school, borrowed $20 from his father at 29% interest compounded continuously. He will pay his father back at his high school graduation, which is 4 years from now. How much will Joey pay back? Round to the nearest cent.

55. The half-life of the radioactive substance P-244 is 80 million years. Write the equation that represents the half-life and then find the decay rate. Round your answer to five decimal places.

LeSSon 5-1 56. Evaluate log6 362,797,056.

A. 6

B. 11

C. 32,981,550.55

D. 60,466,176

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57. a. Graph the function y 5 log3x.

x

y

b. Describe key features of the function, including the domain, range, x-intercept, asymptote, and end behavior.

58. Evaluate 856 5 ln(e2x).

59. Model with mathematics. What is the pH of apple cider that has [H1] of 0.00398 mole/liter? Round your answer to the nearest thousandth.

60. Attend to precision. What is the approximate hydrogen concentration of tomato juice, which has a pH level of about 4.5? Round to the seventh decimal place.

LeSSon 5-2 61. Use appropriate tools strategically. Use tables or

a calculator to evaluate log8 14. Round to three decimal places.

62. Make sense of problems. The number of users of a popular social networking site in the U.S. is growing at an annual rate of 5%. The site currently has 75 million users in the U.S. Explain how you can use a logarithm and the Change of Base Formula to predict how many years it will take for the number of users in the U.S. to double.

63. Write the expression as a single logarithm: 13

log(x 2 12)24 log2.

64. Expand the expression using the properties of logarithms: log (xyz).

65. Which of the following is a true statement?

A. log78

5 log7 1 log8

B. lnxy7 5 7lnx 1 lny

C. lnx

x8

72

1 5 ln(8 2 x)2

12 ln(x 1 7)

D. logxz

87

5 8logx 2 log(7z)

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LeSSon 5-3 66. Model with mathematics. A rabbit population

can begin with 40 rabbits and end with 2,400 rabbits in 20 months, assuming continuous growth. Write and solve an equation to find the annual growth rate to two decimal places.

67. Reason quantitatively. A biologist is studying a colony of bacteria with a doubling time of 30 minutes. The colony initially contains 22 bacteria.

a. Write an equation that can be used to determine t, the number of hours it will take for the population of the bacteria colony to reach 10,000.

b. Solve the equation you wrote in part (a).

c. Interpret your answer in part (b).

d. Explain how you know that your answer in part (b) is reasonable.

Use appropriate tools strategically. For Items 68–70, use tables or a calculator to solve. Round to two decimal places.

68. e2x 2 ex 5 20

69. 9x 2 4 5 0.1

70. log(x 2 5) 1 logx 5 log(2x 2 1)

LeSSon 6-1 71. The graph of g(x) is a vertical stretch of the graph

of f(x) 5 |x 2 2| 1 1 by a factor of 3.

a. Write an equation for g(x).

b. Graph g(x).

x

g(x)

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72. The graph of g(x) is a vertical shift of the graph of f(x) 5 216t2 1 20t by 5.


b. Graph g(x).

x

g(x)

73. The graph of g(x) is a reflection of the graph of f(x) 5 2lnx 1 1 over the y-axis.


b. Graph g(x).

x

g(x)

74. Determine if f(x) 5 x7 2 x2 is odd, even, or neither. Then describe the symmetry of the graph of the function, if any.

75. Make sense of problems. Kayla kicks a soccer ball from a height of 2 feet with an initial vertical velocity of 17 ft/s. The function K(t) 5 216t2 1 17t 1 2 models the height in feet of the ball t seconds after it is kicked. Next, Timmy kicks the ball. The function T(t) models the height in feet of Timmy’s soccer ball t seconds after it is kicked. The graph of T(t) is a translation 1.5 units up of the graph of K(t).

a. Write the equation of T(t).

b. From what height does Timmy kick the soccer ball? What is its initial velocity?

LeSSon 6-2 76. Modeling with mathematics. The income tax

rates for the UK are shown in the table below. [Source: https://www.gov.uk/income-tax-rates/income-tax-rates]

Tax Rates in the United Kingdom

Taxable Income Tax Rate

£0 2 £31,865 20%£31,866 2 £150,000 40%

£150,001 or More 45%

The U.K. tax is a progressive system in which £31,865 is taxed at 20% and every pound earned over £31,865 up to £150,000 is taxed at 40%, etc.

a. Write the equation of the piecewise-defined function f(x) that represents the current tax bracket.

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b. Parliament is considering a 2.5% increase on all tax brackets. Write the equation of the piecewise-defined function g(x) that represents a possible increase. Round to the nearest pound.

