NAME ______________________________________________ DATE
1-1
____________ PERIOD _____
Skills Practice Expressions and Formulas
Find the value of each expression. 1. 18 2 3 27
2. 9 6 2 1 13
3. (3 8)2(4) 3 97
4. 5 3(2 12 2) 7
6. 3
7. (168 7)32 43 152
8. [3(5) 128 22]5 85
Lesson 1-1
6(7 5) 4
1 3
5. [9 10(3)] 7
1 2
Evaluate each expression if r 1, s 3, t 12, v 0, and w . 9. 6r 2s 0
10. 2st 4rs 84
11. w(s r) 2
12. s 2r 16v 1
13. (4s)2 144
14. s2r wt 3
15. 2(3r w) 7
16. 4
3v t 5s t
rv3 s
25 2
17. w[t (t r)]
18. 0 2
19. 9r2 (s2 1)t 105
20. 7s 2v 22
2w r
21. TEMPERATURE The formula K C 273 gives the temperature in kelvins (K) for a given temperature in degrees Celsius. What is the temperature in kelvins when the temperature is 55 degrees Celsius? 328 K 5 9
22. TEMPERATURE The formula C (F 32) gives the temperature in degrees Celsius for a given temperature in degrees Fahrenheit. What is the temperature in degrees Celsius when the temperature is 68 degrees Fahrenheit? 20C
©
Glencoe/McGraw-Hill
3
Glencoe Algebra 2
NAME ______________________________________________ DATE
1-2
____________ PERIOD _____
Skills Practice Properties of Real Numbers
Name the sets of numbers to which each number belongs. 1. 34 N, W, Z, Q, R
2. 525 Z, Q, R
3. 0.875 Q, R
4. N, W, Z, Q, R
5. 9 Z, Q, R
6. 30 I, R
12 3
Name the property illustrated by each equation. 8. 3a 0 3a
Comm. () 9. 2(r w) 2r 2w
Add. Iden. 10. 2r (3r 4r) (2r 3r) 4r
Distributive
5y1
Assoc. ()
11. 5y 1
12. 15x(1) 15x
Mult. Inv.
Mult. Iden.
13. 0.6[25(0.5)] [0.6(25)]0.5
14. (10b 12b) 7b (12b 10b) 7b
Assoc. ()
Comm. ()
Name the additive inverse and multiplicative inverse for each number.
1 15
15. 15 15, 4 4 5 5
5 4
17. ,
16. 1.25 1.25, 0.8 3 4
3 4 4 15
18. 3 3 ,
Simplify each expression. 19. 3x 5 2x 3 5x 2
20. x y z y x z 0
21. (3g 3h) 5g 10h 2g 13h
22. a2 a 4a 3a2 1 2a2 3a 1
23. 3(m z) 5(2m z) 13m 8z
24. 2x 3y (5x 3y 2z) 3x 2z
25. 6(2 v) 4(2v 1) 8 2v
26. (15d 3) (8 10d) 10d 3
©
Glencoe/McGraw-Hill
1 3
9
1 2
Glencoe Algebra 2
Lesson 1-2
7. 3 x x 3
NAME ______________________________________________ DATE
1-3
____________ PERIOD _____
Skills Practice Solving Equations
Write an algebraic expression to represent each verbal expression. 1. 4 times a number, increased by 7
2. 8 less than 5 times a number
4n 7
5n 8
3. 6 times the sum of a number and 5
4. the product of 3 and a number, divided by 9
3n 9
6(n 5)
5. 3 times the difference of 4 and a number 3(4 n) 6. the product of 11 and the square of a number 11n2 Write a verbal expression to represent each equation. 7–10. Sample answers 7. n 8 16
8. 8 3x 5
The difference of a number and 8 is 16.
are given.
The sum of 8 and 3 times a number is 5. y 3
9. b2 3 b
10. 2 2y
Three added to the square of a number is the number.
A number divided by 3 is the difference of 2 and twice the number.
11. If a 0.5b, and 0.5b 10, then a 10.
12. If d 1 f, then d f 1.
Transitive ()
Subtraction ()
13. If 7x 14, then 14 7x.
14. If (8 7)r 30, then 15r 30.
Symmetric ()
Substitution ()
Solve each equation. Check your solution.
1 2
15. 4m 2 18 4
16. x 4 5x 2
17. 3t 2t 5 5
18. 3b 7 15 2b
19. 5x 3x 24 3
20. 4v 20 6 34 5
22 5
2a 5
21. a 3 5
22. 2.2n 0.8n 5 4n 5
Solve each equation or formula for the specified variable.
I rt
23. I prt, for p p xy 2
25. A , for y y 2A x
©
Glencoe/McGraw-Hill
1 4
24. y x 12, for x x 4y 48
A 2 r 2 2 r
26. A 2r2 2rh, for h h
15
Glencoe Algebra 2
Lesson 1-3
Name the property illustrated by each statement.
NAME ______________________________________________ DATE
1-4
____________ PERIOD _____
Skills Practice Solving Absolute Value Equations
Evaluate each expression if w 0.4, x 2, y 3, and z 10. 1. 5w 2
2. 9y 27
3. 9y z 17
4. 17z 170
5. 10z 31 131
6. 8x 3y 2y 5x 21
7. 25 5z 1 24
8. 44 2x y 45
10. 3 1 6w 1.6
9. 24w 3.2
11. 3x 2y 4 4
12. 6.4 w 1 7
Solve each equation. Check your solutions. 13. y 3 2 {5, 1}
43 83
15. 3k 6 2 ,
16. 2g 6 0 {3}
17. 10 1 c {9, 11}
18. 2x x 9 {3, 3}
19. p 7 14
20. 23w 12 {2, 2}
21. 7x 3x 2 18 {4, 4}
22. 47 y 1 11 {4, 10}
1 2
12 56
23. 3n 2 ,
24. 8d 4d 5 13 {2, 2}
5 1 6 6
25. 56a 2 15 ,
©
Glencoe/McGraw-Hill
Lesson 1-4
14. 5a 10 {2, 2}
26. k 10 9
21
Glencoe Algebra 2
NAME ______________________________________________ DATE
1-5
____________ PERIOD _____
Skills Practice Solving Inequalities
Solve each inequality. Describe the solution set using set-builder or interval notation. Then, graph the solution set on a number line. z 4
1. 2 {zz 8} or (∞, 8] 9
8
7
6
5
4
3
2
4
1
3. 16 3q 4 {qq 4} or (4, ∞) 1 0
1
2
3
4
5
6
3
2
1 0
1
2
3
4
2
1 0
1
2
3
4
5
2
1 0
1
2
3
4
3
2
1 0
1
2
3
4
6. 4b 9 7 {bb 4} or (∞, 4] 2
4
7. 2z 9 5z {zz 3} or (3, ∞)
3
4. 20 3s 7s {ss 2} or (∞, 2)
7
5. 3x 9 {xx 3} or [3, ∞) 4
2. 3a 7 16 {aa 3} or (∞, 3]
1 0
1
2
3
4
5
6
8. 7f 9 3f 1 {ff 2} or (2, ∞)
6
4
3
2
1 0
1
2
3
4
7 2
7 2
9. 3s 8 5s {ss 1} or [1, ∞) 10. 7t (t 4) 25 tt or ∞, 4
3
2
1 0
1
2
3
4
4
11. 0.7m 0.3m 2m 4 {mm 4}
3
2
1 0
1 0
1
2
3
4
5
4
or (3.4, ∞)
2
1 0
1
2
3
4
5
3
4
6
13. 1.7y 0.78 5 {yy 3.4}
2
3 4
∞, 34
12. 4(5x 7) 13 xx or
or (∞, 4] 2
1
3
2
1 0
1
2
3
4
14. 4x 9 2x 1 {xx 5} or (5, ∞)
6
1 0
1
2
3
4
5
6
7
Define a variable and write an inequality for each problem. Then solve.
16. The difference of three times a number and 16 is at least 8. 3n 16 8; n 8
1 2
17. One half of a number is more than 6 less than the same number. n n 6; n 12 18. Five less than the product of 6 and a number is no more than twice that same number.
5 6n 5 2n; n 4
©
Glencoe/McGraw-Hill
27
Glencoe Algebra 2
Lesson 1-5
15. Nineteen more than a number is less than 42. n 19 42; n 23
NAME ______________________________________________ DATE
1-6
____________ PERIOD _____
Skills Practice
Write an absolute value inequality for each of the following. Then graph the solution set on a number line. 1. all numbers greater than or equal to 2 or less than or equal to 2 n 2 4
3
2
1 0
1
2
3
8
4
3. all numbers less than 1 or greater than 1 n 1 4
3
2
1 0
1
2
3
2. all numbers less than 5 and greater than 5 n 5 6
4
2 0
2
4
6
8
4. all numbers between 6 and 6 n 6
8
4
6
4
2 0
2
4
6
8
Write an absolute value inequality for each graph.
n 1
5. 4
3
2
1 0
1
2
3
4
4
3
2
1 0
1
2
3
4
n 3
7.
n 4
6. 4
3
2
1 0
1
2
3
4
4
3
2
1 0
1
2
3
4
n 2.5
8.
Solve each inequality. Graph the solution set on a number line. 9. 2c 1 5 or c 0 {cc 2 4
3
2
1 0
1
2
3
4
or c 0}
11. 10 5x 5 {x2 x 1} 4
3
2
1 0
1
2
3
10. 11 4y 3 1 {y2 y 1} 4
2
1 0
1
2
3
4
12. 4a 8 or a 3 {aa 2 4
4
13. 8 3x 2 23 {x2 x 7}
3
3
2
1 0
1
2
3
4
or a 3}
14. w 4 10 or 2w 6 all real
numbers 0
1
2
3
4
5
6
7
15. t 3 {tt 3 or t 3} 4
3
2
1 0
1
2
3
3
2
1 0
1
2
3
4 ©
2 0
2
4
Glencoe/McGraw-Hill
6
4
2
1 0
1
2
3
4
3
2
1 0
1
2
3
4
2
3
4
18. p 2 2 4
4
19. n 5 7 {n2 n 12}
3
16. 6x 12 {x2 x 2}
4
17. 7r 14 {rr 2 or r 2} 4
4
8
3
2
1 0
1
20. h 1 5 {hh 6 or h 4}
8 10 12
8
33
6
4
2 0
2
4
6
8
Glencoe Algebra 2
Lesson 1-6
Solving Compound and Absolute Value Inequalities
NAME ______________________________________________ DATE
2-1
____________ PERIOD _____
Skills Practice Relations and Functions
Determine whether each relation is a function. Write yes or no.
3.
D
R
100 200 300
50 100 150
x
y
1
2
2
4
3
6
yes
2.
D
no
R 1
3 5
yes
4.
Lesson 2-1
1.
no
y
x
O
Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function. 5. {(2, 3), (2, 4), (2, 1)}
6. {(2, 6), (6, 2)} y
y
x
O
x
O
D {2}, R {3, 1, 4}; no 7. {(3, 4), (2, 4), (1, 1), (3, 1)}
D {2, 6}, R {2, 6}; yes 8. x 2 y
y
O O
x
x
D {3, 2, 1, 3}, R {1, 4}; yes
D {2}, R all reals; no
Find each value if f(x) 2x 1 and g(x) 2 x2. 9. f(0) 1
10. f(12) 23
11. g(4) 14
12. f(2) 5
13. g(1) 1
14. f(d) 2d 1
©
Glencoe/McGraw-Hill
59
Glencoe Algebra 2
NAME ______________________________________________ DATE
2-2
____________ PERIOD _____
Skills Practice Linear Equations
State whether each equation or function is linear. Write yes or no. If no, explain your reasoning. 1. y 3x
2. y 2 5x
yes
yes
3. 2x y 10
4. f(x) 4x2
yes
No; the exponent of x is not 1.
3 x
1 3
5. y 15
6. x y 8
No; x is in a denominator. 7. g(x) 8
yes 8. h(x)
No; x is inside a square root.
Write each equation in standard form. Identify A, B, and C. 9. y x x y 0; 1, 1, 0
10. y 5x 1 5x y 1; 5, 1, 1
11. 2x 4 7y 2x 7y 4; 2, 7, 4
12. 3x 2y 2 3x 2y 2; 3, 2, 2
13. 5y 9 0 5y 9; 0, 5, 9
14. 6y 14 8x 4x 3y 7; 4, 3, 7
Find the x-intercept and the y-intercept of the graph of each equation. Then graph the equation. 15. y 3x 6 2, 6
16. y 2x 0, 0 y
y O
x
O
17. x y 5 5, 5
18. 2x 5y 10 5, 2
y
y
O O
©
Glencoe/McGraw-Hill
x
x
x
65
Glencoe Algebra 2
Lesson 2-2
yes
x 3
NAME ______________________________________________ DATE
2-3
____________ PERIOD _____
Skills Practice Slope
Find the slope of the line that passes through each pair of points.
2 3
2. (0, 2), (3, 0)
1. (1, 5), (1, 3) 4
3 4
3. (1, 9), (0, 6) 3
4. (8, 5), (4, 2)
5. (3, 5), (3, 1) undefined 6. (2, 2), (10, 2) 0
7. (4, 5), (2, 7) 1
8. (2, 4), (3, 2)
6 5
9. (5, 2), (3, 2) 0
Graph the line passing through the given point with the given slope. 10. (0, 4), m 1
11. (2, 4), m 1 y
y O
x
12. (3, 5), m 2
13. (2, 1), m 2 y
y
O
O
x
x
Lesson 2-3
O
x
Graph the line that satisfies each set of conditions. 14. passes through (0, 1), perpendicular to 1 a line whose slope is 3
15. passes through (0, 5), parallel to the graph of y 1 y
y O O
x
x
16. HIKING Naomi left from an elevation of 7400 feet at 7:00 A.M. and hiked to an elevation of 9800 feet by 11:00 A.M. What was her rate of change in altitude? 600 ft/h
©
Glencoe/McGraw-Hill
71
Glencoe Algebra 2
NAME ______________________________________________ DATE
2-4
____________ PERIOD _____
Skills Practice Writing Linear Equations
State the slope and y-intercept of the graph of each equation. 2. y x 3 , 3
3 4
2 3
2 3
3 5
3 5
1. y 7x 5 7, 5
4. 3x 4y 4 , 1
3. y x , 0
3 2
4 7
5. 7y 4x 7 , 1
6. 3x 2y 6 0 , 3
7. 2x y 5 2, 5
8. 2y 6 5x , 3
5 2
Write an equation in slope-intercept form for each graph. 9.
10.
y
11.
y
(1, 2)
(0, 3) x
O
y
x
O (–3, –1)
x
O
(4, –1)
(–1, –4) (3, –3)
y 3x 1
y 1
y 2x 3
Write an equation in slope-intercept form for the line that satisfies each set of conditions. 12. slope 3, passes through (1, 3)
13. slope 1, passes through (0, 0)
y x
14. slope 2, passes through (0, 5)
15. slope 3, passes through (2, 0)
y 2x 5 16. passes through (1, 2) and (3, 1)
y 3x 6 17. passes through (2, 4) and (1, 8)
3 7 y x 2
y 4x 4
2
18. x-intercept 2, y-intercept 6
5 2
19. x-intercept , y-intercept 5
y 3x 6
y 2x 5 1 3
20. passes through (3, 1), perpendicular to the graph of y x 4. y 3x 10
©
Glencoe/McGraw-Hill
77
Glencoe Algebra 2
Lesson 2-4
y 3x 6
NAME ______________________________________________ DATE
2-5
____________ PERIOD _____
Skills Practice Modeling Real-World Data: Using Scatter Plots
For Exercises 1–3, complete parts a–c for each set of data. a. Draw a scatter plot. b. Use two ordered pairs to write a prediction equation. c. Use your prediction equation to predict the missing value.
2.
3.
©
1a.
y
x
y
1
1
12
3
5
9
4
7
6
6
11
7
12
8
15
10
?
x
y
5
9
32
10
17
24
20
22
16
25
30
35
38
40
44
50
?
x
y
1
16
2
16
24
3
?
18
4
22
5
30
7
34
8
36
15
3 0
1
2
3
4
5
6
7
8 x
1b. Sample answer using (1, 1) and (8, 15): y 2x 1 1c. Sample answer: 19
2a.
y 40
8 0
5 10 15 20 25 30 35 40 x
2b. Sample answer using (5, 9) and (40, 44): y x 4 2c. Sample answer: 54
3a.
y 36 30
12 6 0
1
2
3
4
5
6
7
8 x
3b. Sample answer using (2, 16) and (7, 34): y 3.6x 8.8 3c. Sample answer: 19.6
Glencoe/McGraw-Hill
83
Glencoe Algebra 2
Lesson 2-5
1.
NAME ______________________________________________ DATE
2-6
____________ PERIOD _____
Skills Practice
Identify each function as S for step, C for constant, A for absolute value, or P for piecewise. 1.
2.
y
x
O
3.
y
y
x
O
x
O
S
C
A
Graph each function. Identify the domain and range. 4. f(x) x 1
5. f(x) x 3 f(x)
f(x)
x
O x
O
D all reals, R all integers
D all reals, R all integers 7. f(x) x 1
6. g(x) 2x
f(x)
g(x)
D all reals, R nonnegative reals 8. f(x)
x2 ifif xx 00
D all reals, R {yy 1} 9. h(x)
3x if x1 if x1> 1 h(x)
f(x)
O
x O
D all reals, R {yy 0 or y 2} ©
Glencoe/McGraw-Hill
x
O
x
O
x
D {xx 1 or x 1}, R {yy 2} 89
Glencoe Algebra 2
Lesson 2-6
Special Functions
NAME ______________________________________________ DATE
2-7
____________ PERIOD _____
Skills Practice Graphing Inequalities
Graph each inequality. 2. y x 2 y
3. x y 4 y
y
O
x
O
x O
4. x 3 y
5. 2 y x y
6. y x y
y
O O
x
8. 9x 3y 6 0
y
10. y 7 9
y
O
x
Glencoe/McGraw-Hill
O
y
y
x
x
12. y x
11. x 5
y
©
x
9. y 1 2x
y
x
O
O
x
7. x y 2
O
x
Lesson 2-7
1. y 1
O
95
x
O
x
Glencoe Algebra 2
NAME ______________________________________________ DATE
3-1
____________ PERIOD _____
Skills Practice Solving Systems of Equations By Graphing
Solve each system of equations by graphing. 2. y 3x 6
y 0 (2, 0)
1 2
y x 1 (2, 2)
y 2x 4 (2, 0)
y
y
y
x
O
3. y 4 3x
4. y 4 x
5. y 2x 2 1 y x 5 3
y x 2 (3, 1) y
6. y x
(3, 4)
y 3x 4 (1, 1) y
y
O
x
O
x
O
x x
O x
O
7. x y 3 x y 1 (2, 1)
8. x y 4 2x 5y 8 (4, 0)
y
9. 3x 2y 4 2x y 1 (2, 5) y
y O
x
x
O x
O
Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. 10. y 3x y 3x 2
11. y x 5 2x 2y 10
y
y
y O
O
12. 2x 5y 10 3x y 15 x
x 2 O
inconsistent ©
Glencoe/McGraw-Hill
consistent and dependent 121
x
2
consistent and independent Glencoe Algebra 2
Lesson 3-1
1. x 2
NAME ______________________________________________ DATE
3-2
____________ PERIOD _____
Skills Practice Solving Systems of Equations Algebraically
Solve each system of equations by using substitution. 1. m n 20 m n 4 (8, 12)
2. x 3y 3 4x 3y 6 (3, 2)
3. w z 1 2w 3z 12 (3, 2)
4. 3r s 5 2r s 5 (2, 1)
5. 2b 3c 4 b c 3 (13, 10)
6. x y 1 2x 3y 12 (3, 2)
Solve each system of equations by using elimination.
