algebra 2 trigonometry - Sewanhaka Central High School

Transcription

ANSWER KEY
Preparing for the
REGENTS EXAMINATION
ALGEBRA 2
and
TRIGONOMETRY
Ann Davidian and Christine T. Healy
AMSCO
AMSCO SCHOOL PUBLICATIONS, INC.
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ANSWER KEY/PREPARING FOR THE REGENTS EXAMINATION:
ALGEBRA 2 AND TRIGONOMETRY
Copyright © 2009 by Amsco School Publications, Inc.
No part of this book may be reproduced in any form without written
permission from the publisher.
Printed in the United States of America
1 2 3 4 5 6 7 8 9 10
14 13 12 11 10 09
Contents
Chapter 1: The Integers
Chapter 2: The Rational Numbers
Chapter 3: Real Numbers and Radicals
Chapter 4: Relations and Functions
Chapter 5: Quadratic Functions and Complex Numbers
Chapter 6: Sequences and Series
Chapter 7: Exponential Functions
Chapter 8: Logarithmic Functions
Chapter 9: Trigonometric Functions
Chapter 10: More Trigonometric Functions
Chapter 11: Graphs of Trigonometric Functions
Chapter 12: Trigonometric Identities
Chapter 13: Trigonometric Equations
Chapter 14: Trigonometric Applications
Chapter 15: Statistics
Chapter 16: Probability
Cumulative Reviews
Practice Algebra 2 and Trigonometry Regents Examinations
1
4
8
11
18
26
30
35
38
42
45
53
57
59
62
69
74
116
CHAPTER
The Integers
1
1.1 Reviewing Real
Numbers
1.2 Writing and Solving
Equations and Inequalities
(page 3)
In 1–15, explanations will vary.
1 Rational
2 Rational
3 Rational
4 Irrational
5 Rational
6 Irrational
7 Irrational
8 Irrational
9 True
10 False
11 True
12 False
13 False
14 True
15 True
16 3, 4
17 1, 2
18 0, 1
19 10, 11
20 0.444, 1.4, 4.4
兹8 , 5, 2 2
21
2 兹兹
7 7
, , 兹7
22
3 3
1
1 1
,
,
23
2
2 兹2
22
,p
24 3.14,
7
1
3
, 0.3333,
25
10
3
(page 7)
1 x⫽2
2 x⫽9
3 x⫽4
4 x⫽8
5 x⫽7
6 x⫽6
7 x ⱖ ⫺4
8 x ⱕ ⫺2
3
9 x⬍
5
10 x ⬎ ⫺7
11 2n ⫹ 8 ⫽ 3n; n ⫽ 8
12 3n ⫹ 8 ⫽ 2n ⫹ 17 ⫺ 5; n ⫽ 4
13 7 ⫹ 2n ⫺ 4 ⫽ n ⫹ 12; n ⫽ 9
14 5n ⫺ 11 ⱖ 3n ⫺ 1; n ⱖ 5
15 Feb. ⫽ Jan. ⫹ 3; Mar. ⫽ Jan. ⫹ 4; Apr. ⫽
1
2(Jan. ⫹ Feb.); May ⫽ Mar.; June ⫽
2
Mar. – 2; Jan. ⫹ Feb. ⫹ Mar. ⫹ Apr. ⫹
May ⫹ June ⫽ 50
Jan: 3; Feb: 6; March: 10; April: 18; May: 5;
June: 8
16 313.50 ⫽ [2(5) ⫹ 2(6) ⫹ 8]8.90 ⫹ 3d;
d ⫽ $15.50
17 c ⫽ 2s; t ⫽ s ⫹ 16; c ⫹ s ⫹ t ⫽ 452
a 109 swimsuits, 125 towels, 218 calendars
b $6,671
18 Co ⫽ 2P ⫹ 8; Ca ⫽ 3P ⫹ 3;
Co ⫹ Ca ⫹ P ⫽ 46
No. Colby: 20; Carlyn: 20; Paolo: 6
冪
冪
1.2 Writing and Solving Equations and Inequalities
1
1.3 Absolute Value
Equations and Inequalities
Absolute Value Equations
(page 10)
1 x ⫽ 6, x ⫽ ⫺2
2 y ⫽ 3, y ⫽ ⫺8
3 z ⫽ 4, z ⫽ 0
4 n ⫽ 5, n ⫽ ⫺1
5 c⫽3
5
6 r ⫽ , r ⫽ ⫺15
7
7 (3) {⫺6, 2}
8 (3) {⫺1, 5}
9 (1) x ⫽ 1, x ⫽ 9
10 (4) 兩4 ⫺ x兩 ⫹ 10 ⫽ 4
11 (2) 兩x ⫺ 7兩 ⫽ 6
3
x ⫺ 6 ⫽ 12
12 (1)
2
13 a 兩x ⫺ 10兩 ⫽ 7
b x ⫽ 3, x ⫽ 17
14 a 兩P ⫺ 0兩 ⫽ 2P ⫹ 5, 兩P兩 ⫽ 2P ⫹ 5
5
b P⫽⫺
3
15 a 兩a ⫺ 16兩 ⫽ 0.5
b a ⫽ 15.5 or a ⫽ 16.5
兩
兩
Absolute Value Inequalities
(page 12)
1 ⫺10 ⱕ a ⱕ 10
2 x ⬍ ⫺5 or x ⬎ 9
3 ⫺4.8 ⱕ d ⱕ 4
4 x ⱕ ⫺19 or x ⱖ 1
5 n ⬍ 2 or n ⬎ 14
6 ⫺2 ⬍ y ⬍ 2
7 m ⱕ ⫺4 or m ⱖ 8
8 (4)
–2 –1
9
10
11
12
13
14
15
2
0
1
2
3
Chapter 1: The Integers
a 兩t ⫺ 350兩 ⬎ 7
b The oven turns off at 357° and turns on
again when the oven temperature
reaches 343°.
1.4 Adding and Subtracting
Polynomials
(pages 14–15)
1 ⫺x 4 ⫺ x 3 ⫹ 5x 2 ⫹ 2x ⫹ 5; degree 4
2 ⫺c 5d 2 ⫺ 5c 4d 3 ⫹ 25c 3d 4; degree 7
3 ⫺c 4 ⫺ c 3 ⫹ 5c ⫹ 3; degree 4
4 ⫺9w 2 ⫹ 9w ⫹ 4; degree 2
5 3x 4 ⫹ 8x 3 ⫺ 8x 2 ⫺ 4x ⫹ 9; degree 4
6 ⫺xy ⫺ 4x 2; degree 2
7 7a2b 2 ⫺ 2a2b ⫹ 3ab 2; degree 4
8 (2) 14x2 ⫺ 4x ⫹ 21
9 (3)
3c 4 ⫺ 5c 2 ⫹ 10
⫹ (⫺3c 4 ⫹ 3c 2 ⫺ 3)
10 (4) 3x2 ⫹ 7x ⫺ 22
11 (1) 27c 2 ⫺ 15c ⫹ 12
12 (3) 4a2 ⫺ 14ab ⫹ 8b2
13 (1) 14c 2 ⫺ 5c ⫹ 2
14 (1) 3x 2 ⫺ 5x ⫹ 13
15 (2) 12m ⫹ 5
1.5 Multiplying
Polynomials
4
5
6
兩4x ⫺ 2兩 ⫺ 6 ⱕ 8
{x : x ⬍ ⫺2 or x ⬎ 3}
{x : ⫺3 ⬍ x ⬍ 4}
兩a ⫺ 16兩 ⱕ 0.5
兩I ⫺ 100兩 ⱕ 15
兩t ⫺ 98.6兩 ⬍ 1.4
Temperatures greater than 100° or less
than 97.6° would be considered
unhealthy.
a 兩a ⫺ .500兩 ⱕ .010
b Con’s batting average is between .490
and .510 inclusive.
(1)
(1)
(4)
(4)
(3)
a
b
16
(pages 17–18)
1 18x 5 ⫺ 12x 4 ⫹ 6x 3 ⫺ 6x 2
2 m4 ⫺ 16
3 12a5b4 ⫺ 9a4b5 ⫹ 15a3b6
4 10c3d 3 ⫹ 6c2d 2 ⫺ 5cd ⫺ 3
5 10c3 ⫹ 17c ⫺ 20
6 36z2 ⫹ 60z ⫹ 25
7 49y2 ⫺ 4
8 2p3 ⫹ p2 ⫹ p ⫺ 6
9 8x3 ⫹ 4x2 ⫺ 2x ⫺ 1
10 14 ⫺ 41y ⫹ 29y2 ⫺ 6y3
11 16c2a ⫹ 18ca ⫺ 9
12 ⫺6h4 ⫺ 5h3k ⫹ 14h2k 2 ⫺ hk 3 ⫺ 2k 4
13 7j 2 ⫹ 64j ⫹ 9
14 x 2 ⫺ 7x ⫺ 4
15 12m3n3 ⫺ 5m2n2
16 2m2b ⫹ 11mb ⫹ 7
17 2y3 ⫹ 7y ⫺ 15
18 4x2 ⫹ 82x
19 12x3 ⫺ 29x2 ⫺ 4x ⫺ 6
20 x3 ⫺ 9x2 ⫺ 8x ⫹ 30
21 (2) 2h2 ⫹ h ⫺ 10
22 (3) 7p3 ⫹ 30p2 ⫺ 27p ⫺ 10
12
(page 21)
1 (x ⫺ 5)(x ⫺ 2)
2 (x ⫹ 5)(x ⫺ 4)
3 (c ⫹ 6)(c ⫹ 2)
4 (y ⫺ 18)(y ⫹ 2)
5 (a ⫺ 6)(a ⫺ 5)
6 (x ⫹ 7)(x ⫺ 5)
7 (p ⫹ 10)(p ⫺ 3)
8 (m ⫹ 9)(m ⫺ 6)
9 (x ⫺ 7)(x ⫹ 2)
10 (2z ⫹ 9)(2z ⫺ 9)
11 2x3(1 ⫺ 2x)(1 ⫺ x)
12 3c2d (3c ⫺ d)(c ⫺ 2d)
13 (4x ⫺ 5)(x ⫹ 1)
5
5
6y ⫹ z 6y ⫺ z
14
7
7
15 (9 ⫹ a)(2 ⫺ a)
16 4p(p ⫹ 4)(p ⫺ 4)
17 2(x ⫺ 5)(x ⫺ 3)
18 (d ⫺ 8)(d ⫹ 6)
19 my(m ⫹ 5)(m ⫺ 5)
20 (3a ⫺ 1)(a ⫺ 3)
21 (ya ⫺ 3)(ya ⫹ 2)
22 (8c ⫺ 5)(c ⫺ 1)
23 (7a ⫺ 3)(a ⫹ 2)
24 4n2 ⫺ 12x ⫹ 9; cannot be factored
冣冢
10
11
1.6 Factoring
冢
9
冣
13
14
15
16
17
18
19
20
21
22
23
24
冦⫺ 34 , 32 冧
冦⫺ 52 , 4冧
{⫺2, 3}
1 3
,
3 2
2
⫺ ,2
5
1
,2
2
{⫺2, 2}
5
⫺ ,3
2
2
⫺ , 38
3
9
⫺ ,4
4
6
⫺ ,4
5
2
⫺ ,5
3
{4}
{2, 6}
{13}
{3, 7}
冦冧
冦冧
冦冧
冦
冦
冦
冦
冦
冧
冧
冧
冧
冧
1.8 Quadratic Inequalities
(pages 26–27)
1 x ⬍ ⫺7 or x ⬎ 8
–8 –6 –4 –2
1.7 Solving Quadratic
Equations with
Integral Roots
(page 24)
1 {2}
2 {⫺6, 8}
3 {⫺3, 8}
4 {⫺2, 7}
5 {⫺3, 3}
6 {⫺7, 4}
7
⫺ ,1
7
3
5
⫺ ,3
8
2
冦
冦
冧
冧
2
4
0
2
6
8
–4
–2
4
6
⫺2 ⬍ x ⬍ 5
–6
4
2
x ⱕ ⫺3 or x ⱖ 3
–6
3
0
–4
–2
0
2
4
6
x ⱕ ⫺4 or x ⱖ ⫺1
–5
–4
–3
–2
5 x ⬍ ⫺3 or x ⬎ ⫺
–5
–4
–3
–1
0
1
–1
0
1
1
2
–2
–1
2
6
⫺6 ⱕ x ⱕ 2
–10 –8
–6
–4
–2
0
2
4
3
7
–3
–2
–1
0
–
8
1
3
x ⱕ ⫺1 or x ⱖ ⫺
⫺
1
–2
–1
0
1
–3
2
3
3
2
2
15
x ⬍ 4 or x ⬎ 8
3
10
3
1
3
3
3
⬍x⬍
2
2
–3
9
2
6
7
8
9
10
11
12
13
14
4
5
6
7
8
9
0ⱕxⱕ5
–1
1
0
2
3
4
5
11 (2) {x : x ⬍ ⫺3 or x ⬎ 6}
12 (4) {x : ⫺6 ⱕ x ⱕ 6}
13 (4) x2 ⫹ 4x ⫺ 21 ⱖ 0
14 (3)
0
3
15
(1)
16
(4) {x : 5 ⬍ x ⬍ 6}
–4
–2
0
2
4
6
8
10
Chapter Review (pages 31–32)
1 x2 ⫹ x ⫺ 20
2 3a2 ⫺ 19a ⫹ 12
3 4c2 ⫺ 6c ⫹ 8
4 2w2 ⫹ 7wy ⫺ 15y2
5 2x3 ⫺ 5x2 ⫺ 8x ⫹ 6
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
9a2 ⫹ 6ab ⫹ b2
6a4b ⫺ 8a3b ⫹ 2a2b2
2x2 ⫹ 16xy ⫹ 5y2
2a2b ⫺ ab2 ⫺ 3b2 ⫹ 4b
r 3 ⫹ 2r 2s ⫺ 4rs2 ⫹ s3
x ⫽ 2, 7
x ⫽ 1, ⫺1
⫺8 ⱕ x ⱕ 2
x ⫽ ⫺2, 6
5
x ⫽ ⫺3,
2
x ⬍ ⫺8 or x ⬎ 7
⫺3 ⱕ x ⱕ 8
x ⱕ 0 or x ⱖ 7
3(a ⫹ 3)(a ⫺ 3)
(b ⫹ 1)(b ⫹ 12)
3(x ⫺ 3)(x ⫹ 1)
(2x ⫹ 3)(x ⫺ 4)
(3cd ⫹ 4)(3cd ⫺ 4)
(2) 0.33333 . . .
(2) 6 feet
(2) ⫺3x2 ⫹ 11x ⫺ 7
(3) ⫺a2 ⫹ a ⫹ 4
(1) ⫺3.14
(4) {1, 6}
(4) 4
(1) {x : x ⱕ ⫺6 or x ⱖ 8}
(2) {x : x ⱕ ⫺3 or x ⱖ 11}
(2) {x : ⫺2 ⬍ x ⬍ 0}
(1) x2 ⫹ x ⱕ 6
(2) 兩x ⫺ 10兩 ⱕ 0.001
CHAPTER
2
2.1 Simplifying Rational
Expressions
(pages 37–38)
1 x⫽0
2 x ⫽2
4
The Rational Numbers
3 x ⫽ 0 or x ⫽ 5
4 x ⫽ ⫺9
5 x ⫽ ⫺2 or x ⫽ ⫺4
6 x ⫽ ⫺5
3a
, (a ⫽ 0, b ⫽ 0, c ⫽ 0)
7
4b 3
8
9
10
11
12
13
14
15
16
17
18
19
20
2xyz10, (x 0, y 0)
1
, (x 0, 2)
2x 4
2, (x 3)
1
, (x 5, 5)
x5
x6
, (x 4, 4)
x4
x5
, (x 0, 3)
x
y3 2y2 4y 8, ( y 2)
x
, (x 3, 7)
x3
7
(4) 2
x 3
2
(1)
x5
x 2 4x
(2)
4
(4) x 0, x 6
x4
(2)
x8
2.2 Multiplying and
Dividing Rational
Expressions
(page 41)
2a3
1
, (a 0, b 0, c 0)
9c 3
9xy 2
, (x 0, y 0, z 0)
2
5
(x 2)2
, (x 2, 0, 2)
3
2x
12
, (x 0, y 0, x y)
4
y
5 2, (a 3, 2, 3)
6 1, (x 1, 0, 1)
1
, (a b, b, 0)
7
2a
3
, (x 3, 2, 4)
8
2
3
9 , (a 0, 3, 6)
a
a(a 3)
, (a 3, 2, 2, 5)
10
a3
1 1
11 2, w 2, ,
2 2
冢
冣
12
13
14
15
16
17
18
19
20
21
22
23
d
, (d 8, 0, 8)
2
2
2, x 1, , 1
3
1
, (y 6, 5, 3, 0, 4)
y
1
, (z 7, 3, 0, 7)
z
2, (x 3, 3)
15, (x 4, 4)
1
, (x 6, 6, 8)
3(x 8)
(1) 4
(4) x 3 or x –1 or x 0
(2) 2
(1) 1
(3) 3
冢
冣
2.3 Adding and Subtracting
Rational Expressions
Expressions with the Same Denominator
(page 44)
3
1
, (a 0)
a
2 5, (x 2)
3 x
4
, (x 0, 2)
4
x
3
, (x 3, 8)
5
x8
y5
, (y 7, 8)
6
y3
z2
, (z 0, 1)
7
z
1
, (a 2, 3)
8
a2
b1
, (x 5)
9
2
10 n 2, (n 0, 2)
Expressions with Different Denominators
(pages 48–49)
x
1
21
14
, (x 0)
2
5x
3 2x
, (x 0)
3
x2
2.3 Adding and Subtracting Rational Expressions
5
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
10 3a
, (a 0)
2a2
3, (c 5)
5x 16
, (x 4, 4)
(x 4)(x 4)
21
, (x 3)
10(x 3)
5x 3
, (x 1, 0, 1)
x(x 1)(x 1)
12
, (x 2, 0)
x(x 2)
4
, ( y 1, 2)
3( y 2)
9
, (x 4, 3, 1)
(x 1)(x 4)
1
, (x 7, 1, 2)
(x 2)(x 1)
x6
, (x 6, 3, 0)
x(x 6)
4
, (x 4, 3)
x4
2
, (x 1, 1)
x1
4n
3
6a 10
5
r 2 2r 3
, (r 0)
r
(3) x2 3x
7
m
(4)
12
x 2 4x 2
(1)
x
4
p
(2)
15
2.4 Ratio and Proportion
(pages 52–54)
1 True, 24 24
2 False, 175 245
3 False, 15 16
4 True, 18 18
5 x 20
6 y 15
7 z 10
8 x2
9 y4
6
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
z5
x 4, 2
x 2, 6
(3) 9 : 25
x
4
(3)
x1
3
(2) x 9
(2) 3.788
(2) 1,000
(4) 8
3a
(3)
b
(3) 40
(4) 14 inches
22.047 inches
240 inches 20 feet
275
13.59375 gallons
400
a 24, 55, 89, 144, 233
b 1.618
c approximately the same
2.5 Complex Rational
Expressions
(pages 58–59)
1 2
5
2
6
3
3
49
9
4
49
a1
, (a 0)
5
3a2
2(6a 5)
, (a 0)
6
7
7 ab, (a, b 0)
w9
, (w 0)
8
18
yx
, (x 0, y 0, x y)
9
yx
3
, (n 0)
10
2
2
b
, a 0, , b 0
11
2a b
2
y
, (y 0, 3)
12
3y
冢
冣
13
14
15
16
17
18
2s
, (s 1)
s1
x4
, (x 0)
x
x7
, (x 0, 2)
x(x 2)
3( y 2)
, ( y 0, 1, 3)
y
R 1R 2
(4)
R1 R2
s • s
(3)
s s
2.6 Solving Rational
Equations
(pages 64–65)
1 a 11
2 r 7 or r 3
3 x 7 or x 7
4 b2
5 x 3 or x 1
6 { }
7 a 13
8 y 4
9 a5
10 x 3
11 y 3 or y 1
12 x 1
13 z 3 or z 6
14 n 5
1
15 b 2
2
3
or
16
3
2
2
17
3
1
18 a
8
5
b
8
1 5
,
c
n n
5
5
1
d
8
n
1
e 13 hr
3
19
a
b
c
20
a
b
c
21
(2)
22
(4)
23
(3)
24
(1)
t
10
t
5
1
3 minutes
3
300 20w
300 20w
w
300 20w
50; 10 weeks
w
{0}
50x 60y
xy
{6}
1,000 10w
w
2.7 Solving Rational
Inequalities
(page 71)
1 a9
2 7 n 4
3 0s2
4 1x5
9
5 y 4
2
6 g 7 or g 5
7 0r4
8 0v9
9 z 0 or z 7
10 d 4 or d 1
11 5 k 4
12 4 q 2 or q 6
13 y 3 or 0 y 4
14 0 w 1 or w 2
15 x 6 or 2 x 3
16 a 25 0.03n
25 0.03n
b
n
c Joey must use more than 1,250 minutes.
