Name class date Algebra 2 Unit 1 Practice - Denton ISD · Algebra 2 Unit 1 Practice. 2 Name class...

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1© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 1 Practice

LeSSon 1-1Use this information for Items 1–3. Aaron has $65 to rent a bike in the city. It costs $15 per hour to rent a bike. The additional fee for a helmet is $3 for the entire ride.

1. Write an equation that can be used to find h, the number of hours Aaron can rent the bike.

2. Solve your equation from Item 1, and interpret the solution.

3. Aaron would like to rent the bike for 5 hours. How much more money will Aaron need? Explain.

4. Aaron and Zelly want to rent a tandem bike so that they can ride together. The rental for a tandem bike is $18.50 per hour plus $3 for each helmet. Aaron and Zelly have $80 to spend on the bike rental. Which equation can be used to find h, the number of hours they can rent a tandem bike?

A. 80 5 18.50h 1 3 B. 80 5 18.50h 1 6

C. 80 5 3h 1 18.50 D. 80 5 6h 1 18.50

5. Make sense of problems. Eliza bought a day pass to rent a bike in the city. The day pass costs $40 from 9 a.m. to 7 p.m. There is an additional fee of $4 per quarter hour if she returns the bike after 7 p.m. Eliza has $50 and plans to return the bike at 8:15 p.m.

a. Does Eliza have enough money? Explain using an equation.

b. When should she return the bike?

LeSSon 1-2 Use this information for Items 6–10. Susan plans to rent a bike in New Orleans to tour the city. The cost of the rental is $20 per day. The cost of a helmet is $7 for as long as Susan needs the bike.

6. Make a table that shows the cost of the rental for 1, 2, 3, 4, and 5 days.

7. Write a linear equation in two variables that models the situation. Tell what each variable in the equation represents.

8. Graph the equation. Be sure to include titles, and use an appropriate scale on each axis.

9. Which of the following statements describes what the slope and y-intercept represent in the situation?

A. The slope represents the cost of the rental. The y-intercept represents the cost of the helmet.

B. The slope represents the cost of the helmet per day. The y-intercept represents the cost of the helmet at day 0.

C. The slope represents the cost of the bike per day. The y-intercept represents the cost of the helmet for the entire time of the rental.

D. The slope represents the rise of a hill in New Orleans. The y-intercept represents rise of the slope.

Algebra 2 Unit 1 Practice

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© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 1 Practice

10. Construct viable arguments. Barbara has $250 to rent a bike for 15 days. Does Barbara have enough money? Explain.

LeSSon 1-3 11. Solve each absolute value equation. Graph the solutions on the number line.

a. |x 1 7| 5 3

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

b. |2x 2 13| 5 1

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

c. 5|x 2 6| 5 20

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

d. |3x 1 15| 5 9

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

12. If the speed of a car on the highway detected by a radar gun is within 7 mph of the speed limit, a police officer will not issue a ticket to a car for driving too fast or too slow. What is the acceptable range of speeds in a 55-mph speed zone?

13. Solve each absolute value inequality. Graph the solutions on the number line.

a. |x 2 5| , 4

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

b. |3x 1 5| $ 8

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

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c. |4x 2 1| # 20

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

d. 5|2x 1 9| . 15

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

14. Which number line shows the solutions of the inequality 2|x 2 1| 1 4 , 6?

A. 210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

B. 210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

C. 210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

D. 210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

15. Model with mathematics. The weight of a pen manufactured at the PEN factory is 8 g. The actual weight can vary by as much as 2 g. Write and solve an inequality to represent all acceptable weights for a pen at this factory. Graph the inequality.

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

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LeSSon 2-1 16. Write the equation of a line with a y-intercept of 2

and a slope of 34

. Graph the equation.

x

y

17. Which is the equation of the line that passes through the point (2, 22) and has a slope of 3?

A. y 5 3x 2 2 B. y 5 3x 2 8

C. y 5 2x 1 3 D. y 5 22x 1 2

Use the following information for Items 18–20. Julian works as a sales representative for a home improvement store. He earns $1200 plus 9 1

2% commission on sales

each month.

18. Write the equation of the function f (e) that represents Julian’s monthly earnings, where e represents the amount of monthly sales.

19. Graph the function using appropriate scales on the axes.

20. Reason quantitatively. What would Julian’s sales have to be for him to earn $2000 in one month? Explain how you determined your answer.