10. 2f 3g 9 f g 2 (3, 1)
8. 2j k 3 3j k 2 (1, 1)
9. 3c 2d 2 3c 4d 50 (6, 8)
11. 2x y 1 x 2y 3 (1, 1)
12. 2x y 12 2x y 6 no solution
Solve each system of equations by using either substitution or elimination. 13. r t 5 2r t 4 (1, 6)
14. 2x y 5 1 4x y 2 , 4
15. x 3y 12 2x y 11 (3, 5)
16. 2p 3q 6 2p 3q 6 (3, 0)
17. 6w 8z 16 3w 4z 8
18. c d 6 c d 0 (3, 3)
19. 2u 4v 6 u 2v 3 no solution
20. 3a b 1 3a b 5 (1, 2)
21. 2x y 6 3x 2y 16 (4, 2)
22. 3y z 6 3y z 6 (2, 0)
23. c 2d 2 2c 5d 3 (4, 1)
24. 3r 2s 1 2r 3s 9 (3, 5)
infinitely many
25. The sum of two numbers is 12. The difference of the same two numbers is 4. Find the numbers. 4, 8 26. Twice a number minus a second number is 1. Twice the second number added to three times the first number is 9. Find the two numbers. 1, 3
©
Glencoe/McGraw-Hill
127
Glencoe Algebra 2
Lesson 3-2
7. 2p q 5 3p q 5 (2, 1)
NAME ______________________________________________ DATE
3-3
____________ PERIOD _____
Skills Practice Solving Systems of Inequalities by Graphing
Solve each system of inequalities by graphing. 2. x 3 y 3
3. x 2 x 4 no solution
y
O
y
y
x
4. y x y x
O
x
5. y 4x y 3x 2 y
O
O
x
6. x y 1 3x y 4 y
y
x
7. y 3 x 2y 12
O
x
8. y 2x 3 yx2
y
O
x
Lesson 3-3
1. x 1 y 1
9. x y 4 2x y 4 y
y
2 O
2
x
O
x
O
x
Find the coordinates of the vertices of the figure formed by each system of inequalities. 10. y 0 x0 y x 1
(0, 0), (0, 1), (1, 0)
©
Glencoe/McGraw-Hill
11. y 3 x y3 x 5
(0, 3), (5, 3), (5, 8)
133
12. x 2 yx2 x y 2 (2, 4),
(2, 4), (2, 0)
Glencoe Algebra 2
NAME ______________________________________________ DATE
3-4
____________ PERIOD _____
Skills Practice Linear Programming
Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. 2. x 1 y6 yx2 f(x, y) x y
3. x 0 y0 y7x f(x, y) 3x y y
y
y
O
x
x
O
x
O
max.: 2, min.: 5
max.: 9, min.: 3 4. x 1 xy6 f(x, y) x 2y
max.: 21, min.: 0
5. y 2x y6x y6 f(x, y) 4x 3y
y
6. y x 2 y 3x 2 yx4 f(x, y) 3x 5y y
y
O O
max.: 13, no min.
x
x
x
O
no max., min.: 20
max.: 22, min.: 2
7. MANUFACTURING A backpack manufacturer produces an internal frame pack and an external frame pack. Let x represent the number of internal frame packs produced in one hour and let y represent the number of external frame packs produced in one hour. Then the inequalities x 3y 18, 2x y 16, x 0, and y 0 describe the constraints for manufacturing both packs. Use the profit function f(x) 50x 80y and the constraints given to determine the maximum profit for manufacturing both backpacks for the given constraints. $620
©
Glencoe/McGraw-Hill
139
Glencoe Algebra 2
Lesson 3-4
1. x 2 x5 y1 y4 f(x, y) x y
NAME ______________________________________________ DATE
3-5
____________ PERIOD _____
Skills Practice Solving Systems of Equations in Three Variables
Solve each system of equations. 1. 2a c 10 (5, 5, 20) b c 15 a 2b c 5
2. x y z 3 (0, 2, 1) 13x 2z 2 x 5z 5
3. 2x 5y 2z 6 (3, 2, 1) 5x 7y 29 z1
4. x 4y z 1 no solution 3x y 8z 0 x 4y z 10
5. 2z 6 (2, 1, 3) 2x 3y z 2 x 2y 3z 9
6. 3x 2y 2z 2 (2, 1, 3) x 6y 2z 2 x 2y 0
7. x 5z 5 (0, 0, 1) y 3x 0 13x 2z 2
8. 3r 2t 1 (1, 6, 2) 4r s 2t 6 r s 4t 3 10. 5m 3n p 4 (2, 3, 5) 3m 2n 0 2m n 3p 8
11. 2x 2y 2z 2 infinitely many 2x 3y 2z 4 x y z 1
12. x 2y z 4 (1, 2, 1) 3x y 2z 3 x 3y z 6
13. 3x 2y z 1 (5, 7, 0) x y z 2 5x 2y 10z 39
14. 3x 5y 2z 12 infinitely many x 4y 2z 8 3x 5y 2z 12
15. 2x y 3z 2 (1, 3, 1) x y z 3 3x 2y 3z 12
16. 2x 4y 3z 0 (3, 0, 2) x 2y 5z 13 5x 3y 2z 19
17. 2x y 2z 2 (1, 2, 3) 3x 3y z 0 xyz2
18. x 2y 2z 1 infinitely many x 2y z 6 3x 6y 6z 3
19. The sum of three numbers is 18. The sum of the first and second numbers is 15, and the first number is 3 times the third number. Find the numbers. 9, 6, 3
©
Glencoe/McGraw-Hill
145
Glencoe Algebra 2
Lesson 3-5
9. x y 3z 3 no solution 2x 2y 6z 6 y 5z 3
NAME ______________________________________________ DATE
4-1
____________ PERIOD _____
Skills Practice Introduction to Matrices
3 2 4 1. 1 4 0 2 3
2. [0 15] 1 2
3 2 3. 1 8 2 2
6 1 2 4. 3 4 5 3 3 2 7 9
9 3 3 6 5. 3 4 4 5 2 4
1 6. 1 1 4 1 3
Lesson 4-1
State the dimensions of each matrix.
Solve each equation. 7. [5x 3y] [15 12] (3, 4)
7x 14 9. 14 2y (2, 7)
8. [3x 2] [7] 3
10. [2x 8y z] [10 16 1] (5, 2, 1)
8 x 4 11. 2y 8 2 (4, 5)
20 10x 12. 56 6y 32 (2, 4)
5x 20 13. 24 8y (4, 3)
3x 2 5x 2 14. 7y 2 3y 10 (0, 2)
4x 1 3x 15. 9y 5 y 3 (1, 1)
16.
x 9 17. 16 4y (9, 4, 3) 3z 9
5x 4x 1 13 (1, 4, 0) 18. 4y 3 8z 4z
2x 6y 19. y 2 x (3, 1)
x 4y 20. 3y x 3 (12, 3)
©
Glencoe/McGraw-Hill
3x 1 18 7 2y 4 12 4z 12 28 (2, 11, 7)
171
Glencoe Algebra 2
NAME ______________________________________________ DATE
4-2
____________ PERIOD _____
Skills Practice Operations with Matrices
Perform the indicated matrix operations. If the matrix does not exist, write impossible. 5] [9
8 3 0 7 8 10 2. 1 1 6 2 7 3
1]
4 3. [3 1 6] 1 impossible 2
5. 3[9 4 3] [27
8 4 5 1 2 9 9 2 14 4. 1 8 6 4 6 4 5 14 2
12 9]
2 5 1 1 7. 2 5 9 1 1
6. [6 3] 4[4 7] [10 31]
5 9 9 17
4 6 6 5 8 40 9. 5 10 1 2 3 2 44 1 1 1 1 0
3
5
8 8. 3 0 3
2 4 2 10
16 8 49
5 3 1 3 1 1 10. 3 4 7 5 2 6 6 3
7 5 1 24 9 21
2 2 2 3 4 Use A 3 4 3 , B 1 2 , and C 3 1 to find the following. 11. A B
5 4 5 1
12. B C
13. B A
1 0 3 5
14. A B C
15. 3B
6 6 3 6
17. A 4C
©
15 14 8 1
Glencoe/McGraw-Hill
16. 5C
5 2 2 3
15 20 15 5
18. 2B 3A
177
2 8 8 2
13 10 14 5
Glencoe Algebra 2
Lesson 4-2
1. [5 4] [4
NAME ______________________________________________ DATE
4-3
____________ PERIOD _____
Skills Practice Multiplying Matrices
Determine whether each matrix product is defined. If so, state the dimensions of the product. 1. A2 5 B5 1 2 1
2. M1 3 N3 2 1 2
3. B3 2 A3 2 undefined
4. R4 4 S4 1 4 1
5. X3 3 Y3 4 3 4
6. A6 4 B4 5 6 5
Find each product, if possible.
5 6 2 5 28 19 8. 2 1 3 1 7 9
2 7. [3 2] 1 [8]
3 5
3 1 3 10. 2 1 1 not possible
0 1 11. [3 4] 2 2 [8 11]
2 3 2 1 12. 3 [2 3 2] 6 9 6
5 4 13. 6 8 not possible 3
2 2 0 3 6 5 14. 4 15 3 1 3 0
4 4 3 3 12 20 15. 2 1 0 8 2 6 2 3
0 1 1 2 4 16. 1 1 0 2 2 4
6
6 12 3 9
0
1 3 2 3 1 , and scalar c 2 to determine whether the Use A 2 2 1 , B 5 1 , C 1 0 following equations are true for the given matrices. 17. (AC)c A(Cc) yes
18. AB BA no
19. B(A C) AB BC no
20. (A B)c Ac Bc yes
©
Glencoe/McGraw-Hill
183
Glencoe Algebra 2
Lesson 4-3
1 3 3 9. 1 1 2
NAME ______________________________________________ DATE
4-4
____________ PERIOD _____
Skills Practice Transformations with Matrices
For Exercises 1–3, use the following information. Triangle ABC with vertices A(2, 3), B(0, 4), and C(3, 3) is translated 3 units right and 1 unit down. 1. Write the translation matrix.
3 3 3 1 1 1
y
x
O
2. Find the coordinates of ABC. A(5, 2), B(3, 3), C(0, 4) 3. Graph the preimage and the image. For Exercises 4–6, use the following information. The vertices of RST are R(3, 1), S(2, 1), and T(1, 3). The triangle is dilated so that its perimeter is twice the original perimeter. 4. Write the coordinates of RST in a vertex matrix.
y
3 2 1 1 1 3
5. Find the coordinates of the image RST.
x
O
R(6, 2), S(4, 2), T (2, 6)
6. Graph RST and RST.
For Exercises 7–10, use the following information. The vertices of DEF are D(4, 0), E(0, 1), and F(2, 3). The triangle is reflected over the x-axis. 7. Write the coordinates of DEF in a vertex matrix.
4 0 2 0 1 3
8. Write the reflection matrix for this situation.
1 0 0 1
y
x
O
Lesson 4-4
9. Find the coordinates of DEF. D(4, 0), E (0, 1), F (2, 3) 10. Graph DEF and DEF. For Exercises 11–14, use the following information. Triangle XYZ with vertices X(1, 3), Y(4, 1), and Z(2, 5) is rotated 180º counterclockwise about the origin. 11. Write the coordinates of the triangle in a vertex matrix.
y
1 4 2 1 5 3
12. Write the rotation matrix for this situation.
1 0 0 1
O
x
13. Find the coordinates of XYZ.
X(1, 3), Y(4, 1), Z(2, 5)
14. Graph the preimage and the image. ©
Glencoe/McGraw-Hill
189
Glencoe Algebra 2
NAME ______________________________________________ DATE
4-5
____________ PERIOD _____
Skills Practice Determinants
Find the value of each determinant. 5 2 1. 1 3 13
10 9 2. 5 8 35
1 6 3. 1 7 1
2 5 4. 3 1 13
0 9 5. 5 8 45
3 12 6. 2 8 0
5 2 7. 8 6 14
3 1 8. 8 7 13
9 2 9. 4 1 1
5 10. 1 1 6 11
1 3 11. 3 4 5
12 12. 1
4 4 52
5 13. 3 6 11 3
1 3 14. 5 2 17
1 14 15. 5 2 68
2 16. 1 0 4 4
2 2 10 17. 1 4
6 18. 1 2 5 17
Evaluate each determinant using expansion by minors. 1 1 1 2
6 1 2 20. 5 1 3
1 1 2 2
2 6 21. 3 5 2 1
1 1 1 2
Evaluate each determinant using diagonals.
2 1 2 22. 3 2 3
©
6 5 3 1
Glencoe/McGraw-Hill
3 1 0 23. 1 3 2
2 4 8 0
195
3 2 24. 1 1 3 1
2 4 40 0
Glencoe Algebra 2
Lesson 4-5
2 1 2 19. 3 2 3
NAME ______________________________________________ DATE
4-6
____________ PERIOD _____
Skills Practice Cramer’s Rule
1. 2a 3b 6 2a b 2 (3, 4)
2. 3x y 2 2x y 3 (1, 1)
3. 2m 3n 6 m 3n 6 (0, 2)
4. x y 2 2x 3y 9 (3, 1)
5. 2x y 4 7x 2y 3 (1, 2)
6. 3r s 7 5r 2s 8 (6, 11)
7. 4g 5h 1 g 3h 2 (1, 1)
8. 7x 5y 8 9x 2y 3 (1, 3)
9. 3x 4y 2 4x 3y 12 (6, 4)
Lesson 4-6
Use Cramer’s Rule to solve each system of equations.
10. 2x y 5 3x y 5 (2, 1)
11. 3p 6q 18 2p 3q 5 (4, 1)
12. x 2y 1 2x y 3 (1, 1)
13. 5c 3d 5 2c 9d 2 (1, 0)
14. 5t 2v 2 2t 3v 8 (2, 4)
15. 5a 2b 14 3a 4b 11 (3, 0.5)
16. 65w 8z 83 9w 4z 0 (1, 2.25)
17. GEOMETRY The two sides of an angle are contained in the lines whose equations are 3x 2y 4 and x 3y 5. Find the coordinates of the vertex of the angle. (2, 1) Use Cramer’s Rule to solve each system of equations. 18. a b 5c 2 3a b 2c 3 4a 2b c 3 (2, 5, 1)
19. x 3y z 5 2x 5y z 12 x 2y 3z 13 ( 3, 2, 4)
20. 3c 5d 2e 4 2c 3d 4c 3 4c 2d 3e 0 (1, 1, 2)
21. r 4s t 6 2r s 3t 0 3r 2s t 4 (1, 1, 1)
©
Glencoe/McGraw-Hill
201
Glencoe Algebra 2
NAME ______________________________________________ DATE
4-7
____________ PERIOD _____
Skills Practice Identity and Inverse Matrices
1 0 1 0 1. X 1 1, Y 1 1 yes
3 2 3 1 2. P 1 1, Q 1 2 yes
0 1 0 1 3. M 0 3, N 0 3 no
2 5 2 5 4. A 1 2, B 1 2 yes
0 1 0 7 7 5. V 7 0, W 1 yes 0 7
1 2 1 4 3 3 6. X 1 2, Y 1 1 yes 6 6
4 3 7. G 1 2, H
2 3 11 11 1 4 yes 11 11
4 4 0.125 0.125 8. D 4 4, E 0.125 0.125 no
Find the inverse of each matrix, if it exists.
1 0 2 0 2 9. 4 0 0 8 4
1 1 2 1 10. 3 2 1 3
9 3 11. 6 2 no inverse exists
4 2 4 1 0 12. 6 0 24 6 2
1 3 1 1 1 13. 3 3 6 3 1
3 6 14. 1 2 no inverse exists
1 1 1 1 1 15. 1 1 2 1 1
1 2 5 4 5 16. 1 2 13 1 4
1 0 7 0 7 17. 7 0 49 7 0
10 8 18. 5 4 no inverse exists
1 8 8 10 8 19. 10 8 160 10 10
2 0 1 2 0 20. 0 2 4 0 2
©
Glencoe/McGraw-Hill
207
Glencoe Algebra 2
Lesson 4-7
Determine whether each pair of matrices are inverses.
NAME ______________________________________________ DATE
4-8
____________ PERIOD _____
Skills Practice Using Matrices to Solve Systems of Equations
Write a matrix equation for each system of equations. 1. x y 5 2x y 1
2. 3a 8b 16 4a 3b 3
1 1 x 5 2 1 y 1
3 8 a 16 4 3 b 3
3. m 3n 3 4m 3n 6
4. 2c 3d 6 3c 4d 7
1 3 m 3 4 3 n 6
2 3 c 6 3 4 d 7
5. r s 1 2r 3s 12
6. x y 5 3x 2y 10
1 1 x 5 3 2 y 10
7. 6x y 2z 4 3x 2y z 10 xyz3
Lesson 4-8
1 1 r 1 3 s 12 2
8. a b c 5 3a 2b c 0 2a 3b 8
6 1 2 x 4 2 1 y 10 3 1 1 z 3 1
1 1 1 a 5 2 1 b 0 3 3 0 c 8 2
Solve each matrix equation or system of equations by using inverse matrices.
