State whether each equation is linear.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

Graph each linear equation.

13. 14. 15.

Determine whether each table represents a linear relationship

between x and y. If so, write the next ordered pair that would

appear in the table.

16. 17. 18.

19. 20. 21.

y � 3x � 2y �45xy �

23x � 4

y � 7x � 2y � �45x2y � 2x � x2

y �38x � 6y �

23x �

25x2y � 9x �

34

y � 3x � 7xy � 6.7 � 6.7x2y � �7 �27 x

y � 3.5x � 7x2y �34 xy � 2x � 1

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Algebra 2 Practice Workbook 1

Practice

1.1 Tables and Graphs of Linear Equations

NAME CLASS DATE

x y

2 11

3 13

4 15

5 17

x y

0 4

1 8

2 12

3 16

x y

�1 5

�2 11

�3 21

�4 35

x y

0 �3

�1 �8

�2 �13

�3 �18

x y

0 3

2 �5

4 �13

6 �21

x y

1 �6

2 �8

3 �10

4 �12

Write the equation in slope-intercept form for the line that has

the indicated slope, m, and y-intercept, b.

1. 2. 3.

4. 5. 6.

Find the slope of the line containing the indicated points.

7. (3, 0) and (–3,4) 8. and

9. (2, 6) and (1, 5) 10. (�1, �5 ) and (2, 4)

Identify the slope, m, and the y-intercept, b, for each line.

11. 12.

13. 14.

Write an equation in slope-intercept form for each line.

15. 16.

x

y

2

4

2

–2

–4

–6

(8, –3)

(0, 3)

O 4 6 8

x

y

2

2

(6, 8)

(0, 3)

4

6

8

4 6 8

15x � 5y � �35�2x � y � 4

34x � 2y � �33x � 4y � 6

( 23, 34 )(�1, �1

5 )

m �14, b � 4m �

16, b � 3m �

45, b � �

25

m � �4, b � 3m � 3, b � 1m � 2, b � �5

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2 Practice Workbook Algebra 2

Practice

1.2 Slopes and Intercepts

NAME CLASS DATE

Write an equation for the line containing the indicated points.

1. (2, 4) and (3, 5) 2. (�1, 3) and (3, �1)

3. (3, 1) and 4. (2, 0) and (�6, 4)

5. (�1, �4) and (�2, 5) 6. and

Write an equation in slope-intercept form for the line that has the

indicated slope, m, and contains the given point.

7. and (3, 3) 8. and (4 ,6)

9. and (4, �2) 10. and (4, 3)

11. and (�2, 3) 12. and (8, 6)

Write an equation in slope-intercept form for the line that

contains the given point and is parallel to the given line.

13. (1, 4); 14. (�2, 3);

15. (4, �2); 16. (�6, 3);

17. (2, �1); 18. (3, �4);

19. (2, �2); 20. (1, �1);

21. (2, �2); 22. (1, 0);

Write an equation in slope-intercept form for the line that

contains the given point and is perpendicular to the given line.

23. (2, 4); 24. (6, �4);

25. (6, �7); 26. (2, �5);

27. ; 28. (3, 5);

29. ; 30. (1, 4);

31. (3, �1); 32. ; y � 4x � 3(�1, �72 )y � 3x �

34

y � �34x � 4y �

34x � 3(1, 23 )

y � �x � 1y � 4x � 6(3, 114 )

y � 2x � 4y � �2x � 5

y � 3x �34y �

12x � 3

y � �3x � 2y �12x � 3

y � 3x � 2y � �12x � 3

y � 4x � 3y � �3x � 6

y � 2x � 2y �34x �

14

y � �4x � 2y � �3x � 2

m � �14m � �2

m � 4m �34

m � �12m � 1

(�2, �12 )( 1

2, 32 )( 1

2, 32 )

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Algebra 2 Practice Workbook 3

Practice

1.3 Linear Equations in Two Variables

NAME CLASS DATE

In Exercises 1–8, y varies directly as x. Find the constant of

variation, and write an equation of direct variation that relates

the two variables.

1. , for 2. , for

3. , for 4. , for

5. , for 6. , for

7. , for 8. , for

Solve each proportion for the variable. Check your answers.

9. 10.

11. 12.

13. 14.

15. 16.

Determine whether the values in each table represent a direct

variation. If so, write an equation for the variation. If not, explain

why not.

17. 18. 19.

5z7 �

z � 314

x � 25 �

525

3y10 �

y � 16

z10 �

60240

2x � 510 �

3x20

1824

y6 �

x � 23 �

3x6

x4 �

912

x �15y � �

35x �

13y �

23

x � 6y � 5x � �7y � �2

x � 12.8y � 3.2x � �3y � 4

x � 3y � 7x � 2y � �10

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4 Practice Workbook Algebra 2

Practice

1.4 Direct Variation and Proportion

NAME CLASS DATE

x y

�2 �6

�1 �3

0 0

1 3

2 6

x y

5 49

4 28

3 20

2 5

1 2

x y

1 2

3 6

5 10

7 14

9 18

Create a scatter plot of the data in the table below. Describe the

correlation. Then find an equation for the least-squares line.

1. 2. 3.

A baseball player has played baseball for several years.

The following table shows his batting average for each year over

a 10-year period.

In Exercises 4–6, refer to the table above.

4. Enter the data in a graphics calculator, and find the equation ofthe least-squares line.

5. Find the correlation coefficient, r, to the nearest tenth.

6. Use the least-squares line to predict the baseball player’s batting average in 1999.

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Algebra 2 Practice Workbook 5

Practice

1.5 Scatter Plots and Least-Squares Lines

NAME CLASS DATE

x y

0 4

2 10

6 22

8 28

x y

0 1.9

2 10

6 �20.15

8 �27.45

x y

1 �2

2 �18

7 �26

9 �34

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997

0.250 0.258 0.262 0.280 0.272 0.278 0.285 0.292 0.316 0.320

Solve each equation.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

Solve each literal equation for the indicated variable.

23.

24.

25.

26.

27.

28. A � (a � b)h, for h

T � To � a(z � z0), for a

q � qp � D � Q, for qp

V1P1 � V2P2, for P1

C � 2πr, for r

L � W � D � V, for W

2x � 2(2x � 3) � �32x � 4(3x � 6) � 12

2x � 3(x � 2)2(2x � 2) � x � 3x � 4

5(x � 0.5) � �1.5(x � 3x)5x � 15 � 4x � 3

7x � 2(x � 3)�4x � 7 � 5(x � 2)

2(x � 3) � 5(x � 3)5x � 10(4x � 3) � 15

6(x � 2) � 5x � 94x � 10 � 3(x � 2)

5x � 3 � 15 � 4x2x � 1 � 7 � 10x

2(x � 3) � 5x � 15�0.4x � 6(3x � 2) � 48.8

3(2x � 4) � 3x � 5(x � 1)3x � 4 � 4(3x � 19)

2x � 5 � 175x � 15 � 10(x � 3)

4x � 20 � 5(x � 3)4x � 4(2x � 1) � 20

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6 Practice Workbook Algebra 2

Practice

1.6 Introduction to Solving Equations

NAME CLASS DATE

Write an inequality that describes each graph.

1.

2.

3.

Solve each inequality, and graph the solution on a number line.

4.

5.

6.

7.

Graph the solution of each compound inequality on a number line.

8.

9.

x190180170160150 200

150 �t � 738

6 and t � 7386 � 155

x

5x � 2 � 3 or 2x � 6 � 4

x

�5(x � 2) � 3x � 6

x

3x � 4 � 3(x � 2)

x

7x � 15 � �2(x � 3)

x

2x � 13 � x � 1

x0 2 4 6–2–4–6

x0 2 4 6–2–4–6

x0 2 4 6–2–4–6

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Algebra 2 Practice Workbook 7

Practice

1.7 Introduction to Solving Inequalities

NAME CLASS DATE

Solve each equation. Graph the solution on a number line.

1.

2.

3.

4.

5.

Solve each inequality. Graph the solution on a number line.

6.

7.

8.

9.

10.

x

�4x � 5� � 15

x

�3x � 6� � 15

x

�5x � 6� � 5

x�6x � 4� � 3

x�5x � 2� � 7

x

�3x � 12� � 18

x

�5x � 3� � 12

x

�2x � 5� � 7

x

�x � 4� � 6

x

�x � 3� � 5

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8 Practice Workbook Algebra 2

Practice

1.8 Solving Absolute-Value Equations and Inequalities

NAME CLASS DATE

x �9

5 and x � �3

Classify each number in as many ways as possible.

1.

2.

3. 3.12112111211112 . . .

4. 801.35

5.

6.

State the property that is illustrated in each statement.

Assume that all variables represent real numbers.

7.

8.

9.

10. , where

11.

12.

13.

Evaluate each expression by using the order of operations.

14. 15.

16. 17.

18. 19.

20. 21.

22. 23. 7 � 12 � 30 � 5(77 � 50) � (13 � 42)

(13 � 7)2 � 5150 � 384 � 4 � 2

12 � 82 � 413 � 3 � 2 � 5

45 � 16 � 827 � 8 � 2

52 � (2 � 11)�2 � 42 � 1

78 � 0 �

78

14(x � 91) � 14x � 14(91)

47y � 3x � 3x � 47y

k � 054k �

k54 � 1

�2 � (33 � 18) � (�2 � 33) � 18

181 � 1 � 181

75 � (�75) � 0

501.07

��900

�91

1317

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Algebra 2 Practice Workbook 9

Practice

2.1 Operations With Numbers

NAME CLASS DATE

Evaluate each expression.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

Simplify each expression, assuming that no variable equals zero.

Write your answer with positive exponents.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30. (3xy )4� x�8

(xy)3 ��2� (x 2y 2 )3

x 5 ��1

(xy3z2

z�2 )�1(m�2p2

2mp3 )�4

(w6

k )3(a�2

b�3 )�2

(wz4

x 2 )�2(5a2b3)3

(�2a3bc6)4(3x3y5)4

w 21w�12

w 9

y14z5

y 9z 4

( 1x�7 )

�5z15

z�2

(x7)2k�11k3

w3y4z � wy�2zd3d�4

�272364

56

( 12 )�4

3215

( 12 )�5( 1

5 )�2

( 35 )2

(�217)1

(�3435)0(2 � 3)2

�(15�1)320

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10 Practice Workbook Algebra 2

Practice

2.2 Properties of Exponents

NAME CLASS DATE

In Exercises 1–8, state whether each relation represents a function.

1. 2. 3.

4. 5. 6.

7. 8.

State the domain and range of each function.

9. 10.

11.

12.

Evaluate each function for the given values of x.

13. , for and

14. , for and

15. , for and

16. , for and

17. , for and x � 0x � 0.5f(x) � 3x � x2

x � �4x � 11f(x) � 3x2 � 2

x � �5x � 7f(x) � 12 � 3x

x � 5x � �3f(x) � 5x2

x � 8x � �2f(x) � 20x � 4

{(�2, 12), (0, 8), (1, 9)(5, 33)}

{(�4.5, 6), (3, �1.5), (6.5, �5), (12, �10.5)}

�(�1, �3), (0, 1), ( 12, 3), ( 3

2, 7)�

{(32, 4), (16, 7), (16, 4)}{(1, 5), (0.5, 8), (0, 3)}

xO

y

2–2–4 4

2

–2

–4

4

xO

y

2–2–4 4

2

–2

–4

4

xO

y

2–2–4 4

2

–2

–4

4

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Algebra 2 Practice Workbook 11

Practice

2.3 Introduction to Functions

NAME CLASS DATE

x �2 �1 0 1 2

y 8 4 0 4 8

x 1 2 3 4 5

y 2 3 2 3 2

x 2 3 2 3 2

y 1 2 3 4 5

xO

y

2–2–4 4

2

–2

–4

4

Find and .

1.

2.

3.

4.

Find and . State any domain restrictions.

5.

6.

7.

Let and . Find each new function, and

state any domain restrictions.

8. 9.

10. 11.

12. 13.

Find and .

14.

15.

16.

Let , and . Evaluate each

composite function.

17. 18. 19.

20. 21. 22.

23. 24. 25. (f � f)(2)(h � h)(�1)(f � g)(0)

(f � h)(5)(g � f)(0)(g � h)(4)

(h � g)(2)(h � f)(�2)(f � g)(�1)

h(x) � 2(x�4)f(x) � 11x, g(x) � x2 � 5

f(x) � �x2 � 1; g(x) � x

f(x) � 4x; g(x) � x2 � 1

f(x) � 3x � 2; g(x) �13(x � 2)

g � ff � g

gf

fg

f � gg � f

f � gf � g

g(x) � x � 10f(x) � �2x � 2

f(x) � x2 � 16; g(x) � x2 � 16

f(x) � x2 � 25; g(x) � 3x � 17

f(x) � 35x � 5; g(x) � 5

fg

f �g

f(x) � �9x2 � 6; g(x) � 12x2

f(x) � x2 �13x � 9; g(x) � �7x � 7

f(x) � 41 � 5x; g(x) � 13x2

f(x) � 7x2 � 5x; g(x) � x2 � 13

f � gf � g

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12 Practice Workbook Algebra 2

Practice

2.4 Operations With Functions

NAME CLASS DATE

Find the inverse of each relation. State whether the relation is a

function and whether its inverse is a function.

1.

2.

3.

4.

5.

For each function, find the equation of its inverse. Then use

composition to verify that the equation you wrote is the inverse.

6. 7.

8. 9.

10. 11.

Graph each function, and use the horizontal-line test to

determine whether the inverse is a function.

12. 13. 14.

x

y

x

y

x

yh(x) � x3 � xg(x) � x3 � 1f(x) � x2 � x

h(x) �x4 � 8g(x) � 8(x � 2)

f(x) �12(x � 2.5)g(x) � 11x � 4

h(x) �2x�1

3f(x) �13(x � 1)

{(�5, 4), (�3, 9), (1, 12), (7, 13)}

{(�5, 7), (�3, 7), (�1, 7), (1, 7)}

{(�2, 16), (�1, 1), (1, 1), (2, 16)}

{(7, 2), (6, 3), (7, 4), (6, 5)}

{(�1, �16), (0, �6), (2, 14)}

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Algebra 2 Practice Workbook 13

Practice

2.5 Inverses of Functions

NAME CLASS DATE

Graph each function.

1. 2. 3.

Write the piecewise function represented by each graph.

4. 5. 6.

Evaluate.

7. 8.

9. 10.

11. 7.8 + �1.88 12.

13. � 14. 2.5 �

15. �12.95 � 16. � � [4.9]��3�[ 6.3]

[2.5]��3.75��5.25�

[�2.22] � ��4.5�

�0.9�13.13�

31.7[�9.23]

xO

y

2–2–4 4

2

–2

–4

4

xO

y

2–2 4

2

–2

–4

4

xO

y

2–2–4 4

2

–2

–4

4

x

y

x

y

x

y

h(x) � �2[x]f(x) � � �x � 4�x � 1

if x � �2if x � �2

g(x) ���1 � xx � 1

if �0if x � 0

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14 Practice Workbook Algebra 2

Practice

2.6 Special Functions

NAME CLASS DATE

Identify each transformation from the parent function to g.

1.

2.

3.

4.

5.

6.

Identify each transformation from the parent function to g.

7.

8.

9.

10.

11.

12.

Write the function for each graph described below.

13. the graph of reflected across the x-axis

14. the graph of translated 7 units to the left

15. the graph of stretched horizontally by a factor of 26

16. the graph of compressed vertically by a factor of

17. the graph of reflected across the y-axis

18. the graph of translated 33 units down f(x) � x3

f(x) �23x � 9

112f(x) � �x�

f(x) � x4

f(x) � x5

f(x) � x3

g(x) � 41�x � 8

g(x) � �3�x

g(x) � �x � 13.7

g(x) � �12x

g(x) � 17�x

g(x) � �x � 21

f(x) � �x

g(x) � 12(x � 7)2

g(x) � 14x2 � 6

g(x) � �2x2

g(x) � (52x)2

g(x) � x2 � 7.5

g(x) � (x � 7.5)2

f(x) � x2

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Algebra 2 Practice Workbook 15

Practice

2.7 A Preview of Transformations

NAME CLASS DATE

Graph and classify each system. Then find the solution from the graph.

1. 2. 3.

4. 5. 6.

Use substitution to solve each system of equations. Check your solution.

7. 8. 9.

10. 11. 12. �3x � 4y � 112x � 4y � 8�x � 2y � 2

2x � 3y � �1� 3x � y � �412x � y � 6

�8x � y2x � y � 5�x � 10y � 2

x � 6y � 6�y � 2x � 11x � y � 5

2O

2

4

–4

–2

4–4 –2x

y

2O

2

4

–4

–2

4–4 –2x

y

2O

2

4

–4

–2

4–4 –2x

y

� x �13y � 3

3x � y � �3�5x � y � 2

2x � y � �1� y � �4x � 10

2x �12y � 6

2O

2

4

–4

4–4 –2x

y

–2

2O

2

4

–4

–2

4–4 –2x

y

2O

2

4

–4

–2

4–4 –2x

y

�3x � 4y � �72x � y � �3� 1

2x � y � 2

2y � x � 4�y � x � 4

y � x � 4

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16 Practice Workbook Algebra 2

Practice

3.1 Solving Systems by Graphing or Substitution

NAME CLASS DATE

Use elimination to solve each system of equations.

Check your solution.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

Use any method to solve each system of linear equations.

Check your solution.

16. 17. 18.

19. 20. 21.

22. 23. 24. �5x � 3y � 462x � 5y � 7�9x � 5y � �4

3x � 2y � 6�12x � y � 24x � 3y � 4

� 2x � 3y � 723 x � y � 9

�y � 7 � xx � y � 13�x � y � 1

2x � 5y � 4

�4x � y � 92y � �8x � 18�y � 6x

2x � 5y � 16�y � 5x � 2y � 2x � 5

�3x � 5y � 45x � 7y � 6�3x � 2y � 31

3x � 2y � �1�11x � 4y � 193x � 2y � 7

�5x � 2y � �9y � 3x � 12�13x � 7y � 19

9x � 2y � 20�x � 9y � �132x � y � �7

� 12x �

34y � 10

2x � y � 8� 6x � y � 26

3x �12y � 13� 2

3x � y � �2

3x � 2y � �35

� 12x � y � 22

2x � 4y � 11�5x � 9y � �7

2x � 3y � �1� 4x � y � 12

3x �14y � 9

�7y � x � 8x � y � 4� 2

3 x � 3y �15

2x � 9y � 4��2x � 9y � �136x � 3y � 15

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Algebra 2 Practice Workbook 17

Practice

3.2 Solving Systems by Elimination

NAME CLASS DATE

Graph each linear inequality.

1. 2. 3.

4. 5. 6.

7. Sheila earns a basic wage of $8 per hour. Under certainconditions, she is paid $12 per hour. The most that she canearn in one week is $400.

a. Write an inequality that describes her total weekly wages for x hours at $8 per hour and for y hours at $12 per hour.

b. Graph the inequality on the grid at right.

c. What is the maximum number of hours that Sheila can workfor $8 per hour? for $12 per hour?

2

2

4

–4

–2

4–4 –2 Ox

y

2

2

4

–4

–2

4–4 –2 Ox

y

O 2

2

4

–4

4–4 –2

–2

x

y

x � �1.5x � y � 42x � y � �3

O 2

2

4

–4

–2

4–4 –2x

y

2

2

4

–4

–2

4–4 –2x

O

y

2O

2

4

–4

–2

4–4 –2x

y

y �14x � 1y � �3x � 2y � �x

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18 Practice Workbook Algebra 2

Practice

3.3 Linear Inequalities in Two Variables

NAME CLASS DATE

302010

24

32

8

16

40Ox

y

Graph each system of linear inequalities.

1. 2. 3.

Write the system of inequalities whose solution is graphed.

4. 5. 6.

7. During the summer, Ryan works 30 hours or less per week mowing lawns and delivering newspapers. He earns $6 per hour mowing lawns and $7 per hour delivering papers. Ryan would like to earn at least $126 per week. Let x be the number of hours mowing, and let y be the number of hours delivering papers. Write a system of inequalities to represent the possible hours and jobs that Ryan can work, and graph this system at right.

2

2 (4, 1)

(2, 3)

(–2, 1)

4

–4

–2

4–4 –2 Ox

y

2

2(–2, 2)

(4, 5)

(4, –4)

4

–4

–2

–4 –2 Ox

y

2

2

(–4, 4) (4, 4)

–4

–2

4–4 –2 Ox

y

2

2

4

–2

4–4

–4

–2 Ox

y

2

2

4

–4

4–4 –2

–2

x

y

O2

2

4

–4

–2

4–4 –2 Ox

y

� y � 4y � �3y � 3x � 2

� y � �3x � 3

y �13x � 1

�y � �xy � 2x � 4

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Algebra 2 Practice Workbook 19

Practice

3.4 Systems of Linear Inequalities

NAME CLASS DATE

xO

y

15

30

15 30

Graph the feasible region for each set of constraints.

1. 2. 3.

The feasible region for a set of constraints has vertices at (2,0),

(10, 1), (8, 5), and (0, 4). Given this feasible region, find the

maximum and minimum values of each objective function.

4. 5. 6.

maximum: maximum: maximum:

minimum: minimum: minimum:

Find the maximum and minimum values, if they exist, of each

objective function for the given constraints.

7. 8.

Constraints: Constraints:

9. 10.