LeSSon 2-2 21. a. Graph the inequality 22x 2 y , 25.

x

y

b. Did you use a solid or dashed line for the boundary line? Explain your choice.

c. Did you shade above or below the boundary line? Explain your choice.

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22. Graph the following inequalities on the same grid and shade the solution region that is common to all of the inequalities.

a. x , 2

b. y # 2x

c. y $ 23x 1 2

x

y

23. Which ordered pair lies in the solution region that is common to all of the inequalities in Item 22?

A. (2, 5) B. (0, 0)

C. (2, 0) D. (1, 22)

24. a. Identify two ordered pairs that do not satisfy the constraints in Item 22.

b. Reason abstractly. For each ordered pair from part a, identify the constraint or constraints that it fails to meet.

25. Model with mathematics. Write a system of inequalities for the graph.

x

y

2223242526272829210 21

210292827262524232221

123456789

10

1 2 3 4 5 6 7 8 9 10

LeSSon 3-1 26. Solve each system by graphing.

a.

y xy x2 3

52

5 2

x

y

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b.

y x

y x

23

5

23

5 2

5

x

y

c.

x yx y3 6

2 21 5

2 5

x

y

d.

y x

y x

12

3

2 6

52 1

52 1

x

y

27. Solve each system by substitution.

a.

x yy x3 8

42 5

5 2

b.

x yx y3 5 11

3 12 5

2 5

c.

x y

x y

35

3

3 5 15

2 5

2 5

d.

x yx y

5 54 5 17

2 5

2 52

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28. Solve each system by elimination.

a.

x yx y

5 2 69 2 22

1 5

1 5 b.

x yx y

4 2 124 2 24

2 5

1 5

c.

x yx y6 5 273 10 24

2 5

1 52 d.

x y

x y

23

15

35

1

15

25

29. Reason abstractly. Explain how you would eliminate one of the variables in this system.

x yx y

0.4 0.5 2.51.2 3.5 2.5

1 5

2 5

30. Persevere in solving problems. Noah’s Café sells special blends of coffee mixtures. Noah wants to create a 20-pound mixture with coffee that sells for $9.20 per pound and coffee that sells for $5.50 per pound. How many pounds of each mixture should he blend to create coffee that sells for $6.98 per pound?

A. 8 pounds of the $5.50-per-pound coffee, 12 pounds of the $9.29-per-pound coffee

B. 10 pounds of the $5.50-per-pound coffee, 10 pounds of the $9.29-per-pound coffee

C. 12 pounds of the $5.50-per-pound coffee, 8 pounds of the $9.20-per-pound coffee

D. 15 pounds of the $5.50-per-pound coffee, 5 pounds of the $9.20-per-pound coffee

LeSSon 3-2 31. Solve each system by substitution.

a.

x y zx y zx y z

3 4 102 6

2 8

2 1 52

2 1 1 5

2 1 52

b.

x y zx y zx y z

6 2 85 3 2

3 3 4 15

2 2 52

2 1 1 5

2 2 5

c.

x y zx y zx y z

3 72 2 33 2 2

1 2 5

2 1 5

1 2 5

d.

x y zx y zx y z

3 92 3 8

2 3 16

1 2 5

1 1 5

1 1 5

32. Solve each system using Gaussian elimination.

a.

x y zx y zx y z

4 3 112 3 2 9

3

1 2 5

2 1 5

1 1 52

b.

x y zx y zx y z

5 3 2 82 4 18

6 4 25

2 1 52

1 2 5

2 1 52

c.

x y zx y zx y z

2 4 33 5 2 4

2 3 20

1 1 5

2 1 52

2 2 52

d.

x y zx y zx y z

3 12 2 2

2 3 1

2 1 5

2 2 5

1 2 52

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33. Use the table for parts a and b.

Smoothie Sales

Time Period Classic no. of cups sold

Fruit no. of cups sold

Veggie no. of cups sold

Sales

8:00 2 12:00 10 12 3 $103.5012:00 2 4:00 6 8 15 $125.504:00 2 8:00 12 6 8 $102.00

a. Write a system of equations that can be used to determine c, f, and v, the price in dollars of a classic, fruit, and veggie smoothie.

b. Solve your system and explain what the solution means in the context of the situation.

34. A food store makes a 10-pound mixture of peanuts, cashews, and raisins. The mixture has twice as many peanuts as cashews. Peanuts cost $1 per pound, cashews cost $2 per pound, and raisins cost $2 per pound. The total cost of the mixture is $16. How much of each ingredient did the store use to make the mixture?