1 3 w 7 9. 4 3 z 1 (2, 3)
4 3 x 6 10. 1 3 y 3 (3, 2)
5 8 a 1 11. 3 1 b 7 (3, 2)
7 3 m 15 (3, 2) 12. 5 4 n 23
1 3 12 c 25 13. 2 6 d 12 7, 3
5 5 6 m 15 14. 12 6 n 2 1, 3
15. p 3q 6 2p 3q 6 (0, 2)
16. x 3y 2 4x 5y 1 (1, 1)
17. 2m 2n 8 6m 4n 18 (1, 3)
18. 3a b 9 5a 2b 14 (4, 3)
©
Glencoe/McGraw-Hill
213
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-1
____________ PERIOD _____
Skills Practice Monomials
Simplify. Assume that no variable equals 0. 2. c5 c2 c2 c 9
1 a
3. a4 a3 7
4. x5 x4 x x 2
5. (g4)2 g 8
6. (3u)3 27u 3
7. (x)4 x 4
8. 5(2z)3 40z 3
9. (3d)4 81d 4 11. (r7)3 r 21 k9 k
1 k
10. (2t2)3 8t 6 s15 s
3 12. 12 s
13. 10
14. (3f 3g)3 27f 9g 3
15. (2x)2(4y)2 64x 2y 2
16. 2gh( g3h5) 2g 4h 6
17. 10x2y3(10xy8) 100x 3y11
18. 3 5 2
6a4bc8 36a b c
c7 6a b
19. 3 7 2
Lesson 5-1
1. b4 b3 b 7
24wz7 8z 2 3w z w
2 10pq4r 2q 5p q r p
20. 3 2 2
Express each number in scientific notation. 21. 53,000 5.3 104
22. 0.000248 2.48 104
23. 410,100,000 4.101 108
24. 0.00000805 8.05 106
Evaluate. Express the result in scientific notation. 25. (4 103)(1.6 106) 6.4 103
©
Glencoe/McGraw-Hill
9.6 107 1.5 10
10 26. 3 6.4 10
241
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-2
____________ PERIOD _____
Skills Practice Polynomials
Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. 1. x2 2x 2 yes; 2
b2c d
1 2
3. 8xz y yes; 2
2. 4 no
Simplify. 4. (g 5) (2g 7)
5. (5d 5) (d 1)
6. (x2 3x 3) (2x2 7x 2)
7. (2f 2 3f 5) (2f 2 3f 8)
3x 2
4x 5
8. (4r2 6r 2) (r2 3r 5)
5r 2 9r 3
4d 4
4f 2 6f 3
9. (2x2 3xy) (3x2 6xy 4y2)
x 2 3xy 4y 2
10. (5t 7) (2t2 3t 12)
11. (u 4) (6 3u2 4u)
12. 5(2c2 d 2)
13. x2(2x 9)
14. 2q(3pq 4q4)
15. 8w(hk2 10h3m4 6k5w3)
2t 2 8t 5
10c 2 5d 2
6pq 2
3u 2 5u 10 2x 3 9x 2
8hk 2w 80h 3m 4w 48k 5w 4
8q 5
16. m2n3(4m2n2 2mnp 7)
17. 3s2y(2s4y2 3sy3 4)
18. (c 2)(c 8)
19. (z 7)(z 4)
4m 4n 5 2m 3n 4p 7m 2n 3
c2
10c 16
20. (a 5)2
a2
10a 25
22. (r 2s)(r 2s)
r2
9 4b 2
©
6s 6y 3 9s3y 4 12s 2y z 2 3z 28
21. (2x 3)(3x 5)
6x 2 19x 15
23. (3y 4)(2y 3)
6y 2 y 12
4s 2
24. (3 2b)(3 2b)
Glencoe/McGraw-Hill
Lesson 5-2
3g 12
25. (3w 1)2
9w 2 6w 1
247
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-3
____________ PERIOD _____
Skills Practice Dividing Polynomials
Simplify. 10c 6 2
2. 3x 5
12x 20 4
3. 5y 2 2y 1
15y3 6y2 3y 3y
4. 3x 1
5. (15q6 5q2)(5q4)1
6. (4f 5 6f 4 12f 3 8f 2)(4f 2)1
1. 5c 3
12x2 4x 8 4x
2 x
3f 2 2
1 q
3q 2 2
f 3 3f 2
7. (6j 2k 9jk2) 3jk
8. (4a2h2 8a3h 3a4) (2a2)
3a 2 2
2j 3k
2h 2 4ah
9. (n2 7n 10) (n 5)
10. (d 2 4d 3) (d 1)
n2
d3
11. (2s2 13s 15) (s 5)
12. (6y2 y 2)(2y 1)1
3y 2
13. (4g2 9) (2g 3)
Lesson 5-3
2s 3
14. (2x2 5x 4) (x 3)
1
2g 3
2x 1 x3
u2 5u 12 u3
2x2 5x 4 x3
15.
16.
12
1
u8 u3
2x 1 x3
17. (3v2 7v 10)(v 4)1
18. (3t4 4t3 32t2 5t 20)(t 4)1
10
3v 5 v4
3t 3 8t 2 5
y3 y2 6 y2
2x3 x2 19x 15 x3
19.
20.
18
3
y 2 3y 6 y2
2x 2 5x 4 x3
21. (4p3 3p2 2p) ( p 1)
22. (3c4 6c3 2c 4)(c 2)1
3
8
4p 2 p 3 p1
3c 3 2 c2
23. GEOMETRY The area of a rectangle is x3 8x2 13x 12 square units. The width of the rectangle is x 4 units. What is the length of the rectangle? x 2 4x 3 units ©
Glencoe/McGraw-Hill
253
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-4
____________ PERIOD _____
Skills Practice Factoring Polynomials
Factor completely. If the polynomial is not factorable, write prime. 1. 7x2 14x
2. 19x3 38x2
3. 21x3 18x2y 24xy2
4. 8j 3k 4jk3 7
19x 2(x 2)
3x(7x2 6xy 8y 2)
prime
5. a2 7a 18
6. 2ak 6a k 3
7. b2 8b 7
8. z2 8z 10
(a 9)(a 2)
(b 7)(b 1)
9. m2 7m 18
(m 2)(m 9)
(2a 1)(k 3)
prime 10. 2x2 3x 5
(2x 5)(x 1)
11. 4z2 4z 15
12. 4p2 4p 24
13. 3y2 21y 36
14. c2 100
15. 4f 2 64
16. d 2 12d 36
17. 9x2 25
18. y2 18y 81
(2z 5)(2z 3)
3(y 4)(y 3)
4(f 4)(f 4)
4(p 2)(p 3)
(c 10)(c 10)
(d 6)2
Lesson 5-4
7x(x 2)
(y 9)2
prime 19. n3 125
(n 5)(n 2 5n 25)
20. m4 1
(m 2 1)(m 1)(m 1)
Simplify. Assume that no denominator is equal to 0. x2 7x 18 x 2
21. x2 4x 45 x 5 x5
x 10x 25 23. 2 x 2
x 5x
©
Glencoe/McGraw-Hill
x2 4x 3 x 1
22. x2 6x 9 x 3 x2 6x 7 x 1
24. x7 x2 49
259
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-5
____________ PERIOD _____
Skills Practice Roots of Real Numbers
Use a calculator to approximate each value to three decimal places. 1. 230 15.166
2. 38 6.164
3. 152 12.329
4. 5.6 2.366
5. 88 4.448
6. 222 6.055
7. 0.34 0.764
8. 500 3.466
3
4
3
5
Simplify. 9. 81 9
10. 144 12
11. (5)2 5
12. 52 not a real number
13. 0.36 0.6
14.
15. 8 2 3
16. 27 3
3
18. 32 2
4 9
3
5
19. 81 3
20. y2 | y |
21. 125s3 5s
22. 64x6 8| x 3|
23. 27a 6 3a 2
24. m8n4 m 4n 2
25. 100p4 q2 10p 2| q |
26. 16w4v8 2| w | v 2
27. (3c)4 9c 2
28. (a b )2 | a b |
4
3
3
©
Glencoe/McGraw-Hill
Lesson 5-5
17. 0.064 0.4
23
4
265
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-6
____________ PERIOD _____
Skills Practice Radical Expressions
1. 24 26
Lesson 5-6
Simplify. 2. 75 53
3
4
3. 16 22
4. 48 2 3
5. 4 50x5 20x 22x
6. 64a4b4 2| ab | 4
3
7.
3
1 8
d 2f 5
3
3 7
4
4
d f 12 f
9.
11.
4
2 2
21
8.
10.
7
g 10gz 5z 2g3 5z
56 |s |t 25 s2t 36
3
3
2 6 9 3
12. (33 )(53 ) 45
13. (412 )(320 ) 4815
14. 2 8 50 82
15. 12 23 108 63
16. 85 45 80
18. (2 3 )(6 2 ) 12 22 63
6
19. (1 5 )(1 5 ) 4
20. (3 7 )(5 2 ) 15 32 57
14
21. (2 6 ) 8 43
22.
12 42 4 7 3 2
24.
17. 248 75 12
2
23.
©
Glencoe/McGraw-Hill
3
5
21 32 3 47 7 2 40 56 5 58 8 6
271
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-7
____________ PERIOD _____
Skills Practice Rational Exponents
Write each expression in radical form. 1
1
6
3
1. 3 6
2
2 122 or (12 ) 3
3. 12 3
5
8
2. 8 5 3
3
4. (s3) 5 s s4 5
5. 51 51
1 2 3
7. 153 15 4 4
3
6. 37 37
Lesson 5-7
Write each radical using rational exponents. 1 3 1
1
2
8. 6xy2 6 3 x 3 y 3 3
Evaluate each expression. 1
1
9. 32 5 2 1
11. 27
3
10. 81 4 3
1 3
3
4
13. 16 2 64 1
1 2
1
12. 42 14. (243) 5 81 5
15. 27 3 27 3 729
3 2
8 27
49
16.
Simplify each expression. 12
3
17. c 5 c 5 c 3
1 2
3
19. q
6
11
21. x
1
q
3 2
5 11
x x 1
y 2 y4 23. 1 y y4 12
25. 64
©
2
Glencoe/McGraw-Hill
2
16
18. m 9 m 9 m 2 4
p5 p
1
5
20. p
2
x3 22. 1 x4
x 1
5 12
2
n3 n3 24. 1 1 n6 n2 n
26. 49a8b2 | a | 7b 8
277
4
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-8
____________ PERIOD _____
Skills Practice Radical Equations and Inequalities
Solve each equation or inequality.
1 25
3. 5j 1
1
2. x 3 7 16
1
4. v 2 1 0 no solution
3
5. 18 3y 2 25 no solution
6. 2w 4 32
7. b 5 4 21
8. 3n 1 5 8
3
9. 3r 6 3 11
11. k 4 1 5 40
1
Lesson 5-8
1. x 5 25
10. 2 3p 7 6 3
1
5 2
12. (2d 3) 3 2
1
13. (t 3) 3 2 11
14. 4 (1 7u) 3 0 9
15. 3z 2 z 4 no solution
16. g 1 2g 7 8
17. x 1 4 x 1 no solution
18. 5 s36 3s4
19. 2 3x 3 7 1 x 26
20. 2a 4 6 2 a 16
21. 2 4r 3 10 r 7
22. 4 3x 1 3 x 0
23. y 4 3 3 y 32
24. 3 11r 3 15 r 2
©
Glencoe/McGraw-Hill
1 3
3 11
283
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-9
____________ PERIOD _____
Skills Practice Complex Numbers
Simplify. 1. 36 6i
2. 196 14i
3. 81x6 9 | x 3 | i
4. 23 46 232
5. (3i)(2i)(5i) 30i
6. i 11 i
7. i 65 i
8. (7 8i) (12 4i) 5 12i 10. (10 4i) (7 3i) 3 7i
11. (2 i)(2 3i) 1 8i
12. (2 i)(3 5i) 11 7i
13. (7 6i)(2 3i) 4 33i
14. (3 4i)(3 4i) 25
6 8i
8 6i 15. 3 3i
3 6i
3i 16. 10 4 2i
Lesson 5-9
9. (3 5i) (18 7i) 15 2i
Solve each equation. 17. 3x2 3 0 i
18. 5x2 125 0 5i
19. 4x2 20 0 i 5
20. x2 16 0 4i
21. x2 18 0 3i 2
22. 8x2 96 0 2i 3
Find the values of m and n that make each equation true. 23. 20 12i 5m 4ni 4, 3
24. m 16i 3 2ni 3, 8
25. (4 m) 2ni 9 14i 5, 7
26. (3 n) (7m 14)i 1 7i 3, 2
©
Glencoe/McGraw-Hill
289
Glencoe Algebra 2
NAME ______________________________________________ DATE
6-1
____________ PERIOD _____
Skills Practice Graphing Quadratic Functions
For each quadratic function, find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. 1. f(x) 3x2
2. f(x) x2 1
3. f(x) x2 6x 15
4. f(x) 2x2 11
5. f(x) x2 10x 5
6. f(x) 2x2 8x 7
1; x 0; 0
11; x 0; 0
15; x 3; 3
5; x 5; 5
7; x 2; 2
Complete parts a–c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 7. f(x) 2x2
8. f(x) x2 4x 4
0; x 0; 0 x
2 1 0
9. f(x) x2 6x 8
4; x 2; 2
1
2
f (x) 8 2 0 2 8
8; x 3; 3
2 0
2
4
f (x) 16 4
0
4 16
x
f (x )
6
x
0
f (x) 8
f (x )
2
3
4
6
0 1 0
8
f (x )
16 O
x
12 8 4 O –2
O
2
4
x
6x
Determine whether each function has a maximum or a minimum value. Then find the maximum or minimum value of each function. 10. f(x) 6x2
min.; 0 13. f(x) x2 2x 15
min.; 14 16. f(x) 2x2 4x 3
max.; 1
©
Glencoe/McGraw-Hill
11. f(x) 8x2
max.; 0 14. f(x) x2 4x 1
max.; 3 17. f(x) 3x2 12x 3
min.; 9
315
12. f(x) x2 2x
min.; 1
15. f(x) x2 2x 3
min.; 4
18. f(x) 2x2 4x 1
min.; 1
Glencoe Algebra 2
Lesson 6-1
0; x 0; 0
NAME ______________________________________________ DATE
6-2
____________ PERIOD _____
Skills Practice Solving Quadratic Equations By Graphing
Use the related graph of each equation to determine its solutions. 1. x2 2x 3 0
2. x2 6x 9 0
f (x )
f (x )
f (x )
f (x ) x 2 6x 9 O
O
3. 3x2 4x 3 0
x
x
f (x ) 3x 2 4x 3 2x 3
3, 1
O
3
x
no real solutions
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 4. x2 6x 5 0
5. x2 2x 4 0
1, 5
6. x2 6x 4 0
no real solutions f (x )
between 0 and 1; between 5 and 6 f (x )
f (x ) O
x
O O
x
x
Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. 7. Their sum is 4, and their product is 0.
8. Their sum is 0, and their product is 36.
x 2 4x 0; 0, 4
x 2 36 0; 6, 6
f (x )
36
f (x )
24 O
12
x –12
©
Glencoe/McGraw-Hill
321
–6
O
6
12 x
Glencoe Algebra 2
Lesson 6-2
f (x )
x2
NAME ______________________________________________ DATE
6-3
____________ PERIOD _____
Skills Practice Solving Quadratic Equations by Factoring
Solve each equation by factoring. 1. x2 64 {8, 8}
2. x2 100 0 {10, 10}
3. x2 3x 2 0 {1, 2}
4. x2 4x 3 0 {1, 3}
5. x2 2x 3 0 {1, 3}
6. x2 3x 10 0 {5, 2}
7. x2 6x 5 0 {1, 5}
8. x2 9x 0 {0, 9}
11. x2 5x {0, 5}
12. x2 14x 49 0 {7}
13. x2 6 5x {2, 3}
14. x2 18x 81 {9}
15. x2 4x 21 {3, 7}
16. 2x2 5x 3 0 , 3
3
1
17. 4x2 5x 6 0 , 2
2
18. 3x2 13x 10 0 , 5
Write a quadratic equation with the given roots. Write the equation in the form ax2 bx c 0, where a, b, and c are integers. 19. 1, 4 x 2 5x 4 0
20. 6, 9 x 2 3x 54 0
21. 2, 5 x 2 7x 10 0
22. 0, 7 x 2 7x 0
1 3
23. , 3 3x 2 10x 3 0
1 3 2 4
24. , 8x 2 2x 3 0
25. Find two consecutive integers whose product is 272. 16, 17
©
Glencoe/McGraw-Hill
327
Glencoe Algebra 2
Lesson 6-3
10. x2 6x 8 0 {2, 4}
9. x2 6x 0 {0, 6}
NAME ______________________________________________ DATE
6-4
____________ PERIOD _____
Skills Practice Completing the Square
Solve each equation by using the Square Root Property. 1. x2 8x 16 1 3, 5
2. x2 4x 4 1 1, 3
3. x2 12x 36 25 1, 11
4. 4x2 4x 1 9 1, 2
2
5. x2 4x 4 2 2
7. x2 6x 9 7 3
7
6. x2 2x 1 5 1
5
8. x2 16x 64 15 8
15
Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 9. x2 10x c 25; (x 5)2
11. x2 24x c 144; (x 12)2
81
9 2
13. x2 9x c ; x
10. x2 14x c 49; (x 7)2
25
1
5 2
12. x2 5x c ; x
1 2
14. x2 x c ; x
15. x2 13x 36 0 4, 9
16. x2 3x 0 0, 3
17. x2 x 6 0 2, 3
18. x2 4x 13 0 2
1
19. 2x2 7x 4 0 4,
3 33 2
17
1
20. 3x2 2x 1 0 , 1
1 13 2
21. x2 3x 6 0
22. x2 x 3 0
23. x2 11 i 11
24. x2 2x 4 0 1 i 3
©
Glencoe/McGraw-Hill
333
Glencoe Algebra 2
Lesson 6-4
Solve each equation by completing the square.