Constraints: Constraints:

� 2x � y � 10x � 2y � 10x � 0y � 0

� 2x � y � 2x � y � 10x � 0y � 0

F � 12x � 5yG � 20x � 10y

� x � y � 9y � x � 7x � 0y � 0

� 2x � y � 10x � y � 4x � 0y � 0

E � 4x � 8yP � x � 5y

M � 3y � xE � 2x � 3yF � 4x � y

2

2

4

6

8

4 6 8Ox

y

2

2

4

6

8

4 6 8Ox

y

2

2

4

6

8

4 6 8Ox

y

� x � 2y � 163x � 4y � 12x � 0, y � 0

� 3x � 4y � 20y � x � 3x � 0, y � 0

� x � y � 92x � y � 5x � 0, y � 0

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20 Practice Workbook Algebra 2

Practice

3.5 Linear Programming

NAME CLASS DATE

Graph each pair of parametric equations for the interval .

1. 2. 3.

Write each pair of parametric equations as a single equation in x and y.

4. 5. 6.

7. 8. 9.

An airplane is ascending at a constant rate. Its altitude

changes at a rate of 12 feet per second. Its horizontal

speed is 150 feet per second.

10. a. Write parametric equations that represent the plane’s flight

path.

b. Graph the equations for the interval 0 ≤ t ≤ 30. Use the gridat right.

11. a. How long will it take the plane to reach an altitude of

300 ft?

b. How far will the plane travel horizontally in that

time?

�x(t) � 3ty(t) � t 2 � 1�x(t) � 3 � 12t

y(t) � 4t � 3�x(t) � 10 � 2ty(t) � t � 11

�x(t) � 4 � 2ty(t) � 6 � t� x(t) �

12t

y(t) � 3t � 2�x(t) � 2t � 12

y(t) � t � 8

xO

y

2

–2

–4

4

6

2–2–4–62

xO

y

2

–2

–4

4

4 6 8

xO

y

2

–2

–4

–6

4

42–2–4

� x(t) �12t � 5

y(t) � t � 3� x(t) � 6 � t

y(t) �12t � 1� x(t) �

13t � 4

y(t) � 2t � 1

�3 � t � 3

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Algebra 2 Practice Workbook 21

Practice

3.6 Parametric Equations

NAME CLASS DATE

xO

y

100

200

300

400

2000 4000

In Exercises 1–13, let , , ,

and .

Give the dimensions of each matrix.

1. D 2. E 3. F 4. G

Find the indicated matrix.

5. 6. 7.

8. 9. 10.

11. 12. 13.

Matrix M at right represents the number

of medals won by athletes from the

United States, Germany, and Russia in

the 1996 Summer Olympic Games.

14. What are the dimensions of matrix M?

15. Find the total number of medals won by the United States.

16. Find the total number of gold medals won by the three nations.

17. Describe the data in location m23.

18. In the 1996 Summer Olympic Games, athletes from China won 16 gold medals, 22 silver medals, and 12 bronze medals. Write a new matrix, , that includes medals for all four countries.M ′

2D � 3EF � G�2G

2F � GE � DD � E

3ED � G�F

� 3

0

6

�3

1

�2

�4

2

2�G �

� 2

3

7

4

3

0

2

6

�5�F �� 0

�1

�2

4 �E �� 7

3

2

1 �D �

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22 Practice Workbook Algebra 2

Practice

4.1 Using Matrices to Represent Data

NAME CLASS DATE

Gold Silver Bronze

� 442026

321821

252716

�United StatesGermany

Russia

Find each product, if it exists.

1. 2. 3.

4. 5. 6.

7. 8. 9.

Matrix , represents triangle XYZ, which

is graphed at right.

10. Find the coordinates of the vertices of the image,triangle , which is formed by multiplying matrix T by the transformation matrix

.

11. Sketch the image, triangle , on the grid at right above.

12. Describe the transformation.

X′Y ′Z′

�32

0

032�

X′Y ′Z′

T, ��2

0

4

4

2

�4 �

� 0.56

4�1 � � 6

208 �� 6

�2�1

114 �� 7

�24

031

151

10

�1�� 2

1 ��01

�3�2

2114�

� 0.56

21 �� 4

10

523�[2 �2 1]� 5

13���3

247 � � 0

120 �

� 69

17

68 �� 4

125

02 � � 6

01

�120�[�4 0 4]� 2

�157 � � 2

�157 �

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Algebra 2 Practice Workbook 23

Practice

4.2 Matrix Multiplication

NAME CLASS DATE

O

y

2–4–6 4 6

2

–4

–6

4

6

Z

X

Y

Determine whether each pair of matrices are inverses of each other.

1. 2.

3. 4. ,

5. 6. ,

Find the determinant and the inverse of each matrix, if it exists.

7. 8.

9. 10.

11. 12.

Find the inverse matrix, if it exists. If the inverse matrix does not

exist, write no inverse.

13. 14.

15. 16.

17. 18. � 1.5�1

�2.52 �� 2

34

2

12�

� 52

31 �� 6

�34

�2 �

� 45

67 �� 2

311 �

� 1316

34 �� 71

212

5

8�

� 117

64 �� 8

755 �

� 95

74 �� 7

453 �

� 34

28 ��

12

�14

�18

316�� 12

1456 �, � 3

�7�2.5

6 �

� 1512

119 �� 3

�4

�3235�� 6

293 �, � 3

�2�9

6 �

� 5�17

�27 �, � 7

1725 �� 4

�5�3

4 �, � 45

34 �

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24 Practice Workbook Algebra 2

Practice

4.3 The Inverse of a Matrix

NAME CLASS DATE

Write the matrix equation that represents each system.

1. 2. 3.

Write the system of equations represented by each matrix equation.

4. 5.

Write the matrix equation that represents each system, and solve

the system, if possible, by using a matrix equation.

6. 7. 8.

9. 10. 11.

12. 13. 14. � 8x � y � z � 05x � 2y � 9z � �312x � y � 5z � 8

� 12x � 7y � z � �53x � 4y � 2z � 35x � 3y � 3z � 12

� x � 2y � z � 153x � y � 2x � 85x � 10y � 5z � 21

� x � 2y � 3z � 33x � y � z � 123x � 2y � 4z � 15

� 3x � 3y � 5z � 135x � 6y � 2z � 107x � 5y � 18

� 4x � y � z � 18x � 4y � 7z � 25y � 9z � 3

�3x � 7y � 255x � 8y � 27�7x � 5y � 14

4x � 3y � 9�8x � 7y � 54x � 9y � 65

��61

20��x

yz� ��3

24

234

�113��1

67��x

yz� �� 2

3�1

34

�1

�112�

� 9x � 5y � z � 63x � y � z � 24x � 3y � 2x � �1

� 5x � 2y � z � 13�x � 4y � z � �14x � 8y � 3z � 6

� 3x � y � z � �19�x � y � 3z � 212x � 2y � z � �7

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Algebra 2 Practice Workbook 25

Practice

4.4 Solving Systems With Matrix Equations

NAME CLASS DATE

Write the augmented matrix for each system of equations.

1. 2. 3.

Find the reduced row-echelon form of each matrix.

4. 5. 6.

7. 8. 9.

Solve each system of equations by using the row-reduction method.

10. 11. 12.

13. 14. 15.

16. 17. 18. � 2x � 4y � 3z � �8x � 3y � 2z � 93x � 2y � z � 13

� 3x � 2y � z � 16x � 3y � 4z � 92x � y � 3z � 15

� x � y � z � 62x � 3y � 5z � �11x � 3y � 4z � 19

� 3x � 6y � 4z � �422x � 2y � 3z � 144x � 3y � 5z � �34

� 2x � 5y � 3z � �113x � 2y � 4z � 72x � 3y � 2x � �10

� x � y � 3z � �212x � y � z � 123x � 2y � 2z � 7

�3x � 4y � 18x � 11y � 4�3x � 11y � 10

2x � 5y � 19�2x � y � 23x � 2y � 7

�513

1�2

4

�1�1

5

M

M

M

9�6

0���3

2�4

�45

�1

532

MMM

4�1�8

��120

103

112

M

M

M

585�

�315

2�4

0

�11

�3

MMM

7�9�1

�� 10

�2

031

�210

MMM

025��1

03

120

0�1�2

M

M

M

�1�3

5�

� x � y � z � �12x � 3y � 3z � 46x � 7y � 3z � 8

� 2x � y � 3z � 45x � 6y � z � 67x � 8y � 3z � 2

� 4x � 5y � z � 27x � 9y � 2z � 7x � y � z � 2

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26 Practice Workbook Algebra 2

Practice

4.5 Using Matrix Row Operations

NAME CLASS DATE

Show that each function is a quadratic function by writing it in

the form and identifying a, b, and c.

1.

2.

3.

4.

5.

Identify whether each function is quadratic. Use a graph to check

your answers.

6. 7.

8. 9.

10. 11.

State whether the parabola opens up or down and whether the

y-coordinate of the vertex is the minimum value or the maximum

value of the function.

12. 13.

14. 15.

Graph each function and give the approximate coordinates of the vertex.

16. 17. 18. p(x) � �(x � 4)(x � 0.5)h(x) � �x2 � x � 6k(x) � 4x2 � 3

q(x) � (4 � x)(2 � 7x)h(x) � (5 � x)(2 � 3x)

g(x) � 4x2 � 7x � 2f(x) � 5x2 � 3x

m(x) � 3x � x(x � 9)b(x) � x2 � 2x(x � 1)

g(x) � 16 � 3xh(x) �2x3 � xx 2 � 1

k(x) �1xf(x) � �4x � x2

d(x) � (x � 3)2 � 4

h(x) � (2x � 5)(3x � 1)

k(x) � �3(x � 11)(x � 1)

g(x) � (7 � x)(9 � x)

f(x) � (x � 3)(x � 5)

f(x) � ax2 � bx � c

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Algebra 2 Practice Workbook 27

Practice

5.1 Introduction to Quadratic Functions

NAME CLASS DATE

Solve each equation. Give both exact solutions and approximate

solutions to the nearest hundredth.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

Find the unknown length in each right triangle. Round answers to

the nearest tenth.

11. 12. 13.

Find the missing side length in right triangle ABC. Round answers

to the nearest tenth.

14. 15 and 7

15. 2.4 and 7.3

16. 2 and

17. 9.1 and 7 b �a �

c � �10b �

c �a �

b �a �

P

R

r

3.56.1

Q

JK

j

10.5

12

L

C

A

B

c

8

5

7(x � 1)2 � 16114 � 0.5x2 � 5

12 � 4(x � 2)2 � 84x2 � 9 � 17

6x2 � 15 � 23x2 � 12 � 4

5x2 � 4 � 96(x � 3)2 � 81

12x2 � 36x2 � 100

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28 Practice Workbook Algebra 2

Practice

5.2 Introduction to Solving Quadratic Equations

NAME CLASS DATE

B

a

A

c

bC

Factor each expression.

1. 2.

3. 4.

5. 6.

Factor each quadratic expression.

7. 8. 9.

10. 11. 12.

13. 4 14. 3 15. 2

Solve each equation by factoring and applying the Zero-Product Property.

16. 17. 5 18.

19. 3 20. 4 21. 6

22. 9 23. 5 24. 7

Use factoring and the Zero-Product Property to find the zeros of

each quadratic function.

25. f(x) 26. g(x) 27. h(x)

28. b(x) 29. k(x) 30. q(x) � 4x2 � 12x � 9� 3x � 2� 2x2� 13x � 42� x2

� 12x � 45� x2� 3x � 5� 2x2� x � 12� x2

� 2 � 5xx2� 1 � 6xx2� 4 � 0x2

� x � 1 � 0x2� 12x � 9 � 0x2� 6x � 3 � 0x2

� 9x � 14 � 0x2� 125 � 0x2� 256 � 0x2

� 3x � 2x2� 5x � 2x2� 4x � 1x2

� 11x � 26x2� x � 90x2� 11x � 28x2

� 7x � 18x2� x � 20x2x2 � 17x � 52

�3x2 � 105x3x2 � 21x

4x(x � 12) � 3(x � 12)(2 � 7x) � 3x(2 � 7x)

�24x � 4x212x � 60

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Algebra 2 Practice Workbook 29

Practice

5.3 Factoring Quadratic Expressions

NAME CLASS DATE

Complete the square for each quadratic expression in order to

form a perfect-square trinomial. Then write the new expression

as a binomial squared.

1.

2.

3.

4.

5.

6.

Solve by completing the square. Round your answers to the

nearest tenth, if necessary.

7. 8. 9.

10. 11. 12.

13. 2 14. 15. 2

Write each quadratic function in vertex form. Find the

coordinates of the vertex and the equation of the axis

of symmetry.

16. f(x) 17. f(x)

18. f(x) 19. f(x)

20. f(x) 21. f(x) � 3x2 � 15x � 2� 10x � 10� x2

� 2x � 10� x2� 12x � 3� x2

� 7 � 3x2� �12x2

� 16x � 3x2� 7x � 2 � 0x2� 13 � 2xx2

� 4 � 6xx2� 1 � 5xx2� 20x � 3x2

� 14x � 1 � 0x2� 8x � 13 � 0x2� 2x � 7 � 0x2

� 19xx2

� 9xx2

� 5xx2

� 20xx2

� 40xx2

� 24xx2

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30 Practice Workbook Algebra 2

Practice

5.4 Completing the Square

NAME CLASS DATE

Use the quadratic formula to solve each equation. Round your

answer to the nearest tenth.

1. 2. 3.

4. 5. 6.

7. 8. 9. 2

10. 3 11. 12. 4

For each quadratic function, find the equation for the axis of

symmetry and the coordinates of the vertex. Round your answers

to the nearest tenth, if necessary.

13.

14.

15.

16.

17.

18. y � 5x2 � 4x � 1

y � 4x2 � 8x � 1

y � 3x2 � 4x � 9

y � 6x2 � 12x � 5

y � �3x2 � 9x � 5

y � 2x2 � 4x � 3

� x � 1 � 0x2�2x2 � 3x � 16 � 0� 6 � 4xx2

� 6x � 9x2(x � 2)(x � 5) � 214 � 2x2 � x

� 7x � 13 � 2xx2� 9 � 4xx2� 3x � 15 � 0x2

� 11 � 0x2� 4x � 1 � 0x2� 10x � 3 � 0x2

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Algebra 2 Practice Workbook 31

Practice

5.5 The Quadratic Formula

NAME CLASS DATE

Find the discriminant, and determine the number of real

solutions. Then solve.

1. 2.

3. 4.

5. 6.

Perform the indicated addition or subtraction.

7. 8. 9.

10. 11. 12.

13. 14. 15.

Write the conjugate of each complex number.

16. 15i 17. 18.

Simplify.

19. 20. 21. 22.

23. 24. 25. 26.�2 � 3i�3 � 2i

5 � 4ii(6 � 3i)(2 � 2i)(2 � 5i)2

14 � 2i3 � i(3 � i)(9 � 3i)5 � i

5 � i4i(�7 � i)

�12 � 19i27 � 4i

(�7 � 13i) � (1 � 6i)(4 � 12i) � 7i(�7 � 2i) � (3 � 3i)

(12 � 16i) � (12 � 11i)(�8 � 4i) � (7 � i)(�8 � 4i) � (7 � i)

(11 � i) � (2 � 8i)(3 � i) � (�4 � 9i)(�6 � 12i) � (4 � i)

4x � 4x2 � 76x2 � 3x � 4 � 0

4x2 � 4 � x3x2 � 2x � 6 � 0

x2 � 3x � 9�2x2 � 5x � 3 � 0

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32 Practice Workbook Algebra 2

Practice

5.6 Quadratic Equations and Complex Numbers

NAME CLASS DATE

Solve a system of equations in order to find a quadratic function

that fits each set of data points exactly.

1. (�2, �20), (0, 2), (3, �25) 2. (1, 6), (2, 13), (�2, 21)

3. (4, 9), (6, 21), (�2, �3) 4. (0, �3), (�1, 0), (1, 4)

5. (�2, 29), (2, 17), (1, 2) 6. (3, 0), (�1, �12), (2, 3)

7. (0, �2), (4, �38), (�2, �20) 8. (�3, 1), (�2, �5), (�1, �7)

9. (4, 24), (6, 52), (8, 92) 10. , (2, �12), (3, –16)

11. (4, 21), 12. (�2, 11), (�1, �3), (4, 77)

A baseball player throws a ball. The table shows the

height, y, of the ball x seconds after it is thrown.

13. Find a quadratic function to model the data.

14. What was the maximum height reached by the ball?

15. How long did it take the ball to reach its maximum height?

16. Use your model to predict the height of the ball 1.25 seconds after it was thrown.

17. Use your model to determine how many seconds it took for the ball to hit the ground.

(3, 1312 ), (�1, 31

2 )

(12, �21

4 )

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Algebra 2 Practice Workbook 33

Practice

5.7 Curve Fitting with Quadratic Models

NAME CLASS DATE

Time (seconds) Height (feet)

0.25 7

0.5 9

1 7

Solve each inequality. Graph the solution on a number line.

Round irrational numbers to the nearest hundredth.

1.

2.

3.

4.

5.

6.

Graph each inequality and shade the solution region.

7. 8. 9.

10. 11. 12. y � 2x2 � x � 1y � �(x � 3)2 � 3y � x2 � 4x � 5

y � �(x � 1)2 � 1y � (x � 3)2 � 2y � x2 �12x

� 10x � 3 � 7xx2

� 3x � 1 � 0x2

� 2x � 4 � 0x2

� 7x � 10 � 0x2

� 2x � 8 � 0x2

� 16 � 0x2

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34 Practice Workbook Algebra 2

Practice

5.8 Solving Quadratic Inequalities

NAME CLASS DATE

–6 –4 –2 0 2 4 6

–6 –4 –2 0 2 4 6

–6–8 –4 –2 0 2 4

–4 –3 –2 –1 0 1 2 3

–2 –1 210 3 4 5

–5 –4 –3 –2 –1 0 1 2

Find the multiplier for each rate of exponential growth or decay.

1. 1% growth 2. 1% decay

3. 7% decay 4. 12% growth

5. 10% growth 6. 3% decay

7. 5.2% decay 8. 7.5% growth

9. 0.4% growth 10. 5.9% decay

Evaluate each expression to the nearest thousandth for the given

value of x.

11. for 0.5 12. for

13. for 14. for

15. for 16. for

17. for 18. for

19. for 20. for

Predict the result in each situation.

21. The population of a city in 1990 was 1,215,112. The population wasgrowing at a rate of about 5% per decade. Predict the population ofthe city

a. in the year 2000. b. in the year 2005.

22. The initial population of bacteria in a lab test is 400. The number ofbacteria doubles every 30 minutes. Predict the bacteria populationat the end of

a. two hours. b. three hours.

x � 6.512(2)x�2x � 0.1512(2)3x

x � 366( 12 )x

x � 215( 12 )2x�1

x � 1.7520 � 22xx �3442 � 2x�1

x � �27(0.5)xx � 2( 12 )3x

x �2310(2x)x �2x

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Algebra 2 Practice Workbook 35

Practice

6.1 Exponential Growth and Decay

NAME CLASS DATE

Identify each function as linear, quadratic or exponential.

1. f(x) 2. g(x) 3. k(x)

4. g(x) 5. w(x) 6. h(x)

7. b(x) 8. f(x) 9. h(x)

Tell whether each function represents exponential growth or decay.

10. f(x) 11. b(x) 12. k(x)

13. m(x) 14. w(x) 15. z(x)

16. h(x) 17. g(x) 18. a(x)

Find the final amount for each investment.

19. $1300 earning 5% interest compounded annually for 10 years

20. $850 earning 4% interest compounded annually for 6 years

21. $720 earning 6.2% interest compounded semiannually for 5 years

22. $1100 earning 5.5% interest compounded semiannually for 2 years

23. $300 earning 4.5% interest compounded quarterly for 3 years

24. $1000 earning 6.5% interest compounded quarterly for 4 years

25. $5000 earning 6.3% interest compounded daily for 1 year

26. $2000 earning 5.5% interest compounded daily for 3 years

� 150(1.1)x� 0.8(3.2)x� 2.5(0.8)x

� 47(0.55)x� 0.72 � 2x� 51(4.3)x

� 22(0.15)x� 13(0.7)x� 5.9(2.6)x

� 450(0.3)�x� ( 23 )3x

� x(x � 4) � (4 � x2)

� 0.42x� x2 � 11� 2x � 11

� 2x � 11� 5x � 42� (x � 1)2 � x

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36 Practice Workbook Algebra 2

Practice

6.2 Exponential Functions

NAME CLASS DATE

Write each equation in logarithmic form.

1. 2. 3.

4. 5. 6.

Write each equation in exponential form.

7. log 8. log 9. log

10. log 11. log 12. log

Solve each equation for x. Round your answers to the nearest hundredth.

13. 10 35 14. 10 91 15. 10 0.2

16. 10 1.8 17. 10 0.08 18. 10 1055

Find the value of v in each equation.

19. log 20. log 21. log

22. 8 log 23. log 24. log

25. log 26. log 27. logv 1256 � �4v 729 � 6v 1

100�2 �

7 v�3 �5 v�4 �2 v�

12 144v �15 225v �10 1000v �

x �x �x �

x �x �x �

625 � �415

11 114,641 � �43600 60 �

12

21 9261 � 35 15,625 � 612 144 � 2

11�3 �1

1331( 37 )3

�27

343( 34 )�3

� 64

337513 � 15203 � 8000192 � 361

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Algebra 2 Practice Workbook 37

Practice

6.3 Logarithmic Functions

NAME CLASS DATE

Write each expression as a sum or a difference of logarithms.

Then simplify, if possible.

1. log 2. log

3. log 4. log

5. log 6. log

Write each expression as a single logarithm. Then simplify, if

possible.