35. Make sense of problems. A company placed $500,000 in three different investment plans. The company placed some money into short-term notes paying 5.5% per year. They placed three times as much into government bonds paying 5% per year. They placed the rest in utility bonds paying 4.5% per year. The income for one year was $25,000. How much more money did the company place in government bonds than in utility bonds?

A. $100,000

B. $200,000

C. $300,000

D. $400,000

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LeSSon 3-3 36. Which entry corresponds to the address c21 for the

following matrix?

5

2

2C1 34 25 0

A. 3 B. 4

C. 21 D. 0

37. A is a 3 3 2 matrix. B is a 2 3 3 matrix. What are the dimensions of the matrix product, AB?

38. Find the sum or difference.

a

5

2

2A232

111

816

5

2

2

B143

329

57

12

5

2

2

2

C3

100

684

1479

a. A 1 B b. A 2 B

c. B 1 C d. C 2 B

39. Find the product.

52

D3 52 4

52

2E

2 13 0

52

2F

1 3 54 2 0

a. DE b. DF

c. EF d. FD

40. Make use of structure. Use the following matrices for parts a and b.

5

2

2G2 41 25 3

52

2H 3

215

62

a. Can you add or subtract the matrices? Explain.

b. Can you multiply the matrices, GH? Explain.

LeSSon 3-4 41. Which of the following is the inverse of 2 1

0 3

?

A.

13

16

0 12

B.

12

16

0 13

2

C.

13

16

0 12

2 D.

12

16

0 13

2

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42. Model with mathematics. Write a matrix equation to model each system. Then use the matrix equation to solve the system.

a.

x yx y

202 10

2 1 5

2 1 5

b.

1 1 5

1 1 5

1 1 5

x y zx y zx y z

2 3 43 3 82 4 5

43. Lori bought cheeses for a party. The cheddar cost $7.95 per pound, and the Swiss cost $9.50 per pound. Lori bought 5

12

pounds of cheese for a total of $47.60.

a. Write a system of equations that can be used to determine how many pounds of each type of cheese Lori bought. Let x be the number of pounds of cheddar cheese, and let y be the number of pounds of Swiss cheese.

b. Write a matrix equation that can be used to solve the system.

c. Use the matrix equation to solve the system.

d. Explain the meaning of the solution.

44. The local bookstore had a sale. The table shows the number of items sold over three days.

Day Sales

Day Paperback Hardcover e-book Amount of Sales

Monday 55 17 33 $970Tuesday 12 10 13 $335

Wednesday 23 18 19 $595

a. Write a matrix equation that can be used to find the price of each item.

b. Solve the equation to find the solution of the system and explain the meaning of the solution.

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45. Mari has 42 coins consisting of nickels, dimes, and quarters. The total value of the coins is $9.35. The number of dimes is one less than three times the number of nickels. How many of each type of coin does Mari have?

LeSSon 4-1 46. The domain of a function is all real numbers

greater than or equal to 23 and less than 12. Write the domain using an inequality, interval notation, and set notation.

47. The range of a function is (2`, 5]. Which of the following shows this range using set notation?

A. y # 5 B. {y | y ∈ , y , 5}

C. y , 5 D. {y | y ∈ , y # 5}

48. Graph each piecewise function. Then write its domain and range using inequalities, interval notation, and set notation.

a. ( )

5

#

2 1 .f x

x x

x x

2 if 212

2 if 2

x

f (x)

b. ( )

51 ,

2 1 .f x

x xx x

3 1 if 12 if 1

x

f (x)

c. ( )

5

2 1 #

2 .f x

x x

x x

1 if 012

2 if 0

x

f (x)

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d. ( )

52 2 ,2

2 .2f x

x xx x3 6 if 1

2 3 if 1

x

f (x)

49. Evaluate the piecewise function for x 5 23, x 5 3, and x 5 30.

( )

h x

x xx x

x x

8 if 35 if 3 5

5 if 52

5

2 ,

2 1 , #

2 , , `

50. Model with mathematics. A water company charges residential customers a quarterly flat fee of $25 for up to 5000 gallons of water. The company charges $10 per thousand for the first 10,000 gallons of water after the minimum allowance. The next 10,000 gallons is $5 per thousand. All other consumption over 25,000 gallons is $2.50 per thousand.

a. Write a piecewise function w(x) that can be used to determine a customer’s quarterly bill for using x gallons of water.

b. Graph the piecewise function.

c. Why are the slopes of each piece of the function different? Explain.

d. On average, a family of four consumes 247.2 gallons of water per day. Calculate the quarterly water bill for a family of four. Explain how you determined your answer.