17 The resistance of the other resistor must be
greater than 6 ohms.
18 The most that Dante can drive is 500 miles.
1 (4) x 0, x 6
x1
2 (4) 2
x 1
2.7 Solving Rational Inequalities
7
3
(1)
4
(2)
5
(1)
6
(4)
7
(2)
8
(3)
9
(1)
10
(2)
11
(3)
12
(2)
13
14
15
16
(4)
(3)
(1)
(3)
2
x5
x4
x8
4
x
x5
1
6
a2
1
x2
x4
2
1
2c 1
x1
1x
{4}
{6}
12 by 21
33
17
18
19
20
21
22
23
24
25
冢
冢
冢
–4
26
冣
c(2c 1)
1
, c 3, 5
2
a(a 3)
, (a 4, 1, 2)
a1
1 9
2y 1
, y 2, ,
2
2 4
5
1
1 5y, y , , 2
3
5
6
, (x 3)
x2 9
x2
, (x 0, 1)
x
x2
x5
{x : x 0 or 1 x 6}
–2
冣
冣
0
2
4
6
–4
–2
0
2
–3.5
4
3.5
CHAPTER
Real Numbers
and Radicals
3.1 Real Numbers and
Absolute Value
– 11
7
2
8
4 3
8
3
2
–3
2
3
33
x
2
2
33
– –––
2
20 x 8
–20
3 x 9 or x –9
(pages 80–81)
11
or x 3
1 x
7
10
冦x : x 2兹3 or 2 x 2 or x 2兹3冧
–6
3
8
3
– ––
2
6
5
c⬍⫺
9
15
or c ⬎
4
4
3.2 Simplifying Radicals
9
– ––
4
6
⫺4 ⬍ x ⬍ 6
–4
7 ⫺
6
6
ⱕcⱕ6
5
6
– ––
5
6
5
3
8 x ⱕ ⫺7 or x ⱖ
5
3
–7
9
⫺11 ⬍ n ⬍ 25
–11
10 y ⬍ ⫺
25
11
or y ⬎ 3
2
11
– –––
2
11
(1)
–2 –1
3
0
1
2
3
12 (3) 兩5x ⫹ 10兩 ⱕ 15
13 (1)
–2
14
(4)
15 (1)
16 (2)
17 a
b
c
18 a
b
c
19
–1
0
冦
1
4
2
5
6
3
7
4
冧
17
a 兩M ⫺ 37兩 ⱕ 8
b 29 ⱕ M ⱕ 45
c
a 兩14 ⫺ C兩 ⱕ 6
b 8 ⱕ C ⱕ 20
8
5
5
⫺ ⬍a⬍7
2
兩S ⫺ 24兩 ⱕ 3
兩w ⫺ 16兩 ⱕ 0.4
兩D ⫺ 132兩 ⬍ 26
106 ⬍ D ⬍ 158
yes
兩H ⫺ 23兩 ⬍ 6
17 ⱕ H ⱕ 29
29
20
(page 84)
1 2兹3
2 3兹6
3 25x 3兹2
4 2兹5
5 2兹7
6 ⫺8a2b 4兹3b
7 4兹2
8 2
9 ⫺3n兹3n
10 3兹3
3
11 8兹2
12 1
13 (4) 4
9
14 (3) ⫺ 兹2
4
15 (2) 8y 4兹5
16 (2) 2兹6
17 (2) 2兹30
18 (4) 250x 6
15
–––
4
29
45
3.3 Operations with
Radicals
Adding and Subtracting Radicals
(page 86)
1 ⫺5p兹7
38
6
2
3 兹
3 2兹5
4 2兹3
5 13兹2
6 ⫺a2b 3兹3c
7 ⫺2 兹6
8 8 兹2
9 (2) 52 兹3
10 (3) 17 兹3
11
12
13
14
3
(1) 12 兹3 ⫹ 3
(4) 兹6
(1) 兹3
1
(4) x ⫽ 兹3
2
3.3 Operations with Radicals
9
3.4 Multiplying Radicals
(page 89)
1 12x兹10 ⫹ 5x
2 12兹3 ⫺ 72
3 ⫺9 ⫺ 兹6
4 40
5 4
6 ⫺13 ⫺ 兹5
7 20 ⫺ 8兹2
8 29
3
3
9 ⫺4 ⫺ 8兹2 ⫹ 30兹4
10 19
11 58 ⫺ 12兹6
12 (3) 45兹2
13 (4) 冢6 ⫹ 3兹2冣冢6 ⫺ 3兹2冣
14 (1) 66 ⫹ 36兹3
3.5 Dividing by Radicals
(pages 93–94)
1 1
2 10, (a ⫽ 0)
3 3
4 4
5 15
6 6兹2
7 6兹2
8 17
9 4
8 ⫺ 2兹7
10
3
6 ⫺ 兹15
11
3
12 18 ⫹ 9兹3
8 ⫺ 2兹2 ⫺ 4兹5 ⫺ 兹10
13
14
⫺5 ⫹ 9兹5
14
20
21 ⫹ 7兹6 ⫹ 3兹2 ⫹ 2兹3
15
3
16 (2) 兹6
17 (2) 5
18 (3) 5
7 ⫹ 3兹5
19 (2)
2
20 (4) 8 ⫹ 2兹10
10
3.6 Solving Radical
Equations
(pages 96–97)
1 x ⫽ 40
2 x⫽6
3 x ⫽ 43
4 x ⫽ 13
5 x ⫽ 58
6 x⫽4
7 x⫽3
8 { }
9 x⫽4
10 x ⫽ 31
11 x ⫽ 6
12 x ⫽ 10
13 { }
14 x ⫽ 4
15 x ⫽ 15
16 (1) 3兹2x ⫺ 5 ⫽ ⫺6
17 (3) 3
18 (4) {4}
19 (3) 5
20 (3) Subtract 2 from each side.
1 (2) {1, 3}
2 (2) 5c兹3c
3 (3) 10n
4 (4)
–6
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
0
(4) 4
(3) 6y2
(4) ⫺6
7 ⫹ 3兹5
(3)
2
(4) from 20 to 40 minutes
(2) {8}
4兹3 ⫹ 9兹2
(4)
6
(1) 4兹6 ⫺ 8
(1) 3x 2y 4兹3y
(2) ⫺3
1
(1) 兩 a ⫺ 12 兩 ⱕ
4
12兹6
⫺25兹3
⫺16 ⫺ 31兹3
8 ⫺ 4兹3
11兹5 ⫺ 26
6兹6 ⫺ 6兹2
3
14
7兹6
3
24 10x 3兹2
3兹2 ⫺ 2兹3
25
6
26 x ⫽ ⫺3, 7
23
68
5
x⫽8
3
⫺ ⱕ x ⱕ 15
2
x ⬍ ⫺5 or x ⬎ 4
27 x ⫽
28
29
30
CHAPTER
Relations and
Functions
4.1 Relations and
Functions
4
7
Relations
(pages 100–101)
1 Domain: {Albany, Bismarck, Juneau}
Range: {New York, North Dakota, Alaska}
Independent variable: capital; Dependent
variable: state
2 Domain: {1, 4, 11, 25}
Range: {January, July, November, December}
Independent variable: date of U.S. holiday;
Dependent variable: month of U.S. holiday
3 Domain: {The Dark Knight, Legally Blonde,
Mr. and Mrs. Smith, Pirates of the Caribbean,
The Bourne Identity, Titanic}
Range: {Christian Bale, Reese Witherspoon,
Angelina Jolie, Johnny Depp, Matt Damon,
Leonardo DiCaprio}
Independent variable: movie title; Dependent variable: movie star
4 Domain: {3, ⫺2, 4, ⫺1, ⫺3, 5};
Range: {9, 4, 16, 1, 9, 25}; Rule: y ⫽ x2
5 Domain: {⫺10, ⫺8, ⫺4, ⫺2, 0};
1
Range: {4, 3, 1, 0, ⫺1}; Rule: y ⫽ ⫺ x ⫺ 1
2
6 Domain: {2, 4, 5, 7, 9}; Range: {1, 5, 7, 11, 15};
Rule: y ⫽ 2x ⫺ 3
8
9
10
11
12
13
14
15
冦(5, rational number), 冢兹7, irrational num1
ber冣, 冢 , rational number冣 (p, irrational
2
1
number)冧; Domain: 冦5, 兹7, , p冧;
2
Range: {rational number, irrational number}
{(0, integer), (0, whole number),
(⫺1, integer), (2, integer), (2, whole
number), (⫺5, integer), (⫺3, integer)};
Domain: {0, ⫺1, 2, ⫺5, ⫺3};
Range: {integer, whole number}
{(⫺6, ⫺5), (⫺1, 5), (4, 15), (9, 25)};
Domain: {⫺6, ⫺1, 4, 9};
Range: {⫺5, 5, 15, 25}
{(⫺4, 4), (⫺2, 0), (⫺2, ⫺1), (0, 3), (1, ⫺4),
(2, 1), (4, 2), (4, ⫺3)};
Domain: {⫺4, ⫺2, 0, 1, 2, 4};
Range: {4, 0, ⫺1, 3, ⫺4, 1, 2, ⫺3}
Domain: {x : ⫺4 ⬍ x ⱕ 5 and x ⫽ 2};
Range: {y : ⫺2 ⱕ y ⱕ 5}
Domain: All real numbers;
Range: {y : y ⱖ ⫺4}
Domain: {x : ⫺2 ⬍ x ⬍ 4};
Range: {y : ⫺4 ⬍ y ⬍ 5}
Domain: {x : ⫺5 ⱕ x ⬍ 4};
Range: {y : ⫺3 ⬍ y ⬍ 4}
Domain: {x : x ⱖ ⫺3};
Range: All real numbers
4.1 Relations and Functions
11
Functions
18
(pages 105–108)
1 Function; it passes the vertical line test.
3 Not a function; it does not pass the vertical
line test.
line test.
line test.
line test.
10 Function; for each x-value in the domain,
there is exactly one y-value.
13 Function; for each value in the domain,
there is exactly one corresponding value in
the range.
14 Function; for each value in the domain,
there is exactly one corresponding value in
the range.
15 Not a function; there are values in the domain that correspond to two distinct values
in the range.
16 (3)
17
54
43
13
32
16
71
(4)
Distance
Time
12
(1)
Height
Time
19
(4)
Money
Time
4.2 The Algebra of
Functions
(pages 111–113)
1 a f(3) ⫽ 10
1
5
⫽⫺
b f
2
4
c f(⫺1) ⫽ ⫺2
2 a g(⫺2) ⫽ 5
b g(1.4) ⫽ ⫺7.92
c g(5) ⫽ ⫺72
3 x⫽6
4 x ⫽ ⫺1, 4
5 a h(⫺2) ⫽ ⫺11
b h(2.5) ⫽ 10.375
1
⫽ 0.625
c h ⫺
2
d x ⫽ 0, 1
6 a 3t ⫹ 9
b 5t ⫹ 15
c t⫹3
d 2
e 2t 2 ⫹ 12t ⫹ 18
7 f (2) ⫽ ⫺2
8 f (⫺1) ⫽ ⫺5
9 f (4) ⫽ 0
10 f (3.5) ⫽ ⫺4
11 2f (0) ⫽ 8
12 f (2) • f (3) ⫽ 10
13 f (0) ⫹ f (1) ⫽ 7
14 3
15 {⫺1, 3}
16 a g(1) ⫽ 9
b g(⫺2) ⫽ 18
冢冣
冢冣
17
22
23
24
25
26
27
a
b
a
b
a
b
a
b
a
b
(3)
(2)
(4)
(4)
(2)
(2)
28
29
(4)
a
18
19
20
21
b
30
c
a
b
c
d
31
a
b
c
d
e
g(1) ⫽ ⫺3
g(⫺2) ⫽ 0
g(1) ⫽ 3
g(⫺2) ⫽ 1.5
g(1) ⫽ 5
g(⫺2) ⫽ 5
g(1) ⫽ 6.25
g(⫺2) ⫽ 8.5
g(1) ⫽ 0
g(⫺2) ⫽ 21
3
(2, ⫺1)
22
⫺13
7
The point (⫺1, 5) would appear on the
graph of this function.
f(x)x ⫹ 6
No, f(1 ⫹ 4) ⫽ f(5) ⫽ 52 ⫽ 25 while
f(1) ⫹ f(4) ⫽ 12 ⫹ 42 = 17.
No, f(a ⫹ b) ⫽ (a ⫹ b)2 ⫽ a2 ⫹ 2ab ⫹ b2
while f(a) ⫹ f(b) ⫽ a2 ⫹ b2.
Yes, if a ⫽ 0 or b ⫽ 0.
In 2007, Mrs. Santiago had 10 girls in her
class.
In 2008, Mrs. Santiago had 5 more girls
than boys in her class.
In year y, Mrs. Santiago had twice the
number of boys in her class as girls.
The function t(y) is the total number of
students, boys and girls, that Mrs. Santiago has in her class in year y.
In three weeks, the Longarzo family
buys 7 gallons of milk.
In three weeks, the Longarzo family
buys the same amount of milk and ice
cream.
In two weeks, the Longarzo family buys
a total of 8 gallons of milk and ice cream.
In w weeks, the Longarzo family buys
three times as much milk as ice cream.
In w weeks, the Longarzo family buys
2 more gallons of milk than they buy of
ice cream.
4.3 Domain and Range
(pages 120–121)
1 (3) ⫺16
2x ⫹ 5
2 (3) y ⫽
x⫺6
3 (2) 84
4 (2) {x: x ⫽ 0, 5}
5 (2) x ⱖ ⫺4
6 (2) {⫺2, 0, 2}
7 (2) ⫺2
4
8 (4) y ⫽
x⫺2
9 Domain: All real numbers; Range: All real
numbers
10 Domain: All real numbers;
Range: { y : y ⱖ ⫺5}
11 Domain: {x : ⫺5 ⱕ x ⬍ 5};
Range: {y : ⫺5 ⱕ y ⱕ 6}
12 Domain: {x : ⫺7 ⱕ x ⬍ 8};
Range: {⫺1, 1, 3, 5, 7}
13 {x : x ⫽ ⫺2, 2}
14 {x : x ⬎ ⫺3}
15 All real numbers
16 Domain: All real numbers; Range: { y : y ⱕ 16}
numbers
18 Domain: All real numbers; Range: { y : y ⱖ 0}
19 Domain: {x : –3 ⱕ x ⱕ 3};
Range: {y : 0 ⱕ y ⱕ 3}
20 Domain: All real numbers; Range: { y : y ⱕ 3}
numbers
4.4 Composition of
Functions
(pages 125–127)
1 2x2 ⫺ 3x ⫺ 20
2 ⫺5x3 ⫹ 2x2 ⫹ 80x ⫺ 32
3 ⫺3x ⫹ 12
x⫹4
, (x ⫽ ⫺1, 4)
4
x⫹1
5 4
6 48
7 62
8 ⫺68
9 234
10 ⫺7
11 兹82
12 x, (x ⱖ ⫺16)
13 兹x 2 ⫺ 3x ⫹ 12
14 82 ⫺ 5x2
15 (2) ⫺1
16 (4) ( f ⴰ g)(4)
17 (2) 0
4.4 Composition of Functions
13
(3)
(4)
(3)
a
b
c
d
22 a
b
c
d
e
18
19
20
21
{⫺0.5, 0.5, 3.5, 4.5}
( f ⴰ g)(2) ⫽ ⫺1
3
t(x) ⫽ 1.0825x
c(x) ⫽ 0.8x
t(c(x)) ⫽ 0.866x
$259.80
f(d) ⫽ 90d
b(d) ⫽ 95d
c(x) ⫽ 1.3849x
c( f(d )) ⫽ 124.641d
At $95 per day, the cost for 5 days would
be $475. At 90 euros per day, the cost for
5 days would be $623.21.
4.5 Inverse Functions
(pages 133–134)
1 A⫺1: {(5, 8), (8, 6), (11, 4), (14, 2)}
2 B⫺1: {(3, ⫺2), (5, ⫺5), (7, ⫺8) (9, ⫺11)}
3 C⫺1: {(&, *), (%, $), (⫹, @), (!, #)}
4 D⫺1: {(9, 2), (⫺5, 4), (8, 13), (⫺10, ⫺1)}
5 Beauty and the Beast
x⫺7
6 y ⫺1 ⫽
4
⫺x ⫺ 1
7 y ⫺1 ⫽
3
5
8 f ⫺1(x) ⫽ x ⫹ 15
2
9 f ⫺1(x) ⫽ x3 ⫹ 4
x⫹6
10 g ⫺1(x) ⫽ 3
3
1
1
11 y ⫺1 ⫽ x ⫺
4
6
1
12 (3)
13
3
13 (2) f(x) ⫽ x 3, g(x) ⫽ 兹x
14 (3) x2 ⫹ 4
15 (3) ⫺1
16 (4) 123
17 (2) 2
18 (4) {(x, y), ( y, z), (z, a)}
1
19 (4) f ⫺1(x) ⫽ ⫺ x ⫹ 5
2
5
20 (3)
2
4.6 Transformations of
Linear, Absolute Value, and
Polynomial Functions
(pages 143–144)
1 (1) (0, 3)
2 (4) (⫺3, 0)
3 (3) 10 units to the right
4 (4) 4
5 (2) 0
6 (4) 5 units to the left and 3 units down
7 (3) 2 units to the right and 1 unit up
8 (1) (1, 5)
9 (1) y ⫽ ⫺x2 ⫹ 6
10 (3) y ⫽ 兩x ⫹ 1兩
11 a
12
zeros at x ⫽ ⫺2, 0, 1, 3
b 4th degree
a
冪
14
b 3
c between ⫺3 and ⫺2, between ⫺2 and
⫺1, between 1 and 2
d zeros at x ⫽ ⫺2.38, ⫺1.27, 1.65
13 a
b 2
c between 1 and 2 and between 3 and 4
d zeros at x ⫽ 1.49, 3.18
14
18
y3
y2
y2
y1
y1
y3
The graph of y2 is the graph of y1 shifted
3 units to the left. The graph of y3 is the
graph of y1 shifted 3 units up.
15
The graph of y2 is the graph of y1 condensed
by a factor of 4; that is, each y-value of y2 is
4 times the corresponding y-value of y1 . The
graph of y3 is the graph of y1 expanded by a
1
factor of 2; that is, each y-value of y3 is
2
the corresponding y-value of y1 .
y3
y1
y2
19
y2
The graph of y2 is the graph of y1 shifted 3
units to the right. The graph of y3 is the
graph of y2 shifted 2 units up.