7. log3 5 log3 6 8. log5 x log5 2 9. log8 2 log8 32

10. log9 5 log9 y log9 4 11. 2 log12 6 log12 4 12. log3 81 log3 15

13. logb m logb 2 logb x 14. 3 logb x (logb 4 logb x) 15. 3 logb z logb y – 4 logb z

Evaluate each expression.

16. 5log5 12 17. 12log12 73 18. log3 32.5

19. log2 24.7 20. log4 43 log3 81 21. 9log 9 15 log3 35

Solve for x and check your answers.

22. log2 log2 23. 2 log3 log3 4

24. log5 log5 25. log7 log7 8

26. log8 log8 27. 2 log2 log2

28. logb 8 logb logb 29. 2 logb logb(�x � 11)(x � 1) �(x � 2)x ��

(3x � 16)(x � 2) �(2x � 6)(x2 � 3x) �

(x2 � 1) �(x � 1)(4x � 3) �

x �(3x � 14)(10x) �

��

�����

�12���

���

9 3a78 64

y

3 15q7(5 � 3 � 4)

5 722510 (4 � 100)

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38 Practice Workbook Algebra 2

Practice

6.4 Properties of Logarithmic Functions

NAME CLASS DATE

Solve each equation. Round your answers to the nearest hundredth.

1. 5 16 2. 6 5.5 3. 2 100

4. 8 12 5. 3 22 6. 9 0.35

7. 5.5 6 8. 7 0.8 9. 3 0.2

10. 12 18 11. 4.22 61 12. 8.2 55

13. 14. 14 33.8 15. 35

Evaluate each logarithmic expression to the nearest hundredth.

16. log7 30.6 17. log3 11 18. log2 13

19. log5 0.4 20. log4 83 21. log9 2.4

22. log6 8 23. log2 8.5 24. log4 6.1

25. log3 0.6 26. log8 0.32 27. log5 10

28. 3 log 20 29. log 30. 1 log7 25 �9 � 514

12

�

722x �x �( 12 )�x

� 17

x�1 �2x �x �

�x �x �x �

x �x �x �

x �x �x �

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Algebra 2 Practice Workbook 39

Practice

6.5 Applications of Common Logarithms

NAME CLASS DATE

Evaluate each expression to the nearest thousandth.

1. e8 2. e2.5 3. e5.2

4. 2e4 5. ln 35 6. ln 12.6

7. ln( ) 8. ln 9. ln 112

Write an equivalent exponential or logarithmic equation.

10. ex � 55 11. ln 44 � 3.78 12. � 0.05

13. ln 10 � 2.30 14. e4 � 54.6 15. ln 125 � 4.83

16. e5 � 148 17. ln 1 � 0 18. � 0.45

Solve each equation for x by using the natural logarithm function.

Round your answers to the nearest hundredth.

19. 33 74 20. 15 19.5 21. 4.8 30

22. 0.7 22 23. 1.5 70 24.

25. 15 24 26. 0.25 41 27. 44 19

28. $1000 is deposited in an account with an interest rate of 6.5%.Interest is compounded continuously, and no deposits or withdrawals are made. Find the amount in the account at the end of three years.

x �2x ��x �

423

x� 0.5x �x �

x �x �x �

e�0.8

e�3

�12�1.4

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40 Practice Workbook Algebra 2

Practice

6.6 The Natural Base, e

NAME CLASS DATE

Solve each equation for x. Write the exact solution and the

approximate solution to the nearest hundredth, when

appropriate.

1. 2.

3. log 4. log

5. log 6.

7. 8. ln ln 3

9. 10. log

11. 3 ln ln 16 ln 4 12. ln ln ln 3

13. log 14. log

15. ln 16.

In Exercises 17 and 18, use the equation log .

17. On January 17, 1994, an earthquake with a magnitude of 6.6 injuredmore than 8000 people and caused an estimated $13–20 billion ofdamage to the San Fernando Valley in California. Find the amount of energy released by the earthquake.

18. On January 17, 1995, an earthquake struck Osaka, Kyoto, and Kobe,Japan, injuring more than 36,000 people and causing an estimated$100 billion of damage. The quake released about ergsof energy. Find the earthquake’s magnitude on the Richter scale.Round your answer to the nearest tenth.

3.98 � 1022

E

1011.8M �23

� 8.25(1 � ex3 )2x � 3

116 � x1

8x 19 � �2

(x � 2) �2x ��x �

3(2x � 1) � 210x � 4 � 32

(2x � 7) �e3x � 15

9x � 6x 1325 �

x � 34 x �12

53x � 257x � 74

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Algebra 2 Practice Workbook 41

Practice

6.7 Solving Equations and Modeling

NAME CLASS DATE

Determine whether each expression is a polynomial. If so, classify

the polynomial by degree and by number of terms.

1. 2. 3.

4. 5. 6.

Evaluate each polynomial expression for the indicated value of x.

7. 8.

9. 10.

11. 12.

Write each sum or difference as a polynomial in standard form.

13. 14.

15. 16.

Sketch the graph of each function. Describe the general shape of

the graph.

17. 18. 19.

xO

y

2–2–4 4

2

–2

–4

4x

O

y

2–4 –2 4

2

–6

–4

–2x

O

y

2–2–4 4

2

–2

–4

4

6

f(x) �34x3 � 2x2 � 1k(x) � 4x4 � 4x3 � 6x2a(x) � �2x4 � 5x3 � 2

(7.1x3 � 3.2x2 � 7x � 8) � (9x2 � 2x3 � 18)(8.8x � 2 � 3x2 � x4) � (5x3 � 10x � 7x2)

(�7x4 � 24x5 � 3x2 � 9) � (2x5 � 6x4 � x � 1)(3x4 � 12x3 � 2x2) � (5x4 � x3 � 7x2)

0.75x3 � 15x2 � 10x, x � 47x2 � 19x, x � 5

0.5x3 � 0.6x2 � 3x, x � 10x5 � x4 � x3 � x2 � 1, x � 2

�x4 � 3x3 � 2x2 � 4, x � �12x3 � 3x2 � 4x, x � �2

43x�6 � 9x�7 � 12x�13�x � 12�x�2x3 � 4x2 � 15x � 7

7x 2 �

13x 3

x2 �

x 2

2 � 135x2 � 22x5 � 17x

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42 Practice Workbook Algebra 2

Practice

7.1 An Introduction to Polynomials

NAME CLASS DATE

Graph each function and approximate any local maxima or

minima to the nearest tenth.

1. 2.

3. 4.

Graph each function. Find any local maxima or minima to the

nearest tenth. Find the intervals over which the function is

increasing and decreasing.

5.

6.

7.

8.

Describe the end behavior of each function.

9. 10.

11. 12.

13. Factory sales of passenger cars, in thousands, in the United Statesare shown in the table below. Find a quartic regression model forthe data by using for 1990. (Source: Bureau of the Census)x � 0

1.1x4 � 2.2x3 � 3.3x2 � 45x3 � 6x4 � x2 � 1

3.3x3 � 2x2 � 5x � 112 � 4.2x3 � x2

P(x) � �x4 � 2.5x3 � x2 � 1, �4 � x � 4

P(x) � x3 � 1.2x2 � 2, �5 � x � 5

P(x) � 0.3x4 � x3 � x, �4 � x � 4

P(x) � 4x3 � 3x2 � 2, �6 � x � 6

P(x) � x4 � x3 � 4x2 � 2x � 2P(x) � 2x3 � 2x2 � 1

P(x) � 6 � x � 3x2P(x) � x2 � 3x � 4

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Algebra 2 Practice Workbook 43

Practice

7.2 Polynomial Functions and Their Graphs

NAME CLASS DATE

1990 1991 1992 1993 1994 1995

6050 5407 5685 5969 6549 6310

Write each product as a polynomial in standard form.

1. 2. 3.

4. 5. 6.

Use substitution to determine whether the given linear

expression is a factor of the given polynomial.

7. 8.

9. 10.

11. 12.

Divide by using long division.

13. 14.

15. 16.

Divide by using synthetic division.

17. 18.

19. 20.

For each function below, use synthetic division and substitution

to find the indicated value.

21. ; P(2) 22. ; P(3)

23. ; P(2) 24. ; P(�2) 3x4 � 4x3 � x2 � 1P(x) �2x4 � 3x3 � 2x2 � 6P(x) �

x3 � 2x � 4P(x) �x2 � 3x � 1P(x) �

(x5 � x3 � 3) � (x2 � 3)(x3 � 4x2 � 4x � 3) � (x2 � x � 1)

(x3 � 5x2 � 20x � 32) � (x � 8)(x3 � x2 � x � 21) � (x2 � 2x � 7)

(5x3 � x2 � x � 3) � (x � 1)(8x3 � 12x2 � 6x � 5) � (2x � 1)

(6x2 � 2x � 5) � (3x � 5)(2x2 � 7x � 30) � (x � 6)

3x3 � 2x2 � 6x � 2; x � 22x3 � 10x2 � 28x; x � 7

2x3 � 11x2 � 8x � 15; x � 5x3 � 9x � 1; x � 3

3x2 � x � 4; x � 1x2 � 2x � 12; x � 4

(3x � 1)3(2x � 5)(x � 1)2(x � 2)(x � 8)(x � 1)

(x � 4)(5x2 � 3x � 7)(x � 10)(2x � 3)0.5x(16x4 � 10x3 � 6x)

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44 Practice Workbook Algebra 2

Practice

7.3 Products and Factors of Polynomials

NAME CLASS DATE

Use factoring to solve each equation.

1. 2. 3.

4. 5. 6.

Use graphing, synthetic division, and factoring to find all of the

roots of each equation.

7. 8. 9.

10. 11. 12.

Use variable substitution and factoring to find all of the roots of

each equation. If necessary, leave your answers in radical form.

13. 14. 15.

16. 17. 18.

Use a graph and the Location Principle to find the real zeros of

each function. Give approximate values to the nearest tenth,

if necessary.

19. 20.

21. 22.

23. 24. 0.5x4 � 2x3 � 5x � 1P(x) �8x3 � 6x2 � 2x � 1P(x) �

12x3 � 15x2 � x � 1P(x) �2.5x4 � 2x2P(x) �

1.5x3 � 2x2 � 0.25P(x) �2x3 � 4x � 1P(x) �

x4 � 20 � 12x2x4 � 17x2 � 16 � 0x4 � 7x2 � �10

x4 � 54 � 15x2x4 � 10x2 � 21 � 0x4 � 10x2 � 24 � 0

x3 � 64 � 4x2 � 16xx3 � 11x2 � 24x � 36 � 0x3 � 29x � 42 � 12x2

3x2 � 2x2 � 37x � 12x3 � 4x2 � x � 6x3 � 3x2 � 4x � 12 � 0

3x3 � x � 4x22x3 � 2x2 � 24x � 0x3 � 2x2 � 15x

2x3 � x2 � x � 0x3 � 11x2 � 10x � 0x3 � 81x � 0

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Algebra 2 Practice Workbook 45

Practice

7.4 Solving Polynomial Equations

NAME CLASS DATE

Find all the rational roots of each polynomial equation.

1. 2.

3. 4.

5. 6.

Find all the zeros of each polynomial function.

7. 8.

9. 10.

11. 12.

Find all real values of x for which the functions are equal. Round

your answers to the nearest hundredth.

13. , 14. ,

15. , 16. ,

Write a polynomial function, P, in standard form by using the

given information.

17. The zeros of P(x) are �3, 2, and 4, and .

18. , and two of the three zeros are �3 and 4i.P(0) � �96

P(0) � 120

x2 � 2x � 1Q(x) �x4 � x3 � 2x � 1P(x) �4x � 1Q(x) �x3 � 2x � 1P(x) �

Q(x) � � x3 � 5x � 3x3 � 5x � 3P(x) �x � 1Q(x) �x4 � 5P(x) �

x4 � 5x3 � 5x2 � 5x � 4P(x) �x3 � 2x2 � 16P(x) �

x3 � 5x2 � 9x � 45P(x) �x3 � 3x2 � 12x � 36P(x) �

x3 � 3x2 � 4x � 12P(x) �x3 � 5x2 � 2x � 10P(x) �

18x4 � 9x3 � 17x2 � 4x � 4P(x) �2x4 � 5x3 � 12x2 � x � 4P(x) �

4x3 � 11x2 � 5x � 2P(x) �6x3 � x2 � 4x � 1P(x) �

3x3 � x2 � 12x � 4P(x) �5x2 � 6x � 1P(x) �

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46 Practice Workbook Algebra 2

Practice

7.5 Zeros of Polynomial Functions

NAME CLASS DATE

For Exercises 1–4, y varies inversely as x. Write the appropriate

inverse-variation equation, and find y for the given values of x.

1. 12 when 7; 5, 10, 16, and 20 2. 0.4 when 2; 0.1, 5, 8, and 20

3. when 45; 6, 15, 20, and 60 4. 12 when 0.4; 0.5, 6, 10, and 16

For Exercises 5–8, y varies jointly as x and z. Write the appropriate

joint-variation equation, and find y for the given values of x and z.

5. 16 when 4 and 0.5; 6. 120 when 2.5 and 2;2 and 0.25 3 and 2

7. 12 when 4 and 5; 8. 192 when 2 and 3 6 and 3 0.6 and 5

For Exercises 9–12, z varies jointly as x and y and inversely as w.

Write the appropriate combined-variation equation, and find z for

the given values of x, y, and w.

9. 320 when 4, 10, and 2.5; 10. 3.2 when 0.2, 8, and 4;5, 6, and 8 3, 6, and 16

11. 3.75 when 6, 12, and 48; 12. 4.8 when 0.2, 10, and 5;

0.05, 40, and 0.5 , 5, and 8

13. The apothem of a regular polygon is the perpendicular distance fromthe center of the polygon to a side. The area, A, of a regular polygonvaries jointly as the apothem, a, and the perimeter, p. A regulartriangle with an apothem of 3 inches and a perimeter of 31.2 incheshas an area of 46.8 square inches. Find the constant of variation andwrite a joint-variation equation. Then find the area of a regulartriangle with an apothem of 2.3 inches and a perimeter of 12 inches.

w �y �213x �w �y �x �

w �y �x �z �w �y �x �z �

w �y �x �w �y �x �w �y �x �z �w �y �x �z �

z �x �z �x �z �x �y �z �x �y �

z �x �z �x �z �x �y �z �x �y �

x �x �y �x �x �313y �

x �x �y �x �x �y �

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Algebra 2 Practice Workbook 47

Practice

8.1 Inverse, Joint, and Combined Variation

NAME CLASS DATE

Determine whether each function below is a rational function.

If so, find the domain. If the function is not rational, state why

not.

1. 2. 3.

Identify all vertical and horizontal asymptotes of the graph of

each rational function.

4.

5.

6.

Find the domain of each rational function. Identify all asymptotes

and holes in the graph of each rational function.

7.

8.

9.

Sketch the graph of each rational function. Identify all

asymptotes and holes in the graph of the function.

10. 11. 12. b(x) �x � x 2

x 2 � 1f(x) �x � 2

2x 2 � 3x � 2a(x) �3x

x � 4

n(x) �3x 2 � 12x

x 2 � 7x � 12

g(x) �x � 1

x 2 � 4x � 5

h(x) �4x � 3

x 2 � 6x

m(x) �3x � 8x 2 � 7

p(x) �2x 2 � 3

(x � 1.5)2

k(x) �2x � 1x � 9

w(x) �12 � 2xx 2 � 1h(x) �

x � 2|x| � 2f(x) �

x 3 � 5x � 7x2 � 3

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48 Practice Workbook Algebra 2

Practice

8.2 Rational Functions and Their Graphs

NAME CLASS DATE

Simplify each rational expression.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

x 2 � y2

5x3y2

4x � 4y15x2y5

x 2 � 493x � 12

x 2 � 14x � 49x � 5

x � 4x � 7 �

�x 2 � 9

x 2 � 3x

x 2 � 5x � 6x 2 � 8x � 15

x � 2x � 5

2x � 1x � 6x � 2x � 2

x � 6x � 2 �

x4 � 813x2 � 27

x 2 � x � 12x

x 2 � 10x � 11x 2 � 6x � 5

x 2 � 9x � 22x 2 � 3x � 10

x � 5x 2 � 100x 2 � 25x � 10

x2 � 16x � 3x � 4x2 � 9

x3 � 9xx 2 � 11x � 24 �

x 2 � 7x � 8x 2 � 4x � 3

4x � 8x 2 � x � 6 �

x 3 � x 2 � 6xx 2 � 9

2x � 35x � 1 �

6x 2 � 13x � 615x 2 � 7x � 2

x 2 � 5x � 6x � 4 �

3x � 12x � 2

4x4

9x �9x3

10x �15x 2

2xx 2 � 7x � 12x 2 � x � 6

3xx10 �

x3

27 �9x4

29x 2 � 12x � 4

9x 2 � 4

x 2 � 8x � 7x 2 � 6x � 7

2x 4

x5 �6xx3 �

x4

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Algebra 2 Practice Workbook 49

Practice

8.3 Multiplying and Dividing Rational Expressions

NAME CLASS DATE

Simplify.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

Write each expression as a single rational expression in simplest form.

16. 17.

18. 19.

20. 21.

22. 23.5

xy � 3y � 2x � 6 �4

x � 3 �2

2 � y2x � 3

3x 2 � 13x � 10 �2x � 15 � x �

13x � 2

�2x 2 � 5xx 2 � 7x �

x � 2x � 7 �

2x � 3x

5xx 2 � 9 �

4x � 3 �

23 � x

x � 1x � 2 �

x � 2x � 4 �

16 � 5xx 2 � 6x � 8

xx � 5 �

x 2 � 2525 � x 2 �

5x � 5

2x � 3x 2 � 9 �

2x � 3x 2 � 9 �

1x � 9

5x � 2x 2 � 49 �

x � 15x 2 � 49 �

3x � 4x 2 � 49

x 2 � 94x 2 � 1�

x � 32x � 12x � 1x � 3

�1

x � 1

xx � 22x2

2 � x

�x 2 � 1

x

103x � 1

5x3x � 1

�3

2x � 1

xx � 6

2x � 1x � 6

�7

x � 2

12x � 2

3x � 2

x � 14x � 3

2

x � 5x 2 � 10x � 25 �

2xx 2 � 25

x � 2x � 8 �

x � 2x 2 � 6x � 16

x � 42x 2 � 2x �

52x � 2

4xx 2 � 16 �

4x � 4

x � 2x � 3 �

x � 3x � 2

x � 2x 2 � 4 �

23x � 6

3x � 43x �

2x � 12x

5x � 4x3 � 1 �

2x � 3x3 � 1

x � 73 �

x � 24

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50 Practice Workbook Algebra 2

Practice

8.4 Adding and Subtracting Rational Expressions

NAME CLASS DATE

Solve each equation. Check your solution.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

Solve each inequality. Check your solution.

13. 14. 15.

16. 17. 18.

19. 20. 21.

Use a graphics calculator to solve each rational inequality.

Round answers to the nearest tenth.

22. 23. 24.

25. 26. 27.2x � 3

x �3

x � 2 � 5x � 4x �

xx � 4 � 1x � 1

x � 2 �1

x � 3

x � 3x � 4 � xx � 2

x � 4 � 1 � x2x � 3 � x � 3

x � 1x � 1 �

xx � 1 �

2x 2 � 1

xx � 3 �

1x � 4 � 1x

x � 1 �2

x � 1 � 1

x � 1x � 2 �

xx � 3 �

7x 2 � 5x � 6

x � 1x � 1 � 2x

x � 3 �4

x � 2

xx � 1 �

xx � 1

xx � 6 � 2x

x � 2 � 2

3x � 2 �

5x � 2 �

4x 2

x 2 � 4x � 3x � 2 �

14x � 2 �

3x � 2x 2 � 4

3x � 1 � 4 �

11 � x 2

xx � 2 �

x � 55 �

x � 25

34 �

1x �

12x

x � 2x � 1 �

2x � 3x

x � 102x � 1 �

4x3x � 4

x 2 � 1x � 2 � 3x � 1x � 8

x � 5 �x � 1

2x � 10

x � 15x � 5 �

x � 12x

x � 5x � 8 �

x � 1x � 5

2x � 14x � 4 �

45

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Algebra 2 Practice Workbook 51

Practice

8.5 Solving Rational Equations and Inequalities

NAME CLASS DATE

Find the domain of each radical function.

1. 2. 3.

4. 5. 6.

Find the inverse of each quadratic function. Then graph the

function and its inverse in the same coordinate plane.

7. 8. 9.

Evaluate each expression. Give exact answers.

10. 11. 12.

13. 14. 15.

16. The volume of a sphere with diameter d is given by the equation

. Solve this equation for d in terms of V. Then use your

equation to find the diameter, to the nearest foot, of a sphere with avolume of 1000 cubic feet.

V �16 πd3

23

3��27�83

��184 3��216

153

��8

12534

4�10,0003

�1923

y � x2 � 2x � 5y � 2 � x2y � x2 � 6x � 8

f(x) � �x2 � 4x � 3f(x) � �x2 � 10x � 25f(x) � �4x2 � 25

f(x) � �x2 � 36f(x) � �7(x � 4)f(x) � �12x � 30

52 Practice Workbook Algebra 2

Practice

8.6 Radical Expressions and Radical Functions

NAME CLASS DATE

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Simplify each radical expression by using the Properties of

nth Roots.

1. 2. 3.

4. 5. 6.

Simplify each product or quotient. Assume that the value of each

variable is positive.

7. 8. 9.

10. 11. 12.

Find each sum, difference, or product. Give your answer in

simplest radical form.

13. 14. 15.

16. 17. 18.