NAME ______________________________________________ DATE
6-5
____________ PERIOD _____
Skills Practice The Quadratic Formula and the Discriminant
Complete parts ac for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. x2 8x 16 0
2. x2 11x 26 0
225; 2 rational roots; 2, 13
0; 1 rational root; 4 3. 3x2 2x 0
4. 20x2 7x 3 0
3 1 289; 2 rational roots; ,
2 4; 2 rational roots; 0, 5. 5x2 6 0
6. x2 6 0
30 120; 2 irrational roots;
24; 2 irrational roots; 6
5
7. x2 8x 13 0
8. 5x2 x 1 0
1 21 21; 2 irrational roots;
12; 2 irrational roots; 4 3 9. x2 2x 17 0
10
10. x2 49 0
72; 2 irrational roots; 1 32 11. x2 x 1 0
196; 2 complex roots; 7i 12. 2x2 3x 2
1 i 3 3; 2 complex roots;
3 i 7 7; 2 complex roots;
2
4
Solve each equation by using the method of your choice. Find exact solutions. 13. x2 64 8
14. x2 30 0 30
15. x2 x 30 5, 6
16. 16x2 24x 27 0 ,
9
17. x2 4x 11 0 2
15
7 17 4 1 i5 24. 2x2 2x 3 0 2
22. 2x2 7x 4 0
25. PARACHUTING Ignoring wind resistance, the distance d(t) in feet that a parachutist falls in t seconds can be estimated using the formula d(t) 16t2. If a parachutist jumps from an airplane and falls for 1100 feet before opening her parachute, how many seconds pass before she opens the parachute? about 8.3 s ©
Glencoe/McGraw-Hill
339
Glencoe Algebra 2
Lesson 6-5
5 3 2
21. 2x2 10x 11 0
1i 4
33
20. 3x2 36 0 2i 3
19. x2 25 0 5i
23. 8x2 1 4x
18. x2 8x 17 0 4
3
NAME ______________________________________________ DATE
6-6
____________ PERIOD _____
Skills Practice
Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. 1. y (x 2)2
y (x 0; (2, 0); x 2; up 2)2
2. y x2 4
y (x 4; (0, 4); x 0; down 0)2
3. y x2 6
y (x 0)2 6; (0, 6); x 0; up
4. y 3(x 5)2
5. y 5x2 9
6. y (x 2)2 18
7. y x2 2x 5
8. y x2 6x 2
9. y 3x2 24x
y 3(x 5)2 0; (5, 0); x 5; down y (x 6; (1, 6); x 1; up 1)2
y 5(x 0)2 9; (0, 9); x 0; down y (x 7; (3, 7); x 3; up 3)2
y (x 2)2 18; (2, 18); x 2; up
y 3(x 4)2 48; (4, 48); x 4; down
Graph each function. 10. y (x 3)2 1
11. y (x 1)2 2
y
12. y (x 4)2 4 y
y O
O
x
1 2
O
y O
x
14. y 3x2 4
13. y (x 2)2
x
15. y x2 6x 4
y
y
x O O
x
x
Write an equation for the parabola with the given vertex that passes through the given point. 16. vertex: (4, 36) point: (0, 20)
y (x 4)2 36
©
Glencoe/McGraw-Hill
17. vertex: (3, 1) point: (2, 0)
y (x 3)2 1 345
18. vertex: (2, 2) point: (1, 3)
y (x 2)2 2 Glencoe Algebra 2
Lesson 6-6
Analyzing Graphs of Quadratic Functions
NAME ______________________________________________ DATE
6-7
____________ PERIOD _____
Skills Practice Graphing and Solving Quadratic Inequalities
Graph each inequality. 2. y x2 4
y
3. y x2 2x 5 y
y O
x
O
O
x
x
Use the graph of its related function to write the solutions of each inequality. 4. x2 6x 9 0
5. x2 4x 32 0
y
6. x2 x 20 0
y
y
5 O
2
x
6 O O
2
x
x
3
8 x 4
x 5 or x 4
Solve each inequality algebraically. 7. x2 3x 10 0
{x2 x 5}
9. x2 18x 81 0
{xx 9}
8. x2 2x 35 0
{xx 7 or x 5}
10. x2 36
{x6 x 6}
11. x2 7x 0
12. x2 7x 6 0
13. x2 x 12 0
14. x2 9x 18 0
15. x2 10x 25 0
16. x2 2x 15 0
{xx 0 or x 7} {xx 4 or x 3}
{x6 x 1}
{x6 x 3} {x5 x 3}
all reals 17. x2 3x 0
18. 2x2 2x 4
19. x2 64 16x
20. 9x2 12x 9 0
{xx 3 or x 0}
{xx 2 or x 1}
all reals ©
Glencoe/McGraw-Hill
351
Glencoe Algebra 2
Lesson 6-7
1. y x2 4x 4
NAME ______________________________________________ DATE
7-1
____________ PERIOD _____
Skills Practice Polynomial Functions
State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. a 8 1; 1
2. (2x 1)(4x2 3) 3; 8
3. 5x5 3x3 8 5; 5
4. 18 3y 5y2 y5 7y6 6; 7
5. u3 4u2v2 v4
6. 2r r2 2
No, this polynomial contains two
No, this is not a polynomial because 1 cannot be written in the form r n, 2 r where n is a nonnegative integer.
variables, u and v.
Find p(1) and p(2) for each function. 7. p(x) 4 3x 7; 2
8. p(x) 3x x2 2; 10 10. p(x) 2x3 5x 3 0; 3
9. p(x) 2x2 4x 1 7; 1
1 3
11. p(x) x4 8x2 10 1; 38
2 3
12. p(x) x2 x 2 3; 2
If p(x) 4x2 3 and r(x) 1 3x, find each value. 13. p(a) 4a2 3
14. r(2a) 1 6a
15. 3r(a) 3 9a
16. 4p(a) 16a2 12
17. p(a2) 4a4 3
18. r(x 2) 7 3x
For each graph, a. describe the end behavior, b. determine whether it represents an odd-degree or an even-degree polynomial function, and c. state the number of real zeroes. 19.
20.
f (x )
O
x
f(x) → as x → , f(x) → as x → ; ©
Glencoe/McGraw-Hill
21.
f (x )
O
x
f(x) → as x → , f(x) → as x → ; 377
f (x )
O
x
f(x) → as x → , f(x) → as x → ; Glencoe Algebra 2
Lesson 7-1
1 r
NAME ______________________________________________ DATE
7-2
____________ PERIOD _____
Skills Practice Graphing Polynomial Functions
Complete each of the following. a. Graph each function by making a table of values. b. Determine consecutive values of x between which each real zero is located. c. Estimate the x-coordinates at which the relative maxima and minima occur. 1. f(x) x3 3x2 1 f(x)
f (x )
x
19 1 3 O x 0 1 1 1 2 3 3 1 4 17 zeros between 1 and 0, 0 and 1, and 2 and 3; rel. max. at x 0, rel. min. at x 2 x
f(x)
3
4. f(x) 2x3 3x2 2
f (x )
7 2 2 O 1 3 0 2 1 25 zero between 1 and 0; rel. max. at x 2, rel. min. at x 1
x
x
6. f(x) 0.5x4 4x2 4 f (x )
x
61 2 6 O x 1 3 0 2 1 3 2 6 3 61 zeros between 2 and 1, and 1 and 2; rel. max. at x 0, at Glencoe/McGraw-Hill
f(x)
f (x )
8.5 2 4 O x 1 0.5 0 4 1 0.5 2 4 3 8.5 zeros between 1 and 2, 2 and 3, 1 and 2, and 2 and 3; rel. max.
3
©
f (x )
1
5. f(x) x4 2x2 2 f(x)
f(x)
3 0 2 O x 1 1 2 6 3 29 zero between 1 and 0; rel. min. at x 1, rel. max. at x 0
3
x
f (x )
17 2 1 O x 1 3 0 1 1 1 2 3 3 19 zeros between 2 and 1, 0 and 1, and 1 and 2; rel. max. at x 1, rel. min. at x 1
2
3. f(x) 2x3 9x2 12x 2
f(x)
3
383
Glencoe Algebra 2
Lesson 7-2
x
2. f(x) x3 3x 1
NAME ______________________________________________ DATE
7-3
____________ PERIOD _____
Skills Practice Solving Equations Using Quadratic Techniques
Write each expression in quadratic form, if possible. 1. 5x4 2x2 8 5(x 2)2 2(x 2) 8
2. 3y8 4y2 3 not possible
3. 100a6 a3 100(a3)2 a3
4. x8 4x4 9 (x 4)2 4(x 4) 9
5. 12x4 7x2 12(x 2)2 7(x 2)
6. 6b5 3b3 1 not possible
7. 15v6 8v3 9 15(v 3) 2 8(v 3) 9
8. a9 5a5 7a a[(a4)2 5(a4) 7]
Solve each equation. 10. x3 3x2 0, 3
11. t4 3t3 40t2 0 0, 5, 8
12. b3 8b2 16b 0 0, 4
13. m4 4 2 ,
14. w3 6w 0 0,
2 , i2 , i2
6 , 6
15. m4 18m2 81 3, 3
16. x5 81x 0 0, 3, 3, 3i, 3i
17. h4 10h2 9 1, 1, 3, 3
18. a4 9a2 20 0 2, 2,
19. y4 7y2 12 0
20. v4 12v2 35 0
21. x5 7x3 6x 0
22. c 3 7c 3 12 0
23. z 5z 6 4, 9
24. x 30x 200 0 100, 400
2, 2, 3 , 3 0, 1, 1, 6 , 6
©
Glencoe/McGraw-Hill
5 , 5
5 , 5 , 7 , 7 2
1
64, 27
389
Glencoe Algebra 2
Lesson 7-3
9. a3 9a2 14a 0 0, 7, 2
NAME ______________________________________________ DATE
7-4
____________ PERIOD _____
Skills Practice The Remainder and Factor Theorems
Use synthetic substitution to find f(2) and f(1) for each function. 1. f(x) x2 6x 5 21, 0
2. f(x) x2 x 1 3, 3
3. f(x) x2 2x 2 2, 1
4. f(x) x3 2x2 5 21, 6
5. f(x) x3 x2 2x 3 3, 3
6. f(x) x3 6x2 x 4 30, 0
7. f(x) x3 3x2 x 2 4, 7
8. f(x) x3 5x2 x 6 8, 1
9. f(x) x4 2x2 9 15, 6 11. f(x) x5 7x3 4x 10
22, 20
10. f(x) x4 3x3 2x2 2x 6 2, 14 12. f(x) x6 2x5 x4 x3 9x2 20
32, 26
Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. 14. x3 x2 5x 3; x 1
x 1, x 2 15. x3 3x2 4x 12; x 3
x 1, x 3 16. x3 6x2 11x 6; x 3
x 2, x 2 17. x3 2x2 33x 90; x 5
x 1, x 2 18. x3 6x2 32; x 4
x 3, x 6 19. x3 x2 10x 8; x 2
x 4, x 2 20. x3 19x 30; x 2
x 1, x 4 21. 2x3 x2 2x 1; x 1
x 5, x 3 22. 2x3 x2 5x 2; x 2
2x 1, x 1 23. 3x3 4x2 5x 2; 3x 1
x 1, 2x 1 24. 3x3 x2 x 2; 3x 2
x 1, x 2
©
Glencoe/McGraw-Hill
Lesson 7-4
13. x3 2x2 x 2; x 1
x2 x 1
395
Glencoe Algebra 2
NAME ______________________________________________ DATE
7-5
____________ PERIOD _____
Skills Practice Roots and Zeros
Solve each equation. State the number and type of roots. 1. 5x 12 0
2. x2 4x 40 0
12 5
; 1 real
2 6i; 2 imaginary
3. x5 4x3 0
4. x4 625 0
0, 0, 0, 2i, 2i; 3 real, 2 imaginary 5. 4x2 4x 1 0
5i, 5i, 5i, 5i; 4 imaginary 6. x5 81x 0
1 2 ; 2 real
0, 3, 3, 3i, 3i; 3 real, 2 imaginary
State the possible number of positive real zeros, negative real zeros, and imaginary zeros of each function. 7. g(x) 3x3 4x2 17x 6
8. h(x) 4x3 12x2 x 3
2 or 0; 1; 2 or 0
2 or 0; 1; 2 or 0
9. f(x) x3 8x2 2x 4
10. p(x) x3 x2 4x 6
3 or 1; 0; 2 or 0
3 or 1; 0; 2 or 0
11. q(x) x4 7x2 3x 9
12. f(x) x4 x3 5x2 6x 1
1; 1; 2
2 or 0; 2 or 0; 4 or 2 or 0
Find all the zeros of each function. 13. h(x) x3 5x2 5x 3
14. g(x) x3 6x2 13x 10
3, 1 2 , 1 2 15. h(x) x3 4x2 x 6
2, 2 i, 2 i 16. q(x) x3 3x2 6x 8
1, 2, 3
2, 1, 4
17. g(x) x4 3x3 5x2 3x 4
18. f(x) x4 21x2 80
4, 4, 5 , 5
1, 1, 1, 4
Write a polynomial function of least degree with integral coefficients that has the given zeros. 19. 3, 5, 1
20. 3i
7x 2
7x 15
22. 1, 3 , 3
21. 5 i
f(x)
x2
10x 26
x4
©
f(x) x 3 x 2 3x 3 24. 1, 1, i 6
23. i, 5i
f(x)
f(x) x 2 9
Lesson 7-5
f(x)
x3
26x 2
Glencoe/McGraw-Hill
25
f(x) x 4 5x 2 6 401
Glencoe Algebra 2
NAME ______________________________________________ DATE
7-6
____________ PERIOD _____
Skills Practice Rational Zero Theorem
1. n(x) x2 5x 3
2. h(x) x2 2x 5
1, 3
1, 5
3. w(x) x2 5x 12
4. f(x) 2x2 5x 3
1 2
1, 2, 3, 4, 6, 12 5. q(x) 6x3 x2 x 2
1 6
Lesson 7-6
List all of the possible rational zeros of each function.
1 3
1 2
3 2
, , 1, 3 6. g(x) 9x4 3x3 3x2 x 27
2 3
1 9
, , , , 1, 2
1 3
, , 1, 3, 9, 27
Find all of the rational zeros of each function. 7. f(x) x3 2x2 5x 4 0
8. g(x) x3 3x2 4x 12
2, 2, 3
1 9. p(x) x3 x2 x 1
10. z(x) x3 4x2 6x 4
1
2
11. h(x) x3 x2 4x 4
12. g(x) 3x3 9x2 10x 8
1
4
13. g(x) 2x3 7x2 7x 12
14. h(x) 2x3 5x2 4x 3
3 2
1 2
4, 1, 15. p(x) 3x3 5x2 14x 4 0
1, , 3 16. q(x) 3x3 2x2 27x 18
1 3
2 3
17. q(x) 3x3 7x2 4
18. f(x) x4 2x3 13x2 14x 24
2 3
, 1, 2 19. p(x) x4 5x3 9x2 25x 70
3, 1, 2, 4 20. n(x) 16x4 32x3 13x2 29x 6
1 3 4 4
2, 7
1, , , 2
Find all of the zeros of each function. 21. f(x) x3 5x2 11x 15
22. q(x) x3 10x2 18x 4
2, 4 14 , 4 14
3, 1 2i, 1 2i 23. m(x) 6x4 17x3 8x2 8x 3
24. g(x) x4 4x3 5x2 4x 4
1 3 1 5 , 1 5 , , 3 2 ©
Glencoe/McGraw-Hill
2, 2, i, i 407
Glencoe Algebra 2
NAME ______________________________________________ DATE
7-7
____________ PERIOD _____
Skills Practice Operations on Functions
f
Find ( f g)(x), (f g)(x), (f g)(x), and (x) for each f(x) and g(x). g 1. f(x) x 5 2x 1; 9; g(x) x 4
x 2 x 20;
3x 1 2x 3
3 2
g(x) 2x 3 , x
x5 , x 4 x4
3. f(x) x2 x 2 x 4; x2 x 4;
0;
3x3 5 x
3x3 5 x
4. f(x) 3x2 , x 0; , x
x2
g(x) 4 x 4x 2 x3; , x 4
3x3 5
5 x
g(x) 15x, x 0; , x 0
For each set of ordered pairs, find f g and g f if they exist. 5. f {(0, 0), (4, 2)} g {(0, 4), (2, 0), (5, 0)}
6. f {(0, 3), (1, 2), (2, 2)} g {(3, 1), (2, 0)}
7. f {(4, 3), (1, 1), (2, 2)} g {(1, 4), (2, 1), (3, 1)}
8. f {(6, 6), (3, 3), (1, 3)} g {(3, 6), (3, 6), (6, 3)}
{(0, 2), (2, 0), (5, 0)}; {(0, 4), (4, 0)}
{(3, 2), (2, 3)}; {(0, 1), (1, 0), (2, 0)}
{(3, 6), (3, 6), (6, 3)}; {(6, 3), (3, 6), (1, 6)}
{(1, 3), (2, 1), (3, 1)}; {(4, 1), (1, 4), (2, 1)} Find [g h](x) and [h g](x). 9. g(x) 2x 2x 4; 2x 2 h(x) x 2
10. g(x) 3x 12x 3; 12x 1 h(x) 4x 1
11. g(x) x 6 x; x h(x) x 6
12. g(x) x 3 x 2 3; x 2 6x 9 h(x) x2
13. g(x) 5x 5x 2 5x 5; h(x) x2 x 1 25x 2 5x 1
14. g(x) x 2 2x 2 1; 2x 2 8x 5 h(x) 2x2 3
If f(x) 3x, g(x) x 4, and h(x) x2 1, find each value. 15. f[ g(1)] 15
16. g[h(0)] 3
17. g[f(1)] 1
18. h[f(5)] 224
19. g[h(3)] 12
20. h[f(10)] 899
©
Glencoe/McGraw-Hill
413
Glencoe Algebra 2
Lesson 7-7
3;
2. f(x) 3x 1 5x 2; x 4; 6x 2 7x
NAME ______________________________________________ DATE
7-8
____________ PERIOD _____
Skills Practice Inverse Functions and Relations
Find the inverse of each relation. 1. {(3, 1), (4, 3), (8, 3)}
2. {(7, 1), (0, 5), (5, 1)}
{(1, 7), (5, 0), (1, 5)}
{(1, 3), (3, 4), (3, 8)} 3. {(10, 2), (7, 6), (4, 2), (4, 0)}
4. {(0, 9), (5, 3), (6, 6), (8, 3)}
5. {(4, 12), (0, 7), (9, 1), (10, 5)}
6. {(4, 1), (4, 3), (0, 8), (8, 9)}
{(2, 10), (6, 7), (2, 4), (0, 4)} {(9, 0), (3, 5), (6, 6), (3, 8)} {(12, 4), (7, 0), (1, 9), (5, 10)}
{(1, 4), (3, 4), (8, 0), (9, 8)}
Find the inverse of each function. Then graph the function and its inverse. 8. f(x) 3x
9. f(x) x 2
1 f 1(x) x 3
x4 y
f 1(x) x 2
f (x )
f (x )
x
O
x
O
x
O
1 4
10. g(x) 2x 1
x1 2
g1(x)
12. y x 2
3 2
h1(x) 4x
g (x )
O
2 3
11. h(x) x
y x 3
h (x )
x
O
y
x
x
O
Determine whether each pair of functions are inverse functions. 13. f(x) x 1 no g(x) 1 x 16. f(x) 2x yes 1 g(x) x 2
©
Glencoe/McGraw-Hill
14. f(x) 2x 3 yes 1 g(x) (x 3) 2
17. h(x) 6x 2 no 1 g(x) x 3 6
419
15. f(x) 5x 5 yes 1 5
g(x) x 1 18. f(x) 8x 10 yes 1 8
5 4
g(x) x
Glencoe Algebra 2
Lesson 7-8
7. y 4
NAME ______________________________________________ DATE
7-9
____________ PERIOD _____
Skills Practice Square Root Functions and Inequalities
Graph each function. State the domain and range of each function. 1. y 2x
2. y 3x
y
3. y 2x
y
x
x
O
x
O
D: x 0, R: y 0
4. y x3
D: x 0, R: y 0
5. y 2x 5
y
6. y x42
y
y
x
O O
D: x 0, R: y 0
x
O
x
D: x 3, R: y 0
D: x 2.5, R: y 0
D: x 4, R: y 2
Graph each inequality. 7. y 4x
8. y x1
y
O
©
Glencoe/McGraw-Hill
9. y 4x 3
y
x
y
x
O
425
O
x
Glencoe Algebra 2
Lesson 7-9
O
y
NAME ______________________________________________ DATE
8-1
____________ PERIOD _____
Skills Practice Midpoint and Distance Formulas
Find the midpoint of each line segment with endpoints at the given coordinates. 1. (4, 1), (4, 1) (0, 0)
2. (1, 4), (5, 2) (2, 3)
3. (3, 4), (5, 4) (4, 4)
4. (6, 2), (2, 1) 4,
1
1
3
5. (3, 9), (2, 3) , 3
6. (3, 5), (3, 8) 3,
7. (3, 2), (5, 0) (1, 1)
8. (3, 4), (5, 2) (4, 1)
5
9. (5, 9), (5, 4) 0,
5
11
10. (11, 14), (0, 4) , 9
9
11. (3, 6), (8, 3) ,
12. (0, 10), (2, 5)
1, 5
Find the distance between each pair of points with the given coordinates. 13. (4, 12), (1, 0) 13 units
14. (7, 7), (5, 2) 15 units
15. (1, 4), (1, 4) 2 units
16. (11, 11), (8, 15) 5 units
17. (1, 6), (7, 2) 10 units
18. (3, 5), (3, 4) 9 units
19. (2, 3), (3, 5)
5 units
21. (5, 5), (3, 10) 17 units
23. (6, 2), (1, 3)
74 units
25. (0, 3), (4, 1) 42 units
©
Glencoe/McGraw-Hill
20. (4, 3), (1, 7) 5 units
22. (3, 9), (2, 3) 13 units
24. (4, 1), (2, 4)
61 units
26. (5, 6), (2, 0)
85 units
457
Glencoe Algebra 2
Lesson 8-1
NAME ______________________________________________ DATE
8-2
____________ PERIOD _____
Skills Practice Parabolas
Write each equation in standard form. 1. y x2 2x 2
2. y x2 2x 4
y [x (1)]2 1
3. y x2 4x 1
y (x 1)2 3
y [x (2)]2 (3)
Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. 5. x (y 2)2 3
y
6. y (x 3)2 4
y
y
x
O
O
x
O
x
vertex: (2, 0);
vertex: (3, 2);
vertex: (3, 4);
1 focus: 2, ;
1 focus: 3 ,2;
3 focus: 3, 3 ;
axis of symmetry: x 2; 1 directrix: y ;
axis of symmetry: y 2; 3 directrix: x 2 ;
axis of symmetry: x 3; 1 directrix: y 4 ;
opens up; latus rectum: 1 unit
opens right; latus rectum: 1 unit
opens down; latus rectum: 1 unit
Write an equation for each parabola described below. Then draw the graph. 7. vertex (0, 0),
1 focus 0, 12 y 3x 2
8. vertex (5, 1),
5 focus 5, 4
Glencoe/McGraw-Hill
x 2(y
y
1
y
x
x
O
©
7 8 3)2
directrix x
y (x 5)2 1
y O
9. vertex (1, 3),
463
O
x
Glencoe Algebra 2
Lesson 8-2
4. y (x 2)2
NAME ______________________________________________ DATE
8-3
____________ PERIOD _____
Skills Practice Circles
Write an equation for the circle that satisfies each set of conditions. 1. center (0, 5), radius 1 unit
2. center (5, 12), radius 8 units
3. center (4, 0), radius 2 units
4. center (2, 2), radius 3 units
5. center (4, 4), radius 4 units
6. center (6, 4), radius 5 units
x 2 (y 5)2 1
(x 5)2 (y 12)2 64
(x 4)2 y 2 4
(x 2)2 (y 2)2 9
(x 4)2 (y 4)2 16
(x 6)2 (y 4)2 25
7. endpoints of a diameter at (12, 0) and (12, 0) x 2 y 2 144 8. endpoints of a diameter at (4, 0) and (4, 6) (x 4)2 (y 3)2 9 9. center at (7, 3), passes through the origin (x 7)2 (y 3)2 58 10. center at (4, 4), passes through (4, 1) (x 4)2 (y 4)2 9 11. center at (6, 5), tangent to y-axis (x 6)2 (y 5)2 36 12. center at (5, 1), tangent to x-axis (x 5)2 (y 1)2 1 Find the center and radius of the circle with the given equation. Then graph the circle. 14. (x 1)2 (y 2)2 4
(0, 0), 3 units
(1, 2), 2 units
y
(5, 8), 7 units O
(0, 2), 6 units
x 6
12
8 4
8
12
x
18. x2 y2 4y 32 0
y
y
6 O
O
x
17. (x 5)2 (y 8)2 49
(0, 3), 9 units
–6
y
O
x
16. x2 (y 3)2 81
–12
(1, 0), 4 units
y
O
12
15. (x 1)2 y2 16
Lesson 8-3
13. x2 y2 9
x
y
4
–4 –8
–8
–4
O
4
8x
–4
–6 –12
–8
–12
©
Glencoe/McGraw-Hill
469
Glencoe Algebra 2
NAME ______________________________________________ DATE
8-4
____________ PERIOD _____
Skills Practice Ellipses
Write an equation for each ellipse. 1.