16
y2
y1
y1
y3
3 units to the right. The graph of y3 is the
graph of y1 shifted 2 units down.
y3
20
y1
The graph of y2 is the graph of y1 condensed
by a factor of 4; that is, each y-value of y2 is
4 times the corresponding y-value of y1 . The
graph of y3 is the graph of y1 expanded by a
1
factor of 2; that is, each y-value of y3 is
2
the corresponding y-value of y1 .
17
y2
y1
y3
3 units to the right. The graph of y3 is the
graph of y1 shifted 2 units to the left and
3 units down.
y3
y2
The graph of y1 is a parabola. The graphs of
y2 and y3 are both 3rd degree. The graph of
y2 is y ⫽ x3 condensed by a factor of 3 and
shifted 1 to the right. The graph of y3 is
y ⫽ x3 expanded by a factor of 3 and shifted
2 down.
4.7 Circles
(pages 146–148)
1 (4) (4, ⫺2)
2 (3) (x ⫺ 1) ⫹ y2 ⫽ 42
3 (4) (⫺4, ⫺3)
4 (1) center (2, ⫺3), radius 4
5 (4) (x ⫺ 2)2 ⫹ (y ⫹ 4)2 ⫽ 25
4.7 Circles
15
6
(4)
–8
7
8
9
10
11
12
13
14
15
16
17
18
19
20
9
y
–6
–4
y
8
8
6
6
4
4
2
2
O
–2
2
4
6
8
x
–8
–6
–4
O
–2
–2
–2
–4
–4
–6
–6
–8
–8
(1) (x ⫺ 4)2 ⫹ ( y ⫹ 6)2 ⫽ 5
(3) (x ⫹ 2)2 ⫹ ( y ⫺ 1)2 ⫽ 36
(4) (x ⫺ 6)2 ⫹ ( y ⫹ 5)2 ⫽ 20.25
(2) 5.6
center (⫺7, 0), r ⫽ 5.4
center (0, 3), r ⫽ 3.7
1
,r⫽4
center ⫺2, ⫺
2
center (5, ⫺2), r ⫽ 4.8
center (⫺1.5, 3.6), r ⫽ 兹10
(x ⫹ 1)2 ⫹ ( y ⫹ 5)2 ⫽ 64
x2 ⫹ ( y ⫺ 3)2 ⫽ 1
(x ⫺ 2)2 ⫹ y2 ⫽ 1.44
(x ⫺ 4)2 ⫹ ( y ⫺ 8)2 ⫽ 8
(x ⫹ 2)2 ⫹ ( y ⫺ 1)2 ⫽ 41
冢
(3)
冣
2
10 Yes, D ⫽ 1.2C
11 Not a direct variation
12 Yes, M ⫽ 3.6K
13 Not a direct variation
14 750
15 11.811
16 a C(g) ⫽ 4.074g
b g
10
15
C(g)
c
$40.74
$61.11
4
6
8
x
20
$81.48
C(g)
85
80
75
70
65
60
4.8 Direct and Inverse
Variation
55
50
45
Direct Variation
(pages 150–152)
1 (3) 15
2 (2) 0.8
x
⫽k
3 (4)
y
4 (4) 263
5 (3) $97.75
6 (2) 28.8
7 (2) 0
y
8 (3)
x
16
40
O 5
17
18
10 15 20
g
a 19,610,620
b No, cities are more densely populated.
The population density is an average
and includes sparsely populated regions
such as forests.
5
p
a w⫽
11
b 10 kg
c 30 mg per day
15
p
d d⫽
11
Yes, all direct variations are a function of
a variable times a constant.
b No, some linear equations are translated
so that they do not go through the origin.
c Answers will vary. Example: y ⫽ 3x ⫹ 6
20 a p
b C ⫽ pd
c 40p in.
19
a
Inverse Variation
(pages 155–156)
1 (4) 8%
2 (2) 16
3 (3) 21
4 (1) 8
5 (4) 18
6 (2) 32
7 (2) 12
8 (4) 4
9 (4) 12
10 (1) 10
11
y
8
6
4
2
–8
–6
–4
O
–2
2
4
6
8
2
4
6
8
x
–2
–4
–6
–8
12
y
8
6
4
2
–8
–6
–4
O
–2
–2
–4
–6
–8
x
16
x
13
xy ⫽ 16 or y ⫽
14
xy ⫽ ⫺6 or y ⫽ ⫺
6
x
1 Function; Domain: {⫺2, 3, 8, 9};
Range: {5, 7, 9, 11}
2 Function; Domain: {8, 5, 3, 0}; Range: {⫺3}
3 Not a function; two different elements of the
range correspond to the element Drew
Carey of the domain.
4 Not a function; two different elements of the
range correspond to the element 9 of the
domain.
5 Function; Domain: {x : ⫺7 ⬍ x ⬍ 7};
Range: { y : 1 ⬍ y ⬍ 7}
6 Not a function; the graph fails the vertical
line test.
7 Function; Domain: {3, 17, 23}; Range: {25, 16}
8 Not a function; the graph fails the vertical
line test.
9 Function; Domain: All real numbers; Range:
All real numbers
10 Function; Domain: All real numbers; Range:
All real numbers
11 (3) 0
12 (1) 1
13 (2) (8, 12)
14 (1) {2, 4}
8 ⫺ 3x
15 (4) g(x) ⫽
x⫹5
16 (4) ⫺16
17 (2) ⫺5
1
18 (4) f ⫺1(x) ⫽ x ⫺ 4
4
19 (1) {(3, ⫺2), (⫺2, 3), (3, ⫺1), (⫺2, 4)}
20 (4) 4
21 (1) ⫺3
22 (1) reflection in y ⫽ x
23 (3) 3x2 ⫹ 5
24 (2) 2
25 (2) divided by 3
26 (2) 2.5
27 (1) ⫺1
28 (3) (g ⴰ f )(⫺4) ⫽ 4
Chapter Review
17
CHAPTER
5
Quadratic Functions
and Complex
Numbers
5.1 Alternate Methods of
Solving Quadratics
Completing the Square
(page 165)
1 x ⫽ ⫺3 ⫾ 兹17
2 x ⫽ 4 ⫾ 兹7
3 x ⫽ 6 ⫾ 2兹6
4 x ⫽ 5 ⫾ 4兹3
5 x ⫽ 3 ⫾ 2兹3
6 x ⫽ 1 ⫾ 兹6
7 x ⫽ 2 ⫾ 兹2
8 x ⫽ ⫺1, 7
9 x ⫽ ⫺1, 2
10 x ⫽ ⫺6 ⫾ 兹31
The Quadratic Formula
(page 168)
1 x ⫽ 2 ⫾ 兹3
1 5
2 x⫽⫺ ,
2 2
3 x ⫽ 4 ⫾ 兹6
4 x ⫽ 3 ⫾ 兹5
1
⫾ 兹2
5 x⫽
2
2
6 x⫽⫺ ,1
5
7 x ⫽ ⫺1 ⫾ 兹3
3 7
1
⫾ 兹
8 x⫽
2
2
2
9 x ⫽ ⫺ ⫾ 兹2
3
10 x ⫽ 12
11 x ⫽ 1 ⫾ 3兹2
4
12 x ⫽ ⫺ , 2
3
13 x ⫽ ⫺3
18
14 x ⫽ ⫺1, ⫺
15
x⫽⫺
2
3
2
14
⫾ 兹
5
5
5.2 The Complex Number
System
Imaginary Numbers
(pages 170–171)
1 7i
2 88i
3 4i
4 ⫺7i
5 4i兹5
6 ⫺8i兹5
7 12i兹11
8 8i兹2
i
9
2
10 2i兹3
11 5i
12 2i兹6
13 ⫺i
14 i
15 i
16 ⫺i
17 1
18 ⫺1
19 i
20 ⫺1
21 12i
22 3i
23 72
24 2i
25 (3) ⫺19
26 (1) ⫺70i
27 (3) ⫺100
28 (2) i 9
29 (4) 48
30 (2) 3i兹3
Complex Numbers
(page 173)
1 a ⫽ 2, b ⫽ 5
2 a ⫽ 0, b ⫽ 7
3 a ⫽ 12, b ⫽ 0
4 a ⫽ 11, b ⫽ 7
5 a ⫽ 6, b ⫽ 2
6 a ⫽ 5, b ⫽ ⫺2
7 a ⫽ ⫺2, b ⫽ 12
8 6 ⫹ 5i
9 ⫺2 ⫹ 7i
10 7 ⫺ 3i兹3
11 5 ⫹ 6i兹2
12 ⫺6 ⫹ 4i
13 True; real numbers are of the form a ⫹ bi
with b ⫽ 0.
14 False; if b ⫽ 0, the number is not real.
15 True; integers are of the form a ⫹ bi with a
an integer and b ⫽ 0.
16 True; if b ⫽ 0, the number is not real.
17 False; an imaginary number is a complex
number when a ⫽ 0.
18 False; an imaginary number is a complex
number when a ⫽ 0.
19 A: ⫺5 ⫹ I, B: 2, C: 4 ⫹ 2i, D: 4 ⫺ 3i,
E: ⫺2 ⫺ 4i
20 A: ⫺4 ⫹ 5i, B: 2i, C: 5 ⫹ 3i, D: 5 ⫺ 4i,
E: ⫺4 ⫺ 2i
5.3 Operations with
Complex Numbers
Addition and Subtraction of Complex
Numbers
(pages 176–177)
1 4i
2 2 ⫹ 3i
3 1⫺i
4 4 ⫹ 10i
5 9 ⫹ 15i
6 ⫺7 ⫺ 40i
7 7 ⫹ 13i兹2
8 ⫺9 ⫺ 16i兹3
9 14 ⫹ 22i兹5
10 ⫺7 ⫹ 5i兹3
11 a ⫽ 5, b ⫽ 5
12 a ⫽ 4, b ⫽ ⫺2
13 a ⫽10, b ⫽ 8
14 a ⫽ 3, b ⫽ 9
15 (4) 5 ⫺ 10i
16 (1) I
17 (1) 20
18 (2) II
19 (1) 4i兹2
20 (4) 8 ⫺ 12i兹5
21 a
yi
6
5
4
Z1 ⫹ Z2
3
Z2
2
1
Z1
–4 –3 –2 –1 O 1
–1
–2
2
3
4
x
b ⫺1 ⫹ 5i
yi
22 a
4
3
2
Z2
Z1 ⫹ Z2
1
–1 O 1 2
–1
Z1
–2
3
4
5
6
7
2
3
4
5
6
x
–3
–4
b 5⫹i
23 a
yi
4
Z2
3
2
1
–2 –1 O
–1
–2
–3
–4
1
x
Z1
Z1 ⫺ Z2
–5
b 6 ⫺ 6i
5.3 Operations with Complex Numbers
19
24
a
5.4 Nature of the Roots
and the Discriminant
yi
4
Z2
3
2
Z1
1
–4 –3 –2 –1 O 1 2 3
–1
Z ⫺ Z2
–2 1
4
–3
–4
b 4 ⫺ 2i
Multiplication and Division of
Complex Numbers
(pages 179–180)
1
47 ⫺ 16i
1
1
⫺ i
2
2
2
3
⫺12 ⫺ 24i
6
4i
⫺ ⫺
4
5
5
5
33 ⫺ 31i
6
40 ⫹ 42i
7
18 ⫺ i
5i
2
⫹
8
29
29
9
8 ⫺ 32i
i
8
⫹
10
13
13
11
8 ⫹ 4i
11
23i
⫹
12
25
25
⫺9 ⫺ 11i兹2
13
3i
⫺1 ⫺
14
2
30 ⫺ i兹3
15
16
100
5
3i
⫺
17
34
34
4i
7
⫹
18
(3)
65
65
19
(4) 85
20
(1) 3 ⫹ 4i
3
3
⫺ i
21
(4)
4
4
3
i兹5
⫹
22
(1)
14
14
23
(2) 47 ⫺ 29i
24
(4) 2i
20
x
(pages 183–184)
1 ⫺80; two imaginary, unequal roots
2 49; two real, rational, unequal roots
3 88; two real, irrational, unequal roots
4 ⫺143; two imaginary, unequal roots
5 81; two real, rational, unequal roots
6 0; two real, rational, equal roots
7 (2) 3x2 ⫺ 2x ⫺ 5 ⫽ 0
8 (2) II and III
9 (2) 2
10 (4) The parabola intersects the x-axis at two
distinct points.
11 (1) 9x2 ⫹ 6x ⫹ 1 ⫽ 0
12 (1) two real, unequal, irrational roots
13 (4) ⫺11
14 (4) 10
15 (2) 6
16 (2) x2 ⫺ 5x ⫹ 2 ⫽ 0
17 (2) a ⬎ 1 only
18 (4) 6
19 (3) x(x ⫹ 6) ⫽ ⫺10
1
20 x ⫽ , 3
2
21 x ⫽ ⫺2, 10
22 x ⫽ 1, 5
3
⫾ 兹2
23 x ⫽
2
24 x ⫽ 2, 12
3
19
⫾ 兹
25 x ⫽
2
2
5.5 Complex Roots of
Quadratic Equations
(page 187)
1 x ⫽ 2 ⫾ i兹3
2 x ⫽ 3 ⫾ 2i
1
1
⫾
i兹19
3 x⫽
4
4
4 x⫽2⫾i
3
i
⫾
5 x⫽
2
2
1
i
⫾
6 x⫽
3
3
1
7 x ⫽ 1 ⫾ i兹2
2
8
9
10
11
12
13
14
15
16
17
18
19
20
1
5
⫾ i
2
2
x⫽1⫾i
2
x ⫽ 2 ⫾ i兹21
3
1
5
⫾ i兹11
x⫽
2
2
x ⫽ 1 ⫾ 4i
1
x⫽1⫾ i
2
4
1
x⫽
⫾ i兹5
3
3
1
x⫽
⫾ 3i
2
1
1
x⫽
⫾ i
2
2
x ⫽ 1 ⫾ i兹3
x⫽7⫾i
3
x⫽
⫾ 2i
2
1
x⫽
⫾ 2i
2
x⫽
5.6 Sum and Product of
the Roots
(pages 189–190)
1 Sum ⫽ 3; Product ⫽ 5
2 x ⫽ 7; x2 ⫺ 12x ⫹ 35 ⫽ 0
3
3 Sum ⫽ ; Product ⫽ ⫺3
4
4 x ⫽ 6 ⫺ 2i; x2 ⫺ 12x ⫹ 40 ⫽ 0
5 x ⫽ 10; k ⫽ 48
6 r2 ⫽ 2 ⫹ 兹5; x 2 ⫺ 4x ⫺ 1 ⫽ 0
7 x ⫽ ⫺5; k ⫽ 9
8 3x2 ⫺ 11x ⫺ 4 ⫽ 0
9 x2 ⫺ 6x ⫹ 25 ⫽ 0
10 81x2 ⫺ 36x ⫹ 5 ⫽ 0
11 (4) 5
12 (3) x2 ⫹ 13 ⫽ 13x
13 (2) x2 ⫺ 14x ⫹ 50 ⫽ 0
14 (3) 7
15
3
15 (1) sum: ⫺ , product: ⫺
4
4
5.7 Solving Higher Degree
Polynomial Equations
Factoring by Grouping
(page 190)
1 (3x ⫹ 5)(x ⫹ 2)(x ⫺ 2)
2 (3a ⫹ 2)(2a2 ⫺ 3)
3 (4x ⫹ 3)(2x2 ⫹ 1)
4 (3y ⫹ 4)(y2 ⫹ 2)
5 (2x ⫹ 7)(x ⫹ 2)(x ⫺ 2)
6 (d ⫺ 2)(d ⫹ 3)(d ⫺ 3)
7 (3x ⫹ 1)(x ⫹ 1)(x ⫺ 1)
8 (3c ⫺ 2)(c ⫺ 2)(c ⫹ 2)
9 (2x ⫺ 3)(x ⫹ 3)(x ⫺ 3)
Factoring the Sum and Difference of Cubes
(page 192)
1 (a ⫹ 4b)(a2 ⫺ 4ab ⫹ 16b2)
1
1
5
⫺ 5a
⫹ a ⫹ 25a2
2
2
4
2
3 (3x2 ⫹ 4)(9x4 ⫺ 12x2 ⫹ 16)
x
x2
7
⫺7
⫹
x ⫹ 49
4
10
100
10
5 (1 ⫹ y2)(1 ⫹ y)(1 ⫺ y)(1 ⫹ y4 ⫹ y8)
3
9 2
15
b⫺5
b ⫹
b ⫹ 25
6
2
4
2
7 (cd ⫹ 5)(c2d 2 ⫺ 5cd ⫹ 25)
x3
3
x6
6⫹
36 ⫺ x 3 ⫹
8
4
2
16
9 (2x2 ⫹ 5)(3x ⫺ 4)
10 (3x ⫹ 2)(x ⫺ 3)(x ⫹ 3)
11 2(2a2 ⫺ b)(4a4 ⫹ 2a2b ⫹ b2)
12 (x2 ⫺ 5)(3x ⫹ 1)
13 (p ⫺ 5q)(p2 ⫹ 5pq ⫹ 25q2)
14 4x(x ⫹ 4)(x ⫺ 1)
15 (4z2 ⫹ 3)(16z4 ⫺ 12z2 ⫹ 9)
冢
冣冢
冢
冣冢
冣
冢
冣冢
冣
冢
冣冢
冣
冣
Algebraic Solutions
(page 194)
1
2
3
4
5
6
7
1
,2
2
x ⫽ ⫺3, ⫺2, 2, 3
5
x ⫽ ⫺2, 0,
3
3
x ⫽ ⫺1, ⫺ , 1
4
x ⫽ ⫺1, 1, 9
x ⫽ ⫺3, 0, 3, ⫾3i
3
x ⫽ 0, , 2
2
x ⫽ 0,
5.7 Solving Higher Degree Polynomial Equations
21
1 1
x ⫽ ⫺1, ⫺ , , 1
2 2
2
9 x ⫽ ⫺ , 0, 2
3
10 x ⫽ 0, 1
11 x ⫽ ⫺4, ⫺1, 0, 1, 4
5
1 1
12 x ⫽ ⫺ , ⫺ ,
3
3 3
Graphic Solutions to Systems
8
(page 202)
1 {(⫺5, ⫺3), (1, 3)}
Graphic Solutions
(page 197)
1 x ⫽ ⫺4, ⫺2, 2,
2
3
4
5
6
7
8
9
10
5
2
x ⫽ ⫺2, ⫺1, 1
1 1 5
x ⫽ ⫺1, ⫺ , ,
2 2 2
4
x ⫽ ⫺3, ⫺2, 0, , 3
3
1
x ⫽ ⫺1, ⫺ , 1, 2
3
3
1
x ⫽ ⫺2, ⫺ , ⫺ , 1
4
2
3
x ⫽ ⫺3, , 3
2
4 4
x ⫽ ⫺2, ,
5 3
2
x ⫽ ⫺3, ⫺2, , 2
3
x ⫽ ⫺4, ⫺3, 4
5.8 Systems of Equations
2
冦冢⫺ 23 , 203 冣, (1, 5)冧
3
{(0, 1), (2, ⫺1)}
4
冦冢 12 , ⫺ 74 冣, (4, 0)冧
5
{(0, ⫺3), (3, 3)}
Algebraic Solutions to Systems
(page 199)
1 {(2, ⫺11), (5, ⫺8)}
11 41
⫺ ,
, (2, ⫺2)
2
4
8
3 {(1, 3), (2, 4)}
4 {(⫺4, 6), (2, 0)}
5 {(⫺3, ⫺10), (2, 0)}
6 {(1, ⫺2), (3, 0)}
7 {(1, 2), (7, 20)}
1 49
⫺ ,
, (3, 1)
8
3 9
9 {(⫺1, 2), (2, 5)}
冦冢
冦冢
22
冣
冣
冧
冧
6
4
{(⫺1, 6), (3, 2)}
5
7 {(1, 4), (3, 8)}
8 {(⫺3, 0)}
9 {(⫺1, 3), (2, 0)}
10 {(⫺2, 5), (10, ⫺19)}
11 {(0, ⫺3), (4, 9)}
2
16
⫺ ,⫺
, (⫺1, ⫺6)
12
3
3
冦冢
冣
冧
6
(pages 208–209)
1
7
2
8
3
9
23
10
1
2
17 x ⱕ ⫺4 or x ⱖ
y
12
10
y ⫽ 2x2 + 7x
8
4
2
–2
14
15
6
y⫽4
11 (4) (4, 5)
12 (4) y ⱕ ⫺2x2 ⫺ 5x ⫹ 2
13 (3)
4
––
3
–8
–6
–4
18
4
x ⬍ ⫺2 or x ⬎ 7
y
y ⫽ x2 – 4x
24
18
y ⫽ x2 + 5x
15
12
8
y ⫽ x + 14
y⫽6
6
9
6
4
3
2
O
2
4
6
8
x
–10 –8 –6 –4 –2
–2
–4
–6
–8
24
O 2
–8
10
–2
x
8
–6
12
–4
6
21
14
–6
4
–4
y
–8
2
–2
(2) (⫺2, 0)
(1) The solution includes all values of (x, y)
that lie within the parabola, including those
values on the parabola.