LeSSon 4-2 51. A step function, known as the greatest integer

function or the floor function, is written as ( ) 5f x x 5 the largest integer not greater than x.

a. Graph this step function.

Floor function ( ) 5f x x

x

f(x)

b. Find f (2.4), f (0.05), and f (23.6).

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52. Use the absolute value function h(x) 5 | x 2 3| for parts a–d.

a. Graph h(x).

x

h(x)

b. What are the domain and range of the function?

c. What are the coordinates of the vertex of the function’s graph?

d. Write the equation for the function using piecewise notation.

53. Model with mathematics. A tutor charges $45 for each hour or for any fraction of an hour.

a. Write a function t(x) to represent this situation using the ceiling function x , the smallest integer greater than or equal to x.

b. Graph the function.

c. Evaluate the function for x 5 1 hour 30 minutes and for x 5 3 hours 2 minutes.

d. Chance hired the tutor to help him prepare for his calculus exam. The tutor and Chance worked together from 4:15 p.m. until 6:45 p.m. How much did the tutor charge Chance for the session?

A. $80 B. $135

C. $180 D. $270

54. Use the step function h(x) for parts a and b.

( )

5

, ,

, ,

.

h x

xx

x

0 if 0 12 if 1 31 if 3

a. Graph the function.

x

h(x)

b. Evaluate the function for x 5 0.3, x 5 3, x 5 1.1, and x 5 100.

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55. Maura gets $8 per hour for the first 40 hours she works in a week. She gets 1.5 times her hourly rate for every hour she works over 40 hours per week. Maura worked 46 hours last week. How much did Maura earn last week? Is this a step function? Explain.

LeSSon 4-3 56. Write the equation g (x) of each transformation of

the parent graph f (x) 5 |x| described below.

a. reflection over the x-axis

b. stretched horizontally by a factor of 3

c. translated right 5 units

d. stretched vertically by a factor of 4, translated up 2 units

e. stretched vertically by a factor of 2, reflected over the x-axis, translated left 1 unit and down 3 units

57. What is the domain and range of g (x) 5 2 |x 1 2| 2 1?

58. Write the equation for the transformation of f (x) 5 |x| as shown in the graph.

x2223242526272829210 21

210292827262524232221

123456789

10

1 2 3 4 5 6 7 8 9 10

f(x)

59. Make use of structure. Find the coordinates of the vertex of g (x) 5 | x 1 1| 2 2.

A. (22, 21) B. (22, 1)

C. (21, 22) D. (1, 22)

60. Graph ( )5 1 2g x x12| 2| 3 as a transformation of

f (x) 5 |x|. Then identify the transformation.

x

g(x)

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LeSSon 5-1 61. For parts a–d, use the following functions.

f (x) 5 x 2 1 1 g (x) 5 x 2 2

Find each function and simplify the function rule. Note any values that must be excluded from the domain.

a. ( f 1 g)(x) b. ( f 2 g)(x)

c. ( f ? g)(x) d. ( f 4 g) (x)

62. Given g (x) 5 (x 2 3)2 and h (x) 5 2x 1 1, which of the following is ( g ? h)(5)?

A. 35 B. 38

C. 44 D. 45

63. Make use of structure. Given h (x) 5 x 2 2 5x, for what values of x is h (x) 5 84? Explain.

64. Marty Besson is a salesperson who earns $25,000 per year plus 8% commission on sales. His wife Phyllis is also a salesperson. She earns $30,000 per year plus 5% commission on sales. Last year, Marty and Phyllis each made sales of $10,000.

a. Write a function m(x) to represent Marty’s annual earnings, and a function p(x) to represent Phyllis’s annual earnings.

b. What does the function (m 1 p)(x) represent in this situation?

c. Add the functions m(x) and p(x) to find (m 1 p)(x). Then simplify the function rule.

d. What does (m 1 p)(10,000) represent in this situation? Find (m 1 p)(10,000).

e. How could you use (m 2 p)(x) in this situation?

65. Shanti has a vegetable garden that is 28 ft long and 14 ft wide. She wants to increase both the length and the width of the garden by x number of feet.

a. Write a function to represent the new area.

b. What is the new area if she increases both the length and width by 3 ft?