2.
y
3.
(0, 5) y
(0, 2)
(0, 5) y
(0, 3) (–4, 2)
(4, 2)
(–3, 0) (3, 0) x
O
O
x
O
x (0, –1)
(0, –2)
(0, –3)
(0, –5)
y2 x2 1
(y 2)2 x2 1 9
y2 x2 1
Write an equation for the ellipse that satisfies each set of conditions.
y2 x2 1 7. major axis 12 units long and parallel to x-axis, minor axis 4 units long, center at (0, 0)
y2 x2 1
5. endpoints of major axis at (2, 6) and (8, 6), endpoints of minor axis at (5, 4) and (5, 8)
6. endpoints of major axis at (7, 3) and (7, 9), endpoints of minor axis at (5, 6) and (9, 6)
8. endpoints of major axis at (6, 0) and (6, 0), foci at ( , 0) 32, 0) and (32
9. endpoints of major axis at (0, 12) and (0, 12), foci at (0, 23 ) and (0, 23 )
(x 5)2 (y 6)2 1 9 4
y2 x2 1
(y 6)2 (x 7)2 1 9 4
y2 x2 1
Find the coordinates of the center and foci and the lengths of the major and minor axes for the ellipse with the given equation. Then graph the ellipse. y2 100
x2 81
x2 81
10. 1
(0, 0); (0, 19 ); 20; 18 8
y2 9
y
8
©
–4
O
x2 25
12. 1
(0, 0); (62 , 0); 18; 6
4 –8
y2 49
11. 1
(0, 0), (0, 26 ); 14; 10
y
8 4
4 4
8x
–8
–4
O
4
8x
–8
–4
O
–4
–4
–4
–8
–8
–8
Glencoe/McGraw-Hill
475
y
4
8x
Glencoe Algebra 2
Lesson 8-4
4. endpoints of major axis at (0, 6) and (0, 6), endpoints of minor axis at (3, 0) and (3, 0)
NAME ______________________________________________ DATE
8-5
____________ PERIOD _____
Skills Practice Hyperbolas
Write an equation for each hyperbola. 1.
8
2.
y
(0, 61 )
(–5, 0) 4 –8
–4
8
(5, 0) O 4
8x
–4 ( 41, 0) (– 41, 0) –8
–8
4
(0, – 61 )
–4
8
(0, 6) (–2, 0)
O
–4
3.
y
4
8x
–8
–4
4
(2, 0)
O
4
(0, –6)
8x
( 29, 0)
(– 29, 0) –4
–8
y2 x2 1
y
–8
y2 x2 1
y2 x2 1
Write an equation for the hyperbola that satisfies each set of conditions.
x2
y2
4. vertices (4, 0) and (4, 0), conjugate axis of length 8 1
y2
x2
y2
x2
5. vertices (0, 6) and (0, 6), conjugate axis of length 14 1 6. vertices (0, 3) and (0, 3), conjugate axis of length 10 1
x2
y2
7. vertices (2, 0) and (2, 0), conjugate axis of length 4 1
x2
y2
y2
x2
8. vertices (3, 0) and (3, 0), foci (5, 0) 1 9. vertices (0, 2) and (0, 2), foci (0, 3) 1
(x 3)2 9
(y 2)2 4
10. vertices (0, 2) and (6, 2), foci (3 13 , 2) 1 Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. y2 36
y2 49
x2 9
x2 16
12. 1
y2 1
13. 1
(3, 0); (35 , 0);
(0, 7); (0, 58 );
(4, 0); (17 , 0);
y 2x
7 y x
1 y x
y
8
y
8
4 O
©
Glencoe/McGraw-Hill
x
–8
–4
O
4 4
8x
–8
–4
O
–4
–4
–8
–8
481
y
4
8x
Glencoe Algebra 2
Lesson 8-5
x2 9
11. 1
NAME ______________________________________________ DATE
8-6
____________ PERIOD _____
Skills Practice
Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. 1. x2 25y2 25 hyperbola 2. 9x2 4y2 36 ellipse
y2
x2
1 4
y2
x2
1
3. x2 y2 16 0 circle
x 2 y 2 16
y
y
y
2 –8
–4
O
4
O
8x
x
O
x
–2 –4
4. x2 8x y2 9 circle
5. x2 2x 15 y parabola 6. 100x2 25y2 400
(x 4)2 y 2 25 8
y (x 1)2 16
y
–4
O
–4
ellipse
y
y –8
4 –8
y2 x2 1
O
4
8x
–4 4
8x
–8
–4
–12
–8
–16
O
x
Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. 7. 9x2 4y2 36 ellipse 9. y x2 2x parabola
8. x2 y2 25 circle 10. y 2x2 4x 4 parabola
11. 4y2 25x2 100 hyperbola
12. 16x2 y2 16 ellipse
13. 16x2 4y2 64 hyperbola
14. 5x2 5y2 25 circle
15. 25y2 9x2 225 ellipse
16. 36y2 4x2 144 hyperbola
17. y 4x2 36x 144 parabola
18. x2 y2 144 0 circle
19. (x 3)2 ( y 1)2 4 circle
20. 25y2 50y 4x2 75 ellipse
21. x2 6y2 9 0 hyperbola
22. x y2 5y 6 parabola
23. (x 5)2 y2 10 circle
24. 25x2 10y2 250 0 ellipse
©
Glencoe/McGraw-Hill
487
Glencoe Algebra 2
Lesson 8-6
Conic Sections
NAME ______________________________________________ DATE
8-7
____________ PERIOD _____
Skills Practice Solving Quadratic Systems
1. y x 2 (0, 2), (1, 1) 2. y x 3 (1, 2), y x2 2 y 2x2 (1.5, 4.5)
3. y 3x (0, 0) x y2
4. y x (2 , 2 ), 5. x 5 (5, 0) 2 2 2 2 x y 4 (2 , 2 ) x y 25
6. y 7 no solution x2 y2 9
7. y 2x 2 (2, 2),
9. y 2 x (0, 2), (3,
1)
y2 2x
1 , 1
8. x y 1 0 (1, 2) y2 4x
y x2 4x 2
10. y x 1 no solution y x2
11. y 3x2 (0, 0) y 3x2
12. y x2 1 (1, 2), y x2 3 (1, 2)
13. y 4x (1, 4), (1, 4) 4x2 y2 20
14. y 1 (0, 1) 4x2 y2 1
15. 4x2 9y2 36 (3, 0), x2 9y2 9 (3, 0)
16. 3( y 2)2 4(x 3)2 12 17. x2 4y2 4 (2, 0), y 2x 2 (0, 2), (3, 4) x2 y2 4 (2, 0)
18. y2 4x2 4 no y 2x solution
Solve each system of inequalities by graphing. 19. y 3x 2 x2 y2 16
20. y x y 2x2 4
y
21. 4y2 9x2 144 x2 8y2 16
y
8
y
4 O
x
O
x
–8
–4
O
4
8x
–4 –8
©
Glencoe/McGraw-Hill
493
Glencoe Algebra 2
Lesson 8-7
Find the exact solution(s) of each system of equations.
NAME ______________________________________________ DATE
9-1
____________ PERIOD _____
Skills Practice Multiplying and Dividing Rational Expressions
Simplify each expression.
3x
8y2(y6)3 4y
(x6)3 (x )
6 3. 3 4 x
18 2x 6
x2 x2 4 (x 2)(x 1)
9
6.
3a2 24a a 8 3a 12a
3m 2n
7. 2
10(ef)3 8e f
5s2 s 4
80y4 49z v
3 11. 2 21g
q2 2q 6q
3x x 4
w2 6w 7 w3
1
17. (3x2 3x)
19.
©
5
Glencoe/McGraw-Hill
q2 4 3q
q2
t2 19t 84 4t 4
2t 2 t 9t 14
t 12
16. 2
(w 8)(w 7)
c2 2d2 c6 5d
32z 7
14. 2
15.
x2 5x 4 2x 8
25y5 14z v
12. 5 7 12 5
x(x 2) 13. 2
w2 5w 24 w1
1
s2 10s
10. 2 5
1
7g y
n3 mn 2 6
8.
6e 9. 2 5
3x2 x2
2
4. 24
5.
24e3 5f
b
5ab3 25a b
2. 2 2
Lesson 9-1
21x3y 14x y
1. 2 2
16a2 40a 25 3a 10a 8
(4a 5)(a 4) 4a 5 a 8a 16
18. 2 2
20.
a2 b2 4a ab 2a
519
ab
Glencoe Algebra 2
NAME ______________________________________________ DATE
9-2
____________ PERIOD _____
Skills Practice Adding and Subtracting Rational Expressions
Find the LCM of each set of polynomials. 1. 12c, 6c2d 12c 2d
2. 18a3bc2, 24b2c2 72a 3b 2c 2
3. 2x 6, x 3 2(x 3)
4. 5a, a 1 5a(a 1)
5. t2 25, t 5 (t 5)(t 5)
6. x2 3x 4, x 1 (x 4)(x 1)
Simplify each expression. 5 5x 3y y
2c 5
2c 7 3
9. 4
12 5y
2 12z 2y 5yz
3 w3
2 w 9
m mn
m nm
13.
3w 7
15. 2
2m
17.
1 x 2x 1
x2 x 1 x x1
19. 2
n n3
2n 2 n 2n 3
21. 2
Glencoe/McGraw-Hill
5 2 5m 2 n
7 4gh
3 4h
7h 3g
5 3b d
2 15bd 6b 2d 3bd
14.
3t 2x
5 3t 5 x2
4z z4
z 4 5z 2 4z 16 z1
16.
18.
2x 1 x5
4 x 3x 10
2x 2 5x 2
20. 2
3 y y 12
2 y 6y 8
22. 2 2
n2
©
2 m n
12. 2
a6
3 2a
5 4p q
10. 2
11. 2
2 a2
13
3 8p q
8. 2 2
Lesson 9-2
3 x
7.
y 12
525
Glencoe Algebra 2
NAME ______________________________________________ DATE
9-3
____________ PERIOD _____
Skills Practice Graphing Rational Functions
Determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function. 10 x 13x 36
3 x 2x 8
2. f(x) 2
x 12 x 10x 24
4. f(x) 2
x2 8x 12 x2
6. f(x)
1. f(x) 2
asymptotes: x 4, x 2
asymptotes: x 4, x 9
x1 x 4x 3
3. f(x) 2
asymptote: x 2; hole: x 12
asymptote: x 3; hole: x 1
x2 x 12 x3
5. f(x)
hole: x 2
hole: x 3
Graph each rational function. 4 x
10 x
8. f(x) f (x )
9. f(x) f (x )
f (x )
2 O
x
2 x1
O
Glencoe/McGraw-Hill
O
12. f(x)
f (x )
x
O
531
x
x2 4 x2
11. f(x)
f (x )
©
x
x x2
10. f(x)
O
2
f (x )
x
O
x
Glencoe Algebra 2
Lesson 9-3
3 x
7. f(x)
NAME ______________________________________________ DATE
9-4
____________ PERIOD _____
Skills Practice Direct, Joint, and Inverse Variation
State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. 4 q
1
1 2
1. c 12m direct; 12
2. p inverse; 4
3. A bh joint;
4. rw 15 inverse; 15
5. y 2rst joint; 2
6. f 5280m direct; 5280
7. y 0.2s direct; 0.2
8. vz 25 inverse; 25
9. t 16rh joint; 16
8 w
10. R inverse; 8
a b
1
1 3
11. direct;
12. C 2 r direct; 2
Find each value. 13. If y varies directly as x and y 35 when x 7, find y when x 11. 55 14. If y varies directly as x and y 360 when x 180, find y when x 270. 540 15. If y varies directly as x and y 540 when x 10, find x when y 1080. 20 16. If y varies directly as x and y 12 when x 72, find x when y 9. 54 17. If y varies jointly as x and z and y 18 when x 2 and z 3, find y when x 5 and z 6. 90
19. If y varies jointly as x and z and y 120 when x 4 and z 6, find y when x 3 and z 2. 30 20. If y varies inversely as x and y 2 when x 2, find y when x 1. 4 21. If y varies inversely as x and y 6 when x 5, find y when x 10. 3 22. If y varies inversely as x and y 3 when x 14, find x when y 6. 7 23. If y varies inversely as x and y 27 when x 2, find x when y 9. 6 24. If y varies directly as x and y 15 when x 5, find x when y 36. 12
©
Glencoe/McGraw-Hill
537
Glencoe Algebra 2
Lesson 9-4
18. If y varies jointly as x and z and y 16 when x 4 and z 2, find y when x 1 and z 7. 14
NAME ______________________________________________ DATE______________ PERIOD _____
9-5
Skills Practice Classes of Functions
Identify the type of function represented by each graph. 1.
2.
y
O
3.
y
O
x
y
x
O
constant
direct variation
x
quadratic
Match each graph with an equation below. 1 x1
A. y |x 1| 4.
B. y
B
y
C. y 1x
5.