Write each expression with a rational denominator and in

simplest form.

19. 20. 21.

22. 23. 24.9

�7 � �2

5

�2 � �3

8

2 � �3

7

�2 � 1

�64

�2

4

�8

8�2(�8 � 3�2 � 7�32)(4 � 7�5)(3 � 2�5)(4 � 3�2)(3 � 6�2)

(11 � 3�18 ) � (6 � 4�8 )(5 � 7�3 ) � (2 � 3�12)(16 � 3�2 ) � (9 � �2)

4�64x10y10

4�2x

3�96x 2y 5z4

3�4yx

5�64x3y7z3

5�2xy

�2x3y � �5x3y3 � �10x2y3

�16x2y5 �3

�4x2y5

�16x6 �5

�2x4

(�16x3y4)13(80x5)

123

�343x5y 9z

�288 x2y44�81

5�32

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Algebra 2 Practice Workbook 53

Practice

8.7 Simplifying Radical Expressions

NAME CLASS DATE

Solve each radical equation by using algebra. If the equation has

no solution, write no solution. Check your solution.

1. 2. 3.

4. 5. 6.

7. 8. 9.

Solve each radical inequality by using algebra. If the inequality

has no solution, write no solution. Check your solution.

10. 11. 12.

13. 14. 15.

16. 17. 18.

Solve each radical equation or inequality by using a graph. Round

solutions to the nearest tenth. Check your solution by any

method.

19. 20. 21.

22. 23. 24.

25. 26. 27. �7x � 1 � x � 2�x3 � 1 � x � 3�x � 9 � x2 � 3x

�x2 � 2 � �7x � 33

�x � 7 � �2x � 3�2x � 3 � x � 2

�x � 4 � x3�3 � x � x2 � 1�3x � 4 � x � 2

�4x � 3 � 7�3 � 2x � 4�x � 5 � �x � 3

�x � 4 � x � 104

�x � 3 � �x � 1�x2 � 2x � 1 � 1.5

�3x � 5�x � 2 � 4�x � 3 � 3

3�x � 4 �

3�3x � 6�3x � 5 � x � 13�2x � 3 � �x � 7

2�x � 2 � x � 2�2x � 3 � x � 1�x � 2 � �x

�x2 � 4 � 2�3�x � 6 � 2�x � 5 � 10

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54 Practice Workbook Algebra 2

Practice

8.8 Solving Radical Equations and Inequalities

NAME CLASS DATE

Solve each equation for y, graph the resulting equation, and

identify the conic section.

1. 2. 3.

Find the distance between P and Q, and find the coordinates of

M, the midpoint of . Give exact answers and approximate

answers to the nearest hundredth when appropriate.

4. P(0, 0) and Q(5, 12) 5. P(4, 1) and Q(12, �5) 6. P(12, 4) and Q(�8, 2)

7. P(7.5, 3) and Q(�1.5, 5) 8. P(�8, �8) and Q(4, 4) 9. P(�1, �1) and Q(1, 2)

Find the center, circumference, and area of the circle whose

diameter has the given endpoints.

10. P(6, 20) and Q(12, 8) 11. P(0, 0) and Q(9, 40) 12. P(4, 16) and Q(�4, 1)

13. P(3, 7) and Q(4, �5) 14. P(10, 5) and Q(20, 6) 15. P(�8, 8) and Q(13, �3)

PQ

9x2 � y2 � 9x2 � y2 � 400x2 � 3y � 0

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Algebra 2 Practice Workbook 55

Practice

9.1 Introduction to Conic Sections

NAME CLASS DATE

Write the standard equation for each parabola graphed below.

1. 2. 3.

Graph each equation. Label the vertex, focus, and directrix.

4. 5. 6.

Write the standard equation for the parabola with the given

characteristics.

7. vertex: (0, 0); focus: (0, 6)

8. vertex: (10 , 0); directrix: x � 8

9. focus: (3, 0); directrix: x � �3

10. vertex: (5, 2); directrix: y � 1

11. vertex: (6, �7); focus: (4, �7)

12. focus: (9, 5); directrix: y � �5

x � 1 �18(y � 2)2y � 4 �

14(x � 1)2x �

14 y2

y =

–2

2

V(3, –1)

4

6

directrix

O

y

x

–54

2–2

2

4

–4

–2

4 6O

y

xFV2

2

4

–4

–2

4

F

6

directrix

8O

y

x

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56 Practice Workbook Algebra 2

Practice

9.2 Parabolas

NAME CLASS DATE

Write the standard equation for each circle graphed below.

1. 2. 3.

Write the standard equation of a circle with the given radius

and center.

4. 5. 6.

Graph each equation. Label the center and the radius.

7. 8. 9.

Write the standard equation for each circle. Then state the

coordinates of its center, and give its radius.

10. 11. x2 � y2 � 22x � 2y � �120x2 � y2 � 10x � 16y � 88 � 0

(x � 3)2 � (y � 3)2 � 9x2 � (y � 5)2 � 16x2 � y 2 � 256

r � 24; C(�3, �3)r � 2.5; C(�2, 1)r �34; C(0, 0)

xO

y

–2

–4

–6

–8

2–2

xO

y

2

–2

4

6

2 4 6 8x

O

y

2

–2

4

6

–4–6

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Algebra 2 Practice Workbook 57

Practice

9.3 Circles

NAME CLASS DATE

Write the standard equation for each ellipse.

1. 2. 3.

Sketch the graph of each ellipse. Label the center, foci, vertices, and co-vertices.

4. 5. 6.

Write the standard equation for the ellipse with the given characteristics.

7. vertices: (�25, 0) and (25, 0); co-vertices: (0, �15) and (0, 15)

8. foci: (�10, 0) and (10, 0); co-vertices: (0, �3), (0, 3)

9. co-vertices: (�20, 0) and (20, 0); foci: (0, �8) and (0, 8)

10. An ellipse is defined by . Write the standard equation, and identify

the coordinates of the center, vertices, co-vertices, and foci.

x2 � 4y2 � 6x � 27 � 0

(x � 4)2

9 �(y � 3)2

25 � 1x2

49 �(y � 1)2

36 � 1x2

4 �y 2

81 � 1

xO

y

–2 2 4

2

6

–2

C (3, 2)

xO

y

–2–4–6 2

2

–2

–4

C (–3,–2)x

O

y

–4 4

4

–4

–8

8

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58 Practice Workbook Algebra 2

Practice

9.4 Ellipses

NAME CLASS DATE

Write the standard equation for each hyperbola.

1. 2.

Graph each hyperbola. Label the center, vertices, co-vertices, foci,

and asymptotes.

3. 4.

For Exercises 5–7, write the standard equation for the hyperbola

with the given characteristics.

5. vertices: and ; co-vertices: and

6. foci: (�5, �2) and (5, �2); vertices: (�3, 0) and (3, 0)

7. center: (1, 1); vertices: (1, �4) and (1, 6); co-vertices: (13, 1) and (�11, 1)

8. A hyperbola is defined by . Write the standard equation,and identify the coordinates of the center, vertices, co-vertices, and foci.

x2 � 4y2 � 28x � 24y � 156 � 0

(0, �15 )(0, ��15 )(�10, 0)(��10, 0)

C (0, 0)

x

y

(x � 1)2

16 �( y � 1)2

9 � 1y 2

9 �x 2

25 � 1

x

y

–2–4–6 2 4 6

2

–2

x

y

–2–4–6 2 4 6

2

–2

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Algebra 2 Practice Workbook 59

Practice

9.5 Hyperbolas

NAME CLASS DATE

Use the substitution method to solve each system. If there are no

real solutions, write none.

1. 2. 3.

Use the elimination method to solve each system. If there are no

real solutions, write none.

4. 5. 6.

Solve each system by graphing. If there are no real solutions,

write none.

7. 8. 9.

Classify the conic section defined by each equation. Write the

standard equation of the conic section, and sketch the graph.

10. 11. 4x2 � 9y2 � 40x � 72y � 80 � 0x2 � 14x � 4y � 61 � 0

�9x2 � 16y2 � 14416x2 � 9y2 � 144�25x2 � 4y2 � 100

4x2 � 9y2 � 36�x2 � 2y2 � 164x2 � y2 � 4

�x2 � 2y2 � 303x2 � 5y2 � 24�x2 � y2 � 8

2x2 � 3y2 � 1�x2 � y2 � 94x2 � 9y2 � 36

�y � xx2 � y2 � 16�y � x � 2

y � x2�y � x2 � 5y � 5x � 1

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60 Practice Workbook Algebra 2

Practice

9.6 Solving Nonlinear Systems

NAME CLASS DATE

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Algebra 2 Practice Workbook 61

Practice

10.1 Introduction to Probability

NAME CLASS DATE

Find the probability of each event.

1. A blue card is drawn at random from a bag containing 2 white cards,1 red card, and 7 blue cards.

2. Frederique, who arrives home at 6:42 P.M., is home to receive a call that can come at any time between 6:40 and 6:50.

3. A letter chosen at random from the letters of the word permutationis a vowel.

4. A card chosen at random from a standard 52-card deck is a heart or a diamond.

5. A card chosen at random from a standard deck is not an 8 or an ace.

6. A number cube is rolled, and a number greater than 3 and less than6 results.

7. A letter chosen at random from the alphabet is not one of the 5 standard vowels.

8. A point on a 12-inch ruler is chosen at random and is located within an inch of an end of the ruler.

A spinner is divided into three colored regions. You spin the

spinner a total of 150 times. The results are recorded in the

table. Find the experimental probability of each event.

9. green

10. yellow 11. pink

12. not pink 13. not yellow

Find the number of possible license plate numbers (with no

letters or digits excluded) for each of the following conditions:

14. 6 digits

15. 2 letters followed by 3 digits

16. 4 letters followed by 3 digits

17. 5 digits followed by 2 letters

18. 2 digits followed by 2 letters followed by 2 digits

green 42

yellow 65

pink 43

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62 Practice Workbook Algebra 2

Practice

10.2 Permutations

NAME CLASS DATE

Find the number of permutations of the first 7 letters of the

alphabet for each situation.

1. taking all 7 letters at a time 2. taking 5 letters at a time

3. taking 4 letters at a time 4. taking 3 letters at a time

In how many ways can 12 books be displayed on a shelf if the

given number of books are available?

5. 12 books 6. 14 books 7. 15 books 8. 20 books

Find the number of permutations of the letters in each word.

9. geometry 10. algebra 11. addition

12. calculus 13. mathematics 14. arithmetic

15. Lizette decorates windows for a department store. She plans todesign a baby’s room with a row of stuffed elephants and monkeysalong one wall. If she has 8 identical elephants and 10 identicalmonkeys, in how many different ways can the stuffed animals be displayed?

16. The 6 candidates for a student government office are invited to speak at an election forum. In how many different orders can they speak?

17. Representatives from 8 schools are represented at a schoolnewspaper workshop. In how many different ways can the 8 representatives be seated around a circular table?

18. Ten colleges are participating in a college fair. Booths will bepositioned along one wall of a high school gymnasium. In how many different orders can the booths be arranged?

Find the number of ways in which each committee can be

selected.

1. a committee of 5 people from a group of 8 people

2. a committee of 2 people from a group of 16 people

3. a committee of 4 people from a group of 7 people

4. a committee of 8 people from a group of 15 people

5. a committee of 3 people from a group of 9 people

At a luncheon, guests are offered a selection of 4 different grilled

vegetables and 5 different relishes. In how many ways can the

following items be chosen?

6. 2 vegetables and 3 relishes 7. 3 vegetables and 2 relishes

8. 4 vegetables and 4 relishes 9. 3 vegetables and 3 relishes

A bag contains 8 white marbles and 7 blue marbles. Find the

probability of selecting each combination.

10. 2 white and 3 blue 11. 3 white and 2 blue 12. 4 white and 1 blue

Determine whether each situation involves a permutation or a

combination.

13. A high school offers 5 foreign language programs. In how many ways can a student choose 2 programs?

14. In how many ways can 20 members be chosen from a 60-member chorus to sing the national anthem at a graduation ceremony?

15. In how many ways can a captain, co-captain, and team manager be chosen from among the 18 members of a volleyball team?

16. First- through fourth-place prizes are to be awarded in an essay contest. In how many ways can the winners be selected from among 125 entries?

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Algebra 2 Practice Workbook 63

Practice

10.3 Combinations

NAME CLASS DATE

A card is drawn at random from a standard 52-card deck. Tell whether

events A and B are inclusive or mutually exclusive. Then find P(A or B).

1. A: The card is a heart. 2. A: The card is a number less than 5.B: The card is an 8. B: The card is a jack, a king, or a queen.

3. A: The card is black. 4. A: The card is not a diamond.B: The card is a number greater than 4. B: The card is a spade.

5. A: The card is red. 6. A: The card is a 2 or a 3.B: The card is the ace of spades. B: The card is not a heart.

A spinner is divided into 8 congruent regions numbered 1 through 8.

The spinner is spun once. Find the probability of each event.

7. The number is even or divisible by 3. 8. The number is odd or greater than 7.

9. The number is less than 2 or greater than 6. 10. The number is odd or divisible by 4.

Two number cubes are rolled, and the

numbers on the top faces are added.

The table at right shows the possible

outcomes. Find each probability.

11. The sum is odd or greater than 11.

12. The sum is less than 6 or greater than 10. 13. The sum is even or less than 5.

14. The sum is less than 8 or a multiple of 6. 15. The sum is less than 4 or a multiple of 5.

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64 Practice Workbook Algebra 2

Practice

10.4 Using Addition with Probability

NAME CLASS DATE

+ 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Events D, E, F and G are independent, and P(D) � 0.2, P(E) � 0.1,

P(F) � 0.4, and P(G) � 0.25. Find the probability of each

combination of events.

1. P(D and E) 2. P(D and F)

3. P(E and F) 4. P(D and G)

5. P(D and E and F) 6. P(E and F and G)

A bag contains 3 white marbles, 2 red marbles, and 7 blue

marbles. A marble is picked at random and is replaced. Then

a second marble is picked at random. Find each probability.

7. Both marbles are blue.

8. The first marble is white and the second marble is red.

9. The first marble is white and the second marble is not white.

10. Neither marble is red.

11. The first marble is blue and the second marble is red.

A number cube is rolled twice. On each roll, the number on the

top face is recorded. Find the probability of each event.

12. The first number is greater than 5 and the second is less than 3.

13. Both numbers are greater than 4.

14. The first number is even and the second number is odd.

15. Both numbers are less than 2.

16. Neither number is greater than 4.

A number cube is rolled, and two coins are tossed. Find the

probability of each event.

17. The number on the cube is 2 and both coins are heads.

18. The number on the cube is even, one coin shows heads, and one shows tails.

19. The number on the cube is greater than 4 and both coins are tails.

20. The number on the cube is greater than 2 and the coins show different sides.

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Algebra 2 Practice Workbook 65

Practice

10.5 Independent Events

NAME CLASS DATE

Two number cubes are rolled, and the first cube shows 6. Find

the probability of each event below.

1. The sum is 9. 2. Both numbers are even.

3. The sum is greater than 8. 4. The sum is greater than 9 and less than 12.

A spinner that is divided into 8 congruent regions, numbered 1

through 8, is spun once. Let A be the event “even” and let B be

the event “6.” Find each of the following probabilities.

5. P(A) 6. P(B) 7. P(A and B)

8. P(A or B) 9. P(A|B) 10. P(B|A)

A spinner that is divided into 5 congruent regions, numbered 1

through 5, is spun once. Let A be the event “odd” and let B be

the event “less than 3.” Find each of the following probabilities.

11. P(A) 12. P(B) 13. P(A and B)

14. P(A or B) 15. P(A|B) 16. P(B|A)

Let A and B represent events.

17. Given P(A and B) 0.25 and P(A) 0.4, find P(B|A).

18. Given P(A and B) and P(A) , find P(B|A).

19. Given P(B|A) and P(A) , find P(A and B).

20. Given P(B|A) 0.4 and P(A) 0.16, find P(A and B).

21. Given P(B|A) 0.5 and P(A and B) 0.2, find P(A).

22. Given P(B|A) 0.8 and P(A and B) 0.45, find P(A).��

��

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�58�

45

�23�

35

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66 Practice Workbook Algebra 2

Practice

10.6 Dependent Events and Conditional Probability

NAME CLASS DATE

Use a simulation with 10 trials to find an estimate for each probability.

1. In 4 tosses of a coin, heads 2. In 5 tosses of a coin, tails 3. In 4 rolls of a number cube,appears exactly 3 times. appears more than 2 times. 3 appears twice.

Of 100 motorists observed at an intersection, 26 turned left, 47 went straight, and 27

turned right. Use a simulation with 10 trials to find an estimate for each probability.

4. Exactly 2 of every 4 5. At least 3 of every 4 6. Fewer than 2 of every 4motorists turn right. motorists go straight. motorists turn left.

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Algebra 2 Practice Workbook 67

Practice

10.7 Experimental Probability and Simulation

NAME CLASS DATE

Trial Result

1

2

3

4

5

6

7

8

9

10

estimate:

Trial Result

1

2

3

4

5

6

7

8

9

10

estimate:

Trial Result

1

2

3

4

5

6

7

8

9

10

estimate:

Trial Result

1

2

3

4

5

6

7

8

9

10

estimate:

Trial Result

1

2

3

4

5

6

7

8

9

10

estimate:

Trial Result

1

2

3

4

5

6

7

8

9

10

estimate:

Write the first six terms of each sequence.

1. 2.

3. 4.

5. 6.

For each sequence below, write a recursive formula, and find the

next three terms.

7. 1, 11, 121, 1331, . . . 8. 81, 78, 75, 72, . . . 9. 2, �6, 18, �54, . . .

10. , �1, 4, �16, . . . 11. 2, 11, 38, 119, . . . 12. �2, �14, �74, �374, . . .

Write the terms of each series. Then evaluate.

13. 14.

Evaluate.

15. 16. 17.

18. 19. 20. �12

n�1(3.5n2 � 5n � 2.2)�

6

a�1(4a2 � 3a � 5)�

10

k�1(2k � 0.5)2

�5

j�1( j � 3)2�

8

n�1(2n � 12)�

6

m�110m

�5

j�1( j 2 � 8j � 2)�

7

n�14.5n

14

a1 � �5; an � 3an�1a1 � 1; an � an�1 � 100

a1 � 20; an � 3an�1 � 10tn � n2 � 12

fn �12n �

12bn � 2.5n

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68 Practice Workbook Algebra 2

Practice

11.1 Sequences and Series

NAME CLASS DATE

Based on the terms given, state whether or not each sequence is

arithmetic. If so, identify the common difference, d.

1. 15,18, 21, 24, . . . 2. 2, 5, 10, 17, . . . 3. 7.2, 9.7, 12.2, 14.7, . . .

4. 4, 6, 9, 13.5, . . . 5. , . . . 6. 8, 5.7, 3.4, 1.1, . . .

Write an explicit formula for the nth term of each arithmetic sequence.

7. 16, 7, �2, �11, . . . 8. �15, �7, 1, 9, . . . 9. 13, 16, 19, 22, . . .

10. �25, �13, �1, 11, . . . 11. 9, 20, 31, 42, . . . 12. 8.6, 7.3, 6, 4.7, . . .

List the first four terms of each arithmetic sequence.

13. 14. 15.

16. 17. 18.

Find the indicated arithmetic means.

19. 3 arithmetic means between �12 and 16 20. 4 arithmetic means between 40 and 100

21. 2 arithmetic means between 50 and 86 22. 3 arithmetic means between 7 and 21

23. 3 arithmetic means between 40 and 16 24. 4 arithmetic means between �8 and 22

tn � �12n � 3tn � 0.5n � 8tn � 40n � 15

t1 � �20; tn � tn�1 � 8t1 � 7.5; tn � tn�1 � 2.5t1 � 50; tn � tn�1 � 100

1, 135, 21

5, 245

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Algebra 2 Practice Workbook 69

Practice

11.2 Arithmetic Sequences

NAME CLASS DATE

Use the formula for an arithmetic series to find each sum.

1. 2.

3. . . . 4. . . .

5. . . . 6. 33 � 38 � 43 � . . . � 123

Find each sum.

7. the sum of the first 225 natural numbers

8. the sum of the first 15 multiples of 3

9. the sum of the first 25 multiples of 4

10. the sum of the multiples of 5 from 75 to 315, inclusive

11. the sum of the multiples of 7 from 84 to 371, inclusive

For each arithmetic series, find S22

.

12. �6 � (�4) � (�2) � 0 � . . . 13. 3 � 7 � 11 � 15 � . . .

14. �24 � (�21)� (�18)� (�15) � . . . 15. � . . .

16. 18 � 8 � (�2) � (�12) � . . . 17. � . . .

Evaluate.

18. 19. 20.

21. 22. 23. �9

i�1(�8i � 1)�

15

b�1(13 � 5b)�

12

m�1(�7 � 4m)

�10

k�1(10k � 4)�

8

j�1(�3j � 3)�

6

n�1(2n � 7)

3�5 � 5�5 � 7�5 � 9�5

3 �334 �41

2 �514

� 6514 � 17 � 20 �

� 305110 � 125 � 140 �� 3�30 � 27 � 24 � 21 �

12 � 21

2 � 312 � 41

2 � 51262 � 66 � 70 � 74 � 78

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70 Practice Workbook Algebra 2

Practice

11.3 Arithmetic Series

NAME CLASS DATE

Determine whether each sequence is geometric. If so, identify the

common ratio, r, and give the next three terms.

1. 9, 25, 49, 81, . . . 2. 200, 80, 32, 12.8, . . . 3. , . . .