LeSSon 5-2 66. Given the functions y (x) 5 2x 2 1 and x (z) 5 3z

1 2, write the equation for the composition function y(x(z)) and evaluate it for z 5 3.

67. The notation j (k(l )) represents a composite function. The domain of j (l ) is all real numbers, and the range is all real numbers greater than or equal to 22. What is the domain of the composite function?

68. Model with mathematics. Tom’s in-home computer service charges a flat fee of $15 plus $75 per hour to service a computer in a home.

a. Write a function c (h) for the cost of servicing a computer or computers for h hours at one location.

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b. What are the units of the domain and range of this function?

c. What is the slope of this function? Interpret the slope as a rate of change.

69. It takes about 45 minutes for one of Tom’s technicians to service one computer.

a. Write a function h(x) for the hours needed to service x computers.

b. Write a function c (h(x)) to represent the cost of servicing x computers in one location.

c. Which is the value of c (h(3))?

A. $101.25 B. $168.75

C. $183.75 D. $213.75

70. Will the cost be the same to service three computers in one location or one computer in each of three different locations? Explain.

LeSSon 5-3 71. Use the tables of values to evaluate each expression.

f f ( x)

1 12 43 94 16

g g ( x)

1 42 73 94 11

a. f (3) b. g (4)

c. f ( g(1)) d. g ( f (2))

72. For parts a–h, use these three functions to evaluate each composite function.

f (x) 5 x 2 2 2 g (x) 5 3x h (x) 5 2x 2 3

a. ( f g)(2) b. ( g f )(1)

c. ( f h)(4) d. ( g h)(3)

e. (h f )(21) f. (h g)(22)

g. ( f f )(2) h. (h g h)(4)

73. Given that a(x) 5 x 2 and b(x) 5 2x, write each composite function in terms of x.

a. (a b)(x) b. (b a)(x)

74. Make use of structure. Given that j(n) 5 n2 1 1 and k(n) 5 n 2 1, for which value(s) of n is ( j k)(n) 5 5?

A. 21 B. 3

C. 23 or 1 D. 21 or 3

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75. Marion wants to buy a laptop at the computer store. Laptops are on sale for 25% off. The sales tax is 7%. Use this information for parts a and b.

a. Write a function s(p) that gives the sale price of a laptop regularly priced at p dollars.

b. Write a function t(p) that gives the total cost including tax for a laptop priced at p dollars.

c. Determine the cost of a laptop regularly priced at $450.

LeSSon 6-1 76. If h and j are inverse functions, what is h( j(15))?

77. Find the inverse of each function.

a. f (x) 5 3x 1 1 b. ( )( )g x x12

65 2

c. ( )h x x34

35 2 d. ( )j x x2 34

51

78. Which is the inverse of the function a(d) 5 5d 2 3?

A. a d d5 31 ( )52 12 B. ( )a d d 35

15

12

C. ( )a d d15

315 22 D. ( )a d d 3

51

522

79. The function k a a5

( )5 gives the map distance in

centimeters between two points on a map that are actually a kilometers apart. Use this information for parts a–c.

a. What is k(15)? What does k(15) represent?

b. What is the inverse of k(a)? What does the inverse represent?

c. Evaluate k21(3). What does k21(3) represent?

80. Make use of structure. Given f (2) 5 8, what is f 21(8)?

LeSSon 6-2 81. Make use of structure. Find the inverse of each

function. Then use the definition of inverse functions to prove that the functions are inverses. Show your work.

a. f (x) 5 3x 2 2 b. g x x13

4( )5 1

82. h(x) and j(x) are inverse functions. Which of the following statements is NOT true?

A. h( j(x)) 5 x

B. The domain of j(x) is the range of h(x).

C. The slopes of h(x) and j(x) are reciprocals.

D. The graphs of h(x) and j(x) are reciprocals.

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83. In parts a and b, graph the inverse of each function shown on the coordinate planes.

a.

x

y

2223242526272829210 21

210292827262524232221

123456789

10

1 2 3 4 5 6 7 8 9 10

f(x) 5 x 11

b.

x

y

2223242526272829210 21

210292827262524232221

123456789

10

1 2 3 4 5 6 7 8 9 10

f (x) 5 2x 1 4

84. Write the inverse of the function defined by the table. Express your answer tabularly or symbolically.

x 0 2 4 6 8f ( x) 25 21 3 7 11

85. Graph each function and its inverse on the same coordinate plane.

a. f (x) 5 | x | 2 3 b. f (x) 5 2x 1 1

x

y

x

y

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