C
y
D. y x 1 6.
A
y
O O
x
x
O
x
Identify the type of function represented by each equation. Then graph the equation. 8. y 2x
inverse variation or rational
9. y 3x
greatest integer y
y
O
©
Glencoe/McGraw-Hill
direct variation
x
O
543
y
x
O
x
Glencoe Algebra 2
Lesson 9-5
2 x
7. y
NAME ______________________________________________ DATE
9-6
____________ PERIOD _____
Skills Practice Solving Rational Equations and Inequalities
x x1
1 2
2. 2
6 2
9 3x
2 z
3. 1
2 d1
4. 3 z 1, 2
s3 5
1 d2
3 2
12 y
7. 3
x2 x4
8. y 7 3, 4
x1 x 10
10. 0 k 0
5 v
12. n n 3 or 0 n 3
3 k
9. 8
3 v
11. 2 0 v 4
1 2m
3 m
5 2
13. 0 m 1
9x 7 x2
15 x
2q q1
5 n 9
2 n3
19. 4 2
x8 2x 2
x 2x 2
2e e 4
1 e2
2x 3 x1
21.
2 e2
23. 6 2
©
Glencoe/McGraw-Hill
3 n
1 2x
12 n
3
2 x
14. 1 0 x
b2 b1
16. 4 4
17. 2 5
1 n3
4 3k
3b 2 b1
15. 9 3
5 2q
8 s
6. 5, 8
5. 5
2x 3 x1
1 12 3
4 n
1. 1
Lesson 9-6
Solve each equation or inequality. Check your solutions.
4 z
8z 8 2 z2
18. 8
1 w2
1 w2
4 w 4
20.
2
12s 19 s 7s 12
3 s3
5 s4
22. 2 2
8 t 9
4 t3
2 t3
24. 5 2
549
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
10-1 Skills Practice Exponential Functions Sketch the graph of each function. Then state the function’s domain and range.
12
1. y 3(2)x
2. y 2
O
y
O
x
domain: all real numbers; range: all positive numbers
x
domain: all real numbers; range: all positive numbers
Determine whether each function represents exponential growth or decay.
109
x
3. y 3(6) x growth
4. y 2
5. y 10x decay
6. y 2(2.5) x growth
decay
Write an exponential function whose graph passes through the given points.
13 1 x 9. (0, 3) and (1, 6) y 3 2
7. (0, 1) and (1, 3) y
x
11. (0, 0.1) and (1, 0.5) y 0.1(5)x
8. (0, 4) and (1, 12) y 4(3)x 10. (0, 5) and (1, 15) y 5(3)x 12. (0, 0.2) and (1, 1.6) y 0.2(8)x
Simplify each expression. 13. (33)3 27
14 14. (x2)7 x
15. 523 543 563
16. x3 x x 2
Solve each equation or inequality. Check your solution. 17. 3x 9 x 2 1 7
18. 22x 3 32 1
1 2
4 3
19. 49x x
20. 43x 2 16
21. 32x 5 27x 5
22. 27x 32x 3 3
©
Glencoe/McGraw-Hill
575
Glencoe Algebra 2
Lesson 10-1
y
x
NAME ______________________________________________ DATE
____________ PERIOD _____
10-2 Skills Practice Logarithms and Logarithmic Functions Write each equation in logarithmic form. 1. 23 8 log2 8 3
2. 32 9 log3 9 2
1
1
13
3. 82 log8 2 64 64
4.
2
1 9
1 9
log1 2 3
Write each equation in exponential form.
1 2
1
7. log9 3 9 2 3
6. log4 64 3 43 64
1 25
8. log5 2 52 1 25
Lesson 10-2
5. log3 243 5 35 243
Evaluate each expression.
1 2
10. log9 3
9. log5 25 2
1 3
12. log125 5
11. log10 1000 3 1 64
1 625
13. log4 3
14. log5 4
15. log8 83 3
16. log27
1 3
1 3
Solve each equation or inequality. Check your solutions. 17. log3 x 5 243
18. log2 x 3 8
19. log4 y 0 0 y 1
20. log14 x 3
1 4
1 64
1 2
21. log2 n 2 n
22. logb 3 9
23. log6 (4x 12) 2 6
24. log2 (4x 4) 5 x 9
25. log3 (x 2) log3 (3x) 1
26. log6 (3y 5) log6 (2y 3) y 8
©
Glencoe/McGraw-Hill
581
Glencoe Algebra 2
NAME ______________________________________________ DATE______________ PERIOD _____
10-3 Skills Practice Properties of Logarithms Use log2 3 1.5850 and log2 5 2.3219 to approximate the value of each expression. 1. log2 25 4.6438
2. log2 27 4.755 5 3
3 5
3. log2 0.7369
4. log2 0.7369
5. log2 15 3.9069
6. log2 45 5.4919
7. log2 75 6.2288
8. log2 0.6 0.7369
1 3
9 5
9. log2 1.5850
10. log2 0.8481
Solve each equation. Check your solutions. 12. 3 log7 4 2 log7 b 8
13. log4 5 log4 x log4 60 12
14. log6 2c log6 8 log6 80 5
15. log5 y log5 8 log5 1 8
16. log2 q log2 3 log2 7 21
17. log9 4 2 log9 5 log9 w 100
18. 3 log8 2 log8 4 log8 b 2
19. log10 x log10 (3x 5) log10 2 2
20. log4 x log4 (2x 3) log4 2 2
21. log3 d log3 3 3 9
22. log10 y log10 (2 y) 0 1
1 25
Lesson 10-3
11. log10 27 3 log10 x 3
23. log2 s 2 log2 5 0
24. log2 (x 4) log2 (x 3) 3 4
25. log4 (n 1) log4 (n 2) 1 3
26. log5 10 log5 12 3 log5 2 log5 a 15
©
Glencoe/McGraw-Hill
587
Glencoe Algebra 2
NAME ______________________________________________ DATE______________ PERIOD _____
10-4 Skills Practice Common Logarithms Use a calculator to evaluate each expression to four decimal places. 1. log 6 0.7782
2. log 15 1.1761
3. log 1.1 0.0414
4. log 0.3 0.5229
Use the formula pH log[H] to find the pH of each substance given its concentration of hydrogen ions. 5. gastric juices: [H] 1.0 101 mole per liter 1.0 6. tomato juice: [H] 7.94 105 mole per liter 4.1 7. blood: [H] 3.98 108 mole per liter 7.4 8. toothpaste: [H] 1.26 1010 mole per liter 9.9 Solve each equation or inequality. Round to four decimal places. 9. 3x 243 {x | x 5}
1 4
1 2
10. 16v v v
12. 7y 15 1.3917
13. 53b 106 0.9659
14. 45k 37 0.5209
15. 127p 120 0.2752
16. 92m 27 0.75
17. 3r 5 4.1 6.2843
18. 8 y 4 15 {y | y 2.6977}
19. 7.6 d 3 57.2 1.0048
20. 0.5t 8 16.3 3.9732
21. 42x 84 1.0888
22. 5x
2
2
1
Lesson 10-4
11. 8 p 50 1.8813
10 0.6563
Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places.
log
7
10 23. log3 7 ; 1.7712
log10 3
log 35 log10 2
10 25. log2 35 ; 5.1293
©
Glencoe/McGraw-Hill
log 66 log10 5
10 24. log5 66 ; 2.6032
log 10 log10 6
10 26. log6 10 ; 1.2851
593
Glencoe Algebra 2
NAME ______________________________________________ DATE______________ PERIOD _____
10-5 Skills Practice Base e and Natural Logarithms Use a calculator to evaluate each expression to four decimal places. 1. e3 20.0855
2. e2 0.1353
3. ln 2 0.6931
4. ln 0.09 2.4079
Write an equivalent exponential or logarithmic equation. 5. ex 3 x ln 3
6. e4 8x 4 ln 8x
7. ln 15 x e x 15
8. ln x 0.6931 x e0.6931
Evaluate each expression. 9. eln 3 3 11. ln e2.5 2.5
10. eln 2x 2x
12. ln e y y
Solve each equation or inequality. 14. ex 3.2 {x | x 1.1632}
15. 2ex 1 11 1.7918
16. 5ex 3 18 1.0986
17. e3x 30 1.1337
18. e4x 10 {x | x 0.5756}
19. e5x 4 34 {x | x 0.6802}
20. 1 2e2x 19 1.1513
21. ln 3x 2 2.4630
22. ln 8x 3 2.5107
23. ln (x 2) 2 9.3891
24. ln (x 3) 1 0.2817
25. ln (x 3) 4 51.5982
26. ln x ln 2x 2 1.9221
©
Glencoe/McGraw-Hill
599
Lesson 10-5
13. ex 5 {x | x 1.6094}
Glencoe Algebra 2
NAME ______________________________________________ DATE______________ PERIOD _____
10-6 Skills Practice Solve each problem. 1. FISHING In an over-fished area, the catch of a certain fish is decreasing at an average rate of 8% per year. If this decline persists, how long will it take for the catch to reach half of the amount before the decline? about 8.3 yr
2. INVESTING Alex invests $2000 in an account that has a 6% annual rate of growth. To the nearest year, when will the investment be worth $3600? 10 yr
3. POPULATION A current census shows that the population of a city is 3.5 million. Using the formula P aert, find the expected population of the city in 30 years if the growth rate r of the population is 1.5% per year, a represents the current population in millions, and t represents the time in years. about 5.5 million
4. POPULATION The population P in thousands of a city can be modeled by the equation P 80e0.015t, where t is the time in years. In how many years will the population of the city be 120,000? about 27 yr
5. BACTERIA How many days will it take a culture of bacteria to increase from 2000 to 50,000 if the growth rate per day is 93.2%? about 4.9 days
6. NUCLEAR POWER The element plutonium-239 is highly radioactive. Nuclear reactors can produce and also use this element. The heat that plutonium-239 emits has helped to power equipment on the moon. If the half-life of plutonium-239 is 24,360 years, what is the value of k for this element? about 0.00002845
7. DEPRECIATION A Global Positioning Satellite (GPS) system uses satellite information to locate ground position. Abu’s surveying firm bought a GPS system for $12,500. The GPS depreciated by a fixed rate of 6% and is now worth $8600. How long ago did Abu buy the GPS system? about 6.0 yr
8. BIOLOGY In a laboratory, an organism grows from 100 to 250 in 8 hours. What is the hourly growth rate in the growth formula y a(1 r) t ? about 12.13%
©
Glencoe/McGraw-Hill
605
Glencoe Algebra 2
Lesson 10-6
Exponential Growth and Decay
NAME ______________________________________________ DATE
____________ PERIOD _____
11-1 Skills Practice Arithmetic Sequences 1. 7, 11, 15, … 19, 23, 27, 31
2. 10, 5, 0, … 5, 10, 15, 20
3. 101, 202, 303, … 404, 505, 606, 707
4. 15, 7, 1, … 9, 17, 25, 33
5. 67, 60, 53, …
6. 12, 15, 18, …
46, 39, 32, 25
21, 24, 27, 30
Find the first five terms of each arithmetic sequence described. 7. a1 6, d 9 6, 15, 24, 33, 42 9. a1 12, d 5 12, 7, 2, 3, 8 11. a1 64, d 11
64, 53, 42, 31, 20
8. a1 27, d 4 27, 31, 35, 39, 43 10. a1 93, d 15 93, 78, 63, 48, 33 12. a1 47, d 20
47, 67, 87, 107, 127
Find the indicated term of each arithmetic sequence. 13. a1 2, d 6, n 12 68
14. a1 18, d 2, n 8 32
15. a1 23, d 5, n 23 133
16. a1 15, d 1, n 25 9
17. a31 for 34, 38, 42, … 154
18. a42 for 27, 30, 33, … 150
Complete the statement for each arithmetic sequence. 19. 55 is the ? th term of 4, 7, 10, … . 18
20. 163 is the ? th term of 5, 2, 9, … . 25
Write an equation for the nth term of each arithmetic sequence. 21. 4, 7, 10, 13, … an 3n 1
22. 1, 1, 3, 5, … an 2n 3
23. 1, 3, 7, 11, … an 4n 5
24. 7, 2, 3, 8, … an 5n 12
Find the arithmetic means in each sequence. 25. 6, ? , ? , ? , 38 14, 22, 30
©
Glencoe/McGraw-Hill
26. 63, ? , ? , ? , 147 84, 105, 126
633
Glencoe Algebra 2
Lesson 11-1
Find the next four terms of each arithmetic sequence.
NAME ______________________________________________ DATE
____________ PERIOD _____
11-2 Skills Practice Arithmetic Series 1. a1 1, an 19, n 10 100
2. a1 5, an 13, n 7 28
3. a1 12, an 23, n 8 44
4. a1 7, n 11, an 67 407
5. a1 5, n 10, an 32 185
6. a1 4, n 10, an 22 130
7. a1 8, d 5, n 12 426
8. a1 1, d 3, n 15 330
9. a1 100, d 7, an 37 685
10. a1 9, d 4, an 27 90
11. d 2, n 26, an 42 442
12. d 12, n 11, an 52 88
Find the sum of each arithmetic series. 13. 1 4 7 10 … 43 330
14. 5 8 11 14 … 32 185
15. 3 5 7 9 … 19 99
16. 2 (5) (8) … (20) 77
5
17. (2n 3) 15 n1
10
19. (4n 1) 225 n2
18
18. (10 3n) 693 n1
12
20. (4 3n) 172 n5
Find the first three terms of each arithmetic series described. 21. a1 4, an 31, Sn 175 4, 7, 10
22. a1 3, an 41, Sn 228 3, 1, 5
23. n 10, an 41, Sn 230 5, 9, 13
24. n 19, an 85, Sn 760 5, 0, 5
©
Glencoe/McGraw-Hill
639
Glencoe Algebra 2
Lesson 11-2
Find Sn for each arithmetic series described.
NAME ______________________________________________ DATE
____________ PERIOD _____
11-3 Skills Practice Geometric Sequences Find the next two terms of each geometric sequence. 3 2
3
3
1. 1, 2, 4, … 8, 16
2. 6, 3, , … ,
3. 5, 15, 45, … 135, 405
4. 729, 243, 81 , … 27, 9
5. 1536, 384, 96, … 24, 6
6. 64, 160, 400, … 1000, 2500
Find the first five terms of each geometric sequence described. 7. a1 6, r 2
8. a1 27, r 3
27, 81, 243, 729, 2187
6, 12, 24, 48, 96 9. a1 15, r 1
10. a1 3, r 4
15, 15, 15, 15, 15 1 2
1 3
11. a1 1, r
12. a1 216, r
1
8
216, 72, 24, 8,
1, , , ,
Find the indicated term of each geometric sequence. 13. a1 5, r 2, n 6 160
14. a1 18, r 3, n 6 4374
15. a1 3, r 2, n 5 48
16. a1 20, r 2, n 9 5120
3
17. a8 for 12, 6, 3, …
80 80 3 9
80
18. a7 for 80, , , …
Write an equation for the nth term of each geometric sequence. 19. 3, 9, 27, … an 3n
20. 1, 3, 9, … an 1(3)n 1
21. 2, 6, 18, … an 2(3)n 1
22. 5, 10, 20, … an 5(2)n 1
Find the geometric means in each sequence. 23. 4, ? , ? , ? , 64 8, 16, 32 ©
Glencoe/McGraw-Hill
24. 1, ? , ? , ? , 81 3, 9, 27
645
Glencoe Algebra 2
Lesson 11-3
1 1 1
3, 12, 48, 192, 768
NAME ______________________________________________ DATE
____________ PERIOD _____
11-4 Skills Practice Geometric Series Find Sn for each geometric series described. 1. a1 2, a5 162, r 3 242
2. a1 4, a6 12,500, r 5 15,624
3. a1 1, a8 1, r 1 0
4. a1 4, an 256, r 2 172
5. a1 1, an 729, r 3 547
6. a1 2, r 4, n 5 410
7. a1 8, r 2, n 4 120
8. a1 3, r 2, n 12 4095 3 8
1 93 2
1 2
21
9. a1 8, r 3, n 5 968
10. a1 6, an , r
127
12. a1 2, r , n 6
1 2
11. a1 8, r , n 7
Find the sum of each geometric series. 13. 4 8 16 … to 5 terms 124
14. 1 3 9 … to 6 terms 364
15. 3 6 12 … to 5 terms 93
16. 15 30 60 … to 7 terms 645
4
5
17. 3n 1 40
18. (2)n 1 11
4
n1
13
19.
n1
Lesson 11-4
n1
9
40
20. 2(3)n 1 9842
n1
n1
Find the indicated term for each geometric series described. 21. Sn 1275, an 640, r 2; a1 5 1 2
23. Sn 99, n 5, r ; a1 144
©
Glencoe/McGraw-Hill
22. Sn 40, an 54, r 3; a1 2 24. Sn 39,360, n 8, r 3; a1 12
651
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
11-5 Skills Practice Infinite Geometric Series Find the sum of each infinite geometric series, if it exists. 2 25 5
1 2
1. a1 1, r 2
2. a1 5, r
3. a1 8, r 2 does not exist
4. a1 6, r 12
1 2
1 2
5. 4 2 1 … 8
6. 540 180 60 20 … 405
7. 5 10 20 … does not exist
8. 336 84 21 … 268.8
9. 125 25 5 … 156.25
3 4
9 4
27 4
11. … does not exist
25
13. 5 2 0.8 …
1 n1 2
15. 10 n1
n1
n1
52
17. 15
1 3
1
1 27
14. 9 6 4 … 27
n1
13
16. 6 n1
25
1 9
12. …
20
81
1 9
10. 9 1 …
n1
n1
43 13
18.
9
2
Write each repeating decimal as a fraction.
4
8
20. 0.8
3
22. 0.6 7
6
24. 0.3 7 5
641
26. 0.1 7 1
21. 0.2 7
23. 0.5 4
25. 0.6 4 1
©
Glencoe/McGraw-Hill
67
125
Lesson 11-5
19. 0.4
57
657
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
11-6 Skills Practice Recursion and Special Sequences 1. a1 4, an 1 an 7
2. a1 2, an 1 an 3
2, 1, 4, 7, 10
4, 11, 18, 25, 32 3. a1 5, an 1 2an
4. a1 4, an 1 6 an
4, 10, 4, 10, 4
5, 10, 20, 40, 80 5. a1 1, an 1 an n
6. a1 1, an 1 n an
1, 2, 0, 3, 1
1, 2, 4, 7, 11 7. a1 6, an 1 an n 1
6, 4, 1, 3, 8
9. a1 3, an 1 2an 7
3, 1, 9, 25, 57
11. a1 0, a2 1, an 1 an an 1
8. a1 8, an 1 an n 2
8, 5, 1, 4, 10
10. a1 4, an 1 2an 5
4, 13, 21, 47, 89
12. a1 1, a2 1, an 1 an an 1
1, 1, 0, 1, 1
0, 1, 1, 2, 3 13. a1 3, a2 5, an 1 4an an 1
3, 5, 23, 97, 411
Lesson 11-6
Find the first five terms of each sequence.