4. 12, 18, 27, 40.5, . . . 5. 54, 36, 24, 16, . . . 6. , . . .

List the indicated terms of each geometric sequence.

7. 8. ; 9. ;first 4 terms first 5 terms first 4 terms

10. 11. 12.

Write an explicit formula for the nth term of each geometric

sequence.

13. 250, 100, 40, 16, . . . 14. �30, 6, �1.2, 0.24, . . . 15. 40, 32, 25.6, 20.48, . . .

16. 2, 5, 12.5, 31.25, . . . 17. 20, 5, , . . . 18. 1.5, �9, 54, �324, . . .

19. Find 2 geometric means between 20. Find 2 geometric means between 7 and 875. �28 and �3.5.

21. Find 3 geometric means between 22. Find 3 geometric means between 12 and 3072. 12.5 and 25.92.

23. Find 3 geometric means between 24. Find 4 geometric means between 12 and 7500. 4 and 972.

114, 5

16

t3 � 2014; t5 � 1821

4; t7t2 � 16; t6 � 64; t5t2 � 40; t4 � 2.5; t5

t1 � 20; tn � 0.5tn�1t1 � �4; tn � 2.5tn�1t1 � 18; tn � �2tn�1;

1, �2, �3, 2

6623, �40, 24, �142

5

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Algebra 2 Practice Workbook 71

Practice

11.4 Geometric Sequences

NAME CLASS DATE

Find each sum. Round answers to the nearest tenth, if necessary.

1. S20 for the geometric series . . .

2. S15 for the geometric series 72 � 12 � 2 � � . . .

3. S6 for the series related to the geometric sequence 7, �14, 28, �56, . . .

4.

5.

For Exercises 6–9, refer to the geometric sequence 3, 6, 12, 24, . . .

6. Find t12. 7. Find t20. 8. Find S12. 9. Find S20.

Evaluate. Round answers to the nearest hundredth, if necessary.

10. 11. 12. 13.

14. 15. 16. 17.

Use mathematical induction to prove that the statement is true

for every natural number, n.

18. . . . � n3 �n 2(n � 1)2

413 � 23 � 33 �

�12

k�13(2)k�

10

k�15k�

10

t�13(�1)t�2�

6

p�12(3)p�1

�15

m�1

23(3m)�

12

j�15(0.25k)�

10

n�14.8n�1�

6

k�16(2k�1)

1.3 � 5.2 � 20.8 � 83.2 � 332.8 � 1331.2

75 �

725 �

7125 �

7625 �

73125

13

4 � 12 � 36 � 108 �

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72 Practice Workbook Algebra 2

Practice

11.5 Geometric Series and Mathematical Induction

NAME CLASS DATE

Find the sum of each infinite geometric series, if it exists.

1. . . . 2. . . .

3. . . . 4. . . .

Find the sum of each infinite geometric series, if it exists.

5. 6. 7.

8. 9. 10.

11. 12. 13.

Write each decimal as a fraction in simplest form.

14. 15. 16.

17. 18. 19.

Write an infinite geometric series that converges to the

given number.

20. 0.0707070707… 21. 0.9393939393…

22. 0.1515151515… 23. 0.358358358…

24. 0.011011011… 25. 0.445445445…

0.3700.2250.753

0.490.370.1

��

n�120(0.1)n�1�

�

b�149(0.02)b�1�

�

k�17.3k�1

��

x�10.92x�

�

t�10.45t�1�

�

j�10.75 j

��

k�111 � (1

9 )k�1

��

m�1(11

9 )m�1

��

n�10.8n

5 � 4 � 3.2 � 2.56 �78 �

712 �

718 �

727 �

45 �

415 �

445 �

4135 �60 � 84 � 117.6 � 164.64 �

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Algebra 2 Practice Workbook 73

Practice

11.6 Infinite Geometric Series

NAME CLASS DATE

State the location of each entry in Pascal’s triangle. Then give the

value of each expression.

1. 2. 3.

4. 5. 6.

Find the indicated entries in Pascal’s triangle.

7. fourth entry, 8. seventh entry, 9. ninth entry, 10. third entry,row 10 row 13 row 15 row 18

A fair coin is tossed the indicated number of times. Find the

probability of each event.

11. 5 tosses; exactly 3 heads

12. 6 tosses; no more than 3 heads

13. 10 tosses; exactly 1 head

14. 8 tosses; fewer than 5 heads

15. 5 tosses; no fewer than 3 heads

16. 7 tosses; 2 or 3 heads

A student guesses the answers to 6 questions on a true-false

quiz. Find the probability that the indicated number of guesses

are correct.

17. exactly 4 18. fewer than 5 19. no more than 2

20. exactly 5 21. at least 3 22. at least 4

12C513C1010C5

8C66C37C5

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74 Practice Workbook Algebra 2

Practice

11.7 Pascal’s Triangle

NAME CLASS DATE

Expand each binomial raised to a power.

1.

2.

For Exercises 3–5, refer to the expansion of .

3. How many terms are in the expansion?

4. What is the exponent of r in the term that contains ?

5. Write the term that contains .

Expand each binomial.

6.

7.

8.

9.

Ty Cobb was the American League batting champion for 12 of his

24 years in baseball. His lifetime batting average was 0.367.

Use the Binomial Theorem to find each theoretical probability for

Ty Cobb.

10. exactly 3 hits in 5 at bats

11. at least 3 hits in 5 at bats

12. no more than 2 hits in 5 at bats

13. exactly 4 hits in 6 at bats

14. fewer than 3 hits in 6 at bats

(2m � 3q)6

(34a � d)5

(12w � 2z)4

(5x � y)5

r5

s12

(r � s)15

(b � w)6

(s � t)5

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Algebra 2 Practice Workbook 75

Practice

11.8 The Binomial Theorem

NAME CLASS DATE

Find the mean, median, and mode of each data set. Round

answers to the nearest thousandth, when necessary.

1. 62, 54, 63, 92, 62, 79, 54, 62

2. 12, 28, 40, 22, 33, 28, 9

3. 3.6, 6.3, 1.3, 3.6, 1.0, 5.9

4. 277, 725, 920, 835, 255, 725

5. 1828, 1008, 1600, 7309, 2215, 1600

Find the mean, median, and mode of the data, and compare them.

6. percent of total social welfare expenditures for educationin 1985–1992: 22.8, 23.2, 24.5, 24.8, 25.0, 25.0, 24.0, 23.1

Make a frequency table for the data, and find the mean.

7. ages (in years) of members of the swim team:14, 15, 17, 17, 18, 16, 15, 14, 16, 17, 17, 18, 17,16, 16, 15, 14, 17

mean:

Make a grouped frequency table for the data, and estimate the mean.

8. number of books read by the members of a class in the past year:5, 4, 12, 22, 30, 5, 7, 3, 1, 10, 12, 16,26, 15, 10, 2, 5, 3, 10, 21

estimated mean:

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76 Practice Workbook Algebra 2

Practice

12.1 Measures of Central Tendency

NAME CLASS DATE

Age (years) Tally Frequency

14

15

16

17

18

Number of books Class mean Frequency Product

1–5

6–10

11–15

16–20

21–25

25–30

Make a stem-and-leaf plot for each data set. Then find the median

and the mode, and describe the distribution of the data.

1. 40, 64, 54, 38, 42, 45, 33, 37, 56, 58, 64 2. 3.6, 4.8, 3.9, 1.7, 4.3, 2.3, 4.8, 3.1, 4.0, 2.3

Make a frequency table and a histogram for the data.

3. 1.0, 1.3, 1.1, 1.4, 1.4, 1.2, 1.1, 1.0, 1.0, 1.3, 1.3, 1.4, 1.3, 1.2, 1.0, 1.3, 1.4, 1.2, 1.0, 1.3

Complete the table, and make a relative frequency histogram for the data.

4.

Make a circle graph for the data.

5.

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Algebra 2 Practice Workbook 77

Practice

12.2 Stem-and-Leaf Plots, Histograms, and Circle Graphs

NAME CLASS DATE

Stem Leaf Stem Leaf

Number Frequency

Number Frequency Relative frequency

10 8

11 12

12 7

13 11

14 12

Number

01.0 1.1 1.2 1.3 1.4

123456

Freq

uen

cy

Number10 11 12 13 14

0

0.1

0.2

0.3

Pro

babi

lity

Motor Vehicle Registration by Type, 1994

Passenger cars Motorcycles Buses Trucks

33.2% 0.9% 15.9% 50%

Find the quartiles, the range, and the interquartile range for each

data set.

1. 9, 5, 2, 8, 2, 8, 7, 8, 3, 2, 8, 1, 9, 1, 5, 8, 9, 7, 9, 6

2. 13, 14, 15, 19, 16, 19, 8, 17, 10, 7, 5, 18, 10, 16, 17, 12

3. 35.1, 40.3, 13.8, 15.3, 42.7, 40.8, 15.5, 38.5, 28.4, 11.0, 11.7, 12.1, 23.9, 8.9, 24.0

Find the minimum and maximum values, quartiles, range, and

interquartile range for each data set. Then make a box-and-

whisker plot for each data set.

4. 17, 14, 5, 8, 15, 4, 11, 6, 13, 17, 17, 13, 7, 9, 5, 3

5. 35, 35, 23, 20, 29, 13, 26, 21, 39, 22, 14, 35, 10, 16, 36

The box-and-whisker plots at right compare

birth rates (per 1000 population) for the fifty

states for 1992 and 1993.

6. Which set of data has

the greater median?

7. Does it appear that, in general, the birth rates increased or decreased from 1992 to 1993?

8. What percent of the data are less than Q3 for each year?

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78 Practice Workbook Algebra 2

Practice

12.3 Box-and-Whisker Plots

NAME CLASS DATE

12 13 14 15 16 17 18 19 20 21

1992

1993

Find the range and mean deviation for each data set.

1. 24, 20, 38, 36, 52 2. 12, 11, 15, 18, 22, 30

3. 71, 56, 88, 82, 40, 95 4. 120, 142, 167, 188, 167, 200

5. 5.8, 3.4, 7.2, 10.5, 8.6 6. 38, 52, 40, 61, 53, 90, 100

Find the variance and standard deviation for each data set.

7. 13, 13, 17, 11, 22, 20 8. 82, 44, 67, 52, 120

9. 1215, 1805, 1715, 2010, 1875 10. 12, 14.5, 18, 16, 11.5, 15

11. 30, 40.2, 40.8, 22.6, 18 12. 19.4, 19, 19.2, 19.6, 19.8, 19

The table shows the winning scores in the United States

Women’s Open Golf Championships from 1977 to 1996. Refer to

the data in the table for Exercises 13–16.

13. Find the range. 14. Find the mean deviation.

15. Find the variance. 16. Find the standard deviation.

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Algebra 2 Practice Workbook 79

Practice

12.4 Measures of Dispersion

NAME CLASS DATE

292 289 284 280 279 283 290 290 280 287

285 277 278 284 283 280 280 277 278 272

A coin is flipped 6 times. Find the probability of each event.

1. exactly 1 head 2. exactly 5 heads

3. more than 3 heads 4. fewer than 2 heads

A spinner is divided into 5 congruent segments, each labeled with

one of the letters A–E. Find the probability of each event.

5. exactly 3 As in 3 spins 6. fewer than 2 Bs in 4 spins

7. exactly 4 Cs in 5 spins 8. fewer than 3 Ds in 5 spins

9. more than 3 Es in 5 spins 10. exactly 3 As in 10 spins

At one university, the probability that an entering student will

graduate is 40%. Find the probability of each event.

11. Exactly 4 out of 5 randomly selected entering students will graduate.

12. Fewer than 3 out of 5 randomly selected entering students will graduate.

13. Exactly 2 out of 6 randomly selected entering students will graduate.

14. More than 3 out of 6 randomly selected entering students will graduate.

The probability that any given person is left-handed is about 10%.

Find each of the following probabilities:

15. Exactly 3 out of 7 randomly selected people are left-handed.

16. More than 3 out of 7 randomly selected people are left-handed.

17. Fewer than 4 out of 7 randomly selected people are left-handed.

18. Exactly 4 out of 10 randomly selected people are left-handed.

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80 Practice Workbook Algebra 2

Practice

12.5 Binomial Distributions

NAME CLASS DATE

Let x be a random variable with a standard normal distribution.

Use the area table for a standard normal curve, given on page 807

of the textbook, to find each probability.

1. 2. 3.

4. 5. 6.

7. 8. 9.

The time required to finish a given test is normally distibuted

with a mean of 40 minutes and a standard deviation of 8 minutes.

10. What is the probability that a student chosen at randon will finish

in less than 32 minutes?

11. What is the probability that a student chosen at random will take

more than than 56 minutes to finish?

12. What is the probability that a student chosen at random will take

between 24 minutes and 48 minutes?

The owners of a restaurant determine that the number of

minutes that a customer waits to be served is normally

distributed with a mean of 6 minutes and a standard deviation

of 2 minutes.

13. What is the probability that a randomly selected customer will be

served in less than 4 minutes?

14. During a survey, 500 customers are served. How many would you

expect to be served in less than 8 minutes?

15. If 1000 customers are served, how many would you expect to

wait between 4 minutes and 10 minutes?

P(�0.4 � x � 1.2)P(�0.2 � x � 0.2)P(1.0 � x � 2.0)

P(�0.2 � x � 0)P(0 � x � 2.0)P(0 � x � 0.4)

P(x � �1.8)P(x � 1.2)P(x � 0)

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Algebra 2 Practice Workbook 81

Practice

12.6 Normal Distributions

NAME CLASS DATE

Refer to the triangle at right to find each value listed.

Give exact answers and answers rounded to the

nearest ten-thousandth.

1. sin � 2. cos � 3. tan �

4. sin � 5. cos � 6. tan �

7. sec � 8. csc � 9. cot �

Solve each triangle. Round angle measures to the nearest degree

and side lengths to the nearest tenth.

10. 11. 12.

XZ ≈ UV ≈ RT ≈

m�X ≈ m�U ≈ m�R ≈

m�Z ≈ m�W ≈ m�T ≈

13. 14. 15.

m�Q ≈ m�L ≈ m�J ≈

QN ≈ LM ≈ HJ ≈

NP ≈ MN ≈ HK ≈

2.9

34°H J

K6 54°

L M

N

2.5

31°N

Q

P

12

18R S

T

21

24

V

U W

4.5

7

X

Y Z

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82 Practice Workbook Algebra 2

Practice

13.1 Right-Triangle Trigonometry

NAME CLASS DATE

41

�

� 9

40

For each angle below, find all coterminal angles, �, such that

�360˚ � � � 360˚. Then find the corresponding reference angle.

1. 47˚ 2. �123˚ 3. 218˚

4. 512˚ 5. �222˚ 6. 307˚

7. 1122˚ 8. �185˚ 9. 645˚

Find the reference angle.

10. 105˚ 11. �213˚ 12. 715˚

13. �144˚ 14. 860˚ 15. �72˚

16. �2˚ 17. 1000˚ 18. �420˚

Find the exact values of the six trigonometric functions of �,

given each point on the terminal side of � in standard position.

19. (12, 8) 20. (�5, 10) 21. (4, 9)

sin � � sin � � sin � �

cos � � cos � � cos � �

tan � � tan � � tan � �

csc � � csc � � csc � �

sec � � sec � � sec � �

cot � � cot � � cot � �

Given the quadrant of � in standard position and a trigonometric

function value of �, find exact values for the indicated

trigonometric function.

22. IV, sin � ; tan � 23. I, tan � ; csc � 24. II, cos � ; sin �

25. III, csc � 1.25; tan � 26. II, cot � ; sin � 27. IV, sec � ; cot ��43� �2.4� �

� �58�

58� �

35

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Algebra 2 Practice Workbook 83

Practice

13.2 Angles of Rotation

NAME CLASS DATE

Point P is located at the intersection of a circle with a radius of r

and the terminal side of angle �. Find the exact coordinates of P.

1. � � 45˚; r � 5 2. � � 60˚; r � 12 3. � � �120˚; r � 15

4. � � 330˚; r � 40 5. � � 135˚; r � 10 6. � � 750˚; r � 4

Point P is located at the intersection of the unit circle and the

terminal side of angle � in standard position. Find the coordinates

of P to the nearest thousandth.

7. � � 42˚ 8. � � 129˚ 9. � � 244˚

10. � � 305˚ 11. � � �41˚ 12. � � �105˚

Find the exact values of the sine, cosine, and tangent of each

angle.

13. 2160˚ 14. 315˚ 15. �240˚ 16. 1770˚

sin: sin: sin: sin:

cos: cos: cos: cos:

tan: tan: tan: tan:

Find each trigonometric function value. Give exact answers.

17. sin 420˚ 18. cos(�150˚) 19. csc(�480˚)

20. tan 300˚ 21. cos 1035˚ 22. sin 1470˚

23. cot(�120˚) 24. tan 495˚ 25. csc 210˚

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84 Practice Workbook Algebra 2

Practice

13.3 Trigonometric Functions of Any Angle

NAME CLASS DATE

Convert each degree measure to radian measure. Give exact

answers.

1. 135˚ 2. 300˚ 3. 36˚ 4. 150˚

5. 105˚ 6. �85˚ 7. 70˚ 8. 75˚

Convert each radian measure to degree measure. Give answers to

the nearest hundredth of a degree.

9. radians 10. radians 11. radians 12. radians

13. 8.25 radians 14. 1.8 radians 15. 3 radians 16. 0.5 radian

A circle has a diameter of 20 feet. For each central angle measure

below, find the length in feet of the arc intercepted by the angle.

17. radians 18. radian 19. radians 20. radian

21. 2.5 radians 22. 4 radians 23. 7.3 radians 24. 10 radians

Evaluate each expression. Give exact values.

25. sin 3π 26. cos 27. tan 28. csc

29. tan 30. sin 31. cos 32. sec 5π3

5π2

7π4

13π6

(�π2 )5π

32π3

π6

2π3

π12

3π4

13π12

7π9

11π12

5π2

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Algebra 2 Practice Workbook 85

Practice

13.4 Radian Measure and Arc Length

NAME CLASS DATE

Complete the table of values for the function and graph the

function along with its parent function.

1.

Identify the amplitude, if it exists, and the period of each

function.

2. 4.5 cos 2� 3. 3 tan 4. 1.2 cos

Identify the phase shift and vertical translation of each function

from its parent function. Then graph at least one period of the

function for 0° � � � 360°, or 0 � x � 2π.

5. 2 cos(� � 45°) � 1.5 6. sin 2(x � π) � 1y �y �

(x � π)y �(x �π2 ) � 1y �y �

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86 Practice Workbook Algebra 2

Practice

13.5 Graphing Trigonometric Functions

NAME CLASS DATE

� 0° 30° 45° 60° 90°

cos 2�

120° 135° 150° 180°

Evaluate each trigonometric expression. Give exact answers in

degrees.

1. Sin�1 2. Cos�1 3. Tan�1 0

4. Cos�1 1 5. Sin�1 6. Tan�1

Evaluate each trigonometric expression.

7. cos(Sin�1 1) 8. tan Cos�1 9. sin(Tan�1 )

10. Cos�1 11. Tan�1(cos(�720˚)) 12. Sin�1

Find each value. Give answers in radians, rounded to the nearest

ten-thousandth.

13. Tan�1 40.2356 14. Sin�1 0.0345 15. Cos�1 (�0.8114)

16. Cos�1 0.7756 17. Tan�1 (�38.2004) 18. Sin�1 (�0.5454)

Use inverse trigonometric functions to solve each problem.

19. A ramp that is 18 feet long rises to a loading platform that is 3 feet above the ground. Find, to the nearest tenth of a degree, the angle that the ramp makes with the ground.

20. At one point in the day, a tower that is 150 feet high casts a shadow that is 210 feet long. Find, to the nearest tenth of a degree, the angle of elevation of the sun at that point.

21. A kite is flying 67 meters above the ground, and its string is 90meters long. Find the angle, to the nearest tenth of a degree, that the kite string makes with the horizontal.

( tan 5π4 )(sin π3 )

�3(��2

2 ))(

(��3)(��2

2 )

(�12 )(�

�32 )

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Algebra 2 Practice Workbook 87

Practice

13.6 Inverses of Trigonometric Functions

NAME CLASS DATE

Use the given information to find the indicated side length in

. Round answers to the nearest tenth.

1. Given , , and , find b.

2. Given , , and , find b.

3. Given , , and , find b.

4. Given , , and , find c.

5. Given , , and , find c.

6. Given , , and , find a.

Solve each triangle. Round answers to the nearest tenth.

7. 8.

9. 10.

11. 12.

State the number of triangles determined by the given

information. If 1 or 2 triangles are formed, solve the triangle(s).

Round answers to the nearest tenth, if necessary.

13.

14.

15.

16. Find, to the nearest tenth of a foot, the length of fence needed to enclose the triangular piece of land shown in the diagram.

m� B � 28, a � 40, b � 26

m� B � 98, a � 10.5, b � 8.8

m� A � 64, b � 16, a � 20

m� B � 39, m�C � 66, b � 54m� A � 46, m� B � 52, b � 17

m� A � 72, m�C � 64, c � 5.2m� A � 100, m� B � 35, b � 15

m� B � 65, m�C � 80, b � 20m� A � 82, m� B � 60, a � 5

b � 24.5m� B � 55m� A � 75

a � 5.6m� A � 82m�C � 100

b � 25m�C � 62m� B � 48

a � 18m� B � 64m� A � 105

c � 30m�C � 70m� B � 51

a � 12m� B � 95m� A � 28

ABC

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88 Practice Workbook Algebra 2

Practice

14.1 The Law of Sines

NAME CLASS DATE

55°

45°

500 ft

Classify the type of information given, and then use the law of

cosines to find the missing side length of . Round answers

to the nearest tenth.