Make use of structure. For Items 77 and 78,

use the function f(x) 5 x x

x x

2 if 0if 0

3

2

≤

>

2 2.

77. The graph of g(x) is a reflection of the graph of f(x) over the y-axis.


b. Graph g(x).

x

g(x)

78. The graph of g(x) is a reflection of the graph of f(x) over the x-axis.


b. Graph g(x).

x

g(x)

For Items 79 and 80, find the sum, difference, product, and quotient of each pair of functions. State the domain.

79. f(x) 5 x2 1 2x, g(x) 5 2x2 2 1

80. f(x) 5 3x 1 2, g(x) 5 32x

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LeSSon 7-1 Modeling with mathematics. Use the table for Items 81–85. In a local town, the speed limit was studied along a stretch of highway. The following table shows the number of accidents with respect to the speed limit.

Speed Limits and Accidents on Hwy. 123

Speed Limit(mph)

Accidentsper Year

15 561 14563 16765 22566 45671 98172 1,679

81. Attend to precision. Make a scatterplot of the data.

x

y

82. a. Construct viable arguments. Do you think a linear model is necessarily the best model for the data in the scatterplot? Support your answer.

b. What transformation can you perform on the data to check whether a power function is a good model for the data?

c. Transform the data using the transformation you chose. Find a linear regression equation for the transformed data. Round to three decimal places.

d. Find the correlation coefficient. What does this indicate about the transformed data?

83. What power function models the original data set?

A. y 5 9.916 3 1024 (3.085)x

B. y 5 3.085(9.916 3 1024)x

C. y 5 3.085x9.916 3 1024

D. y 5 (9.916 × 1024)x3.085

84. Use your model to predict the number of accidents with a speed limit of 90 mph.

85. Use your model to predict the speed limit needed to reduce the number of accidents to less than 100.

LeSSon 7-2 86. Modeling with mathematics. The function

f(x) 5 0.000739x4.0971 models the number of accidents at various speed limits x on Highway 456 in a local town. Graph the function.

x

y

87. Find the approximate value for f(25). A. 0.757 B. 1225 C. 400 D. 1.52 3 1012

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88. Reason quantitatively. Explain why the answer to Item 87 makes sense for the scenario represented by the model.

For Items 89 and 90, describe the key features of the graphs of the following power functions, including domain, range, symmetry, maximum/minimum, end behavior, and decreasing/increasing intervals.

89. y 5 45

x3

90. y 5 23x45

LeSSon 8-1 Find f(g(x)) and g(f(x)) for each pair of functions. State the domain.

91. f(x) 5 x4

, g(x) 5 x4 2 16

92. f(x) 5 x52 , g(x) 5 x2 1 1

93. Find f(x) and g(x) such that f(g(x)) = h(x) and h(x) 5 | x 1 12 | 2 5.

A. f(x) 5 | x 1 12 | and g(x) 5 x 2 5

B. f(x) 5 | x | and g(x) 5 x 1 12

C. f(x) 5 | x | 2 5 and g(x) 5 x 1 12

D. f(x) 5 | x | 2 5 and g(x) 5 x

94. Modeling with mathematics. Loam is a spongy soil used in gardening and has a density of 80 lb./ft3. At a certain nursery, loam costs $0.67 per lb. If f(x) represents the value in dollars of loam with a mass of x lbs. and g(x) represents the mass in lbs. of a cubic foot of loam with a side length of x ft., find the composition f(g(x)). State its reasonable domain. What does the composition represent in this situation?

95. Attend to precision. What is the value of a cube of loam having a side length of 10 ft.?

LeSSon 8–2 96. Which is the inverse function of f(x) 5 log (x 2 3)

and the domain of the inverse function?

A. f –1(x) 5 10x 1 3; D:

B. f –1(x) 5 10x 1 3; D:

C. f –1(x) 5 log(10x 2 3); {x| x $ log 3}

D. f –1(x) 5 log(10x 2 3); {x| x $ 3}

97. Make use of structure. Find the inverse function. Restrict the domain of f if needed, and describe the restriction. Then state the domain and range of the inverse function.

f(x) 5 127 (x 1 1)3 1 3

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Reason quantitatively. For Items 98 and 99, use composition to determine whether each pair of functions are inverses.

98. f(x) 5 x1

213 and g(x) 5

x3

3 11

99. f(x) 5 7(x 2 5)2 1 12 and g(x) 5 5 2 x 127

2

100. a. Find the inverse function of f(x) 5 3| x 1 2 | 2 1. Restrict the domain of f if needed, and describe the restriction. Then state the domain and range of the inverse function.

b. Graph the inverse function.

x

y

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Name class date Precalculus Unit 1 Practice