In 16–20, graphs will vary. The graphs in 16–18
represent one type of acceptable solution; the
graphs in 19–20 represent another.
16 ⫺6 ⱕ x ⱕ 1
O
–2
–3
6
8 10
x
19
1
⬍x⬍4
3
4
6
7
8
9
10
11
12
y ⫽ 3x2 – 11x – 4 2
13
⫺
y
8
6
–8
–6
–4
O
–2
2
4
6
8
x
14
–2
15
–4
16
17
18
19
20
21
22
23
–6
–8
–10
–12
–14
20
⫺8 ⬍ x ⬍ 3
y
28
24
24
20
y ⫽ x2 – 5x + 24
16
25
12
8
4
–12 –9
O 3
–6 –3
6
9
x
–4
–8
–12
21
22
23
26
–6
1
–4
1
––
2
–2
7
1 (3) real, irrational, and unequal
2 (3) ⫺2
3 (1) 1
4 (4) 25
5 (1) ⫺12
27
28
(2)
(3)
(2)
(3)
(1)
(4)
(4)
II
1⫹i
x⫺5
real, rational, and unequal
⫺10
4
{⫺5i, 5i}
6 ⫺ 9i
(1)
13
(3) 2 ⫺ 兹5
12 ⫹ 3i
(3)
153
(4) 0
(1) 5
(4) ⫺3
(1) x ⫹ 4
(1) ⫺36
(1) 18i兹2
(3) III
a x ⫽ 1, 5
b x ⫽ 2 ⫾ 兹7
c x ⫽ ⫺1 ⫾ 2兹2
a x ⫽ 5 ⫾ 2兹5
1
15
⫾ 兹 i
b x⫽
2
2
7
3
⫾ 兹 i
c x⫽
4
4
a x ⫽ 1, 2
1
兹19 i
b x⫽⫺ ⫾
2
2
c x ⫽ ⫺2 ⫾ 兹3
d x ⫽ 2 ⫾ 兹6
e x ⫽ ⫺5, 8
f x ⫽ ⫺1 ⫾ i
10
1
g x⫽
⫾ 兹
3
3
h x ⫽ 0, 3, 4
i x ⫽ ⫺4, ⫺2, 2
a 1,224 ft
b 1,288 ft
c approximately 2 seconds
d approximately 10.972 seconds
2 ⫺ 20i
a {(⫺3, 10), (2, 15)}
b {(5, 6)}
c {(⫺5, 23), (–2, ⫺1)}
Chapter Review
25
29
30
31
32
33
34
35
The coordinates of P are (0, 228). This
tells us that the bridge is suspended 228
feet above the water at its center.
b A(⫺2,130, 700), B(2,130, 700)
c 100 feet from the center of the bridge,
the cables are 229.04 feet above the
water.
d y ⫽ 332; at a distance of 1,000 feet from
the center of the bridge, the cables are
332 feet above the water.
e y ⫽ 332; at a distance of 1,000 feet from
the center of the bridge in the opposite
direction, the cables are 332 feet above
the water.
f x ⫽ 1,617.21508 or ⫺1,617.21508; the cables are suspended 500 feet above the
water at 1,617.2151 feet from the center
of the bridge, in either direction.
7
4
⫺ ⫺ i
5
5
x ⫽ ⫺1 ⫾ i 兹2
(3, ⫺12)
x2 ⫹ 2x ⫺ 48 ⫽ 0
a x ⫽ ⫺3.3, ⫺0.4, 1.7
b x ⫽ ⫺2.6, 1.4
x2 ⫺ 4x ⫹ 29 ⫽ 0
a
36
37
38
x ⫽ 7.4
⫺2
a, b
yi
1
–4 –3 –2 –1 O 1
–1
–2
Z2 = –3 – 5i
–3
–4
–5
–6
–7
39 ⫺2 ⱕ x ⱕ
3
4
5
6
x
Z1 = 5 – 2i
Z1 + Z2 = 2 – 7i
5
4
–2
40
2
5
––
4
⫺2 ⬍ x ⬍ 3.5
CHAPTER
6
Sequences and Series
6.1 Sequences
4
(pages 217–218)
1 Each term is 5 times the number of the term;
35, 40, 45
2 Each term is 5 more than the preceding
term, starting with 7; 37, 42, 47
3 Each term is twice the preceding term, starting with 11; 352, 704, 1,408
5
26
6
7
8
9
10
Each term is the cube of the number of the
term; 216, 353, 512
Each term is 1 more than the cube of the
number of the term; 217, 354, 513
3, 7, 11
6, 9, 12
⫺7, 0, 9
2, 5, 14
6, 10, 17
a1 ⫽ 4, an ⫽ an⫺1 ⫹ 2
a1 ⫽ 5, an ⫽ an⫺1 ⫹ 2
a1 ⫽ 5, an ⫽ 2an⫺1
a1 ⫽ 2, an ⫽ 2an⫺1 ⫹ 1
a1 ⫽ 1, an ⫽ 3an⫺1 ⫺ 1
an ⫽ 2n
an ⫽ 2n ⫺ 1
an ⫽ n2
an ⫽ n2 ⫺ 1
an ⫽ n(n ⫹ 1)
a 144
b a1 ⫽ 1, a2 ⫽ 1, an ⫽ an⫺1 ⫹ an⫺2
22 a 100
b an ⫽ n2
11
12
13
14
15
16
17
18
19
20
21
6
14.25 or
7
8
9
121
110
124
10
6
13
兺 (1 ⫹ 4k)
k⫽1
10
14
兺 ( j 2 ⫺ 1)
j⫽6
6
6.2 Arithmetic Sequences
兺冢2 ⫺
n⫽2
兺冢
5
16
k⫽2
冣
冣
1
n
2
⫺k
2k ⫹ 1
5
17
(3)
兺 ( 3j ⫺ 12)
j⫽2
18
19
(2) 20
(3) 91
20
(2)
(n 2 ⫺ 10)
兺
n⫽4
21
(4)
兺 (20,000 ⫹ 5,000(k ⫺ 1))
k⫽1
22
(1)
兺 (4 ⫹ 0.5i)
i⫽0
23
(3) 300
7
4
51
6.4 Arithmetic Series
(page 228)
In 1–5, answers may vary.
8
1
兺 (8i ⫺ 1) ⫽ 280
i⫽1
7
2
兺 (1.5n) ⫽ 42
n⫽1
5
3
6.3 Sigma Notation
(pages 224–225)
1 20
2 65
3 11
4 42
5 30
15
4
11 12
12 5
15
(pages 221–222)
1 Arithmetic; d ⫽ 6; 30, 36
2 Not arithmetic
3 Arithmetic; d ⫽ ⫺4; ⫺9, ⫺13
4 Arithmetic; d ⫽ 4n; 17n, 21n
5 Not arithmetic
6 Arithmetic; d ⫽ 16; 56, 72
7 Arithmetic; d ⫽ b; 3b, 4b
8 Arithmetic; d ⫽ ⫺6; ⫺36, ⫺42
9 Arithmetic; d ⫽ ⫺4; 15, 11
10 Not arithmetic
11 12, 9, 6, 3
12 d ⫽ 6
13 a20 ⫽ 83
14 18, 22, 26, 30; an ⫽ 18 ⫹ 4(n ⫺ 1)
15 13
16 an ⫽ 102 ⫺ 4(n ⫺ 1)
17 $900
18 61
19 8
20 No, the nth term’s value would be increased
by a1 .
3.75 or
57
4
兺 (150 ⫺ 10k) ⫽ 750
k⫽0
10
4
兺 (2n ⫺ 1)k ⫽ 100k
n⫽1
9
5
兺冢⫺ 2 i冣 ⫽ ⫺22.5
i⫽1
6
7
8
9
5, 6, 7; 126
1, 5, 9; 435
28, 32, 36; 228
32, 38, 44; 1,232
1
6.4 Arithmetic Series
27
10
11
12
13
14
15
16
17
18
19
20
3
5
, 2, ; 51
2
2
3, 10, 17; 498
3, 6, 9; 15,150
Sodds ⫽ 2,500; Sevens ⫽ 2,550; S ⫽ 5,050; this
is the sum of the first 100 integers.
405
15; starting at the bottom: 15, 13, 11, 9, 7, 5
$825
day 1: 120 miles, day 2: 180 miles, day 3:
240 miles, day 4: 300 miles, day 5: 360 miles
500,500
615
No, they will have only 1,500 calculators.
6.5 Geometric Sequences
(page 232)
1 Not geometric
2 Geometric; r ⫽ 3; 81, 243
1 11 11
,
3 Geometric; r ⫽ ;
3 3
9
4 Geometric; r ⫽ n; n4, n5
1 1
1
5 Geometric; r ⫽ ⫺ ; , ⫺
2 4
8
16 32
,
6 Geometric; r ⫽ 2;
3
3
7 No
8 Geometric; r ⫽ 0.4; ⫺0.0512, ⫺0.02048
3 243
729
,
9 Geometric; r ⫽ ,
4 1,024 4,096
10 Geometric; r ⫽ 兹2; 4兹2, 8
11 12, ⫺36, 108
12 r ⫽ 0.2
1
13
512
14 13, 91, 637, 4,459
15 12
16 an ⫽ 10(⫺4)n⫺1
17 $1,310,720
18 Yes
19 $51,201; $46,080.90; $41,472.81; $37,325.53;
$33,592.98
20 1st bounce: 9 ft, 5th bounce: 2.8 ft
21 $39,871.04
28
6.6 Geometric Series
(page 235)
1 510
2 ⫺510
364
3
81
4 ⫺6,825
5,461
5
64
6 5, 25, 125; 12,207,030
7 21, 63, 189; 5,580,120
8 4,096, ⫺16,384, 65,536; 839,680
45 135 405 35,145
,
,
;
9
4
16
64
1,024
5 25 125 278,835
,
;
10 ⫺ ,
2 16 8
256
1,093
11 9, 3, 1;
81
12 5, 30, 180; 7,775
7
13
兺 7(3i) ⫽ 22,960
i⫽0
14
15
16
17
18
19
20
19,682
243
31
0.333333
172.479963089 in.
9
$37,044,180.04
$4,682.89
6.7 Infinite Series
(page 240)
320
1
3
2 No finite sum
3 No finite sum
2
4
3
5 10
25
6
3
7 No finite sum
8 No finite sum
9
10
11
12
13
14
15
16
17
18
19
20
21
8
17
9
340
2
3
7
9
31
99
52
99
41
333
634
999
17
3
17
11
11
45
86
165
325 in.
1 Arithmetic; d ⫽ 7; 25, 32, 39
2 Geometric; r ⫽ ⫺0.5; 9, ⫺4.5, 2.25
64
2
3 Geometric; r ⫽ ; 48, 32,
3
3
4 Neither
5 Arithmetic; d ⫽ ⫺9; 38, 29, 20
1
1 1 1
,
6 Geometric; r ⫽ ; ,
2 9 18 36
7 Arithmetic; d ⫽ 11; 57, 68, 79
3 162
243 729
,⫺
,
8 Geometric; r ⫽ ⫺ ;
2
7
7
14
9
10
11
12
13
14
Geometric; r ⫽ 兹2; 4兹3, 4兹6, 8兹3
Neither
(4) an ⫽ 11 ⫹ 6n
(2) a1 ⫽ 9 d ⫽ 4
(3) 616
(2) an ⫽ 25 ⫹ 6(n ⫺ 1)
15
(1)
16
(2) 30
17
(2) ⫺
5
18
19
20
21
22
23
24
25
26
27
28
29
30
兺 (11 ⫹ 8n)
n⫽1
1
3
(4) 3兹3
(3) 0.0512
1
(2) ⫺
2
(3) 4 ⫹ 5 ⫹
25
125
⫹
⫹ ⭈⭈⭈
4
16
2,401
512
14
(4)
37
9 12
3
,⭈ ⭈ ⭈
(2) , 2, ,
2
4 5
a an ⫽ 150 ⫹ 10n; a52 ⫽ $660
b Yes, she will have $550 in the bank.
4,960 grains
92
a 11 ⫹ 17 ⫹ 23 ⫹ 29 ⫹ 35 ⫹ 41 ⫹ 47 ⫹
53 ⫹ 59 ⫹ 65
b 380
c 20%
a an ⫽ 48(0.6)n⫺1
b a3 ⫽ 17.28 ft
c 116.64 ft
156
(2)
Chapter Review
29
CHAPTER
7
Exponential Functions
7.1 Review of Exponents
(pages 250–251)
1 5
2 1
3 7.5a⫺2b5c2
8
4 ⫺
9
5 8
2s 13
6
3t 4
7 ⫺3
8 1
1
9
4
1
10 ⫺
4
10
11
c 7d 8
1
12 ⫺
27
1
13
27
14 4.5r 7
15 ⫺0.07x5z14
2
16
3
17 12
18 4
1
19 ⫺
9
2
20
27
3
21 4兹x 3 or 4冢兹x冣
5
5
4
22 兹(3xy)4 or 冢兹3xy冣
3
3
2
23 ⫺兹642 or ⫺冢兹64冣
30
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
1
兹93
or
1
3
兹(⫺8)4
5
1
冢兹9冣3
or
1
冢兹⫺8冣4
3
x6
1
2 ⫺1 ⫺2
2
a 5 b 5 or (ab 2)⫺5
3 2
3
12533
(de)2
2
⫺(cd)⫺3
1
(3)
16
(3) ⫺1
(4) 4
1
(4) 10
9
(1) 46
1
(4) 4
81
(2) 2
4
(4)
3
(2) 5
(1) 3x⫹1
7.2 Exponential Functions
and Their Graphs
(pages 256–258)
1 (1) I and II
2 (4) They will never intersect.
3 (3) 1 and 2
4 (2) II only
5 (4) x2 ⫹ y2 ⫽ 5
6 (4) They are all in the domain.
7
8
9
10
11
13
(2) ⫺2
1
(3)
25
(3) 3
(2) y-axis
a
x
⫺2
y
0
1
2
1
4
1
4
16
1
16
b, c
y
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
y
B
–4
12
⫺1
a, b
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
A
–4
14
–2 –1
2
d y ⫽ 4⫺x
a
–2 –1
c x ⫽ ⫺1
a, b
8
7
6
5
4
3
2
1
–4
–2 –1
y
9
8
6
15
5
2
x
y
4
y⫽
2
4
B
c y ⫽ ⫺2x
a, b
3
x
A
–2
–3
–4
–5
–6
–7
–8
7
4
y
x
4
2
x
( 12 )
8
1
–4 –3 –2 –1
–1
1
2
3
4
6
x
4
b f(⫺1.6) ⫽ 0.172427286
c
2
–4 –3 –2 –1
y⫽
( 12 )
x
1
2
3
4
x
– 3 –2
–4
f(x) ⫽ 1.6; x ⫽ 0.42781574
c x-intercept: none; y-intercept: 1
d x-intercept: ⫺1.584963; y-intercept: ⫺2
7.2 Exponential Functions and Their Graphs
31
16
f(x) ⫽
冢 23 冣 ; g(x) ⫽ 冢 32 冣 ; in functions of
x
x
19
a, b
6
the form y ⫽ bx, the graph will increase
from left to right if b ⬎ 1 and decrease from
left to right if 0 ⬍ b ⬍ 1.
17
y
y⫽
3 x
4
()
2
1
y⫽
16
1 x
3
()
y⫽
1 x
4
()
–4 –3 –2 –1
–1
14
c
12
20
10
y⫽
y⫽
()
a, b
4
x
B
9
8
6
7
6
5
y ⫽ 2x
4
–2
2
4
x
3
2
All three graphs decrease from left to right.
1 x
For x ⬍ 0, the graph of y ⫽
decreases
4
1 x
, which decreases faster
faster than y ⫽
3
1 x
.
than
2
冢冣
冢冣
冢冣
a, b
y
8
6
4
y ⫽ 3x
2
–4
3
y
2
18
2
x
4
–4
1
冢 43 冣
8
1 x
2
B
4
3
y
18
5
–2
2
4
x
–2
–4
–6
B
1
–4 –3 –2 –1
1
2
3
4
x
c y ⫽ 2x ⫹ 3
21 f(x) ⫽ 2⫺x, g(x) ⫽ 3⫺x, h(x) ⫽ 3x, j(x) ⫽ 2x,
k(x) ⫽ ⫺3x
22 a x ⫽ 0
b g(x); 4 raised to any positive power is
greater than 2 raised to the same power.
c f(x); 2 raised to any negative power is
greater than 4 raised to the same power.
23 a x ⫽ 0
b h(x); 2 raised to any positive power is
1
greater than
raised to the same
2
power.
c j(x); a negative exponent raises the reciprocal of the base to the positive power.
d j(x) ⫽ 2⫺x
e ry-axis
–8
c
y ⫽ ⫺3x
7.3 Solving Equations
Involving Exponents
(pages 262–263)
5
1 x ⫽ 5兹5
2 y ⫽ 625
1
3 z⫽
27
32
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
1,296
a ⫽ 243
1
b⫽
125
1
r⫽
81
1
s⫽⫺
32
g ⫽ 26
7
w⫽⫺
8
145
r⫽
36
v ⫽ 17
x ⫽ 1,042
z⫽2
1
m⫽
2
2
y⫽
3
2
(1) y 3 ⫽ ⫺4
33
(4)
16
3
(4) (x ⫹ 1)2 ⫽ 4
(2) irrational
a⫽
7.4 Solving Exponential
Equations
(pages 267–268)
1 x⫽3
2 z⫽5
3 r⫽1
4 x ⫽ ⫺3
5 x ⫽ ⫺2
6 x ⫽ ⫺1
7 s⫽6
8 x ⫽ ⫺2
3
9 x⫽⫺
4
10 x ⫽ ⫺5
3
11 n ⫽
2
12 p ⫽ ⫺3
13 x ⫽ 5
14 x ⫽ ⫺1, 2
2
15 x ⫽ ⫺
5
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
(3) 3
(4) {0, 3}
8
(4)
7
3
(2)
2
1
(1) ⫺
4
1
r⫽
2
w ⫽ 2.16
p⫽3
x ⫽ 9.17
v ⫽ ⫺2
w ⫽ ⫺2.37
x ⫽ ⫺2
r ⫽ 1.89
x ⫽ ⫺1.26
m ⫽ 5.17
7.5 Applications of
Exponential Equations
(pages 273–275)
1 (4) P(t) ⫽ 15,000(1.02)t
2 (4) P(t) ⫽ 15,000(0.98)t
3 (1) P(t) ⫽ 15,000e0.02t
4 (2) P(t) ⫽ 15,000e⫺0.02t
5 (3) y ⫽ 2x
6 (2) 2012
7 (3) P(t) ⫽ 3t
8 (4) $168.07
9 a $6,381.41
b approximately 14.2 years
10 a $6,420.13
11 a $25,809.23
12 a 2 weeks
b 25
13 a 20,000 people
b 15% per year
14 $11,034.39
15 a 20
b Each day, there are 90% of the fish left in
the tank. The fish are dying at a rate of
10% per day.