1. 2.

3. 4.

5. 6.

Solve each triangle by using the law of cosines and, where

appropriate, the law of sines. Round answers to the nearest

tenth.

7.

8.

9.

10.

11.

Classify the type of information given, and then solve .

Round answers to the nearest tenth. If no such triangle exists,

write not possible.

12.

13.

14.

15.

16.

17. Find, to the nearest tenth of a degree, the measures of the angles ofan isosceles triangle in which the base is half as long as each side.

m�C � 95, a � 8, c � 6

m� A � 50, b � 3, c � 8

a � 200, b � 100, c � 150

a � 6.2, b � 8, c � 4.2

m�B � 110, a � 75, c � 85

ABC

a � 10, b � 15, c � 13

a � 4.5, b � 3.2, c � 6.1

a � 12, b � 11, c � 9

a � 50, b � 31, c � 46

a � 17, b � 15, c � 24

b � 8, c � 14, m� A � 73m�C � 110, a � 16, b � 22

a � 10, c � 12, m� A � 52m� A � 32, b � 8, c � 10

a � 25, b � 28, m�C � 64m� A � 46, b � 24, c � 18

ABC

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Algebra 2 Practice Workbook 89

Practice

14.2 The Law of Cosines

NAME CLASS DATE

Use definitions to prove each identity.

1. 2.

3. 4.

Write each expression in terms of a single trigonometric function.

5. 6.

7. 8.

9. 10.

Write each expression in terms of sin �.

11.

12.

13. sin � � (cos �)(cot �)

(csc � � cot �)(1 � cos �)

csc � � sin �cot2 �

(sin �)(tan �) � cos �csc � � (cos �)(cot �)

(sin �)(cos �)1 � sin2 �cot2 � � csc2 �

(sec � � 1)(sec � � 1)1 � cos2 �sin �

1 � tan2 � � sec2 �1 � cot2 � � csc2 �

(sin � � 1)(sin � � 1) � �cos2 �cot �cos � � csc �

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90 Practice Workbook Algebra 2

Practice

14.3 Fundamental Trigonometric Identities

NAME CLASS DATE

Find the exact value of each expression.

1. sin 2. sin 3. cos

4. cos 5. sin 6. cos

7. sin 8. sin 9. cos

Find the exact value of each expression.

10. sin(�285) 11. cos(�285) 12. sin 135

13. cos 210 14. cos(�75) 15. sin 345

16. cos 345 17. sin 240 18. sin(�75)

Find the rotation matrix for each angle. Round entries to the

nearest hundredth, if necessary.

19. 120 20. 135 21. 225

22. 65 23. �40 24. 112

25. A rectangle has vertices at (3, 5), (3, 10), (7, 10), and (7, 5). Find thecoordinates of the vertices after a 135 counterclockwise rotationabout the origin. Round coordinates to the nearest hundredth.

(π6 �

π4 )(π

6 �π4 )(π

6 �π4 )

(3π4 �

2π3 )(3π

4 �2π3 )(3π

4 �π3 )

(3π4 �

π3 )(3π

4 �π3 )(3π

4 �π3 )

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Algebra 2 Practice Workbook 91

Practice

14.4 Sum and Difference Identities

NAME CLASS DATE

Verify that the double-angle identities and the half-angle

identities are true for the sine and cosine of each angle.

1. 90

2. 120

3.

4.

Write each expression in terms of trigonometric functions of �rather than multiples of �.

5. 6. 7.

Simplify.

8.

9.

10.

11.

12. The angle of elevation of a flagpole was measured at distances of45 feet and 14.4 feet from the flagpole. The second measure of the angle of elevation was twice the first. Find the height of the flagpole.

(sin(�2 ) � cos(�

2 ))2

1 � cos 2�sin 2�

1 � sin � � cos 2�cos � � sin 2�

1 � cos 2�1 � cos 2�

sin 2�tan �cos2(�

2 )sin2(�2 )

2π3

π3

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92 Practice Workbook Algebra 2

Practice

14.5 Double-Angle and Half-Angle Identities

NAME CLASS DATE

Find all solutions of each equation.

1. 2.

3. 4.

Find the exact solutions of each equation for .

5. 6.

7. 8.

Find the exact solutions of each equation for .

9. 10.

11. 12.

Solve each equation to the nearest hundredth of a radian for

.

13. 14.

15. The equation describes the altitude of a ballt seconds after it was hit at an angle of � degrees. Determine, to thenearest tenth of a degree, the measure of the angle at which the ballwas hit if it had an altitude of 20 feet after 2.8 seconds.

y(t) � 122t sin � � 16t 2

6 sin2 x � 5 sin x � 1 � 09 cos2 x � 1 � 0

0 � x � 2π

2 sin2 x � sin x � 1 � 02 cos 3x � 1 � 0

cos x � sin x � 02 sin x2 � 1 � 0

0 � x � 2π

2 sin2 � � cos 2�4 cos2 � � 2 � 0

2 sin2 � � 3 sin � � 1 � 02 cos2 � � 3 cos � � 2

0 � � � 360

2 cos �2 � 1 � 0sec �2 � 2

tan � � 1 � 0sin2 � �14

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Algebra 2 Practice Workbook 93

Practice

14.6 Solving Trigonometric Equations

NAME CLASS DATE

Practice — Chapter 1Lesson 1.1

1. yes 2. yes 3. no 4. yes 5. no6. yes 7. yes 8. no 9. yes 10. no

11. no 12. yes13.

14.

15.

16. (6, 19) 17. (4, 20) 18. not linear19. (�4, 23) 20. (8, �29) 21. (5, �14)

Lesson 1.21. 2.3. 4.5. 6. 7.8. 9. 1 10. 3

11. 12.13. 14.15. 16.

Lesson 1.31. 2.3. 4.5. 6.7. 8. 9.

10. 11.12. 13.14. 15.16. 17.18. 19.20. 21.22. 23.24. 25.26. 27.28. 29.30. 31.32. y� �

14x�154

y� �13xy�

43x�83

y� �43x� 2y� x� 2

y� �14x�

72y� �12x� 4

y�12x� 10y� �

13x� 2y� �2x� 8y� �3x� 3

y�12x� 3y� 3x� 4

y� �12x� 1y� 4x� 16

y� �3x� 5y� 2x� 15y�

34x� 5y� �4x� 5y� �3x� 7y� �

14x� 8y� �2x� 1y� 4x� 13

y�34x� 5y� �

12x� 1y� xy�

45x�1110y� �9x� 13

y� �12x� 1y� �

15x�85

y� �x� 2y� x� 2

y� �34x� 3y�

56x� 3m� �3, b� �7m� �2, b� �4

m� �38 , b� �

32m� �34 , b�

3257100

�23y�

14x� 4y�16x� 3

y�45x�

25y� �4x� 3y� 3x� 1y� 2x� 5

x

y

22468

4 6 8

22468

4 6 8 x

y

x

y

22468

4 6 8O

Algebra 2 1

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Lesson 1.41. 2.3. 4.5. 6.7. 8.9. 10. 11.

12. 13. 14.15. 16. 17. yes;18. no; there is no constant, k, such that19. yes;Lesson 1.5

1. ; positive

2. ; negative

3. ; negative

4. 5. 6. 0.360Lesson 1.6

1. 2. 3. 4.5. 6. 7.8. 9. 10.

11. 12. 13.14. 15. 16.17. 18. 19.20. 21. 22.23. 24. 25.26. 27.28.

Lesson 1.71. 2. 3.4.

5.x0 1 2 3–1–2–3

x � 1–2 –1 0 1–3–4–5 x

x � �4x � 2x � 2x � �3

h �Aa � b

a �T0 � Tz � z0qp �

qD � QP1 �

V2P2V1r �C2πW �

VLDx �

12x � �67x � �6

x � �4x � �522x � 18

x � �65x � �

13x � 7x � �

13x � �21x � 16x � 2x �

12x � �3x � �2x �

78x � 8x � 6x � 9x � 5x � 2

r � 0.96y � 0.01x � 0.24

x

y

42 6 8 1010

–10–20–30–40

(1, –2)O(2, –18) (7, –26)

(9, –34)

y � �3.26x � 4.5

x

y

2 4 6 8 1010

–10–20–30

(2, 10)O

(8, –27.45)(6, –20.15)

(0, 1.9)

y � �4.44x � 8.85x

y

10O

91827

20(0, 4)

(2, 10)

(6, 22)(8, 28)

y � 3x � 4

y � 2xy � kx

y � 3xz �13x � �1

y � �54z �

52x � 10y �

92x � 4x � 3k � �3; y � �3xk � 2; y � 2xk �

56 ; y �56xk �

27 ; y �27x

k �14 ; y �

14xk � �43 ; y � �

43xk �

73 ; y �73xk � �5; y � �5x

2 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

6. no solution7.

8.

9.

Lesson 1.81. x � 2 and x � �8

2. x � 10 and x � �2

3. x � �6 and x � 1

4. and x � �3

5. and x � 2

6. x � 1 and

7. and

8. and

9. or

10. or

Practice — Chapter 2Lesson 2.1

1. rational, real2. irrational, real3. irrational, real4. rational, real5. integer, rational, real6. rational, real7. Inverse Property of Addition8. Identity Property of Multiplication9. Associative Property of Addition

10. Inverse Property of Multiplication11. Commutative Property of Multiplication12. Distributive Property13. Identity Property of Addition14. �33 15. 4 16. 43 17. 43 18. 4919. 28 20. 26 21. 7.2 22. 56 23. 90

6 x0 2 4–2–4–6x � 5x � �2.5

6 x0 2 4–2–4–8 –6x � 3x � � 7

x2 310x �

15x �115

x0 1 2 3–1–2–3x �

16x �76

x2 4 6 80–2–4x � �

95

x0 2–10 –8 –6 –4 –2x � �10

x–2 0 2 4–4–6–8x �

95x–2 0 2 4–4–6–8

x4 6 8 1020–2

x–2 0 2 4–4–6–8

x190180170160150 200162 � t � 192

x0 2 4 6–2–4–6x � 5

x–1 0 1 2–2–3–4x � �2

Algebra 2 3

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Lesson 2.21. 1 2. 3. 36 4. 1 5. �2176. 7. 25 8. 32 9. 2 10. 16

11. 32 12. 9 13. 14. w4y2z2 15.16. x14 17. z17 18. 19. y5z 20. 121. 81x12y20 22. 16a12b4c24 23. 125a6b924. 25. 26. 27. 16m12p428. 29. 30. 81x26y2

Lesson 2.31. yes 2. yes 3. no 4. yes 5. yes6. no 7. yes 8. no9. domain: ; range: {�3, 1, 3, 7}

10. domain: x � �5; range: y � �211. domain: {�4.5, 3, 6.5, 12};range: {6, �1.5, �5, �10.5}12. domain: {�2, 0, 1, 5}; range: {12, 8, 9, 33}13.14.15.16.17.

Lesson 2.41. ( f � g)(x) � 8x2 � 5x � 13;( f � g)(x) � 6x2 � 5x � 132. ( f � g)(x) � 13x2 � 5x � 41;( f � g)(x) � �13x2 � 5x � 413. ( f � g)(x) � x2 � x � 2;

( f � g)(x) � x2 � x � 16

4. ( f � g)(x) � 3x2 � 6;( f � g)(x) � �21x2 � 65. ( f � g)(x) � 175x � 25;

(x) � 7x � 16. ( f � g)(x) � 3x3 � 17x2 � 75x � 425;

(x) � , x � �57. ( f � g)(x) � x4 � 256;

(x) � , x � �4 and x � 48. ( f � g)(x) � �x � 8 9. ( f � g)(x) � �3x � 12

10. (g � f )(x) � 3x � 1211. ( f � g)(x) � �2x2 � 22x � 2012. (x) �

13. (x) �

14.15.16.17. �44 18. �52 19. �10 20. �521. �5 22. 22 23. �55 24. �28 25. 44Lesson 2.5

1. {(�16, �1), (�6, 0), (14, 2)}; yes; yes2. {(2, 7), (3, 6), (4, 7), (5, 6)}; no; yes3. {(16, �2), (1, �1), (1, 1), (16, 2)}; yes; no4. {(7, �5), (7, �3), (7, �1), (7, 1)}; yes; no5. {(4, �5), (9, �3), (12, 1), (13, 7)}; yes; yes6. f �1(x) � 3x � 17. h�1(x) �

3x � 12

( f � g)(x) � �x2 � 1; (g � f )(x) � �x2 � 1( f � g)(x) � 4x2 � 4; (g � f )(x) � 16x2 � 1( f � g)(x) � x ; (g � f )(x) � x

�x � 102x � 2 , x � �1(gf ) �2x � 2x � 10 , x � �10( fg)

x2 � 16x2 � 16( fg)23x 2 � 253x � 17( fg)

( fg)

223203

f(0.5) � 1.25; f(0) � 0f(11) � 361; f(�4) � 46f(7) � �9; f(�5) � 27f(�3) � 45; f(5) � 125f(�2) � �44; f(8) � 156

��1, 0, 12 , 32�

1xy61xy3z4

w18k3a4b6x4w2z8

1x 35

1k81d925

�115

4 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

8.9. f �1(x) � 2x � 2.5

10.11. h�1(x) � 4x � 3212.

not a function13.

function14.

function

Lesson 2.61.

2.

3.

4. ifif5. f(x) � x � 3 if x � �3, f(x) � 0 if

�3 � x � 0, f(x) � x if x � 06. f(x) � x � 2 if x � 0, f(x) � 0 if x � 0,f(x) � x � 2 if x � 0

7. �10 8. 32 9. 13.13 10. 0 11. 712. �7.5 13. 1.5 14. 1 15. �19 16. �7

x � 2f(x) � 1�4 � x � 2, f(x) �12xf(x) � �2 if x � �4,

xO

y

2–2–4 42

–2–4

4

xO

y

–2–4 422

–2–4

4

xO

y

2–2–4 42

–2–4

4

xO

y

1–2 24

–4–8

8

xO

y

1–1–2 24

–4–8

8

xO

y

2–2–4 448

1216

g�1(x) �x8 � 2

g�1(x) �x � 411

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Lesson 2.71. a horizontal translation 7.5 units to the left2. a vertical translation 7.5 units up3. a horizontal compression by a factor of 524. a vertical stretch by a factor of 2 and areflection across the x-axis5. a vertical stretch by a factor of 14 and avertical translation 6 units up6. a vertical stretch by a factor of 12 and ahorizontal translation 7 units to the right7. a translation 21 units to the left8. a vertical stretch by a factor of 179. a horizontal stretch by a factor of 2

10. a vertical translation 13.7 units up11. a vertical stretch by a factor of 3 and areflection across the x-axis12. a vertical stretch by a factor of 41 and ahorizontal translation 8 units to the right13. 14.15. 16.17. 18.

Practice — Chapter 3Lesson 3.1

1. independent; (0, 4)2. dependent; infinitely many solutions3. independent; (�1, �1)4. inconsistent; no solution5. independent; (1, 3)

6. independent; (1, 6)7. (6, �1) 8. (12, 1) 9.

10. (�4, 8) 11. (�8, 5) 12.

Lesson 3.21. (2, �1) 2. no solution 3. (6, 2)4. (3, 0) 5. (4, �3) 6. no solution7. (�9, �4) 8. infinitely many solutions9. (8, 8) 10. (�4, 1) 11. (2, �1)

12. (�3, 3) 13. (1, �2) 14. (5, �8)15. 16. (1, 7) 17.18. infinitely many solutions19. 20. (3, 10) 21. no solution22. 23. 24. (11, �3)Lesson 3.3

1.

2.

224

–4–2 4–4 –2 x

y

2O24

–4–2 4–4 –2 x

y

(23 , 2)(14 , 1)(13 , 23 )(12 , 3)(12 , 12 )

(3, 12 )(12 , 4)

g(x) � x3 � 33g(x) � �23x � 9

g(x) � �x12�g(x) � (26x)4

g(x) � (x � 7)5g(x) � �x 3

6 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

3.

4.

5.

6.

7a. 8x � 12y � 400b.

c. 50 hours; 33 hoursLesson 3.4

1.

2.

224

–44–4 –2 x

y

224

–4–2 4–4 –2 O x

y

13302010

2432

816

40O x

y

224

–4–2 4–4 –2 O x

y

2O24

–4–2 4–4 –2 x

y

2O24

–44–4 –2 x

y

2O24

–4–2 4–4 –2 x

y

Algebra 2 7

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3.

4.

5.

6.

7.

Lesson 3.51.

2.

3.

4. maximum: 41; minimum: 45. maximum: 17; minimum: �126. maximum: 12; minimum: �77. maximum � 50 8. maximum � 68minimum � 0 minimum � 09. maximum � 200 10. maximum � 62minimum � 20 minimum � �25

22468

4 6 8O x

y2

2468

4 6 8O x

y2

2468

4 6 8O x

y

xO

y

1530

15 30

� x � y � 306x � 7y � 126

� y � 1y �12 � 2

y � �x � 5

� x � 4y � � xy �

12x � 3

� y � 4y � 2x � 4y � �2x � 4

224

–24–4 –2 O x

y

8 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

Lesson 3.61.

2.

3.

4.5.6.7.8.9.

10a.b.

11a. 25 secondsb. 3750 feet

Practice — Chapter 4Lesson 4.1

1. 2 � 2 2. 2 � 2 3. 3 � 3 4. 3 � 35.

6. not possible; the matrices do not havethe same dimensions.7.

8.

9.

10.

11. � �60�12

6�24

8�4�4 �

� 762057

�2014

�8 ���7

�4 �43 �� 74 4

�3 �� 0

�3 �612 �

��2�3�7

�4�30

�2�65 �

xO

y

100200300400

2000 4000

� x(t) � 150ty(t) � 12t

y �x29 � 1

y �13x � 2

y � �12x � 16

y � �12x � 8

y � 6x � 2y �

12x � 14

xO

y

2

–2–4

46

2–2–4–6

xO

y

2

–2–4

4

2 4 6 8

xO

y

2

–2–4–6

4

42–2–4

Algebra 2 9

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12.

13.14. 3 � 3 15. 101 medals16. 90 gold medals17. the number of bronze medals, 27, won byGermany18. Gold Silver Bronze

Lesson 4.21.2.3. does not exist4.

5.

6.

7.

8. does not exist

9.10. X�(�3, 0), Y�(6, 6), Z�(3, �6)11.

12. an enlargement by a factor ofLesson 4.3

1. yes 2. yes 3. no 4. yes5. yes 6. yes7.

8.

9.

10.11. D � 0; no inverse12.

13.

14.15. no inverse

��3.52.5 3�2 �

��13 1�2 �

D� 4; inverse: � 1�4 �0.753.25 �

D� 2; inverse: � 2�3.5 �35.5 �

D� 5; inverse: � 1�1.4 �11.6 �

D� 1; inverse: � 4�5 �79 �

D� 1; inverse: � 3�4 �57 �

32Z'

Y'

X' x

y

2–4–6 62

–4–6

46

Z

XY

�1134 32�8 �

�23

�50 �

32�12.5181343 �

�1026�10�2�6

513 ��47 �64 �

[�20 4]��1

�9 4544 �

�44202616

32182122

25271612 �United StatesGermanyRussiaChina

� 143 �214 ���131

72264

�7 �

10 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

16.17. no inverse18.

Lesson 4.41.

2.

3.

4.

5.

6.

7.

8.

9.

10. ; (4, �2, �1)

11. (3, 9, �6)

12. no solution

13. (�3, 5, �4)

14. (�1, 10, 2)

Lesson 4.51.

2.

3.

4.

5.

6.

7. � 100010

001MMM

2�14 �

� 100010

001MMM

132 �� 100

010001

MMM

�21�1 �

� 100010

001M

M

M

1�2�1 �

� 126137

133MMM

�148 �� 257

168�31�3

MMM

462 �� 471

591121

MMM

272 �

� 8512121

�1�95 ��

xyz �� � 0�38 �;

� 1235743

12�3 ��

xyz �� ��5312 �;� 135

�21�10

1�25 ��

xyz �� �15821 �;� 133

�21�2

�31�4 ��

xyz �� � 31215 �;

� 357�365

5�20 ��

xyz �� � 131018 �� 480

1�45

1�79 ��

xyz �� � 123 �; (0.5, �3, 2)� 35 �7

�8 �� xy �� � 2527 �; (�1, �4)� 74 53 �� xy �� � 149 �; (�3, 7)� 84 7

�9 �� xy �� � 565 �; (5, �5)� 3x � 2y � z � �62x � 3y � z � 14x � 4y � 3z � 20

� 2x � 3y � z � 13x � 4y � z � 6�x � y � 2z � 7

� 934�51�3

1�1�2 ��

xyz �� � 62�1 �

� 5�14

�24�8

1�13 ��

xyz �� � 13�16 �

� 3�12

1�12

�131 ��xyz �� ��1921

�7 �

�42 53 �

��12 3�5 �

Algebra 2 11

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8.

9.

10. (�3, 8) 11. (7, �1) 12. (�5, 4)13. (1, �4, 6) 14. (�3, 2, 5) 15. (2, �4, 6)16. (3, 4, �1) 17. (4, �1, 2) 18. (1, 4, 2)Practice — Chapter 5Lesson 5.1

1. ; a � 1, b � �8, c � 152. ; a � 1, b � �16,c � 633. ; a � �3,b � 30, c � 334. ; a � 6, b � 13,c � �55. ; a � 1, b � �6, c � 56. yes 7. no 8. no 9. no

10. yes 11. yes12. up; minimum value13. up; minimum value14. up; minimum value15. down; maximum value

16.