14. a1 3, a2 2, an 1 an 1 an
3, 2, 5, 7, 12
Find the first three iterates of each function for the given initial value. 15. f(x) 2x 1, x0 3 5, 9, 17
16. f(x) 5x 3, x0 2 7, 32, 157
17. f(x) 3x 4, x0 1 1, 7, 25
18. f(x) 4x 7, x0 5 13, 45, 173
19. f(x) x 3, x0 10 13, 10, 13
20. f(x) 3x 6, x0 6 12, 42, 120
21. f(x) 3x 4, x0 2 2, 10, 26
22. f(x) 6x 5, x0 1 1, 1, 1
23. f(x) 7x 1, x0 4
24. f(x) x2 3x, x0 5
27, 188, 1315
©
Glencoe/McGraw-Hill
10, 70, 4690 663
Glencoe Algebra 2
NAME ______________________________________________ DATE______________ PERIOD _____
11-7 Skills Practice The Binomial Theorem Evaluate each expression. 1. 8! 40,320
2. 10! 3,628,800
3. 12! 479,001,600
4. 210
15! 13!
6! 3!
6. 45
9! 3!6!
8. 15,504
7. 84
10! 2!8!
Lesson 11-7
5. 120
20! 15!5!
Expand each power. 9. (x y)3
x 3 3x 2y 3xy 2 y 3 11. (g h)4
g 4 4g 3h 6g 2h 2 4gh 3 h4 13. (r 4)3
r 3 12r 2 48r 64 15. ( y 7)3
y 3 21y 2 147y 343 17. (x 1)4
x 4 4x 3 6x 2 4x 1 19. (c 4d)3
c 3 12c 2d 48cd 2 64d 3
10. (a b)5
a 5 5a 4b 10a 3b 2 10a 2b 3 5ab4 b 5 12. (m 1)4
m4 4m 3 6m 2 4m 1 14. (a 5)4
a 4 20a 3 150a 2 500a 625 16. (d 2)5
d 5 10d 4 40d 3 80d 2 80d 32 18. (2a b)4
16a 4 32a 3b 24a 2b 2 8ab3 b4 20. (2a 3)3
8a3 36a 2 54a 27
Find the indicated term of each expansion. 21. fourth term of (m n)10 120m7n 3
22. seventh term of (x y)8 28x 2y 6
23. third term of (b 6)5 360b 3
24. sixth term of (s 2)9 4032s4
25. fifth term of (2a 3)6 4860a 2
26. second term of (3x y)7 5103x 6y
©
Glencoe/McGraw-Hill
669
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
11-8 Skills Practice Proof and Mathematical Induction Prove that each statement is true for all positive integers. 1. 1 3 5 … (2n 1) n2
Step 1: When n 1, 2n 1 2(1) 1 1 12. So, the equation is true for n 1. Step 2: Assume that 1 3 5 … (2k 1) k 2 for some positive integer k. Step 3: Show that the given equation is true for n k 1. 1 3 5 … (2k 1) [2(k 1) 1] k 2 [2(k 1) 1] k 2 2k 1 (k 1)2 So, 1 3 5 … (2n 1) n 2 for all positive integers n. Step 1: When n 1, 2n 2(1) 2 12 1. So, the equation is true for n 1. Step 2: Assume that 2 4 6 … 2k k 2 k for some positive integer k. Step 3: Show that the given equation is true for n k 1. 2 4 6 …. 2k 2(k 1) k 2 k 2(k 1) (k 2 2k 1) (k 1) (k 1)2 (k 1) So, 2 4 6 … 2n n 2 n for all positive integers n. 3. 6n 1 is divisible by 5.
Step 1: When n 1, 6n 1 61 1 5. So, the statement is true for n 1. Step 2: Assume that 6k 1 is divisible by 5 for some positive integer k. Then there is a whole number r such that 6k 1 5r. Step 3: Show that the statement is true for n k 1. 6k 1 5r 6k 5r 1 6(6k ) 6(5r 1) 6k 1 30r 6 6k 1 1 30r 5 6k 1 1 5(6r 1) Since r is a whole number, 6r 1 is a whole number, and 6k 1 1 is divisible by 5. The statement is true for n k 1. So, 6n 1 is divisible by 5 for all positive integers n. Find a counterexample for each statement. 4. 3n 3n is divisible by 6.
n(n 1)(2n 1) 6
5. 1 4 8 … 2n
Sample answer: n 2 ©
Glencoe/McGraw-Hill
Sample answer: n 3 675
Glencoe Algebra 2
Lesson 11-8
2. 2 4 6 … 2n n2 n
NAME ______________________________________________ DATE
____________ PERIOD _____
12-1 Skills Practice The Counting Principle State whether the events are independent or dependent. 1. finishing in first, second, or third place in a ten-person race dependent 2. choosing a pizza size and a topping for the pizza independent
4. The 232 members of the freshman class all vote by secret ballot for the class representative to the Student Senate. independent
Solve each problem. 5. A surveying firm plans to buy a color printer for printing its maps. It has narrowed its choice to one of three models. Each of the models is available with either 32 megabytes of random access memory (RAM), 64 megabytes of RAM, or 128 megabytes of RAM. From how many combinations of models and RAM does the firm have to choose? 9 6. How many arrangements of three letters can be formed from the letters of the word MATH if any letter will not be used more than once? 24 7. Allan is playing the role of Oliver in his school’s production of Oliver Twist. The wardrobe crew has presented Allan with 5 pairs of pants and 4 shirts that he can wear. How many possible costumes consisting of a pair of pants and a shirt does Allan have to choose from? 20 8. The 10-member steering committee that is preparing a study of the public transportation needs of its town will select a chairperson, vice-chairperson, and secretary from the committee. No person can serve in more than one position. In how many ways can the three positions be filled? 720 9. Jeanine has decided to buy a pickup truck. Her choices include either a V-6 engine or a V-8 engine, a standard cab or an extended cab, and 2-wheel drive or 4-wheel drive. How many possible models does she have to choose from? 8 10. A mail-order company that sells gardening tools offers rakes in two different lengths. Customers can also choose either a wooden, plastic, or fiberglass handle for the rake. How many different kinds of rakes can a customer buy? 6 11. A Mexican restaurant offers chicken, beef, or vegetarian fajitas wrapped with either corn or flour tortillas, and topped with either mild, medium, or hot salsa. How many different choices of fajitas does a customer have? 18
©
Glencoe/McGraw-Hill
701
Glencoe Algebra 2
Lesson 12-1
3. Seventy-five raffle tickets are placed in a jar. Three tickets are then selected, one after the other, without replacing a ticket after it is chosen. dependent
NAME ______________________________________________ DATE
____________ PERIOD _____
12-2 Skills Practice Permutations and Combinations Evaluate each expression. 1. P(6, 3) 120
2. P(8, 2) 56
3. P(2, 1) 2
4. P(3, 2) 6
5. P(10, 4) 5040
6. P(5, 5) 120
7. C(2, 2) 1
8. C(5, 3) 10
9. C(4, 1) 4
10. C(8, 7) 8
11. C(3, 2) 3
12. C(7, 4) 35
Determine whether each situation involves a permutation or a combination. Then find the number of possibilities. 13. seating 8 students in 8 seats in the front row of the school auditorium
14. introducing the 5 starting players on the Woodsville High School basketball team at the beginning of the next basketball game
permutation; 120 15. checking out 3 library books from a list of 8 books for a research paper
combination; 56 16. choosing 2 movies to rent from 5 movies
combination; 10 17. the first-, second-, and third-place finishers in a race with 10 contestants
permutation; 720 18. electing 4 candidates to a municipal planning board from a field of 7 candidates
combination; 35 19. choosing 2 vegetables from a menu that offers 6 vegetable choices
combination; 15 20. an arrangement of the letters in the word rhombus
permutation; 5040 21. selecting 2 of 8 choices of orange juice at a store
combination; 28 22. placing a red rose bush, a yellow rose bush, a white rose bush, and a pink rose bush in a row in a planter permutation; 24 23. selecting 2 of 9 kittens at an animal rescue shelter
combination; 36 24. an arrangement of the letters in the word isosceles
permutation; 30,240 ©
Glencoe/McGraw-Hill
707
Glencoe Algebra 2
Lesson 12-2
permutation; 40,320
NAME ______________________________________________ DATE
____________ PERIOD _____
12-3 Skills Practice Probability Ahmed is posting 2 photographs on his website. He has narrowed his choices to 4 landscape photographs and 3 portraits. If he chooses the two photographs at random, find the probability of each selection.
1 7
2 7
1. P(2 portrait)
2. P(2 landscape)
4 7
3. P(1 of each)
The Carubas have a collection of 28 video movies, including 12 westerns and 16 science fiction. Elise selects 3 of the movies at random to bring to a sleep-over at her friend’s house. Find the probability of each selection.
55 819
20 117
5. P(3 science fiction)
40 91
88 273
6. P(1 western and 2 science fiction)
7. P(2 westerns and 1 science fiction)
8. P(3 comedy) 0
9. P(2 science fiction and 2 westerns) 0
For Exercises 10–13, use the chart that shows the class and gender statistics for the students taking an Algebra 1 or Algebra 2 class at La Mesa High School. If a student taking Algebra 1 or Algebra 2 is selected at random, find each probability. Express as decimals rounded to the nearest thousandth. 10. P(sophomore/female) 0.208 11. P(junior/male) 0.143
Class/Gender Freshman/Male
Number 95
Freshman/Female
101
Sophomore/Male
154
Sophomore/Female
145
Junior/Male
100
Junior/Female
102
12. P(freshman/male) 0.136 13. P(freshman/female) 0.145
Find the odds of an event occurring, given the probability of the event. 14. 5:3
5 8
15. 2:5
2 7
16. 3:2
1 10
18. 5:1
5 6
19. 5:7
17. 1:9
3 5
5 12
Find the probability of an event occurring, given the odds of the event.
2 3 1 23. 1:9 10 20. 2:1
©
Glencoe/McGraw-Hill
8 17 2 24. 2:7 9
4 5 5 25. 5:9 14
21. 8:9
22. 4:1
713
Glencoe Algebra 2
Lesson 12-3
4. P(3 westerns)
NAME ______________________________________________ DATE
____________ PERIOD _____
12-4 Skills Practice Multiplying Probabilities A die is rolled twice. Find each probability.
1 36
1 36
25 36
1. P(5, then 6)
2. P(no 2s)
5 6
4. P(any number, then not 5)
3. P(two 1s)
5 36
25 36
5. P(4, then not 6)
6. P(not 1, then not 2)
A board game uses a set of 6 different cards. Each card displays one of the following figures: a star, a square, a circle, a diamond, a rectangle, or a pentagon. The cards are placed face down, and a player chooses two cards. Find each probability.
1 30
7. P(circle, then star), if no replacement occurs
1 36
8. P(diamond, then square), if replacement occurs
25 36
9. P(2 polygons), if replacement occurs
2 3
10. P(2 polygons), if no replacement occurs 11. P(circle, then hexagon), if no replacement occurs 0
Determine whether the events are independent or dependent. Then find each probability. 12. A mixed box of herbal teabags contains 2 lemon teabags, 3 orange-mango teabags, 3 chamomile teabags, and 1 apricot-ginger teabag. Kevin chooses 2 teabags at random to bring to work with him. What is the probability that he first chooses a lemon teabag and 1 then a chamomile teabag?
dependent; 12
Type of Oil
Domestic Imported
Pure
2
5
Cold Pressed
4
8
First Cold Pressed
7
15
dependent; 820
For Exercises 14 and 15, two thirds of the area of the spinner earns you 50 points. Suppose you spin the spinner twice. 14. Sketch a tree diagram showing all of the possibilities. Use it to find the probability of spinning 50 points, then 100 points. 2
50
100
9
15. What is the probability that you get 100 points on each spin? 1
9
©
Glencoe/McGraw-Hill
719
Glencoe Algebra 2
Lesson 12-4
13. The chart shows the selection of olive oils that Hasha finds in a specialty foods catalog. If she randomly selects one type of oil, then randomly selects another, different oil, what is the probability that both selections are domestic, 21 first cold pressed oils?
NAME ______________________________________________ DATE
____________ PERIOD _____
12-5 Skills Practice Adding Probabilities Eli has 10 baseball cards of 10 different players in his pocket. Three players are pitchers, 5 are outfielders, and 2 are catchers. If Eli randomly selects a card to trade, find each probability.
4 5
1 2
1. P(pitcher or outfielder)
2. P(pitcher or catcher)
7 10
3. P(outfielder or catcher)
A die is rolled. Find each probability.
1 3
2 3
4. P(5 or 6)
5. P(at least a 3)
1 2
6. P(less than 4)
Determine whether the events are mutually exclusive or inclusive. Then find the probability.
1 3
7. A die is rolled. What is the probability of rolling a 3 or a 4? mutually exclusive;
1 2
8. A die is rolled. What is the probability of rolling an even number or a 4? inclusive; 9. A card is drawn from a standard deck of cards. What is the probability of drawing a king 2 or a queen? mutually exclusive;
13
10. A card is drawn from a standard deck of cards. What is the probability of drawing a jack 4 or a heart? inclusive;
13
11. The sophomore class is selling Mother’s Day plants to raise money. Susan’s prize for being the top seller of plants is a choice of a book, a CD, or a video. She can choose from 6 books, 3 CDs, and 5 videos. What is the probability that Susan selects a book or a CD?
9 14
mutually exclusive; A spinner numbered 110 is spun. Find each probability.
7 10
12. P(less than 5 or even)
13. P(even or odd) 1
4 5
14. P(prime or even)
Two cards are drawn from a standard deck of cards. Find each probability.
55 221
25 51
16. P(both aces or both red)
11 221
17. P(both 2s or both less than 5)
188 663
18. P(both black or both less than 5)
For Exercises 19 and 20, use the Venn diagram that shows the number of participants in two different kinds of aerobic exercise classes that are offered at a health club. Determine each probability if a person is selected at random from the participants.
49 62
Step Aerobics 22
13 Jazzercise 27
19. P(step aerobics or jazzercise, but not both)
13 62
20. P(step aerobics and jazzercise) ©
Glencoe/McGraw-Hill
725
Glencoe Algebra 2
Lesson 12-5
15. P(both red or both black)
NAME ______________________________________________ DATE
____________ PERIOD _____
12-6 Skills Practice Find the variance and standard deviation of each set of data to the nearest tenth. 1. {32, 41, 35, 35, 46, 42} 23.6, 4.9 2. {13, 62, 77, 24, 38, 19, 88} 763.8, 27.6 3. {89, 99, 42, 16, 42, 71, 16} 959.1, 31.0 4. {450, 400, 625, 225, 300, 750, 650, 625} 30,537.1; 174.7 5. {17, 23, 65, 94, 33, 33, 33, 8, 57, 75, 44, 12, 11, 68, 39} 630.7, 25.1 6. {7.2, 3.1, 3.8, 9.5, 8.3, 8.4} 5.8, 2.4 7. {1.5, 2.5, 3.5, 4.5, 4.5, 5.5, 6.5, 7.5} 3.5, 1.9 For Exercises 8 and 9, use the table that shows the profit in billions of dollars reported by U.S. manufacturers for the first quarter of the years from 1997 through 2001. Year
1997
1998
1999
2000
2001
Seasonally-Adjusted $61.4 $75.6 $60.9 $78.5 $45.3 Profit ($ billions) Source: U. S. Census Bureau
8. Find the mean and median of the data to the nearest tenth. $64.3 billion, $61.4 billion 9. Which measure of central tendency best represents the data? Explain.
The median is more representative because the value 45.3 is not close to the other data points, and it lowers the mean. For Exercises 10 and 11, use the table that shows the percent of fourth grade students reading at or above the proficiency level in a nationally-administered reading assessment. Year
1992 1994 1998 2000
Percent at or above 29% 30% 31% 32% proficiency level Source: National Center for Education Statistics
10. Find the mean, median, and standard deviation of the data to the nearest tenth.
30.5%, 30.5%, 1.1 11. What do the statistics from Exercise 11 tell you about the data?
Sample answer: Since the median and mean are equal and the standard deviation is small, the percent of students reading at or above the proficiency level has not varied much from 1992 to 2000. ©
Glencoe/McGraw-Hill
731
Glencoe Algebra 2
Lesson 12-6
Statistical Measures
NAME ______________________________________________ DATE
____________ PERIOD _____
12-7 Skills Practice The Normal Distribution Determine whether the data in each table appear to be positively skewed, negatively skewed, or normally distributed. 2. Speeches Given Political Candidates
0–4
3
0–5
1
5–9
4
6–11
2
10–14
7
12–17
3
15–19
5
18–23
8
20–23
2
24–29
8
normally distributed
negatively skewed
For Exercises 3 and 4, use the frequency table that shows the average number of days patients spent on the surgical ward of a hospital last year.
4. Do the data appear to be positively skewed, negatively skewed, or normally distributed? Explain.
Positively skewed; the histogram is high at the left and has a tail to the right.
Frequency
3. Make a histogram of the data.
Patient Stays
20 18 16 14 12 10 8 6 4 2 0–3
Days
Number of Patients
0–3
5
4–7
18
8–11
11
12–15
9
16
6
4–7 8–11 12–15 16
Days
DELIVERY For Exercises 5–7, use the following information. The time it takes a bicycle courier to deliver a parcel to his farthest customer is normally distributed with a mean of 40 minutes and a standard deviation of 4 minutes. 5. About what percent of the courier’s trips to this customer take between 36 and 44 minutes?
68% 6. About what percent of the courier’s trips to this customer take between 40 and 48 minutes?
47.5% 7. About what percent of the courier’s trips to this customer take less than 32 minutes? 2.5%
TESTING For Exercises 8–10, use the following information. The average time it takes sophomores to complete a math test is normally distributed with a mean of 63.3 minutes and a standard deviation of 12.3 minutes. 8. About what percent of the sophomores take more than 75.6 minutes to complete the test?
16% 9. About what percent of the sophomores take between 51 and 63.3 minutes? 34% 10. About what percent of the sophomores take less than 63.3 minutes to complete the test?
50% ©
Glencoe/McGraw-Hill
737
Glencoe Algebra 2
Lesson 12-7
1. Miles Run Track Team Members
NAME ______________________________________________ DATE
____________ PERIOD _____
12-8 Skills Practice Binomial Experiments Find each probability if a coin is tossed 4 times.
1 16
1 16
1. P(4 heads)
2. P(0 heads)
1 4
3 8
3. P(exactly 3 heads)
4. P(exactly 2 heads)
5 16
1 4
5. P(exactly 1 head)
6. P(at least 3 heads)
Find each probability if a die is rolled 3 times.