7.5 Applications of Exponential Equations
33
P(x) ⫽ 31,443,790(1.346)x
251,681,000 people
In approximately 8.6 decades or 1946
The population was 452,190,734 fewer
than predicted by the formula.
17 a 88.6025 micrograms
b approximately 2,377 years
c 5,728 years
18 i ⫽ C, ii ⫽ A, iii ⫽ B, iv ⫽ D, v ⫽ E
19 Answers will vary.
y
20
16
a
b
c
d
y ⫽ ex
24
22
y⫽
3x
10
11
12
13
14
15
16
17
18
19
20
20
21
22
23
24
25
18
16
14
12
y ⫽ 2x
10
8
6
4
26
2
–8 –6 –4 –2
21
2
4
6
8
x
a 9 days
b At 9.96 days, the two options are worth
the same amount. So, on the 10th day
the second option is worth more. The
second option then continues to be the
better choice.
1 (2) ⫺2
2 (2) ⫺12x7
3 (4) 48
4 (3) 2ab13
5 (3) 15
6 (1) 4x⫺2
7 (3) 28
3
8 (2)
4
4
9 (3) (x 2 ⫹ 3x)3 ⫽ 16
34
27
28
29
30
31
(4)
(3)
(3)
(4)
(2)
(4)
(3)
6
9
3
{3, ⫺1}
3兹x
8b
{⫺3, ⫺1}
5
(4)
4
(3) the amount of a person’s salary if she
gets a $1,500 raise every year
(2) $952
8
(4)
7
(1) N(t) ⫽ 5(1.08)t
(2) 41
(2) y ⫽ c(1.05)x
(2) $7.50
a C(t) ⫽ 10(1.079)t
b 521
c week 61
a 720,500 is the population of Halycon in
January 2007; 1.022 indicates the population is increasing by 2.2% per year.
b 2022
a y ⫽ 39,389(0.82)x
b $32,299
c in year 10: 2019
a $4,073.58
b 9.4 years
a $11,472
b 25.8 years
a 47,827 is the initial population in 2008;
e⫺0.1779 indicates the population is decreasing at an annual rate of 17.79%
compounded continuously.
b 6.5 years
a 1.55 billion
b 8.006%
c 7.8155 billion
d 2041
CHAPTER
Logarithmic Functions
8.1 Inverse of an
Exponential Function
18
4
g(x)
2
1
c
g(x) ⫽ log3 x
3
4
5
6
7
8
5
10
15
20
25
x
8.2 Logarithmic Form of an
Exponential Equation
f(x)
2
O
b (1, 0)
7
–2 –1 O 1
–1
–2
–3
y = log5 x
–4
8
3
y
–2
9
5
a
4
2
(pages 282–283)
1 (4) y ⫽ log3 x
2 (2) I and IV
3 (3) f(x) ⫽ logb x
4 (4) 0
5 (2) y ⫽ 2x
6 (1) reflection in the y-axis
7 (4) 10 ⬍ x ⬍ 100
8 (4) g(x) ⫽ log3 x
9 (4) It will not intersect the y-axis.
10 (1) y ⫽ 2x
11 y ⫽ log6 x
12 f ⫺1(x) ⫽ 4x
13 y ⫽ log3 x
14 f ⫺1(x) ⫽ 10x
15 y ⫽ 2x
16 f ⫺1(x) ⫽ log10 x
y
17 a, b
6
8
9
x
(pages 285–286)
1 5 ⫽ log3 243
1
⫽ log 16 4
2
2
3 2 ⫽ log6 36
1
4 ⫺2 ⫽ log 2
4
25
5 2 ⫽ log 56
36
6 ⫺2 ⫽ log10 0.01
1
⫽ log 49 7
7
2
8 a ⫽ logb c
9 26 ⫽ 64
10 53 ⫽ 125
11 10⫺3 ⫽ 0.001
1
12 42 ⫽ 2
13 112 ⫽ 121
1
14 2⫺2 ⫽
4
15 x ⫽ 4
1
16 x ⫽ ⫺
2
3
17 x ⫽
2
8.2 Logarithmic Form of an Exponential Equation
35
18 x ⫽ 4
19 x ⫽ ⫺2
20 x ⫽ 3
21 x ⫽ 64
1
22 x ⫽
81
23 x ⫽ 5
24
x ⫽ 0.001 or
25
x ⫽ 27
1
x⫽
7
x⫽2
x⫽9
x ⫽ 10
x⫽2
x⫽6
x⫽2
(4) 4
1
(3)
4
(4) ⫺4
(4) 7
(3) 36
b
(3)
2
(4) b y
1
(3)
125
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
11
12
13
14
15
16
1
1,000
18
19
20
8.4 Common and Natural
Logarithms
8.3 Logarithmic
Relationships
(pages 290–291)
1 (2) m ⫹ 2n
2
(4) log 10 a ⫹
3
4
5
7
(2) 5.6
(2) 8.48
(1) 1.935
1
(2) ⫺
2
(2) 2
8
(4) 2log c 8 ⫹
9
(3)
6
10
36
17
1
(p ⫺ q)
2
(2) product and power rules
(1) 216
(4) 5
(3) {6}
(4) {12}
p3 r
x⫽ 兹
q
3
兹qr
x⫽ 2
p
logb x ⫽ 2logb p ⫹ 3logb q ⫹ logb r
1
1
logb x ⫽ logb p ⫹ logb r ⫺ 3logb q
2
2
(4)
1
log 10 b ⫺ log 10 c
2
1
log c 5 ⫺ log c 21
3
a2
b 3兹d
(2) 2
Chapter 8: Logarithmic Functions
(pages 294–295)
1 (1) 3,781.8126
2 (3) 8.6365
3 (2) 3 ⫹ a
4 (3) 5
5 (3) 6.752270376
6 (4) 2w ⫹ 1
7 (3) b ⫹ 1
log n
8 (1) 1 ⫺
2
ln n
9 (1) 1 ⫺
2
10 (2) 5.3804
11 a 100.5171959 ⫽ 3.29
b e2.0347056 ⫽ 7.65
c 10b ⫽ a
d ed ⫽ c
12 a log 13,182.56739 ⫽ 4.12
b ln 906.870869 ⫽ 6.81
c log y ⫽ x
d ln z ⫽ x
13 a True; product rule
b False; log A ⫹ log B ⫽ log AB
c True; power rule
1
3
d False; ln 兹e ⫽
3
e True; log 1 ⫽ ln 1 ⫽ 0
14 a 6
b ⫺4
c 5
d ⫺3
8.5 Exponential Equations
(pages 299–301)
1 2.26
2 2.13
3 5.56
4 2.14
5 1.29
6 ⫺25.04
7 3.28
8 5.31
9 0.585
10 ⫺12.9801
11 2.8966
12 33.9709
13 a Yes
b 2012
c 5.0 ounces
14 9.64 hours
15 9.64 hours; yes
16 a V(t) ⫽ 32,640(0.82)t
b $26,765
c 2017
17 2013
18 2011
19 a $14.91
b N(t) ⫽ 14.91e 0.054t
c 2010
20 10.9%; this is not a plausible interest rate.
Plausible interest rates will vary.
21 2012
22 a W(t) ⫽ 17,420(0.97)t
b 22.8 hours
c 5,644.2 gallons
23 a 0.995 ⫽ e ⫺0.00012101x
b x ⬇ 41.4; the painting is a forgery since it
is only about 41 years old.
24 9.0 years
25 a 5.83 years
b $73,637.65
c Deidre: 9.7 years or early 2018; Alan:
10.7 years or early 2019
8.6 Logarithmic Equations
(page 305)
1 x ⫽ 17.78
2 x ⫽ 7.39
3 x ⫽ 4.57
4 x⫽2
5 x ⫽ 134.48
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
x ⫽ 3.21
x ⫽ 547.72
x ⫽ 148.41
x⫽6
x⫽3
1
x⫽
2
x⫽5
x⫽2
x⫽3
x⫽1
x⫽2
x ⫽ ⫺4, 5
x ⫽ 10
x⫽e
x ⫽ 3, 6
1
1 (4) log a ⫹ log b ⫺ log c ⫺ log d
2
2 (1) y ⫽ x
3 (4) IV and I
1
1
x⫹ y
4 (3)
2
2
5 (1) 81
6 (2) x ⫽ log a y
7 (2) a ⫹ 2
8 (1) 64
9 (4) The range is { y : y 僆⺢}
10 (2) y ⫽ 4x
11 (1) 6logb 6
12 (4) 2e4x⫺1
13 (3) 10
14 (3) 2a ⫹ b
15 (2) logb 7
16 (4) ⫺4
17 (2) 2a ⫹ 1
1
18 (2)
512
1
19 (1)
5
20 (4) ⫺4
21 (2) 0
22 (3) 100 ⬍ x ⬍ 1,000
23 (2) log10 0 ⫽ 1
24 (2) 1.271
25 (2) h ⫹ 1
26 x ⫽ 3.25
27 x ⫽ 0.456
28 x ⫽ 2
Chapter Review
37
29
30
31
32
33
34
35
1
2x 3
log
兹b
x 2.585
Domain: x 0; Range: All real numbers
x2
x7
x6
36
37
38
39
40
0
x 2.74
9.6 years
a 659
b 6.2 minutes
3.7 hours
CHAPTER
9
9.1 Trigonometry of the
Right Triangle
(pages 312–313)
1 6.4 cm
2 52.6°
3 14.7 cm
4 26.1 cm
5 16.2°
6 33.7 cm
7 64.6°
8 8.8 cm
9 28.5°
10 11.4 cm
11 32.9°
12 29.5 cm
13 77 in.
Trigonometric
Functions
14
68.2°
150 ft
x
60 ft
15
7.3 ft
x
36°
x
96 in.
53°
38
10 ft
16
9.3 The Unit Circle
11.8 ft
(pages 320–322)
1
Quadrant Sin u
10 ft
x
58°
17
2
70°
15 ft
x
18
a
b
16 ft
23°
17.5 ft
x
4
a
b
5
6
7
8
9
10
(2)
(4)
(1)
(4)
(2)
(3)
11
(3)
12
a
41 ft
9.2 Angles as Rotations
(page 316)
1 III
2 III
3 IV
4 I
5 II
6 IV
7 III
8 II
9 I
10 II
11 IV
12 I
13 II
14 I
15 IV
16 (2) 68°
17 (2) 34°
18 (1) 104°
b
13
14
15
I
II
III
IV
兹3
a
b
3
Cos u
a
b
1
2
2
1
2
兹2
2
兹2
2
III
Negative; y-values are negative in Quadrant III.
II
303°
I
IV
II
210°
3
sin 120° 兹
2
兹5
3
2
3
AB
OB
兹2
2
9.4 The Tangent Function
(pages 325–326)
1
Quadrant sin u
2
cos u
tan u
I
II
III
IV
(2) II
9.4 The Tangent Function
39
3
4
5
6
7
8
9
10
11
(3)
(1)
(3)
(4)
(1)
(4)
(2)
(3)
(4)
12
(1)
13
14
15
(3)
(3)
(4)
III
I
I and III
313°
sin u 0
IV
1
tan 240° 兹3
Sin u may be positive or negative.
12
5
PR
III
Sin u can equal cos u only in Quadrant I
of the unit circle.
5
6
7
8
9
10
11
12
13
14
15
9.5 Special Angles and
Reference Angles
16
(pages 331–332)
1 In the following table, U = undefined.
u
0°
30°
45°
60°
sin u
0
1
2
兹2
兹3
2
2
cos u
1
兹3
兹2
2
0
兹3
tan u
3
1
0
1
0
2
1
2
0
1
0
1
1
兹3
U
2
3
4
40
90° 180° 270° 360°
0
U
Angle
Quadrant
Reference
Angle
Formula
210°
III
210° 180°
30°
330°
IV
360° 330°
30°
135°
II
180° 135°
45°
300°
IV
360° 300°
60°
120°
II
180° 120°
60°
240°
III
240° 180°
60°
225°
III
225° 180°
45°
0
Reference
Angle
320°
30°
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
80°
320°
240°
兹3
2
1
2
2
兹
2
兹3
兹2
2
1
13
12
a 1
b 兹2
c 2
a sin 40°
b tan 50°
c cos 20°
d tan 40°
e sin 40°
(2) 90°
(3) 1
1
(2) 2
(3) 240°
(1) cos 70°
(3) reflection in the x-axis
(3) tan 240°
(3) sin 30°
5
(4)
4
(4) 4
(3) tan 135°
(2) 48°
(3) 150°
(2) sin x
(4) sin 180°
(4) Sin u cos u only in Quadrant I.
9.6 Reciprocal
Trigonometric Functions
(pages 334–335)
1 In the following table, U = undefined.
u
0°
30°
45°
60°
90°
180°
270°
360°
sec u
1
2兹3
3
兹2
2
U
1
U
1
csc u
U
2
兹2
2兹3
3
1
U
1
U
cot u
U
兹3
1
兹3
0
U
0
U
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
3
a 90°, 270°
b 0°, 180°, 360°
c 0°, 90°, 180°, 270°, 360°
a II
b IV
c IV
d II
e I
2
兹2
0
兹3
3
2
1
1
1.179
1.556
1.428
0.700
1.743
(4) varies depending upon the value of u
1
(1)
sin 45°
4兹3
(4)
3
(2) 2
(3) cot 135°
1 (2) II
2 (2) 32°
3 (1) I
4 (4) tan 30°
5 (3) 0
6 (4) 兹3
7
7 (1)
25
8 (3) 330°
9 (2) decreases from 1 to 0
10 (4) 225°
11 (4) 315
3
12 (4)
4
13 (2) cos 90°
14 (4) 45° and 225°
兹3
15 (2)
3
16 (4) II or IV
17 (2) 300°
18 (1) 150°
19 (4) 0
20 (2) 450°
21 (2) 23
22 (3) 28°
23 (1) 1
24 (1) 1
12
25 (4)
13
26 (2) 150°
27 (4) 9
28 (3) 680°
29 (3) csc ␪ 兹2
30 (2) 120° and 240°
31 (3) 210°
32 (2) 34°
33 (1) 2兹2
Chapter Review
41
CHAPTER
10
More Trigonometric
Functions
10.1 Radian Measure
(pages 343–344)
2p
1
3
3p
2
2
5p
3 ⫺
18
7p
4
4
3p
5 ⫺
4
4p
6
9
11p
7
6
8 ⫺p
p
9 ⫺
4
4p
10
3
11 270°
12 120°
13 225°
14 ⫺90°
15 150°
16 ⫺315°
17 36°
18 300°
19 ⫺30°
20 180°
21 2 m
22 6 ft
23 1 degree, 1 radian, 3 radians, 1 revolution
42
24
25
26
27
28
29
30
105°
7p
b
12
6p ⬇ 18.85 in.
(2) II
(3) 25
(3) III
(4) 300°
(4) 2p
a
10.2 Trigonometric
Functions with Radian
Measure
(pages 348–349)
p
1 ⫺sin
3
p
2 ⫺cos
4
p
3 ⫺cot
4
p
4 ⫺sec
3
p
5 ⫺csc
6
p
6 ⫺cos
6
p
7 ⫺tan
4
p
8 ⫺sin
6
9 1
10 ⫺1
11 ⫺兹3
12 2
13
14
15
16
17
18
19
20
21
22
23
24
25
2
兹7 , cos u ⫽ 3 , tan u ⫽ 兹7 ,
13 sin u ⫽
4
4
3
3 7
4兹7
, cot u ⫽ 兹
csc u ⫽
7
7
2
14 (3) sec u
3
15 (2) ⫺
5
16 (1) sec2 u
17 (2) ⫺0.75
18 (3) k2
3
19 (3)
4
1
20 (4)
2
sin u cos2 u
2
a ⫺兹
2
兹3
b ⫺
2
c 0
⫺3
(3) 0
(3) 3
1
(3)
2
3p
(2)
4
3p
(3) sin
4
(3) 0
11p
(4)
6
5p
(2)
6
(1) 1
(4) sin p
10.4 Range and Domain of
10.3 Basic and
Pythagorean Trigonometric
Identities
(pages 354–355)
1 sin2 u
2 1, (sin u ⫽ 0)
3 1, (sin u ⫽ 0, cos u ⫽ 0)
4 tan2 u
5 sec u, (sin u ⫽ 0)
6 cot u
7 sec u, (sin u ⫽ 0)
8 csc2 u
9 cos2 u, (cos u ⫽ 0)
10 sin2 u, (sin u ⫽ 0, cos u ⫽ 0)
12
12
13
11 sin u ⫽
, tan u ⫽ ⫺ , csc u ⫽
,
13
5
12
13
5
sec u ⫽ ⫺ , cot u ⫽ ⫺
5
12
24
7
25
12 cos u ⫽ ⫺ , tan u ⫽
, csc u ⫽ ⫺ ,
25
24
7
25
24
sec u ⫽ ⫺ , cot u ⫽
24
7
(page 357)
3p
1
3p
⫽ 0 so sec
⫽ , which is
1 cos
2
2
0
undefined.
1
2 sin 0° ⫽ 0 so csc 90° ⫽ , which is
0
undefined.
1
3 sin p ⫽ 0 so cos p ⫽ , which is undefined.
0
1
4 cos 90° ⫽ 0 so sec 90° ⫽ , which is
0
undefined.
兹3
5
3
6 ⫺1
7 ⫺2
8 Undefined; sin p ⫽ 0
兹3
9 ⫺
2
10 Undefined; tan 2p ⫽ 0
兹3
11 ⫺
2
兹3
12 ⫺
3
13 ⫺2
14 ⫺兹2
15 ⫺兹3
16 1
1
17
2
10.4 Range and Domain of Trigonometric Functions
43
18
19
20
2 3
⫺ 兹
3
p
(2)
2
(3) The domain of the tangent function is all
real numbers.