(0, �3)17.

(�0.5, 6.25)18.

(�1.75, 5.0625)Lesson 5.2

1. �10 or 102.3. �12 or 64. �2�5 � �4.47 or 2�5 � 4.47

��3 � �1.73 or �3 � 1.73

xO

y

2–2–6 42

–2–4

6

xO

y

–2–4 42

–2

4

x

y

2–2–4 4246

d(x) � x2 � 6x � 5h(x) � 6x2 � 13x � 5k(x) � �3x2 � 30x � 33g(x) � x2 � 16x � 63f(x) � x2 � 8x � 15

� 100010

001M

M

M

05�4 �

� 100010

001M

M

M

3�21 �

12 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

5. �4 or 46. or7. or8. or9. or

10. or11. c � 9.4 12. j � 5.8 13. r � 5.014. c � 16.6 15. b � 6.9 16. a � 2.417. c � 11.5Lesson 5.3

1. 12(x � 5) 2. �4x(6 � x)3. (2 � 7x)(1 � 3x) 4. (4x � 3)(x � 12)5. 3x(x � 7) 6. �3x(x � 35)7. (x � 4)(x � 13) 8. (x � 5)(x � 4)9. (x � 9)(x � 2) 10. (x � 4)(x � 7)

11. (x � 10)(x � 9) 12. (x � 13)(x � 2)13. (2x � 1)2 14. (3x � 1)(x � 2)15. (2x � 1)(x � 2) 16. �16 or 1617. �5 or 5 18. �7 or �219. x � 1 20. 1.5 21.22. 23. 24.25. �3, 4 26. �2.5, 1 27. �15, 328. 6, 7 29. �2, 0.5 30. �1.5Lesson 5.4

1. x2 � 24x � 144; (x � 12)22. x2 � 40x � 400; (x � 20)23. x2 � 20x � 100; (x � 10)24. x2 � 5x � 6.25; (x � 2.5)2

5. x2 � 9x � 20.25; (x � 4.5)26. x2 � 19x � 90.25; (x � 9.5)27. �1.8 or 3.8 8. 2.3 or 5.79. �0.1 or 14.1 10. �20.1 or 0.1

11. 0.2 or 4.8 12. �0.6 or 6.613. �2.1 or 3.1 14. �6.7 or �0.315. �8.2 or 0.216.17. f(x) � �3x2 � 7; (0, 7); x � 018. f(x) � (x � 6)2 � 39; (6, �39); x � 619. f(x) � (x � 1)2 � 11; (1, �11); x � 120. f(x) � (x � 5)2 � 35; (5, �35); x � 521. f(x) � 3(x � 2.5)2 � 20.75;(�2.5, �20.75); x � �2.5Lesson 5.5

1. 0.3 or 9.7 2. 0.3 or 3.73. �3.3 or 3.3 4. �5.7 or 2.75. �5.6 or 1.6 6. �6.9 or 1.97. �2.9 or 2.4 8. 1.4 or 5.69. �1.1 or 4.1 10. �0.9 or 2.2

11. �2.2 or 3.7 12. �0.6 or 0.413. x � �1; (�1, �5)14. x � 1.5; (1.5, 11.75)15. x � 1; (1, �1)16. x � ;17. x � 1; (1, �3)18. x � ; (�

25 , �145 )�25

(�23 , �1013 )�

23

f(x) � �12x2; (0, 0); x � 0

�27 or 11 or 15�

23 or 23�

13 or 12

�1 � �23 � 3.80�1 � �23 � �5.803�2 � 4.24�3�2 � �4.242 � �5 � 4.242 � �5 � �0.24�6.5 � 2.55��6.5 � �2.55

2�33 � 1.15�2�33 � �1.15

Algebra 2 13

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Lesson 5.61. 1; two; x � 1.5 or x � 12. �27; none; x � or

3. �68; none; or

4. �63; none; or

5. �87; none; or

6. �96; none; or

7. �2 � 11i 8. 7 � 10i 9. 13 � 9i10. �1 � 3i 11. �15 � 5i 12. 5i13. �4 � i 14. 4 � 5i 15. �8 � 7i16. �15i 17. 27 � 4i 18. �12 � 19i19. �4 � 28i 20. 21. 3022. 4 � 2i 23. �21 � 20i 24. 18 � 6i25. �4 � 5i 26. �iLesson 5.7

1.2.3.4.5.6.7.

8.9.

10.11.12.13.14. 9.25 feet 15. about 0.6 second16. 3 feet 17. about 1.4 secondsLesson 5.8

1.

2.

3.

4. no solution5.

6.

7.

xO

y

2–2–4 42

–2

46

–5 –4 –3 –2 –1 0 1 2�3.79 � x � 0.79

–2 –1 210 3 4 5x � �0.30 or x � 3.30

–6–8 –4 –2 0 2 4�5 � x � �2

–6 –4 –2 0 2 4 6�4 � x � 2

–6 –4 –2 0 2 4 6x � �4 or x � 4

f(x) � �16x2 � 20x � 3f(x) � 5x2 � x � 7f(x) � x2 �

12x � 3f(x) � x2 � 9x � 2f(x) �

32x2 � x � 4f(x) � 2x2 � 4x � 5

f(x) � �3x2 � 3x � 2f(x) � �2x2 � 7x � 3f(x) � 6x2 � 3x � 1f(x) � 5x2 � 2x � 3f(x) � 0.5x2 � x � 3f(x) � 3x2 � 2x � 5f(x) � �4x2 � 3x � 2

1213 �5i13

x �12 �

i�62x �

12 �i�62

x �14 �

i�8712x �

14 �i�8712

x �18 �

3i�78x �

18 �3i�78

x �13 �

i�173x �

13 �i�173

x � �32 �

3i�32�

32 �3i�32

14 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

8.

9.

10.

11.

12.

Practice — Chapter 6Lesson 6.1

1. 1.01 2. 0.9 3. 0.93 4. 1.12 5. 1.16. 0.97 7. 0.948 8. 1.075 9. 1.004

10. 0.941 11. 1.414 12. 15.874 13. 0.01614. 28 15. 35.318 16. 226.274 17. 1.87518. 8.25 19. 630.346 20. 271.529

21a. 1,275,868b. 1,307,375

22a. 6400b. 25,600

Lesson 6.21. quadratic 2. linear 3. linear4. exponential 5. quadratic6. exponential 7. linear8. exponential 9. exponential

10. exponential growth11. exponential decay12. exponential decay13. exponential growth14. exponential growth15. exponential decay

x

y

42–2–4

42

6

–2

xO

y

642–2–2

24

–4–6

xy

642–2

–6–8

x

y

2 4 6–22

–4–6

xO

y

2–62

–2

Algebra 2 15

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16. exponential decay17. exponential growth18. exponential growth19. $2117.56 20. $1075.52 21. $977.0622. $1226.08 23. $343.10 24. $1294.2225. $5325.11 26. $2358.76Lesson 6.3

1. log19 361 � 22. log20 8000 � 33. log3375 15 �

4.5.6.7. 122 � 144 8. 56 � 15,6259. 213 � 9261 10.

11.12.13. x � 1.54 14. x � 1.96 15. x � �0.7016. x � 0.26 17. x � �1.10 18. x � 3.0219. v � 3 20. v � 2 21. v � 222. v � 256 23. 24.25. v � 10 26. v � 3 27. v � 4Lesson 6.4

1. log10 4 � log10 100 � log10 4 � 22. log5 72 � log5 25 � log5 72 � 23. log7 5 � log7 3 � log7 44. log3 15 � log3 q

5. log8 64 � log8 4 � 2 � log8 46. log9 3 � log9 a � log9 7 �0.5 � log9 a � log9 77. log3 308. log59. log8 64 � 2

10. log911. log12 144 � 212. log3 13513. logb14. logb � logb15. logb � logb16. 12 17. 73 18. 2.5 19. 4.7 20. �121. 10 22. 2 23. 2 24. 25. �3, 326. �1, 6 27. 3 28. 4 29. 2Lesson 6.5

1. x � 1.72 2. x � 0.95 3. x � 6.644. x � 1.19 5. x � 2.81 6. x � �0.487. x � 1.05 8. x � �0.11 9. x � 1.46

10. x � 1.16 11. x � 1.43 12. x � 0.9013. x � 4.09 14. x � 1.33 15. x � 0.4216. x � 1.76 17. x � 2.18 18. x � 3.7019. x � �0.57 20. x � 3.19 21. x � 0.4022. x � 1.16 23. x � 3.09 24. x � 1.3025. x � �0.46 26. x � �0.55 27. x � 1.4328. x � 7.32 29. x � 3.42 30. x � �0.65

43

z2y4z3y4z

x24x34x2mx

5y4

x2

v �1343v �

1625

(15 )�4� 625

11�4 �114,641

360012 � 60log11 11331 � �3log37

27343 � 3log34

64 � �313

16 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

Lesson 6.61. 2980.958 2. 12.182 3. 181.2724. 109.196 5. 3.555 6. 2.5347. not defined 8. 1.242 9. 4.718

10. ln 55 � x 11. e3.78� 44

12. ln 0.05 � �3 13. e2.30� 10

14. ln 54.6 � 4 15. e4.83� 125

16. ln 148 � 5 17. e0 � 118. ln 0.45 � �0.8 19. x � 1.2320. x � 1.10 21. x � 2.17 22. x � �8.6723. x � 10.48 24. x � �0.7525. x � �1.17 26. x � �1.3427. x � 0.78 28. $1215.31Lesson 6.7

1. x � 4 2. 3. x � 24. x � 1000 5.6.7.8. x � 59. x � log 28 � 1.45

10. x � 4 11. x � 4 12. x � 613. x � 3 14. 15.16. x � 3 ln 0.64 � �1.34 17. 5.0 � 1021 ergs18. 7.2

Practice — Chapter 7Lesson 7.1

1. yes; quintic trinomial2. yes; quadratic trinomial3. no 4. yes; cubic polynomial5. no 6. no 7. �36 8. �29. 61 10. 410 11. 80 12. �152

13. 8x4 � 11x3 � 5x214. 22x5 � 13x4 � 3x2 � x � 815. �x4 � 5x3 � 10x2 � 1.2x � 216. 5.1x3 � 12.2x2 � 7x � 2617.

upside-down W-shape18.

W-shape

x

y

2–4 –2 42

xO

y

2–2–4 424

–4–2

6

x �e32 � 10.0x �

43

x �13 ln 15 � 0.90

x �ln 6ln 9 � 0.82

x �12

x �23 � 0.67

Algebra 2 17

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19.

S-shapeLesson 7.2

1. local minimum at (�1.5, 1.75)2. local maximum at (0.2, 6)3. local maximum at (0, 1); local minimumat (0.7, 0.7)4. local maximum at (�0.2, 2.2); localminima at (�1.7, 2.7) and (1.3, �2.3)5. local maximum at (0, 2); local minimumat (0.5, 1.8); increasing for �6 � x � 0and 0.5 � x � 6; decreasing for 0 � x �0.56. local maximum at (�0.7, 0.4); localminima at (�2.3, �1.5) and (0.5, �0.4);increasing for �2.3 � x � �0.7 and 0.5 � x � 4; decreasing for

�4 � x � �2.3 and �0.7 � x � 0.57. local maximum at (�0.8, �1.7); localminimum at (0, �2); increasing for

�5 � x � �0.8 and 0 � x � 4;decreasing for �0.8 � x � 08. local maxima at (0, 1) and (1.6, 2.1); localminimum at (0.3, 1.0); increasing for

�4 � x � 0 and 0.3 � x � 1.6; decreasing for 0 � x � 0.3 and 1.6 � x � 49. rises on the left, falls on the right

10. falls on the left, rises on the right11. falls on the left and on the right12. rises on the left and on the right13. f(x) � �4.17x4 � 30.83x3 � 473.42x2 �1019.27x � 6039.64Lesson 7.3

1. 8x5 � 5x4 � 3x22. 2x2 � 17x � 303. 5x3 � 17x2 � 5x � 284. x3 � 7x2 � 10x � 165. 2x3 � x2 � 8x � 56. 27x3 � 27x2 � 9x � 17. no 8. yes 9. no 10. yes

11. yes 12. no13. 2x � 514. 2x � 1 �

15. 4x2 � 4x � 1 �

16. 5x2 � 4x � 3 17. x � 318. x2 � 3x � 4 19. x � 320. x3 � 2x �

21. 11 22. 25 23. 10 24. 83Lesson 7.4

1. �9, 0, and 9 2. 0, 1, and 103. , 0, and 1 4. �5, 0, and 35. �3, 0, and 4 6. 0, , and 17. �2, 2, and 3 8. �3, �2, and 19. 3, �4, and 10. �1, 6, and 7

11. �1 and 6 (multiplicity 2)13

13�

12

6x � 3x2 � 3

42x � 15x3x � 5

xO

y

2–2–4 42

–2–4

4

18 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

12. �4 and 4 (multiplicity 2)13. , and14. , and15. , and 316. , and17. �4, 4, �1, 118. , and19. 1.3, 0.3, and �1.520. �1.2, �0.4, and 0.321. �0.9, 0 (multiplicity 2), and 0.922. �0.2, 0.4, and 1.123. �0.4, 0.3, and 0.924. �2.9, �2.4, �0.2, and 1.4Lesson 7.5

1. and 12. �2, , and 23. �1, , and 4. , 1, and 25. �1 (multiplicity 2), , and 46. , and 17. �5, , and 8. �3, 2i, and �2i9. 3, , and

10. 5, 3i, and �3i11. 2, �2 � 2i, and �2 � 2i12. 1, 4, i, and �i13. �1.46 and 1.66

14. �2.24, 0, and 2.2415. �1.41, 0, and 1.4116. �1.85 and 1.1617. P(x) � 5x3 � 15x2 � 50x � 12018. P(x) � �2x3 � 6x2 � 32x � 96Practice — Chapter 8Lesson 8.1

1. ; 16.8, 8.4, 5.25, 4.22. ; 8, 0.16, 0.1, 0.043. ; 25, 10, 7.5, 2.54. ; 9.6, 0.8, 0.48, 0.35. y � 8xz; 46. y � 24xz; 1447. y � 0.6xz; 10.88. y � 32xz; 969. ; 75

10. ; 911. ; 1012. ; 17.513. 0.5; A � 0.5ap; 13.8 in.2Lesson 8.2

1. yes; all real numbers except and 2. no; |x| � 2 is not a polynomial.3. yes; all real numbers except �1 and 14. vertical: x � 9; horizontal: y � 2

�3��3

z �12xyw

z �5xy2w

z �8xyw

z �20xyw

y �4.8x

y �150x

y �0.8x

y �84x

� 2i�32i�3�i�2i�2

�23 , �

12 , 2312

�14

121313

15

�10��2, �2, ��10�5��2, �2, ��5

��6, �6, �3�7��3, �3, ��7

�6�2, 2, ��6

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5. vertical: x � 1.5; horizontal: y � 06. vertical: and ;horizontal: y � 07. all real numbers except 0 and 6;vertical: x � 0, x � 6; horizontal: y � 08. all real numbers except 1 and �5;vertical: x � �5; horizontal: y � 0;hole at x � 19. all real numbers except �3 and �4;vertical: x � �3, horizontal: y � 3;hole at x � �4

10.

11.

12.

Lesson 8.31. 2. 3. 4.5. 6. 3x6 7. 3x � 9 8. 19. 10. x 11. x2 � x � 12

12. 13. 14.15. 16. 17.18.

Lesson 8.41. 2. 3. 4.5. 6. 7.8. 9. 10.

11. 12. 13.14. 15. 0 16. 17.18. 19. 20.21. 22.23.

Lesson 8.51. 3.5 2. 11 3. 7.5 4. 15 5. , �36. �2, �4 7. �2.5 8. 2 9. 7, 1

10. 0, 11. 6 12. 113. x � 2 or x � 4 14. x � 6 or x � 1215. �1 � x � 0 or x � 1 16. x � 2 or x � 317. 1 � x � 3 18. x � �3 or �2 � x � 219. x ��1 or � x � 120. �3 � x � 4 or x � 152

�13

�34

12

3 � 4y � 2x(x � 3)(y � 2)� 2(3x 2 � 2x � 5)(3x � 2)(x � 5)x � 3x

� x � 6(x � 3)(x � 3)2(x � 2)x � 42xx � 51x � 33x � 7x � 12x(x � 1)

x 2 � 3xx � 32x � 14x � 1x � 22x � 28x � 3�

1x � 5x � 3x � 8�

2x16x 2 � 162x 2 � 2x � 13x 2 � x � 65x � 23x 2 � 1256x3x � 7x 3 � 17x � 2212

3y3 (x � y)4xx � 53x � 21xx � 32x � 1x � 2

x 2 � 3x3x � 12x � 1x � 11x 2 � 5x � 504x � 2

x � 4x � 212x 23x � 23x � 2x � 7x � 73x 2

xO

y

2

–2–4

4

y = 1

x = 1

hole2–2–4 4

xO

y

12

y = 0x = 12

hole

xO

y

–4 84

–4–8

8

x = 4

y = 3

x � �7x � ��7

20 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

21. x � �1 or x � 122. �2.2 � x � 3 or x � 323. �4 � x � 0.5 or x � �4.524. x � 0.7 or 4 � x � 4.325. 0 � x � 226. x � �6.5 or 0 � x � 2.5 or x � 427. �1.4 � x � 0 or 1.4 � x � 2Lesson 8.6

1. x � 2.5 2. x � 4 3. x � �6 or x � 64. x � �2.5 or x � 2.5 5. x � 56. x � �3 or x � �17.

8.

9.

10. 4 11. 7.5 12. �6 13. �24 14. 4 15. �2 16. ; 12.4 feetLesson 8.7

1. 2 2. 3 3. 12|x|y2 4. 7xy35. 6. 7. 2x28. 4xy2 9. 10x4y2 10.

11. 12.13. 14. 15.16. 17.18. 432 19. 20. 21.22. 23.24.

Lesson 8.81. 95 2. 10 3. �4, 44. no solution 5.6. 2, 6 7. no solution 8. 2, 39. 5 10. 3 � x � 12 11. x � 14

12. 0 � x � 13. x � �0.5 or x � 2.514. x � 1 15. 8 � x � 13 16. no solution

813

��2 , �2

9�7 � 9�255�3 � 5�216 � 8�3

7�2 � 74�2�2�58 � 13�5�24 � 15�2

5 � �23 � 13�325 � 2�22x2y2 4�2xy22yz 3�3xyz

2y 5�x2yz3�y3�x

�2xy(2y)134x2(5x)123

�x2z�2

3�6V

πd �

xO

y

22

y = x2 – 2x – 5

y = 1 – √x + 6

y = 1 + √x + 6

y � 1 � �x � 6, y � 1 � �x � 6

x

y

–46 4 2

–2

4

y = 2 – x2

Oy = √2 – x

y = –√2 – x

y � ��2 � x, y � �2 � xxO

y

2 4 6

68

y = x2 – 6x + 8y = 3 + √x + 1

y = 3 – √x + 1

y � 3 � �x � 1, y � 3 � �x � 1

Algebra 2 21

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17. x � �6.5 18. � x � 13 19. x � 5.620. �1.8 � x � 1.5 21. �4 � x � 1.322. x � 5.8 23. x � 0.4 24. 0.4 � x � 0.8 or x � 6.225. �9 � x � �0.8 or x � 3.926. x � 3.4 27. x � 10.5Practice — Chapter 9Lesson 9.1

1.

parabola2.

circle

3.

hyperbola4. 13; (2.5, 6) 5. 10; (8, �2) 6.7.8.9.

10. (9, 14); 10π; 25π

11. (4.5, 20); 41π;12. (0, 8.5); 17π;13. (3.5, 1); ;14. (15, 5.5); ;15. (2.5, 2.5); ;Lesson 9.2

1.2.3. y � 1 � (x � 3)2x � 1 �

14 y2y �

112(x � 3)2

562π4π�562101π4π�101

145π4π�145289π4

1681π4

�13 � 3.61; (0, 0.5)12�2 � 16.97; (�2, �2)�85 � 9.22; (3, 4)2�101 � 20.10; (2, 3)

2

4

–44–4 –2 O x

yy � �3�x2 � 1, y � 3�x2 � 1

1010

–10–10 O x

yy � ��400 � x2, y � �400 � x2

22468

4–4 –2 O x

yy �

13x2

34

22 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

4.

5.

6.

7. 8.9. 10.

11. 12.

Lesson 9.31. (x � 5)2 � (y � 2)2 � 162. (x � 6)2 � (y � 3)2 �

3. x2 � (y � 4)2 � 254. x2 � y2 �

5. (x � 2)2 � (y � 1)2 � 6.256. (x � 3)2 � (y � 3)2 � 5767.

8.

9.

10. (x � 5)2 � (y � 8)2 � 1; C(5, 8); r � 111. (x � 11)2 � (y � 1)2 � 2; C(�11, 1);

Lesson 9.41.2. (x � 3)24 �

(y � 2)216 � 1x264 �

y236 � 1

r � �2

xO

y2

–2–4–6

2 6(3, 3) r = 3

xO

y

2468

2 4–2–4

r = 4 (0, 5)

xO

y

–8

8(0, 0) r = 168–8

916

254

y �120(x � 9)2x � 6 � �

18(y � 7)2y � 2 �

14(x � 5)2x �112 y2

x � 10 �18 y2y �

124x2

2246

4F(3, 2)

V(1, 2)x = –1 6

directrixO

y

x

22468

4

F(1, 5)V(1, 4)y = 3

–2–4directrix

O

y

x

224

–24F(1, 0)

x = –16

directrix

O

y

xV(0, 0)

Algebra 2 23

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3.4.