5 72
25 72
8. P(exactly two 2s)
1 216
9. P(exactly three 2s)
25 27
10. P(at most one 2)
A town that presents a fireworks display during its July 4 celebration found the 3 probability that a family with two or more children will watch the fireworks is . 5 If 5 of these families are selected at random, find each probability. 11. P(exactly 3 families watch the fireworks) 12. P(exactly 2 families watch the fireworks)
216 625
144 625
13. P(exactly 5 families watch the fireworks) 14. P(no families watch the fireworks)
243 3125
32 3125
15. P(at least 4 families watch the fireworks) 16. P(at most 1 family watches the fireworks)
272 3125
1053 3125
One section of a standardized English language test has 10 true/false questions. Find each probability when a student guesses at all ten questions.
45 1024
17. P(exactly 8 correct)
63 256
19. P(exactly half correct)
1 1024
21. P(0 correct)
©
Glencoe/McGraw-Hill
45 1024
18. P(exactly 2 correct)
1 1024
20. P(all 10 correct)
7 128
22. P(at least 8 correct)
743
Glencoe Algebra 2
Lesson 12-8
7. P(exactly one 2)
NAME ______________________________________________ DATE
____________ PERIOD _____
12-9 Skills Practice Sampling and Error Determine whether each situation would produce a random sample. Write yes or no and explain your answer. 1. calling households at 3:30 P.M. on Tuesday to determine a political candidate’s support
No; since most registered voters are likely to be at work at this time, this sample would not be representative of all registered voters. 2. polling customers as they exit a sporting goods store about their attitudes about exercise
No; these customers are likely to value exercise more than those who do not shop at sporting goods stores, who are not represented in this survey. 3. recording the number of sit-ups performed by 15-year old girls in the high schools of a large school district to determine the fitness of all high-school girls in the district
No; 15-year old girls may not have the same abilities as 18-year old seniors, for example. 4. selecting two of a city’s 20 apartment buildings for a survey to determine the desire of apartment dwellers in the city to own a home No; the residents of the two
buildings selected might, for example, have nicer apartments or be in a nicer area of town, and thus would not well represent the desires of people in other buildings.
teachers from all levels was selected at random, the sample should well represent the population of teachers in the district. 6. For seven consecutive days, one hour each in the morning, afternoon, and evening, every tenth customer who enters a mall is asked to choose her or his favorite store. Yes;
because the sample is chosen over the course of a whole week, during hours when different consumer groups shop, and because the selection is systematic, the sample should well represent the general population that shops at the mall stores. Find the margin of sampling error to the nearest percent. 7. p 85%, n 100 about 7%
8. p 78%, n 100 about 8%
9. p 15%, n 100 about 7%
10. p 37%, n 500 about 4%
11. p 12%, n 500 about 3%
12. p 93%, n 500 about 2%
13. p 23%, n 1000 about 3%
14. p 56%, n 1000 about 3%
15. HEALTH In a recent poll of cigarette smokers, 67% of those surveyed said they had tried to quit smoking within the last year. The margin of error was 3%. About how many people were surveyed? about 983 ©
Glencoe/McGraw-Hill
749
Glencoe Algebra 2
Lesson 12-9
5. In a large school district, the superintendent of schools interviews two teachers at random from each school to determine whether teachers in the district think students are assigned too much or too little homework. Yes; since a cross section of
NAME ______________________________________________ DATE
____________ PERIOD _____
13-1 Skills Practice Right Triangle Trigonometry Find the values of the six trigonometric functions for angle . 2.
3. 5
2
6
8
3
13
13 3 13 13 2 cos , 13 3 13 tan , csc , 2 3 2 13 sec , cot 3 2
5 12 13 13 5 13 tan , csc , 12 5 13 12 sec , cot 12 5
4 3 5 5 4 5 tan , csc , 3 4 5 3 sec , cot 3 4
sin , cos ,
sin , cos ,
sin ,
Write an equation involving sin, cos, or tan that can be used to find x. Then solve the equation. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. 4.
5.
6. 60
8
x
x
5
10 22
30
x
8 x
7.
cos 60 , x 10 8.
60
x 10
5 x
tan 30 , x 13.9
5
9.
x 8
tan 22 , x 4.0 x 2
5 4
x
x 5
sin 60 , x 4.3
5 8
cos x , x 51
4 2
tan x , x 63
Solve ABC by using the given measurements. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. 10. A 72, c 10
a 9.5, b 3.1, B 18
a 41.2, c 43.9, A 70
13. A 58, b 12
14. b 4, c 9
15. a 7, b 5
b 1.6, c 9.1, B 10 a 8.1, A 64, B 26
©
Glencoe/McGraw-Hill
b
11. B 20, b 15
12. A 80, a 9
A
C
c
a
B
a 19.2, c 22.6, B 32 c 8.6, A 54, B 36
777
Glencoe Algebra 2
Lesson 13-1
1.
NAME ______________________________________________ DATE
____________ PERIOD _____
13-2 Skills Practice Angles and Angle Measure Draw an angle with the given measure in standard position. 2. 810
3. 390
y
y
x
O
y
x
O
5. 50
4. 495
6. 420 y
y
x
O
x
O
O
y
x
O
x
Rewrite each degree measure in radians and each radian measure in degrees.
13 18
7. 130
8. 720 4
7 6
2
9. 210
10. 90
6
3 2
11. 30
12. 270
13. 60
3
14. 150
5 6
2 3
16. 225
5 4
15. 120 3 4
17. 135
7 6
18. 210
Find one angle with positive measure and one angle with negative measure coterminal with each angle. 19–26. Sample answers are given. 19. 45 405, 315
20. 60 420, 300
21. 370 10, 350
22. 90 270, 450
2 8 3 3
4 3
24. ,
13 6 6
11 6
26. ,
23. , 25. ,
©
Glencoe/McGraw-Hill
5 9 2 2
3 5 4 4
783
2
3 2
Glencoe Algebra 2
Lesson 13-2
1. 185
NAME ______________________________________________ DATE
____________ PERIOD _____
13-3 Skills Practice Trigonometric Functions of General Angles Find the exact values of the six trigonometric functions of if the terminal side of in standard position contains the given point. 1. (5, 12)
2. (3, 4)
5 12 12 sin , cos , tan , 13 13 5 5 13 13 csc , sec , cot 12 12 5
4 3 4 5 5 3 5 5 3 csc , sec , cot 4 3 4
sin , cos , tan ,
3. (8, 15)
4. (4, 3)
8 15 15 17 17 8 8 17 17 csc , sec , cot 15 15 8
3 4 3 5 5 4 5 5 4 csc , sec , cot 3 4 3
sin , cos , tan ,
5. (9, 40)
sin , cos , tan ,
6. (1, 2)
25 5
9 40 40 sin , cos , tan , 41 41 9
2,
41 40
5
5
9 40
41 9
5
sin , cos , tan 1 2
csc , sec 5 , cot
csc , sec , cot
2
Sketch each angle. Then find its reference angle. 8. 200 20
y
y
y
x
O
x
O
5 3 3
9.
O
Lesson 13-3
7. 135 45
x
Find the exact value of each trigonometric function.
1 2
10. sin 150 4
14. tan 1
12. cot 135 1
11. cos 270 0 4 3
3 13. tan (30)
3 2 3 16. cot () 17. sin 4 2 undefined
1 2
15. cos
Suppose is an angle in standard position whose terminal side is in the given quadrant. For each function, find the exact values of the remaining five trigonometric functions of . 4 5
12 5
18. sin , Quadrant II
3 4 5 3 5 3 sec , cot 3 4
19. tan , Quadrant IV
5 4
cos , tan , csc ,
©
Glencoe/McGraw-Hill
5 12 13 13 5 13 sec , cot 12 5
13 12
sin , cos , csc ,
789
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
13-4 Skills Practice Law of Sines Find the area of ABC to the nearest tenth. 1.
36.9 cm2
B
2.
10.0 ft2
A 7 ft
10 cm
C
125
35
B
A
9 cm
5 ft
C
3. A 35, b 3 ft, c 7 ft 6.0 ft2
4. C 148, a 10 cm, b 7 cm 18.5 cm2
5. C 22, a 14 m, b 8 m 21.0 m2
6. B 93, c 18 mi, a 42 mi 377.5 mi2
Solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. 7. A
8. B
9.
12
B
15 18
C
B
B 93, a 102.1, b 393.8 10. C 10
30
A
121
A
C 150, a 31.5, b 21.2 11.
C
B 29, C 30, c 124.6 12. B
C
109
20
B
B 60, C 90, b 17.3
A
119
C 105
A
37
75 22
B
C 68, a 14.3, b 22.9
70
A
B 65, C 45, c 82.2
Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. 13. A 30, a 1, b 4
14. A 30, a 2, b 4 one solution;
15. A 30, a 3, b 4 two solutions;
16. A 38, a 10, b 9 one solution;
17. A 78, a 8, b 5 one solution;
18. A 133, a 9, b 7 one solution;
19. A 127, a 2, b 6 no solution
20. A 109, a 24, b 13 one solution;
no solution
B 42, C 108, c 5.7; B 138, C 12, c 1.2 B 38, C 64, c 7.4
©
Glencoe/McGraw-Hill
B 90, C 60, c 3.5
B 34, C 108, c 15.4 B 35, C 12, c 2.6
B 31, C 40, c 16.4
795
Glencoe Algebra 2
Lesson 13-4
375
212
51
72 C
NAME ______________________________________________ DATE
____________ PERIOD _____
13-5 Skills Practice Law of Cosines Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. 1. B
2. B
7
3.
C
4 34
5
B
9
A 10 18
C
41 3
A
A
cosines; B 23, C 116, a 5.1
4.
B 4
C
sines; A 27, C 119, c 7.9
5.
6.
C
2
cosines; A 143, B 20, C 18
C
4
130
B
4 20
A 3
cosines; A 104, B 47, C 29
A
5
B
cosines; A 41, C 54, b 6.1
7. C 71, a 3, b 4
sines; B 30, a 2.7, c 6.1
8. A 11, C 27, c 50
cosines; A 43, B 66, c 4.1
9. C 35, a 5, b 8
A
sines; B 142, a 21.0, b 67.8
10. B 47, a 20, c 24
cosines; A 37, B 108, c 4.8
cosines; A 55, C 78, b 17.9
11. A 71, C 62, a 20
12. a 5, b 12, c 13
13. A 51, b 7, c 10
14. a 13, A 41, B 75
15. B 125, a 8, b 14
16. a 5, b 6, c 7
sines; B 47, b 15.5, c 18.7
cosines; A 23, B 67, C 90
cosines; B 44, C 85, a 7.8
sines; A 28, C 27, c 7.8
©
Glencoe/McGraw-Hill
sines; C 64, b 19.1, c 17.8
cosines; A 44, B 57, C 78
801
Glencoe Algebra 2
Lesson 13-5
C
85
NAME ______________________________________________ DATE
____________ PERIOD _____
13-6 Skills Practice The given point P is located on the unit circle. Find sin and cos .
4 5
35 45
153
1. P , sin ,
12 13
12 13
9 41
40 41
2. P , sin , 3. P , sin
5 13
3
cos
cos
4. P(0, 1) sin 1,
5. P(1, 0) sin 0,
cos 0
cos 1
9 41
40 41
, cos
12
3
6. P , sin
2
2
1 2
, cos
Find the exact value of each function.
10. cos 330
2
13. sin 5 0 7 3
16. sin
1 2 1 11. cos (60) 2
12. sin (390)
14. cos 3 1
15. sin 1
8. sin 210
7. cos 45
7 3
1 2
5 2
1 2
17. cos
2
1 2
9. sin 330
5 6
18. cos
2
Determine the period of each function. 19.
4
y 2
O
1
2
3
4
5
6
7
8
9
10
2
20.
2
y 2
O
1
2
3
4
5
6
7
8
9
10 x
2
21.
2
y 1 O
2
3
4
1
©
Glencoe/McGraw-Hill
807
Glencoe Algebra 2
Lesson 13-6
Circular Functions
NAME ______________________________________________ DATE
____________ PERIOD _____
13-7 Skills Practice Inverse Trigonometric Functions Write each equation in the form of an inverse function. 1. cos cos1
2. sin b a sin1 a b
3. y tan x x tan1 y
2 2 4. cos 45 cos1 45 2 2
5. b sin 150 150 sin1 b
6. tan y tan1 y
Lesson 13-7
4 5
4 5
Solve each equation by finding the value of x to the nearest degree. 7. x Cos1 (1) 180
9. Tan1 1 x 45
11. x Arctan 0 0
8. Sin1 (1) x 90 3
10. x Arcsin 60 2 1 2
12. x Arccos 60
Find each value. Write angle measures in radians. Round to the nearest hundredth. 2
3
13. Sin1 0.79 radians 2
14. Cos1 2.62 radians 2
15. Tan1 3 1.05 radians
16. Arctan 0.52 radians 3
2
3
17. Arccos 2.36 radians 2
18. Arcsin 1 1.57 radians
19. sin (Cos1 1) 0
20. sin Sin1 0.5
3
1 2
21. tan Arcsin 1.73 2
22. cos (Tan1 3) 0.32
23. sin [Arctan (1)] 0.71
24. sin Arccos 2
©
Glencoe/McGraw-Hill
813
2
0.71 Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
14-1 Skills Practice Graphing Trigonometric Functions Find the amplitude, if it exists, and period of each function. Then graph each function. 2. y 4 sin
y
y
y
2
4
4
1
2
2
O
90 180 270 360
O
90 180 270 360
O
1
2
2
2
4
4
1 2
4. y tan
5. y sin 3
y
y
y 2
4
1
1
2
90 180 270 360
O
90 180 270 360
O
1
1
2
2
2
4
7. y tan 2
y 4
2
1
2
90 135 180
O
45
90 135 180
O
2
1
2
4
2
4
Glencoe/McGraw-Hill
150
y
2
45
90
9. y 4 sin
4
O
30
1 2
8. y cos 2
y
90 180 270 360
6. y csc 3
2
O
©
3. y 2 sec
839
180 360 540 720
Glencoe Algebra 2
Lesson 14-1
1. y 2 cos
NAME ______________________________________________ DATE
____________ PERIOD _____
14-2 Skills Practice Translations of Trigonometric Graphs State the amplitude, period, and phase shift for each function. Then graph the function. 2. y cos ( 45)
y
y 2
4
1
1
2
90 180 270 360
O
y
2
O
2
3. y tan
90 180 270 360
O
1
1
2
2
2
4
2
3 2
2
State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. 4. y csc 2
5. y cos 1
y
y
6. y sec 3 y 6
2 2
4
O
180 360 540 720
1
2
2 O
4
180 360 540 720
1
O
90 180 270 360
2
6
State the vertical shift, amplitude, period, and phase shift of each function. Then graph the function. 7. y 2 cos [3( 45)] 2
8. y 3 sin [2( 90)] 2
y 6
4
4
4
2
2
2
90 180 270 360
2
©
Glencoe/McGraw-Hill
O
2
y
y
6
O
4
43
9. y 4 cot
O 2
2
3 2
2
90 180 270 360 4
2
845
Glencoe Algebra 2
Lesson 14-2
1. y sin ( 90)
NAME ______________________________________________ DATE
____________ PERIOD _____
14-3 Skills Practice Trigonometric Identities Find the value of each expression. 4 5
1. sin , if cos and 90 180
2. cos , if tan 1 and 180 270
3. sec , if tan 1 and 0 90
4. cos , if tan and 0 90
1 2
2
5. tan , if sin and 180 270 6. cos , if sec 2 and 270 360 2
3 2
9. cos , if cot and 90 180
11. cot , if csc 2 and 180 270
25
8. tan , if cos and 180 270 5
8 17
10. csc , if cos and 0 90
5 13
12. tan , if sin and 180 270
Simplify each expression. 13. sin sec
14. csc sin
15. cot sec
16.
17. tan cot
18. csc tan tan sin
1 sin2 sin 1
20. csc cot
sin2 cos2 1 cos
22. 1
19.
21. 2
©
cos sec
Glencoe/McGraw-Hill
tan2 1 sec
851
Glencoe Algebra 2
Lesson 14-3
7. cos , if csc 2 and 180 270
NAME ______________________________________________ DATE
____________ PERIOD _____
14-4 Skills Practice Verifying Trigonometric Identities Verify that each of the following is an identity. 1. tan cos sin
2. cot tan 1
3. csc cos cot
4. cos
5. (tan )(1 sin2 ) sin cos
6. cot
1 sin2 cos
sin2 1 sin
cos2 1 sin
7. tan2 2
©
Glencoe/McGraw-Hill
Lesson 14-4
csc sec
2
8. 1 sin
857
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
14-5 Skills Practice Sum and Difference of Angles Formulas Find the exact value of each expression. 1. sin 330
2. cos (165)
3. sin (225)
4. cos 135
5. sin (45)
6. cos 210
7. cos (135)
8. sin 75
9. sin (195)
Verify that each of the following is an identity. 10. sin (90 ) cos
11. sin (180 ) sin
12. cos (270 ) sin
13. cos ( 90) sin
2
Lesson 14-5
14. sin cos
15. cos ( ) cos
©
Glencoe/McGraw-Hill
863
Glencoe Algebra 2
NAME ______________________________________________ DATE
____________ PERIOD _____
14-6 Skills Practice 2
7 25
2 4 2. sin , 180 270 5
40 41
4. cos , 270 360
Find the exact values of sin 2, cos 2, sin , and cos for each of the following. 1. cos , 0 90
3. sin , 90 180
3 5
5. cos , 90 180
3 7
5 13
6. sin , 0 90
Find the exact value of each expression by using the half-angle formulas. 1 2
7. cos 22
8. sin 165
8
10. sin
9. cos 105
15 8
11. sin
12. cos 75
Verify that each of the following is an identity. 2 tan 1 tan
13. sin 2 2
©
Glencoe/McGraw-Hill
14. tan cot 2 csc 2
869
Glencoe Algebra 2
Lesson 14-6
Double-Angle and Half-Angle Formulas
NAME ______________________________________________ DATE
____________ PERIOD _____
14-7 Skills Practice Solving Trigonometric Equations Find all solutions of each equation for the given interval. 2
2. 2 cos 3 , 90 180
3. tan2 1, 180 360
4. 2 sin 1, 0
5. sin2 sin 0, 2
6. 2 cos2 cos 0, 0
Lesson 14-7
1. sin , 0 360 2
Solve each equation for all values of if is measured in radians. 7. 2 cos2 cos 1
9. sin sin cos 0
11. 4 cos 1 2 cos
8. sin2 2 sin 1 0
10. sin2 1
1 2
12. tan cos
Solve each equation for all values of if is measured in degrees. 13. 2 sin 1 0
14. 2 cos 3 0
15. 2 sin 1 0
16. 2 cos2 1
17. 4 sin2 3
18. cos 2 1
Solve each equation for all values of . 19. 3 cos2 sin2 0
20. sin sin 2 0
21. 2 sin2 sin 1
22. cos sec 2
©
Glencoe/McGraw-Hill
875
Glencoe Algebra 2