10.5 Inverse Trigonometric
Functions
(pages 360–361)
1 ⫺45°
2 90°
3 ⫺45°
4 41.2°
5 ⫺60°
6 41.4°
7 22.6°
8 ⫺60°
9 0
p
10 ⫺
6
5p
11
6
p
12 ⫺
6
p
13 ⫺
3
1
14 (1) ⫺ p
4
15 (3) 300°
16 (1) 0°
p
17 (2) ⫺
3
(3) arccos ⫺
19
(4)
20
兹2
(1) arcsin ⫺
2
5p
3
冢
9
10
11
12
13
14
15
16
17
18
19
20
21
23
24
25
26
冣
10.6 Cofunctions
(page 363)
1 cos 25°
p
2 ⫺tan
6
3 ⫺sec 15°
44
8
22
冢 12 冣 ⫽ u
18
4
5
6
7
⫺cot 20°
csc 43°
⫺cos 16°
tan 27°
p
sin
15
⫺cot 19°
⫺sec 33°
u ⫽ 22
u ⫽ 11
u ⫽ 35
u⫽6
u ⫽ 10
u ⫽ 17
u⫽3
u ⫽ 18
u ⫽ 24
u ⫽ 12
(3) 35
7p
(1) csc
30
兹3
(2)
3
(3) 14
(1) 34°
(2) ⫺csc 12°
2p
1 (2)
3
3
2 (3)
2
兹3
3 (2)
2
4 (2) 2
5 (1) 1
5p
6 (1) csc
3
7p
7 (3)
6
8 (3) sec2 x
p
9 (2)
4
5p
10 (4)
6
11 (4) 13.5
12 (4) 315
1
13 (2)
4
14
(4)
15
16
(1)
(4)
17
(3)
18
(1)
19
(1)
20
(4)
21
22
23
(1)
(1)
(4)
24
(3)
12
5
10
csc2 a
兹3
3
sin u
p
⫺
6
4p
3
⫺150°
1
20 centimeters
5p
6
25
(1) ⫺
26
27
28
29
30
31
(4)
(1)
(1)
(2)
(4)
(1)
32
(1)
33
34
(1)
(1)
35
(2)
4
5
0
sin x and cos x
1
11
11
1 radian
5p
6
57°
0
p
⫺
4
CHAPTER
Graphs of
Trigonometric
Functions
11.1 Graphs of the Sine
and Cosine Functions
(pages 371–372)
1
11
2
2
1
–2p 3p
–
2
2
–p
–p
y = cos x
p
2
2
p
3p
2
2p
–1
y = sin x
1
–2
–2p 3p
–
2
–p
–p
p
2
2
–1
–2
p
3p
2
2p
a 1
b ⫺1
c
冦⫺ 3p2 , ⫺ p2 , p2 , 3p2 冧
a 1
b ⫺1
c {⫺2p, ⫺p, 0, p, 2p}
11.1 Graphs of the Sine and Cosine Functions
45
3
2
y = sin x
1
p
p
2
2p
3p
2
6
b
1
2
c
x⫽⫺
2
–1
y = sin x
1
y = cos x
p
,0
2
–2
冦
冤
冤
冤
冤
冧
冥
b
c
d
e
–1
a
b
c
d
e
f
冥
冥
–p
y = sin x
1
–p
p
p
2
2
–1
–2
p
2
b The two graphs are identical.
x⫽
a
2
1
–p
–p
p
p
2
2
–1
–2
5
2
y = cos x
4p
y = cos x
2
2
4
Answers will vary. Example: [0, 2p]
Answers will vary. Example: [2p, 4p]
The length of the interval is 2p.
11.2 Amplitude, Period,
and Phase Shift
2
y = cos x
3p
–2
冥
4
2p
p
p 5p
,
or {45°, 225°}
4 4
3p
p,
2
p
0,
2
3p
, 2p
2
p
,p
2
a
(pages 381–383)
1 (3) Cos x increases from 0 to 1.
2 (3) 3
3 (3) line y ⫽ x
4 (3) 3
5 (4) p
p
6 (3)
units to the left
3
7 (2) 2
8 (3) coincide
9 (1) y ⫽ sin (⫺x)
10 (4) y ⫽ ⫺2cos 2x
11 (3) ⫺2 ⱕ y ⱕ 4
12 (1) p
13 (4) 4
14 (2) y ⫽ sin (⫺x)
y
15 a
y = –3cos 2x
1
3
2
–p
2
p
2
y = sin x
–1
–2
a
1
2
1
O
–1
p
2
3p
2
–2
–3
b {0, p, 2p}
46
p
y = 3 sin x
2
2p
x
16
a, b
The two graphs have the same amplitude (1), frequency (1), and period (2p).
However, y ⫽ cos (x ⫹ p) is y ⫽ cos x
shifted p units to the left, while y ⫽ cos
x ⫹ p is y ⫽ cos x shifted p units up.
a, b
2
c
y
y = 3 sin 1 x
2
3
2
1
O
p
2
–1
3p
2
p
–2
20
x
2p
y = sin (x – p)
1
–2p 3p
–
y = 2 cos x
–p
2
–p
2
–3
–1
p
2
p
3p
2
2p
–2
–3
17
c
a
p
–4
–5
y
2
1
O
p
2
–
p
3
–
p
6
p
6
p
3
x
p
2
21
–1
y = 3 sin 2x
However, y ⫽ sin (x ⫺ p) is y ⫽ sin x
shifted p units to the right, while y ⫽ sin
x ⫺ p is y ⫽ sin x shifted p units down.
a, b
4
c
y = –2cos 2x
–
y = sin x + 2
3
–2
2
y = sin x – p
2
1
b {⫺0.46, 1.11}
18 a, b
y = 2sin 2x
–2p
y
–2
2
O
p
2
p
2
p
x
–1
However, y ⫽ sin (x ⫹ 2) is y ⫽ sin x
shifted 2 units to the left, while y ⫽
sin x ⫹ 2 is y ⫽ sin x shifted 2 units up.
a, b
2
22
y = cos (x – 1)
1
–2
–2p 3p
–
c
d
19
2
4
冦
p p
⫺ ,
2 2
冧
a, b
–p
2
–1
p
2
p
3p
2
2p
y = cos x – 1
–3
4
c
y = cos x + p
3
2
1
–p
y = cos (x + p)
–p
–2
5
–2p
y = sin (x + 2)
c
y = 3 cos (x + p)
1
–
2p
p
–1
2
–p
–p
p
2p
However, y ⫽ cos (x ⫺ 1) is y ⫽ cos x
shifted 1 unit to the right, while y ⫽ cos
x ⫺ 1 is y ⫽ cos x shifted 1 unit down.
–1
–2
11.2 Amplitude, Period, and Phase Shift
47
11.3 Writing the Equation
of a Sine or Cosine Graph
5
4
2
5
4
3
2
1
1
p
p
2
冢冢
冣冣
6
(2) y ⫽ 3sin
7
(3) y ⫽
8
9
1
10
p
p
2
2p
3p
2
–1
11
12
–2
13
y ⫽ 4sin
冢冢
p
5
x⫹
5
2
冣冣
5
6
1
y ⫽ 2sin
2
3
4
5
6
7
8
9 10
冢冣
j(x) ⫽ ⫺1.5cos (x ⫹ p)
p
15 g(x) ⫽ 1.5sin x ⫺
2
p
16 h(x) ⫽ 1.5cos x ⫺
2
17 f(x) ⫽ 1.5sin (x ⫹ p)
冣
冣
18 y ⫽ ⫺6sin 2x
p
x
19 y ⫽ 5cos
2
p
20 y ⫽ 2sin 4 x ⫹
6
冢冢
3
2
21
1
2p
1
sin 2x
2
amplitude ⫽ 3.5, period ⫽ p, frequency ⫽ 2
2p
, frequency ⫽ 3
amplitude ⫽ 2, period ⫽
3
p
amplitude ⫽ 6, period ⫽ 8, frequency ⫽
4
y ⫽ ⫺3.5sin 2x
y ⫽ 2sin 3x
p
y ⫽ 6sin
x
4
冢
冢
冢 12 (x ⫺ 3)冣
p
1
x
2
14
5
4
3
2
1
3p
4p
–1
–2
–3
48
4
–6
2
4
3
–4
p
y ⫽ sin 3 x ⫺
3
–1
–2
–3
–4
–5
2
–2
–1
–2
–3
–4
–5
3
冢 2p3 (x ⫺ 1)冣
6
(pages 386–389)
1 y ⫽ 4sin 6x
2
y ⫽ 5sin
22
冣冣
冢冢 p6 冣冣
p
y ⫽ 1.5cos 3x, y ⫽ ⫺1.5sin 冢3冢x ⫺ 冣冣
6
y ⫽ ⫺1.5sin 3x, y ⫽ 1.5cos 3 x ⫹
11.4 Graph of the Tangent
Function
5
y = tan x
(pages 391–392)
1 No, the tangent function does not have an
amplitude because its range is [⫺⬁, ⬁].
5
2
y = tan x
p
2
y = 2 tan x
a p
b Both curves have the same basic shape
and have the same domain and range.
However, the graph of y ⫽ 2tan x is
stretched vertically, so all of the y-values
are double those of y ⫽ tan x.
p
2
2
–5
6
a
1
b
冢
p
–5
y = cos x
–p
5
5
y = tan 1 x
冥
p
⫺ ,0
2
2
3
5
p
y = tan x
2p
y = 1 tan x
2
y = sin x
–5
–2p
–p
p
2p
Both curves are the same basic shape and
have the same range. However, the period
1
of y ⫽ tan x is p and its domain is
2
p
x:x⫽
⫹ np for n an integer , while the
2
1
period of y ⫽ tan x is 2p and its domain
2
is {x : x ⫽ p ⫹ 2np for n an integer}.
–5
a 2
b 4
c 5
d
4
冦
冤⫺2p, ⫺ 3p2 冣, 冢⫺ p2 , p2 冣, 冢 3p2 , 2p冥
5
7
y = tan 2x
a p
b {x : x ⫽ np for n an integer}
y = tan x
p
2
c
p
8
–5
p
2
b Both curves are the same basic shape
and have the same range. However, the
p
period of y ⫽ tan 2x is
and its domain
2
p
np
⫹
for n an integer ,
is x : x ⫽
4
2
while the period of y ⫽ tan x is p and
its domain is
p
x:x⫽
⫹ np for n an integer .
2
a
冦
冦
冧
冧
冧
The function y ⫽ tan x was shifted
units to the left.
3p
p
p
,x⫽⫺ ,x⫽ ,x⫽
a x⫽⫺
2
2
2
7p
5p
3p
,x⫽⫺
,x⫽⫺
,
b x⫽⫺
4
4
4
p
p
3p
x⫽⫺ ,x⫽ ,x⫽
,
4
4
4
5p
7p
x⫽
,x⫽
4
4
p
p
3p
,x⫽⫺ ,x⫽ ,x⫽
c x⫽⫺
2
2
2
d x ⫽ ⫺p, x ⫽ p
p
p
3p
,x⫽⫺ ,x⫽ ,x⫽
e x⫽⫺
2
2
2
p
2
3p
2
3p
2
3p
2
11.4 Graph of the Tangent Function
49
9
3p
p
p
3p
,x⫽⫺ ,x⫽ ,x⫽
2
2
2
2
11p
3p
7p
,x⫽⫺
,x⫽⫺
,
b x⫽⫺
6
2
6
5p
p
p
p
x⫽⫺
,x⫽⫺ ,x⫽⫺ ,x⫽ ,
6
2
6
6
p
5p
7p
3p
x⫽ ,x⫽
,x⫽
,x⫽
,
2
6
6
2
11p
x⫽
6
3p
p
p
3p
,x⫽⫺ ,x⫽ ,x⫽
c x⫽⫺
2
2
2
2
3p
3p
, x⫽
d x⫽⫺
2
2
a
x⫽⫺
4
Graph each function using information from
its reciprocal. Find the asymptotes where
the reciprocal function equals zero. Plot
points based on known values of the function’s reciprocal. Pay special attention to
maxima, minima, and whether the function
is increasing or decreasing.
a The reciprocal of y ⫽ 2sec x is
1
y⫽
cos x. Asymptotes are at
2
p
⫹ pn for n an integer. When
x⫽
2
y ⫽ 2sec x is decreasing, the maximum
value is ⫺0.5, which occurs at x ⫽ p ⫹
2pn for n an integer. When y ⫽ 2sec x is
increasing, the minimum value is 0.5,
which occurs at x ⫽ 2pn for n an integer.
b The reciprocal of y ⫽ ⫺csc x is y ⫽ ⫺sin x.
Asymptotes are at x ⫽ pn for n an
integer. When y ⫽ ⫺csc x is decreasing,
the maximum value is ⫺1, which occurs
p
at x ⫽
⫹ 2pn for n an integer. When
2
y ⫽ ⫺csc x is increasing, the minimum
3p
value is 1, which occurs at x ⫽
⫹
2
2pn for n an integer.
1
c The reciprocal of y ⫽ cot x is y ⫽
2
2tan x. Asymptotes are at x ⫽ pn for n
p
1
an integer. cot x ⫽ 0 at x ⫽
⫹ pn
2
2
1
for n an integer. The graph of y ⫽ cot x
2
decreases from ⬁ to ⫺⬁ for each interval
[2pn, (2p ⫹ 1)n] for n an integer.
5
Answers may vary.
a y ⫽ tan x, y ⫽ sec x
b y ⫽ cot x, y ⫽ csc x
c y ⫽ tan x, y ⫽ sec x
d y ⫽ cot x, y ⫽ csc x
11.5 Graphs of the
Reciprocal Functions
(pages 398–399)
1 a y ⫽ tan x
b y ⫽ cot x
c y ⫽ sec x
d y ⫽ csc x
p p 3p
3p
,⫺ , ,
2 a x⫽⫺
2
2 2 2
b x ⫽ ⫺2p, ⫺p, 0, p, 2p
7p
5p
3p
p p 3p
,⫺
,⫺
,⫺ , ,
,
c x⫽⫺
4
4
4
4 4 4
5p 7p
,
4
4
3 a They are undefined when their reciprocal functions, cosine, sine, and tangent,
equal zero.
b The contangent function equals zero
when the tangent function is undefined.
The secant and cosecant functions never
equal zero because their reciprocal functions are always defined.
c Asymptotes occur when the functions
are undefined, that is, when their reciprocal functions equal zero.
50
6
a
16
Function
Interval
冢0, p2 冣
冢 p2 , p冣
y ⫽ sin x
Increasing
Decreasing
Decreasing
Increasing
y ⫽ cos x
Decreasing
Decreasing
Increasing
Increasing
y ⫽ tan x
Increasing
Increasing
Increasing
Increasing
y ⫽ csc x
Decreasing
Increasing
Increasing
Decreasing
y ⫽ sec x
Increasing
Increasing
Decreasing
Decreasing
y ⫽ cot x
Decreasing
Decreasing
Decreasing
Decreasing
7
冢p, 3p2 冣冢 3p2 , 2p冣
b When a function increases, its reciprocal
function decreases. When a function decreases, its reciprocal function increases.
p
⫹ np
a Period: 2p; Domain: x : x ⫽
2
冧
冦
for n an integer ; Range: {y : 兩y兩 ⱖ 1}
b Period: 2p; Domain: {x : x ⫽ np for n an
integer}; Range: {y : 兩y兩 ⱖ 1}
8 Period: p; Domain: {x : x ⫽ np for n an integer}; Range: All real numbers
11.6 Graphs of the Inverse
(pages 403–405)
1 (3) y ⫽ arccos x
2 (3) III
3 (3) ⫺75°
4 (1) Unless the domain of y ⫽ sin x is restricted, the reflection is not a function.
2 兹5
5 (2) ⫺
5
6 (4) I and III
7 (1) y ⫽ arcsin x
p
8 (1) ⫺
6
2
兹
9 (2)
2
1
10 (3) ⫺ ⫽ arccos x
2
11 (4) sin 0
12 (1) ⫺60°
13 (3) 210°
14 (4) y ⫽ arccos x
p
15 (1)
2
(1) An inverse will exist if the original
p
p
function has a domain ⫺ ⱕ x ⱕ .
2
2
11.7 Trigonometric Graphs
and Real-World Applications
(pages 407–409)
1 (4) 4 seconds
2 (2) 15 seconds
3 (2) 262
4 (2) 2 seconds
5 a Lowest in July; highest in January
b $58,934 on 746 snow blowers
c Highest in June; lowest in December
d during March and September
e $31,752 (highest monthly profit)
p
t ⫹ 300; we ex6 a (2) S(t) ⫽ ⫺200cos
6
pect swimming pool sales to be lowest
in the winter and highest in the summer.
b 126
c June; 500
d March and October
7 a 10.8
b March 20 and September 23 (assuming
that it’s not a leap year)
c December 21, the first day of winter;
9.4 hours of daylight
8 a 0.75; the tide varies 0.75 foot above and
below the average depth of 1.5 feet.
b 12; every 12 hours, the tides complete
one full cycle.
c 6 hours
d 1.875 feet
e at approximately 1:47 A.M., 10:47 A.M.,
1:37 P.M., and 10:47 P.M.
9 a January: 23°; February: 26°
b July; 71°
c 47°
d April and October
e 24; the average monthly temperature
varies 24 degrees above and below the
average temperature for a year in
Syracuse.
10 a ⫺503.90 ⱕ P(t) ⱕ 933.94; Captain Freeze
can lose at most $503.90 in a week and
have a maximum profit of $933.94 in a
week.
冢冣
11.7 Trigonometric Graphs and Real-World Applications
51
b 52; these are the number of weeks in the
year from January through December.
c Week 29 (t ⫽ 28); $933.94
d Weeks 1 through 13 (t ⫽ 0 through 12)
and weeks 45 through 52 (t ⫽ 44 through
51)
e Yes, it is profitable. Explanations will vary
f Answers will vary.
兹3
1 (3)
2
2 (4) y ⫽ ⫺3cos 2x
3 (2) y ⫽ cos x
4 (4) There is no minimum value.
5 (1) p
6 (2) 2
7 (3) 60°
8 (4) ⫺5 ⱕ y ⱕ 5
p
p
9 (3) ⫺ ⱕ x ⱕ
2
2
10 (3) y ⫽ ⫺cos x
11 (2) y ⫽ ⫺cos x
12 (1) 1
1
13 (2)
cos x
14 (2) decreases only
p
15 (2)
2
16 (3) y ⫽ tan x
17 (1) 1
p
18 (3)
4
19 (1) 1.4 inches
20 (3) the range increases
21 (2) y ⫽ sec x
p
22 (2) It is undefined at .
2
23 (3) 15
3
24 a
25
c
26
p
3p
2
y = 3cos 2x
p 3p
,
4 4
y ⫽ ⫺3cos 2x
5
4
3
2
1
y = 2cos x
y = tan x
p
p
2
2
1 y = sin 1 x
2
–p
–
p
2
p
2
–1
p
b
–2
c
28
29
2p
–3
b (p, 0)
c x⫽p
y ⫽ sin
冢 12 (x ⫺ p)冣 or y ⫽ ⫺cos 12 x
June
80
March 1: 50; May 1: 76
trigonometric (sinusoidal) function
800
p
c
400
d high point: 200 feet; midline: 100 feet
e 100
p
x ⫹ 100
f y ⫽ 100cos
400
p
px
⫹5
a ⫽ 4, b ⫽ , d ⫽ 5; y ⫽ 4sin
6
6
a
b
c
a
b
冢
y = –sin x
–2
52
e
a
p
p
2
b x⫽0
c x⫽p
p
d x⫽
2
27 a, b
2
p
2
c
–1
–2
–3
–4
–5
1
–1
4
3
2
1
–1
–2
–3
–4
y = 2cos 1 x
2
a p
b, d
30
冣
CHAPTER
Trigonometric
Identities
12
12.1 Proving Trigonometric
Identities
6
(1 ⫹ cos u)(1 ⫺ cos u) ⱨ sin2 u
1 ⫺ cos2 u ⱨ sin2 u
sin2 u ⫽ sin2 u ✔
(page 420)
Answers may vary.