5.

6.

7.8.9.

10. ; center: (�3, 0)foci: (�3 � , 0) and (�3 � , 0);vertices: (�9, 0) and (3, 0); co-vertices:(�3, �3), and (�3, 3)

Lesson 9.51. 2.3.

4.

5. 6.7.8. ; center: (14, �3);vertices: (12, �3) and (16, �3);co-vertices: (14, �4) and (14, �2);foci: and

Lesson 9.61. (1, 6) and (4, 21) 2. none3. (�2.83, �2.83) and (2.83, 2.83)4. (�3, 0) and (3, 0)

(14 � �5, �3)(14 � �5, �3)

(x � 14)24 � (y � 3)2 � 1

(y � 1)225 �

(x � 1)2144 � 1

x29 �(y � 2)2

16 � 1x210 �y215 � 1

xO

y

–2–4–6 2 4 6

C (1, 1)

y = – x –34 14y = x +34 14

(1, 4)

(5, 1) (6, 1)(–4, 1) (–3, 1)

(1, –2)

x

y

–2–4 2 4C (0, 0)y = – x53 2

4

–2y = x53

(0, –√34)

(0, 3)

(0, –3)(–5, 0) (5, 0)

(0, √34)

y216 �

x225 � 1x29 �y216 � 1

3�33�3(x � 3)2

36 �y29 � 1

x2400 �y2

464 � 1x2109 �

y29 � 1

x2625 �y2

225 � 1

xO

y1

–2–4–6

2 4 6 8(1, –3) (7, –3)

(4, 2)(4, 1)

(4, –3)(4, –7)(4, –8)

xO

y

4

–8

4–4–8 8(7, 0)(0, 3)(0, 1)(1, –3)(–7, 0)

(0, –5)

(0, 7)

x

y

44–4–8 8

(0, √77)

(0, –√77)

(0, 9)

(0, 0)(–2, 0) (2, 0)

(0, –9)

(x � 3)225 �

(y � 2)29 � 1

24 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

5. (�2.24, �1.73), (�2.24, 1.73),(2.24, –1.73), and (2.24, 1.73)6. (�4.24, �2.45), (�4.24, 2.45),(4.24, �2.45), and (4.24, 2.45)7. none8. (�2.08, �1.44), (�2.08, 1.44),(2.08, �1.44), and (2.08, 1.44)9. none

10. parabola;

11. hyperbola;

Practice — Chapter 10Lesson 10.1

1. 2. 3.4. 5. 6.

7. 8. 9.10. 11.12. 13.14. 1,000,000 15. 676,00016. 456,976,000 17. 67,600,00018. 6,760,000Lesson 10.2

1. 5040 2. 2520 3. 840 4. 840 5. 479,001,6006. about 4.36 � 10107. about 2.18 � 10118. about 6.03 � 10139. 20,160 10. 2520 11. 10,080 12. 5040

13. 4,989,600 14. 907,200 15. 43,75816. 720 17. 5040 18. 3,628,800Lesson 10.3

1. 56 2. 120 3. 35 4. 6435 5. 846. 60 7. 40 8. 5 9. 40

10. about 33% 11. about 39%12. about 16% 13. combination14. combination 15. permutation16. permutationLesson 10.4

1. inclusive;2. mutually exclusive;3. inclusive;4. inclusive; 3952 � 75%

1926 � 73%613 � 46%

413 � 31%

85150 � 57%107150 � 71%43150 � 29%65150 � 43%

42150 � 28%16 � 17%2126 � 81%

13 � 33%1113 � 85%12 � 50%511 � 45%45 � 80%710 � 70%

xO

y

246

42 86

(x � 5)29 �

(y � 4)24 � 1

xO

y

2468

–242 86

y � 3 �14(x � 7)2

Algebra 2 25

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5. mutually exclusive;6. inclusive;7. 8.9. 10.

11. 12.13. 14.15.

Lesson 10.51. 0.02 � 2% 2. 0.08 � 8%3. 0.04 � 4% 4. 0.05 � 5%5. 0.008 � 0.8% 6. 0.01 � 1%7. 8.9. 10.

11. 12.13. 14.15. 16.17. 18.19. 20.

Lesson 10.61. 2. 3.4. 5. 6.7. 8. 9. 1 � 100%

10. 11. 12.13. 14. 15.

16. 17. 0.625 � 62.5%18. 19.20. 0.064 � 6.4% 21. 0.4 � 40%22. 0.5625 � 56.25%Lesson 10.71–6. Simulation results will vary.Practice — Chapter 11Lesson 11.1

1. 2.5, 5, 7.5, 10, 12.5, 152. 0, , 1, 1 , 2, 23. 13, 16, 21, 28, 37, 484. 20, 70, 220, 670, 2020, 60705. 1, 101, 201, 301, 401, 5016. �5, �15, �45, �135, �405, �12157. a1 � 1; an � 11an � 1; 14,641; 161,051;1,771,5618. a1 � 81; an � an � 1 � 3; 69, 66, 639. a1 � 2; an � �3an � 1; 162, �486, 1458

10. a1 � ; an � �4an � 1; 64, �256, 102411. a1 � 2; an � 3an � 1 � 5; 362, 1091, 327812. a1 � �2; an � 5an � 1 � 4; �1874; �9374;

�46,87413. 4.5 � 9 � 13.5 � 18 � 22.5 � 27 � 31.5;12614. 11 � 22 � 35 � 50 � 67; 18515. 210 16. �24 17. 1018. 1432.5 19. 397 20. 1911.4

14

121212

12 � 50%910 � 90%13 � 33%

12 � 50%45 � 80%15 � 20%25 � 40%35 � 60%14 � 25%

12 � 50%18 � 12.5%18 � 12.5%12 � 50%13 � 33%23 � 67%12 � 50%16 � 17%

13 � 33%112 � 8%18 � 12.5%124 � 4%49 � 44%136 � 3%14 � 25%19 � 11%

118 � 6%772 � 10%2536 � 69%316 � 18.75%

124 � 4%49144 � 34%

518 � 28%2236 � 61%59 � 56%

1336 � 36%1936 � 53%34 � 75%38 � 37.5%

58 � 62.5%58 � 62.5%4152 � 79%

2752 � 52%

26 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

Lesson 11.21. yes; d � 3 2. no 3. yes; d � 2.54. no 5. yes; d � 6. yes; d � �2.37. tn � 25 � 9n 8. tn � 8n � 239. tn � 3n � 10 10. tn � 12n � 37

11. tn � 11n � 2 12. tn � 9.9 � 1.3n13. 50, 150, 250, 350 14. 7.5, 10, 12.5, 1515. �20, �12, �4, 4 16. 25, 65, 105, 14517. 8.5, 9, 9.5, 10 18. �15, �27, �39, �5119. �5, 2, and 9 20. 52, 64, 76, and 8821. 62 and 74 22. 10.5, 14, and 17.523. 34, 28, and 22 24. �2, 4, 10, and 16Lesson 11.3

1. 350 2. 3. �165 4. 29055. 711 6. 1482 7. 25,425 8. 3609. 1300 10. 9555 11. 9555 12. 330

13. 990 14. 165 15. 239.25 16. �191417. 18. 84 19. �13220. 590 21. 228 22. 795 23. �351Lesson 11.4

1. no 2. yes; r � 0.4 3. yes; r �

4. yes; r � 1.5 5. yes; r � 6. no7. 18, �36, 72, �144 8. �4, �10, �25, �62.5, �156.259. 10, 5, 2.5, 1.25 10. 0.625 or �0.625

11. or12. or13.14. tn � �30(�0.2)n � 1

15. tn � 40(0.8)n � 116. tn � 17. tn �

18. tn �

19. 35 and 175 20. �14 and �721. 48, 192, and 768 or �48, 192, and �76822. 15, 18, and 21.6 or �15, 18, and �21.623. 60, 300, and 1500 or �63, 300, and

�150024. 12, 36, 108, and 324Lesson 11.5

1. 6,973,568,800 2. 86.4 3. �1474. 5. �1064.76. 6144 7. 1,572,864 8. 12,2859. 3,145,725 10. 378 11. 1,708,554.00

12. 1.67 13. 14,348,90614. Show the statement is true for n � 1:

13 � 1 and . Assume that thestatement is true for a natural number k.Then13 � 23 � 33 � … � k3 � and13 � 23 � 33 � … � k3 � (k � 1)3 �

.Thus, the statement is true for k � 1.(k � 1)2[(k � 1) � 1]2

4(k � 1)2(k � 2)24 �

(k � 1)2(k2 � 4k � 4)4 �

(k � 1)2[k2 � 4(k � 1)]4 �

k2(k � 1)2 � 4(k � 1)34 �

k2(k � 1)24 � (k � 1)3 �

k2(k � 1)24

12(1 � 1)24 � 1

54673125 �1.7

�14(�6)n

80(14 )n2(52 )n�1

tn � 250(25 )n�1�164014164014�32�232�2

23�

35

528�5 � 1180.64

1612

35

Algebra 2 27

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Lesson 11.61. does not exist 2. 1.2 3. 2.6254. 25 5. 4 6. does not exist7. 12.375 8. 3 9. 10. 11.5

11. does not exist 12. 50 13.14. 15. 16. 17. 18. 19.20. 21.

22. 23.

24. 25.

Lesson 11.71. sixth entry, row 7; 212. fourth entry, row 6; 203. seventh entry, row 8; 284. sixth entry, row 10; 2525. eleventh entry, row 13; 2866. sixth entry, row 12; 7927. 120 8. 1716 9. 6435 10. 153

11. 0.3125 12. about 0.66 13. about 0.0114. about 0.64 15. 0.5 16. 0.437517. about 0.23 18. about 0.8919. about 0.34 20. about 0.0921. about 0.66 22. about 0.34

Lesson 11.81. s5 � 5s4t � 10s3t 2 � 10s2t3 � 5st4 � t52. b6 � 6b5w � 15b4w2 � 20b3w3 �15b2w4 � 6bw5 � w63. 16 terms 4. 3 5. 3003r5s106. 3125x5 � 3125x4y � 1250x3y2 �250x2y3 � 25xy4 � y57. w4 � w3z � 6w2z2 � 16wz3 � 16z48. a5 � a4d � a3d2 � a2d3 �

ad4 � d59. 64m6 � 576m5q � 2160m4q2 �4320m3q3 � 4860m2q4 � 2916mq5 �729q6

10. about 0.20 11. about 0.26 12. about 0.74 13. about 0.11 14. about 0.61Practice — Chapter 12Lesson 12.1

1. 66; 62; 622. 24.571; 28; 283. 3.617; 3.6; 3.64. 622.833; 725; 7255. 2593.333; 1714; 16006. 24.05; 24.25; 25; all are reasonablemeasures because they are so close, butthe mean and median are better than themode because 6 of the 8 numbers are lessthan the mode.

154458135324052562431024

116

445 � ( 11000 )k��

k�111 � ( 11000 )k��

k�1

358 � ( 11000 )k��

k�115 � ( 1100 )k��

k�1

93 � ( 1100 )k��

k�17 � ( 1100 )k��

k�1

1027251112513334999379919

22291.81

28 Algebra 2

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7.

mean:8.

estimated mean: 10.75Lesson 12.2

1.

45; 64; flat2.

3.75; 2.3 and 4.8; mound-shaped

3.

4.

5.33.2%Passengercars

50%Trucks

15.9%Buses0.9%Motor-cycles

Motor VehicleRegistration by Type, 1994Number10 11 12 13 140

0.10.20.3

Probabili

ty

Number0 1.0 1.1 1.2 1.3 1.4123456

Frequen

cy

Stem Leaf 3|1 � 3.11 72 3, 33 1, 6, 94 0, 3, 8, 8

Stem Leaf 3|8 � 383 3, 7, 84 0, 2, 55 4, 6, 86 4, 4

16.05

Algebra 2 29

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Age (years) Tally Frequency14 315 316 417 618 2

No. of books Class mean Freq. Product1–5 3 8 24

6–10 8 4 3211–15 13 3 3916–20 18 1 1821–25 23 2 4625–30 28 2 56

Number Frequency1.0 51.1 21.2 31.3 61.4 4

Number Frequency Rel. Freq.10 8 16%11 12 24%12 7 14%13 11 22%14 12 24%

Lesson 12.31. Q1 � 2, Q2 � 7, Q3 � 8; range � 8;IQR � 62. Q1 � 10, Q2 � 14.5, Q3 � 17;range � 14; IQR � 73. Q1 � 12.1, Q2 � 23.9, Q3 � 38.5;range � 33.8; IQR � 26.44. minimum � 3, maximum � 17;Q1 � 5, Q2 � 10, Q3 � 15;range � 14; IQR � 10

5. minimum � 10, maximum � 39;Q1 � 16, Q2 � 23, Q3 � 35; range � 29;IQR � 19

6. the data for 19927. The quartiles and the extremes alldecreased, so it appears that the birthrates decreased in general.8. about 75%

Lesson 12.41. 32; 9.6 2. 19; 3. 55;4. 80; 22 5. 7.1; 2 6. 62; about 18.97. 16; 4 8. 721.6; about 26.99. 74,064; about 272.1 10. 5; about 2.2

11. 83.7856; about 9.2 12. ; about 0.313. 20 14. 4.3 15. 26.24 16. about 5.1

Lesson 12.51. 9.375% 2. 9.375% 3. 34.375%4. 10.9375% 5. 0.8% 6. 81.92%7. 6.4% 8. 94.208% 9. 0.672%

10. about 20.1% 11. 7.68% 12. 68.256%13. 31.104% 14. 17.92% 15. about 2.3%16. about 0.27% 17. about 99.7%18. about 1.1%Lesson 12.6

1. 0.5 2. 0.8849 3. 0.96414. 0.1554 5. 0.4772 6. 0.07937. 0.1359 8. 0.1586 9. 0.5403

10. 16% 11. 2% 12. 82% 13. 16%14. about 420 customers15. about 820 customersPractice — Chapter 13Lesson 13.1

1. ; 0.9756 2. ; 0.2195 3. ; 4.44444. ; 0.2195 5. ; 0.9756 6. ; 0.2257. ; 4.5556 8. ; 4.5556 9. ; 4.4444

10. 8.3; 57; 33 11. 11.6; 61; 29

12. 21.6; 34; 56 13. 59; 4.9; 4.214. 36; 4.9; 3.5 15. 56; 5.2; 4.3Lesson 13.2

1. –313; 47 2. 237; 57 3. �142; 38

4. 152; 28 5. 138; 42 6. �53; 53

7. 42; 42 8. 175; 5 9. 285; 75

10. 75 11. 33 12. 5

4094194199404041941

4099414041

0.08

16.35.3

8 12 16 20 24 28 32 36 40

2 4 6 8 10 12 14 16 18

30 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

13. 36 14. 40 15. 72 16. 2

17. 80 18. 60

19.20.21.22. 23. 24. 25.26. 27.

Lesson 13.31. 2.3. 4.5. 6.7. (0.743, 0.669) 8. (�0.629, 0.777)9. (�0.438, �0.899) 10. (0.574, �0.819)

11. (0.755, �0.656) 12. (�0.259, �0.966)13. 0; 1; 0 14.15. 16.17. 18. 19. 20.21. 22. 23. 24. �1 25. �2Lesson 13.4

1. radians 2. radians 3. radian4. radians 5. radians 6. radians 7. radians8. radians 9. 450 10. 165

11. 140 12. 195 13. 472.69

14. 103.13 15. 171.89 16. 28.65

17. 47.1 feet 18. 5.2 feet 19. 41.9 feet20. 10.5 feet 21. 50 feet 22. 80 feet23. 146 feet 24. 200 feet 25. 026. 27. 28. �1 29.30. 31. 0 32. 2Lesson 13.5

1. 1; 0.5; 0; �0.5; �1; �0.5; 0; 0.5; 1

2. 4.5; π radians 3. does not exist; π radians4. 1.2; 2π radians5. shift of 45 right;translation of 1.5 units up

�90° 180° 270°

1

–1

y = cos �

y = cos 2�

��22

�33��3�12

5π127π18�

17π367π125π6

π55π33π4

�3312�22��3�

2�33��32�32

�12 ; �32 ; �

�33�32 ; �12 ; ��3

��22 ; �22 ; �1

(2�3, 2)(�5�2, 5�2)(20�3, �20)(�

152 , �15�32 ) (6, 6�3)(5�22 , 5�22 )

3�7751343�398�895�

349�9797 ; 4�9797 ; 2.25; �979 ; �974 ; 492�55 ; �

�55 ; �2; �55 ; ��5; �12

2�1313 ; 3�1313 ; 23 ; �132 ; �133 ; 32

Algebra 2 31

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�90° 180° 270° 360°

–10123

y

6. shift of π units left;translation of 1 unit down

Lesson 13.61. �60 2. 120 3. 0 4. 0 5. �45

6. �60 7. 0 8. �1 9.10. radians 11. �45 12. radians13. 1.5459 radians 14. 0.0345 radians15. 2.5173 radians 16. 0.6831 radians17. �1.5446 radians 18. �0.5769 radians19. 9.6 20. 35.5 21. 48.1

Practice — Chapter 14Note: Throughout Chapter 14, answers mayvary slightly due to rounding, method ofcalculation, or order in which solutions were found.Lesson 14.1

1. 25.5 2. 24.8 3. 16.7 4. 29.7 5. 5.6 6. 28.97. m�C � 38, b � 4.4, c � 3.18. m�A � 35, c � 21.7, a � 12.79. m�C � 45, a � 25.8, c � 18.5

10. m�B � 44, a � 5.5, b � 4.011. m�C � 82, a � 15.5, c � 21.412. m�A � 75, a � 82.9, c � 78.413. one; m�B� 46.0, m�C� 70.0, c � 20.914. none15. two; (1) m�A � 46.2, m�C � 105.8,c � 53.3; (2) m�A � 133.8,m�C � 18.2, c � 17.316. 1532.7 feetLesson 14.2

1. SAS; a � 17.3 2. SAS; c � 28.23. SAS; a � 5.3 4. SSA; b � 9.8 5. SAS; c � 31.3 6. SAS; a � 13.9 7. m�A � 44.7, m�B � 38.3,m�C � 97.0

8. m�A � 78.3, m�B � 37.4,m�C � 64.3

9. m�A � 73.0, m�B � 61.2,m�C � 45.8

10. m�A � 45.8, m�B � 30.7,m�C � 103.5

11. m�A � 41.1, m�B � 80.3,m�C � 58.6

12. SAS; b � 131.2, m�A � 32.5,m�C � 37.5

13. SSS; m�A � 50.0, m�B � 98.7,m�C � 31.3

14. SSS; m�A � 104.5, m�B � 29.0,m�C � 46.5

15. SAS; a � 6.5, m�B � 20.7,m�C � 109.3

16. SSA; not possible17. 29.0, 75.5, and 75.5

π2π6�32

32 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.

xπ 2π

–2–1012y

π2 3π2

Lesson 14.31.2.

3. cos2 �

4.5. sin � 6. tan2 � 7. �1 8. tan �

9. sin � 10. sec �

11.

12. (1 � cos �) � sin �

13. �

Lesson 14.41. 2.3. 4.5. 6.7. 8.9. 10.

11. 12. 13.14. 15.16. 17. 18.

19.

20.

21.

22.

23.

24.

25. (�5.66, �1.41), (�9.19, �4.95),(�12.02, �2.12), (�8.49, 1.41)Lesson 14.51–4. Check students’ work.5–11. Answers may vary. Sample answers:

5. 6. 7. 2 cos2 �

8. tan2 � 9. tan � 10. cot �

11. 1 � sin � 12. 27 feetLesson 14.6

1. 30 � n(360), 150 � n(360),210 � n(360), and 330 � n(360)2. 45 � n(360) 3. 120 � n(360) 4. 120 � n(360) 5. 120 and 240

6. 210, 270, and 330

7. 45, 135, 225, and 315

8. 30, 150, 210, and 330

9. 10. π4 and 5π4π3 and 5π3

1 � cos �21 � cos �2

��0.370.93 �0.93�0.37 �

� 0.77�0.64 0.640.77 �

� 0.420.91 �0.910.42 ���0.71

�0.71 0.71� 0.71 �

��0.710.71 �0.71� 0.71 �

��0.50.87 �0.87�0.5 �

��24 �

�64��32�24 �

�64�24 �

�64��24 �

�64�

�32�22��24 �

�64�24 �

�64�64 ��24

�24 ��64�24 �

�64�24 �

�64��24 �

�64�

�24 ��64�

�24 ��64

�24 ��64�24 �

�64

1sin �sin � � (cos �)( cos �sin � ) � csc �

( 1sin ��

cos �sin � )� sin �

1sin �� sin �

cos2 �sin2 �

1�tan2 � � 1�y2x2 �

x2 � y2x2 �

r2x2 � sec2 �

x2 � y2y2 �

r2y2 �1 � cot2 � � 1 �x2y2 �

�x2r2 � �cos2 �

y2 � r2r2 �

y2r2 � 1 �(yr � 1)(yr � 1) �

(sin � � 1)(sin � � 1) �

cot �cos ��

xy xr �

xy �rx �

ry � csc �

Algebra 2 33

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11.12.13. 1.23, 1.91, 4.37, and 5.0514. 0.34, 0.52, 2.62, and 2.8015. 25.2

π2 , 7π6 , and 11π6π9 , 5π9 , 7π9 , 11π9 , 13π9 , and 17π9

34 Algebra 2

AnswersCopyright © by Holt, Rinehart and Winston. All rights reserved.