1 sin2 u ⫹ cot2 u ⫹ cos2 u ⱨ csc2 u
1 ⫹ cot2 ⱨ csc2 u
csc2 u ⫽ cos2 u ✔
7
cos2 u (tan2 u ⫹ 1) ⫹ cot2 u ⱨ csc2 u
cos2 u (sec2 u) ⫹ cot2 u ⱨ csc2 u
1 ⫹ cot2 u ⱨ csc2 u
csc2 u ⫽ csc2 u ✔
sec2 u (1 ⫺ cos2 u) ⱨ tan2 u
sec2 u (sin2 u) ⱨ tan2 u
sin2 u
ⱨ tan2 u
cos2 u
tan2 u ⫽ tan2 u ✔
8
2
3
4
5
1 ⫺ sin2 u
• sec u ⱨ cot u
sin u
cos2 u
1
•
ⱨ cot u
sin u
cos u
cos u
ⱨ cot u
sin u
cot u ⫽ cot u ✔
sin2 u ⫺ cos2 u ⫹ 1
ⱨ tan u
2sin u cos u
2sin2 u
ⱨ tan u
2sin u cos u
sin u
ⱨ tan u
cos u
tan u ⫽ tan u ✔
sec u
sin u
⫺
ⱨ cot u
sin u
cos u
1 ⫺ sin2 u
ⱨ cot u
sin u cos u
cos2 u
ⱨ cot u
sin u cos u
cos u
ⱨ cot u
sin u
cot u ⫽ cot u ✔
9
10
1
1
⫹
ⱨ sec 2 csc 2 2
sin cos2 cos2 ⫹ sin2 ⱨ sec 2 csc 2 sin2 cos2 1
ⱨ sec 2 csc 2 2
sin cos2 sec2 b csc2 b ⫽ sec2 b csc2 b ✔
csc u ⫺ (cos u)(cot u) ⱨ sin u
1
cos2 u
⫺
ⱨ sin u
sin u
sin2 u
sin2 u
ⱨ sin u
sin u
sin u ⫽ sin u ✔
1 ⫺ cos2 u
ⱨ sin2 u ⫹ 1
tan2 u
sin2 u
2⫺
ⱨ sin2 u ⫹ 1
tan2 u
sin2 u cos2 u
2⫺
ⱨ sin2 u ⫹ 1
sin2 u
2 ⫺ cos2 u ⱨ sin2 u ⫹ 1
1 ⫺ cos2 u ⫹ 1 ⱨ sin2 u ⫹ 1
sin2 u ⫹ 1 ⫽ sin2 u ⫹ 1 ✔
2⫺
12.1 Proving Trigonometric Identities
53
12.2 Sum and Difference
of Angles
(pages 424–426)
117
1 a
125
4
b
5
117
c
44
2 0.96
兹6 ⫹ 兹2
3
4
4 a Answers may vary. Example: 60° and
45°
兹6 ⫹ 兹2
b
4
5
sin (p ⫺ u) ⱨ sin u
sin p cos u ⫺ cos p sin u ⱨ sin u
sin u ⫽ sin u ✔
6
7
cos (360° ⫺ A) ⱨ cos A
cos 360° cos A ⫹ sin 360° sin A ⱨ cos A
cos A ⫽ cos A ✔
sin (180° ⫹ x) ⱨ ⫺sin x
sin 180° cos x ⫹ cos 180° sin x ⱨ ⫺sin x
⫺sin x ⫽ ⫺sin x ✔
cos
8
cos
9
(3)
10
11
12
13
14
15
16
17
(2)
(1)
(4)
(4)
(1)
(3)
(2)
(3)
18
(2)
19 (3)
20 (1)
21 (1)
22
(3)
23
(2)
54
24
(1)
25
(4)
26
(3)
27
(2)
28
(4)
12.3 Double-Angle
Formulas
(pages 429–431)
24
1
25
1
2 ⫺
2
3 0.28
3
4
5
5
6
7
冢 3p2 ⫺ 冣 ⱨ ⫺sin 3p
3p
cos ⫹ sin
sin ⱨ ⫺sin 2
2
⫺sin b ⫽ ⫺sin b ✔
16
65
⫺tan y
sin 300°
cos 270°
sin 120°
⫺1
tan 40°
⫺cos y
0
1
⫺
2
k
sin x
cos y
兹2 ⫹ 兹6
4
⫺sin u
84
85
⫺cos x
16
65
兹3
2
2sin A sin B
8
9
2sin 60° cos 60° ⫽
兹3
2
2cos2 90 ⫺ 1 ⫽ ⫺1
1 ⫺ tan u • sin 2u ⱨ cos 2u
sin u
1⫺
• 2sin u cos u ⱨ cos 2u
cos u
1 ⫺ 2sin2 u ⱨ cos 2u
cos 2u ⫽ cos 2u ✔
cos 2
⫹ 2sin a ⱨ csc a
sin 1 ⫺ 2sin2 2sin2 ⫹
ⱨ csc a
sin sin 1
ⱨ csc a
sin csc a ⫽ csc a ✔
2 tan u
1 ⫹ tan2 u
2 tan u
sin 2u ⱨ
sec 2 u
2 sin u
sin 2u ⱨ
• cos2 u
cos u
sin 2u ⱨ 2sin u cos u
sin 2u ⫽ sin 2u ✔
sin 2u ⱨ
10
11
12
1
cos2 u
1
(2cos2 u ⫺ 1)(sec2 u) ⱨ 2 ⫺
cos2 u
1
2 ⫺ sec2 u ⱨ 2 ⫺
cos2 u
1
1
✔
2⫺
⫽2⫺
2
cos u
cos2 u
cos 2u (1 ⫹ tan2 u) ⱨ 2 ⫺
sin 2u sec u ⱨ 2sin u
1
2sin u cos u •
ⱨ 2sin u
cos u
2sin u ⫽ 2sin u ✔
a
b
c
13
14
15
16
17
(1)
(1)
(2)
(1)
(4)
18 (4)
19 (2)
20 (3)
21 (2)
22 (3)
23 (2)
24 (3)
25
(4)
26
(1)
27
(2)
cos 2A ⫽ cos (A ⫹ A)
⫽ cos A cos A ⫺ sin A sin A
⫽ cos2 A ⫺ sin2 A
cos 2A ⫽ cos2 A ⫺ sin2 A
⫽ cos2 A ⫺ (1 ⫺ cos2 A)
⫽ 2cos2 A ⫺ 1
cos 2A ⫽ cos2 A ⫺ sin2 A
⫽ (1 ⫺ sin2 A) ⫺ sin2 A
⫽ 1 ⫺ 2sin2 A
sin u
⫺1 ⱕ y ⱕ 1
sin 2x
1
cos 30°
527
625
1
⫺
9
tan (2u)
⫺0.96
3
28
⫺
100
1 ⫺ cos 2x
sin2 x ⫽
2
兹30
6
⫺1
5
⫺
8
12.4 Half-Angle Formulas
(pages 435–436)
2兹5
1
5
1
2
2
兹10
3
10
兹3
4
3
5
cos 30° ⫽
6
a
冪
1 ⫹ cos 60⬚
⫽
2
sin 15° ⫽
⫽
b 0.259
c 0.259; yes
7
a
9
10
11
12
13
14
15
1
2
2
冪
1 ⫺ cos 30⬚
2
冪
3
1⫺ 兹
2
2 ⫺ 兹3
⫽ 兹
2
2
cos 22.5° ⫽
⫽
8
冪
1⫹
b 0.924
c 0.924; yes
(2) II
2 兹13
(2) ⫺
13
兹2 ⫹ 兹3
(1)
2
(2) ⫺1
兹5
(3)
3
兹5
(3)
5
兹7
(1)
4
(3) 60°
3
⫽ 兹
2
冪
1 ⫹ cos 45⬚
2
冪
2
1⫹ 兹
2
兹2 ⫹ 兹 2
⫽
2
2
12.4 Half-Angle Formulas
55
1 (4) 0
2 (4) ⫺cos x
3 (4) csc u
1
4 (3) ⫺
2
5 (2) sin 120°
7
6 (1)
6
7 (2) 5 ⫺ sin2 A
8 (1) sin2 4x ⫹ cos2 4x ⫽ 1
1⫺c
9 (1)
2
10 (3) 3
11 (1) 1
13
12 (3) ⫺
85
13 (2) ⫺sin u
14 (3) cos 90°
7
15 (3)
25
sin 2u
ⱨ tan u
16
1 ⫹ cos 2u
2sin u cos u
ⱨ tan u
1 ⫹ 2cos2 u ⫺ 1
sin u
ⱨ tan u
cos u
tan u ⫽ tan u ✔
17
56
sin 2u
2sin2 u
2 sin u cos u
cot u ⱨ
2sin2 u
cos u
cot u ⱨ
sin u
cot u ⫽ cot u ✔
cot u ⱨ
18
19
20
1 ⫹ cos u cos 2u
ⱨ cot u
sin u ⫹ sin 2u
1 ⫹ cos u ⫹ 2cos2 u ⫺ 1
ⱨ cot u
sin u ⫹ 2sin u cos u
cos u(1 ⫹ 2cos u)
ⱨ cot u
sin u(1 ⫹ 2cos u)
cos u
ⱨ cot u
sin u
cot u ⫽ cot u ✔
2sin ⱨ sec 2 sin 2 cos 2sin ⱨ sec 2 2sin cos cos 1
ⱨ sec 2 cos2 sec2 b ⫽ sec2 b ✔
sec x ⫺ sin x tan x ⱨ cos x
sin x
1
⫺ sin x
ⱨ cos x
cos x
cos x
1 ⫺ sin2 x
ⱨ cos x
cos x
cos2 x
ⱨ cos x
cos x
cos x ⫽ cos x ✔
CHAPTER
Trigonometric
Equations
13.1 First-Degree
Trigonometric Equations
(pages 443–444)
1 u 35°, 315°
2 u 30°, 150°
3 u 0°, 180°, 360°
4 u 150°, 210°
5 u 30°, 210°
6 u 90°
p 5p
7 u ,
4 4
2p 4p
,
8 u
3
3
5p 7p
,
9 u
4
4
5p 7p
,
10 u 6
6
3p
11 u 2
p 3p
12 u ,
2 2
13 b 11.5°, 168.5°
14 b 76.0°, 256.0°
15 b 78.5°, 281.5°
16 b 14.5°, 165.5°
17 b 60°, 240°
18 m⬔B 225
19 x 180°
7p
20 u 4
21 x {70.5°, 180°, 289.5°}
22 u 90°
23 (4) 74°
7p
24 (3)
6
25 (3) III and IV
13
26
(3)
冦 3p4 , 5p4 冧
7p
6
28 (3) 5csc u 2 1
29 (3) 150°
p 2p
,
30 (2)
3 3
27
(3)
冦
冧
13.2 Second-Degree
Trigonometric Equations
(pages 448–449)
1 u 45°, 135°, 225°, 315°
2 u 45°, 135°, 225°, 315°
3 u 270°
4 u 30°, 150°, 210°, 330°
5 u 60°, 90°, 270°, 300°
2p 4p
,
, 2p
6 u 0,
3
3
p 2p 4p 5p
,
,
7 u ,
3 3
3
3
p 5p
8 u ,
3 3
9 u 0, , 2
p
10 u 0, , p, 2p
2
11 b 21.8°, 135°, 201.8°, 315°
12 b 81.8°, 180°, 278.2°
13 b 193.4°, 346.6°
14 b 109.5°, 120°, 240°, 250.5°
15 b 57.9°, 122.1°
16 b 63.4°, 161.6°, 243.4°, 341.6°
17 m⬔B 180
18 x 135°
13.2 Second-Degree Trigonometric Equations
57
19
20
21
22
23
24
25
26
27
u
11p
6
17
a
2
–1
18
p
2p
b p
c p
a 2
y = –cos x
–1
b
c
19
a
p
,
2
p
,
2
7p
,
6
7p
,
6
3p 11p
,
2
6
3p 11p
,
2
6
2
y = sin 1 x
1
2
2p
p
–1
y = –cos x
–2
20
21
22
23
1
p
y = sin 2x
–2
y = 2sin x
2p
y = cos 2x
–2
58
2p
y = –sin x
1
(pages 452–453)
1 u 0°, 90°, 180°, 270°, 360°
2 u 0°, 120°, 240°, 360°
3 u 90°, 210°, 330°
4 u 180°
5 u 90°, 210°, 270°, 330°
2p
4p
, p,
, 2p
6 a 0,
3
3
p
5p
7 a , p,
3
3
8 0, 2
4p
9 a 0,
3
p
5p
, 2p
10 a 0, , p,
3
3
11 u 25.7°, 154.3°, 230.1°, 309.9°
12 u 38.2°, 141.8°
13 u 56.0°, 153.3°, 206.7°, 304.0°
14 u 90°, 221.8°, 318.2°
15 u 102.7°, 257.3°
16 a 2
c
p
–2
13.3 Trigonometric
Equations That Use
Identities
b
y = 2cos 1 x
1
(3) 3
(3) 120
5p
(3)
6
(2) cos2 x 1 0
4p
(4)
3
(2) 2
(3) 3
(4) 180°
–1
2
p
2
p
2
Chapter 13: Trigonometric Equations
b
c
(2)
(4)
(3)
(4)
p
p
2
p
90°
4
1 (4) 180°
p
2 (4)
6
3 (1) 1
4 (1) 1
5 (3) 180°
6 (3) Quadrant I or Quadrant III
7 (2) 90°
4p
8 (4)
3
9 (3) 3
10
11
12
13
14
p
2
66.4°, 180°, 293.6°
19.5°, 160.5°
60°, 300°
33.7°, 45°, 213.7°, 225°
(1)
15
16
17
18
19
20
55.5°, 155.4°, 235.5°, 335.4°
221.8°, 270°, 318.2°
180°
48.6°, 131.4°, 270°
30°, 150°, 199.5°, 340.5°
228.6°, 311.4°
CHAPTER
Trigonometric
Applications
14.1 Law of Cosines
(pages 460–462)
1 (2) 18.0
2 (1) 11.87
3 (3) 12.3 centimeters
4 (4) obtuse
5 (1) 135.6°
6 (4) 0.8647
7 (2) 90°
8 (1) 17.2 inches
9 (4) 9.63 inches
10 (3) 22.2 inches
11 73.61°, 62.95°, 43.43°
12 6.87 miles
13 a 66.4° or 66°25
b 47.2° or 47°9
14 6.3 miles
15 6.1 miles
14.2 Area of a Triangle
(pages 466–467)
1 60.79 in.2
2 25.95 cm2
3 81.27 cm2
4 131.06 cm2
5 11.66 in.2
14
6
7
8
9
10
11
12
13
14
15
16
17
(1)
(3)
(4)
(3)
(3)
(2)
(3)
(4)
(2)
(2)
a
b
a
b
8 inches
9 兹3
228.8
0.6697
170.9 square centimeters
29.53
52.94
12.2 inches
88.16
385.28
47.4°
257.1 cm2
13.2 in.
361.1 in.2
14.3 Law of Sines and the
Ambiguous Case
(pages 472–474)
1 a 28.5, m⬔A 135.2
2 a 30.3, m⬔A 45.4
3 a 38.2
4 m⬔A 57.8
5 m⬔A 37.7
6 12.1 cm
7 13.8 in.
8 5.5 ft
14.3 Law of Sines and the Ambiguous Case
59
9
10
11
9.40 in.
m⬔AMH ⫽ 77.9
a 2
b 49°, 131°
c 22, 38
d
16
17
18
T
114°
31
C
H
38
T
32°
31
C
12
13
14
15
16
17°
22
131°
51.3°
77.4°
607 ft
3,065 ft
107.2° and 72.8°
252.2 ft2
12
49°
17°
a
b
a
b
a
b
14.5 Forces and Vectors
(pages 482–483)
1 50.5 lb
12
(2) 49.8°
(1) 10.4
sin 61⬚
sin 58⬚
⫽
(2)
12.5
x
(1) cannot be determined
(4) 0
x
21.8 lb
H
52.6°
34.2 lb
2
62.5 lb
14.4 Mixed Trigonometric
Applications
(pages 477–478)
1 (1) ⫺0.1141
2 (2) 5.4
3 (3) 99.6°
4 (1) 21
5 (1) 7.2
6 (3) 40 centimeters
7 (1) 䉭MAT is a right triangle.
8 (2) 2
9 (3) 0.32
10 (1) 1
11 a 49°
b 218 square inches
12 No, the Law of Sines does not hold for these
measurements:
sin 69.98⬚
sin 43.74⬚
⫽
120
96
13 228 ft and 386 ft
14 a 57.5°
b 14.1 cm
c 66 cm2
15 a 16 cm
b 228 cm2
60
Chapter 14: Trigonometric Applications
73.4 lb
56°
47°
x
3
128.7°
37 N
48.4 N
x
62 N
4
34.2 lb
x
21.8 lb
110.6°
35.1 lb
5
81.0 lb
37 lb
29°15’
52 lb
x
6
79°2⬘
41.6 N
83.4 N
x
64.8 N
7
x
48.3 N
18°25’
29°40’
y
a 72.6 N
b 30.8 N
8
124.7 lb
78 lb
31.5°
y
x
a 71.0 lb
b 35.0°
9
10.2 mph
8 mph
y
x
12 mph
1 36 sq units
2 0.306
3 90°
4 8.7
5 16兹3 sq units
6 19.9
7 48
8 84兹3 in.2
9 13.16
10 (1) 24
11
11 (2)
18
12 (2) 2兹19
13 (3) 84兹2
29
14 (3)
36
s2
3
15 (2)
4 兹
16 (4) 0
17 (3) must be an obtuse triangle
n2
18 (2)
2
19 a 84.5 lb
b 27.6°
20 a 87.3°
b 83.9 ft2
c 4
21 68°
22 3.9 mi
23 10.6°
24 a 658.9 ft
b $607,279.90
25 a 5.5 兹2 in.
b Two of the angles are 11° too large, two
of the angles are 11° too small.
c 7.0 in.
a 122.78°
b 81.53°
Chapter Review
61
CHAPTER
Statistics
15.1 Gathering Data:
Univariate Statistics
(pages 488–489)
1 (4) shoppers at a mall
2 (1) people who received free samples of the
candy
3 (2) the post office
4 (3) the first fifty people encountered on a
city street
5 (3) calling the tenth person listed on each
page of the city telephone directory
15
5
6
7
8
9
10
In 6–12, explanations will vary.
6 controlled experiment
7 observation
8 sample survey
9 population survey
10 controlled experiment
11 simulation
12 sample survey
13 sample survey
14 sample survey
15 population survey
16 sample survey
17 sample survey
18 Answers will vary.
11
15.2 Measures of Central
Tendency
14
(pages 495–497)
1 (2) 74
2 (3) mean median
3 (4) 100
4 (2) 6
62
12
13
15
(4) mean mode
(3) 3.5
(3) 34
(3) an equal number of players are taller
than 6 feet 4 inches and shorter than
6 feet 4 inches
mean $39,858,881.50;
median $34,271,570.50; no mode
a mean $188; median $200;
mode $200
b Mode or mean; this value represents the
highest-priced gown that will still be
sellable to the majority of the girls.
a mean 222.81; median 222.35;
no mode
b The mean and median are both representative since the distribution of values
if fairly even on both sides of the mean.
c 7.6
a mean $20.40; median $19;
mode $17.50; Andrew should use the
mean as the “average” price for yard
work.
b His dad could counter with the mode,
saying “most” of the neighborhood pays
this lesser amount.
Answers will vary. Example: The median
age will not be as skewed by outliers such
as the very young and the very old.
a mean $922.61 million;
median $863.45 million
b The median is a better average since the
earnings of Gone With the Wind represent
an outlier that skews the mean
upward.
Answers will vary.
16
17
Arrange the data in ascending order. The
median is then either the middle number, if
the number of data values is odd, or the
average of the two middle numbers, if the
number of data values is even. Repeat this
procedure on the half of the data greater
than the median to find the third quartile
and again on the half of the data less than
the median to find the first quartile.
Changing any data value greater than the
median such that it continues to be greater
than the median will not affect the median.
Likewise, changing any data value less than
the median such that it continues to be less
than the median will not affect the median.
Any data value greater than the median
may be changed to be less than the median
without affecting the median itself so long
as a second data value less than the median
is changed to be greater than the median.
The median is also unaffected if a pair of
data points, one greater than the median
and one less than the median, is changed to
equal the median. Similarly, if a pair of data
values shares the value of the median, one
can be changed to be greater than the median and the other changed to be less than
the median without affecting the median. It
is also possible to change data values in a
list without changing the mode or median.
15.3 Measures of
Dispersion: Range,
Interquartile Range,
Variance, and Standard
Deviation
(pages 502–506)
1 (3) 10
2 (4) 4
3 (1) 256
4 (2) {14, 76, 56, 42, 86}
5 (3) 8
6 (2) 30
7 (3) there is no change
8 (2) 7.6
9 (3) 27.276
10 (1) be multiplied by 2
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
a 5
b 4
c 2
a 81.2
b 4.2
25
a 90
b 0
B; the data have smaller deviations from the
mean.
a 78.8 in.
b 16 in.
c median 80 in., Q1 75.5 in.,
Q3 82.5 in., IQR 7 in.
a mean 16, median 17, Q1 8.5,
Q3 21
b range 31, IQR 12.5
c Answers will vary. Either the mean, 68.6,
or the median, 70, would be a good
average.
a 53.84
b 10.31
a 190
b Maurice appears to be a less consistent
bowler than Monica. His interquartile
range is 50, so his range will be greater
than 50.
a 2:33:09.4
b 19 minutes, 27 seconds
160°F
5,344 ft
a 262.0
b 47.3
a 255.8 lb
b 46.6 lb
a 15.58
b 2.97
c Answers will vary. Examples: advertising agencies, concert promoters, vendors
a $6.67
b $4.12
a 3.1 in.
b 0.4 in.
15.4 Normal Distribution
(pages 510–512)
1 (3) 80.5
2 (2) 2.3
3 (3) 163
15.4 Normal Distribution
63
4
5
6
7
8
9
10
11
(3)
(4)
(2)
(1)
(2)
(4)
(1)
a
b
c
d
12
13
14
a
b
c
a
b
c
a
b
c
d
12.6
90
29,120
50
300
40
5
15 days
2 days
50%: 15 days; 85%: between 17 and 18
days; 98%: between 19 and 20 days
After 21 days, a person could be almost
completely certain (99.9%) that he or she
will not get the disease from the single
exposure.
68.2
97.7
191
32.3°
5.2°
(1) 32.3°
(2) 21.9°–42.7°
(3) close to 50%
(4) 24.5°
(5) 27.1° and 37.5°
x 20.8, ␴ 0.24
21.28
20.56 and 21.04
443,300
15.5 Bivariate Statistics,
Correlation Coefficients,
and the Line of Best Fit
(pages 517–518)
1 a 1
2 d 0.1
3 f 0.9
4 e 0.6
5 b 0.8
6 c 0.3
7 (4) 0.9
8 (4) If the correlation coefficient is negative,
the line of best fit has a negative slope.
9 (1) 0.89
10 (3) 0.15
64
15.6 Linear Regression
(pages 521–523)
1 (4) 0.9017
2 (3) y 1.65x 122.4
3 a y 806.4x 93892.2
b r 0.9965093864
c 99,537
d Answers will vary. Example: The model
is not a close fit for years beyond 1994;
the pattern of adoptions is changing.
4 a y 442.35x 743.27
b r 0.9597033424
c approximately 442
d 6,494
e Answers will vary. Example: Russian
government restrictions on foreign
adoptions slowed American adoptions
beginning in 2005.
f The model does not hold.
5 a
b y 217.14x 2,363.37
c r 0.991
d 1982: 9,312 stations; 2010: 15,392 stations
6 a
b y 0.526x 8.012
c r 0.9976339247; this means that there
is a strong positive correlation between
the year and the percentage of women
over 25 in the population who have
completed four or more years of college.
d 24.9%
e 2010: 29.1%; 2020: 34.3%; the greater the
extrapolation, the less accurate the
model.
f 1977
7
a
12
a
13
b y 10.17515021(0.8453967676)x
c 10.175 grams; the model does not exactly
fit all data points.
d 3.7145 grams
e It will never disappear entirely, but will
continue to get smaller and smaller.
a, c
b y 3.6469x 27.4449
c 63.9%
d 2009
15.7 Curve Fitting
(pages 527–529)
1 (4) y 2(1.7)x
2 (1) y 5(0.4)x
3 (1) y 20log x
4 (3) logarithmic
5 (3) 23
6 (4) 3.5
7 (1) linear
8 (2) 0
9 a Approximately 4.68 hours per day
b Approximately 5.77 hours per day
c 24.48 weeks after school starts
10 f(x): vii; the graph appears linear with a
positive slope and a positive y-intercept.
g(x): i; the graph appears logarithmic without any reflections.
h(x): iv; the graph appears exponential, and
since the y-values decrease and the
x-values increase, 0 x 1.
11 a
b
c
d
e
f
y 1,014.2434(1.0683)
6.83%
$5,647.37
$4,440.73
Keep the money where it is.
x
b logarithmic; the shape is comparable to a
logarithmic function.
c Approximately 26.8 weeks
15.8 More Curve Fitting
(pages 535–537)
1 (2) y 0.73x 2.969
2 (4) cubic
3 y 3.93218146x 2.762206751
4 a
b y 1.184961951x 0.4183021777
c Approximately 2.32 seconds
2p
d T
兹5 ⬇ 2.48 seconds, which is
兹32
only slightly longer than the time found
in part c.
15.8 More Curve Fitting
65
5
a
9
b Exponential: y 99,348.09(1.03)x
Power: y 99,764.55x 0.13
c Exponential: 2008; Power: 2024
6 a
a
b Exponential
c y 19.50607505(1.026824147)x
d y 0.0162946429x2 0.3163690476x 20.13333333
e Exponential: 50.59%; Quadratic: 52.64%
FYI
7
b y 7.81952384x 0.6644168582
c 102,382
d 2004
a
8
b y 11.414 22.308ln x
c y 13.489x 0.626
d The power regression more closely fits
the data.
e 85.3
a
b
c
d
e
f
66
y 9.136(1.072)x
y 0.006x3 0.028x2 0.058x 10.258
$22.50
$28.26
Exponential: December 2008; Cubic: July
2008
(page 539)
a y 22.22sin (0.53x 2.37) 55.42
b Midline 55.42. This represents the average
temperature in Central Park.
c Amplitude 22.22. This represents the
maximum variation in temperature from the
average.
1 (i) strong positive linear correlation
2 (iv) weak negative linear correlation
3 (ii) strong negative linear correlation
4 (v) linear correlation close to zero
5 (2) 30
6 (3) 83
7 (3) logarithmic
8 (4) 81.75–84.5
9 (1) 5
10 (3) 420
11 (3) 150
12 (1) 3.58
13 (3) y 3x
14 (4) 32.49
15 (2) ask drivers in the neighborhood
16 (2) median mean
17 (1) The median salary would not be
affected.
18 (3) multiplied by 兹2
19 (2)
20
21
(3) divided by 4
22
23
a
24
b C(t) 199.001t 28.934
c C(0) 28.93, C(8) 1620.94, C(18) 3610.95; these would be the expected
cost of tuition given the linear model.
d Answers will vary.
e C(t) 926.184(1.073)t
f C(0) 926.18, C(8) 1628.77, C(18) 3298.47
g Answers will vary.
h The exponential function more closely
fits the data.
i The linear function would mean a lower
tuition than the exponential function.
a
b
c y 2.398601399x 31.42715618
d Approximately 55.41 inches, close to the
actual data
e Approximately 65 inches
f y 2.236013986x 32.68822844
g Girls; according to the equations, they
grow approximately 2.399 inches per
year, while boys grow approximately
2.236 inches per year.
h Approximately 48.34 inches
i Approximately 79.64 inches; no, this
does not make sense because the
average 21-year-old male is not
6 feet 8 inches tall.
25 a 7.9
b 1.1
c 1.1
d
y
4.64 5.18 5.72 6.27 6.81 7.36 7.9 8.44 8.99 9.53 10.08 10.62 11.16
e 77
f 77%
g No, the mean and the median are different from each other. However, the data
are close to being normally distributed.
Chapter Review
67
x
26
a
b y 0.9995x0.6667
c 39.47 au
d The regression equation is approximately the same as the statement of
Kepler’s Third Law, i.e., x2 ky3 or
2
y cx 3.
27 a mean 81.79, median 85, mode 85
b No, the mean, median and mode are different from each other.
c v 91.45; 9.56
d 20
e If the scores were normally distributed,
68.2% of them should lie within one
standard deviation of the mean. As it
stands, 71.4% of the scores lie within one
standard deviation of the mean.
28 a y 0.2340282448x 0.4959650303
b Approximately 4.5 hours
c Approximately 12.83 pounds
d C(20) 5.1765299; it takes about 5 hours
to cook a 20-pound turkey.
e w 83.340517; a turkey weighing approximately 83.3 pounds needs to be
cooked for 20 hours.
f Part d makes sense because we have
20-pound turkeys. Part e does not make
sense because we don’t have turkeys
weighing 83 pounds. Additionally, we
don’t have enough data to know if the
trend holds for hypothetical turkeys of
that size.
68
29
a
30
b
c
d
e
f
a
Exponential
y 102.3521(1.0379)x
6,095,719
Answers will vary.
Answers will vary.
b Logarithmic or power
c Since log 1 0, the 15.3902 represents
the fuel efficiency when x 1, the year
1975.
d 1993
e approximately 29.958 mpg
f y 15.6226x0.1891
g 1993
h approximately 30.759 mpg
CHAPTER
Probability
16
16.1 Fundamental
Counting Principle
4
(pages 547–549)
1 (4) 12
2 (3) 72
3 (2) 105
4 (3) 720
5 (4) 192
6 (1) 36
7 (4) 20
8 (4) 1,440
9 (2) 192
10 (4) 420
11 480
12 a 42
b 35
13 432
14 24
15 144
16 12
17 150
18 9
5
6
8
9
10
11
12
13
14
15
16
17
18
16.2 Permutations
16.3 Combinations
(pages 554–556)
1 a 24
b 6
c 18
d 18
e 4: 4,267; 4,627; 6,247; 6,427
2 a 362,880
b 15,120
c 1,680
d 210
3 720
(pages 559–561)
1 (3) 15P7
2 (2) 495
3 (3) 3,003
4 (2) 35
5 (1) 66
6 (2) 560
7 (4) 58,212
8 (4) 600
9 (3) 840
10 (4) 11,088
7
a 11! ⫽ 39,916,800
b 10! ⫽ 3,628,800
c 9! ⫽ 362,880
5,040
a 840
b 1,680
c 210
a 210
b 6,720
c 180
d 34,650
e 3,780
(3) 30
(2) 60,480
(1) 12
(1) 7P1
(2) 6
(3) 120
(3) 6,720
(1) 552
(4) 210
(2) 24
(4) 58,500
16.3 Combinations
69
11
12
13
14
15
16
17
18
19
20
21
22
1,400
a 7C3 • 11C3
b 7C2 • 11C4
c 11C6
d 1 • 17C5
a Combination; 330
b Permutation; 24
Permutation; 56
a Combination; 120
b Permutation; 5,040
Combination; 324
a Combination; 55
b Combination; 36
c Permutation; 9! ⫽ 362,880
a Permutation; 120
b P: 72
c P: 72
d P: 54
a Permutation; 200P3
b Permutation; 199P2
10!
⫽ 151,200
Permutation;
3!2!2!
Combination; 6,000
a Combination; 5,400
b Combination; 1,080
c Permutation; 9! ⫽ 362,880
10
11
12
13
14
15
16
16.4 Probability
(pages 567–569)
1
1 (1)
12
1
2 (4)
1,776
3
3 (4)
28
1
4 (2)
220
1
5 (1)
33
y ⫺ x2
6 (3)
y
8C3 • 6C2
7 (1)
14C5
1
8 (4)
210
1
9 (1)
504
70
17
18
19
20
1
4
11 ⫺ 2p
(2)
11
5
a
8
3
b
4
1
a
28
1
b
28
3
c
7
1
5
2
a
15
2
b
15
28
c
45
7
d
15
9
16
29
8C6 ⫹ 6C6
⫽
134,596
24C6
C
⫹
C
•
C
13
3 3
3 2
12 1
⫽
455
15C3
1 • 15C4
5
⫽
16
16C5
1
1,320
(3)
16.5 The Binomial Theorem
(page 574)
1 z3 ⫹ 15z2 ⫹ 75z ⫹ 125
2 a4 ⫺ 12a3b ⫹ 54a2b2 ⫺ 108ab3 ⫹ 81b4
3 1,024p5 ⫹ 2,560p4 ⫹ 2,560p3 ⫹ 1,280p2 ⫹
320p ⫹ 32
4 41 ⫹ 38i
5 1,296x4 ⫺ 864x3 ⫹ 216x2 ⫺ 24x ⫹ 1
3 5
1 6
z ⫹
z ⫹ 15x 4 ⫹ 160z 3 ⫹ 960z 2 ⫹
6
64
4
3,072z ⫹ 4,096
7 ⫺86,016
8
9
10
11
12
13
14
15
16
17
18
19
20
4,860x4y2
540
1,176c2d2
(1) 10p 2
(3) ⫺8sin3 x
(2) ⫺720
(2) 256
(4) 216tan2 u
(4) 35
(4) 8C5(4x)3(3)5
(1) 84
(3) the middle term
(4) 160i
5
16.6 Binomial Probability
(pages 577–580)
3
1 a
8
1
b
8
3
c
8
2 a .189
b .02835
c .036756909
d .2401
25
3 a
216
5
b
324
125
c
324
1
d
1,296
625
e
1,296
1
4 a
8
1
b
2
1
c
2
21
d
512
5
e
16
1
f
16
6
1
8
1
b
6
1
c
3
1
d
8
1
e
4
21
f
512
4
g
9
15
h
1,024
25
i
216
1
j
4,096
Answering 3 out of 5 on the test where each
question has 5 choices has the greater probability, assuming you are guessing randomly.
1 4 3 1
P(4 of 5, 4 choices) ⫽ 5C4
4
4
15
⫽
⬇ .015
1,024
1 3 4 2
P(3 of 5, 5 choices) ⫽ 5C3
5
5
160
⫽
⬇ .051
3,125
125
(3)
216
4
(2)
9
3
(3)
8
(3) 10C1(.999)9(.001)
1 10
(3) 10
2
32
(2)
243
1
(1)
125
4 17 1 3
(4) 20C3
5
5
9
(4)
64
a
冢冣冢冣
冢冣冢冣
7
8
9
10
11
12
13
14
15
冢冣
冢冣冢冣
16.6 Binomial Probability
71
16
17
18
19
20
216
625
108
(3)
343
(3)
16
冢冣冢冣
1 2 5 4
6
6
4
(1) 5C1(.99) (.01)1
16
(1)
625
(3) 6C2
17
16.7 At Least or at Most r
Successes in n Trials
(pages 582–584)
19
1
144
15
2
16
5
3
16
15
4
16
16
5
27
6 (2) 1 ⫺ 2x
3,125
7 (3)
8,192
14,375
8 (4)
16,807
215
9 (4)
216
5
10 (2)
16
11 (4) .973
12 (1) .94
13 (3) .457
189
14 (3)
256
15 The probability of randomly guessing the
correct answer on a given question is .25 for
the first-period class and .5 for the fifthperiod class. Therefore, students who randomly guess will tend to do better on the
fifth-period quiz.
72
18
19
20
This is a wise strategy. Without the marketing strategy, the resort would have 120 fullprice stays for February. The probability
that there is snow on at least 2 days of a
20
3-day stay is
. That means that the resort
27
can expect 115 full-price stays and 20 halfprice stays, which is the same as 125 fullprice stays.
2
a
27
3,773
b
4,096
13
c
16
20
a
27
19
b
27
5
a
16
1
b
2
a 56
b 120
c 24
16.8 Normal Approximation
to the Binomial Distribution
(page 586)
1 .052
2 .029
3 .922
4 .026
5 .804
6 a .215
b .207
c The normal approximation slightly underestimates the probability that the coin
will land on heads.
d .448
FYI
(page 587)
a .3292
b .4219
c .2995
d .9997
e .0579
1 (4) 720
2 (3) 960
3 (4) 24x2y2
1
4 (1)
2
67
5 (2)
256
6 (1) 28
2x
7 (2)
2x ⫹ 1
8 (4) ⫺20a3b3i
9 (4) 1,320
10 (4) .015625
11 (2) cos3 u
98
12 (1)
125
13 (2) 72
14 (1) ⫺40
297
15 (4)
625
16 (4) 54a2b2
27
17 (3)
64
2,125
18 (3)
4,096
19 (3) ⫺2,035 ⫹ 828i
347
20 (2)
2,048
6,250
21 (2)
16,807
32
22 (1)
243
23 (3) 336
24 a 19C5
b 8C2 • 11C3
C • C
c 8 2 11 3
19C5
1 • 8C2 • 10C2
d
19C5
25 a3 ⫺ 6a2b ⫹ 12ab2 ⫺ 8b3
26 a 0.6561
b 0.0001
c 0.0037
d 0.9999
27
16
28
a
b
c
d
29
a
b
c
d
1
5
2
5
12
125
98
125
4
13
2
13
冢冣冢冣
11
2
C冢冣冢冣⫹ C冢冣冢冣
13
13
11
2
⫹ C冢冣冢冣
13
13
12
1
12
1
C 冢冣冢冣 ⫹ C 冢冣冢冣
13
13
13
13
23
26
11
13
3C2
3
1
2
2
3
26
2
13
3
0
3
3
32
33
0
3
3
3
2
3
⫺7 ⫹ 24i
99
31 a 6C1
100
99
b 10C0
100
30
2
2
2
e
1
3
冢冣冢冣
冢冣冢冣
5
1 1
100
10
1
100
0
c 1⫺
99
99
冣冢 1001 冣 ⫹ C 冢 100
冣冢 1001 冣冥
冤 C 冢 100
d
10C1
冢冣冢冣
a
b
c
d
a
b
c
.552
.186
.138
.990
.878
.199
.344
7
8
1
1
99
100
8
8
9
1
100
1
0
0
冢冣冢冣
⫹ 10C0
99
100
10
Chapter Review
1
100
0
73
Cumulative Reviews
Each review has a total of 48 points. Use the following chart, adapted from the Regents Examinations,
to convert the student’s raw score to a scaled score.
Raw Score
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
Raw Score
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
Scaled Score
100
99
97
96
94
92
91
89
87
86
84
82
80
79
77
76
74
Raw Score
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Scaled Score
72
70
68
67
65
63
61
60
58
56
54
53
51
49
47
46
44
Scaled Score
42
41
39
37
35
34
32
30
28
26
21
16
10
5
0
Chapters 1–2
(pages 592–594)
Part I
Allow a total of 26 credits, 2 credits for each of the following. Allow credit if the student has written the correct
answer instead of the numeral 1, 2, 3, or 4.
1 (2) {4, 6}
2 (1) x2 ⫹ 2x ⬍ 8
3 (3) {0, 4}
4 (2) ⫺1 or 1
5 (4) x3 ⫹ 3x2 ⫹ 3x ⫹ 1
2x ⫹ 1
6 (3) ⫺
2⫹x
74
Cumulative Reviews
冦冧
1
,1
2
8 (4) 2a ⫹ 3b
9 (2) Its factors are
(2x ⫹ 3)(x ⫹ 2)(x ⫺ 2).
4x ⫹ 3
10 (2) 2
x ⫺ 3x
7
(3)
11
(2)
–1
12
13
0
(3) x2 ⫹ 15x ⫹ 17
x⫺1
(2)
2
3
2
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
14
15
16
17
Score
Explanation
2
For a fraction to be undefined, the denominator must equal zero. Since x2 ⱖ 0; the
denominator x2 ⫹ 3 is always greater than 0.
1
Student attempts substitution of various values.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
Score
Explanation
x⫹2
x⫹2
miles, while Efrim traveled
3
4
miles. Dividing the same number by a smaller value produces a larger quotient.
2
Cody traveled farther. Cody traveled
1
Cody, but no explanation is given.
0
Score
Explanation
2
x ⫽ 5 and appropriate work is shown.
1
Appropriate work is shown but one computational error is made or
1
x ⫽ 5 but no work is shown.
0
Score
Explanation
2
x ⫽ ⫺1,
7
and appropriate work is shown.
3
1
x ⫽ ⫺1,
7
but no work is shown or
3
1
Only one solution is correct with appropriate work shown.
0
Chapters 1–2
75
Part III
18
19
Score
Explanation
4
x ⫽ 2, 6 and appropriate work is shown.
3
Appropriate work is shown with one computational error.
2
Appropriate work is shown with two computational errors or
2
Only one solution is correct with appropriate work shown.
1
x ⫽ 2, 6 but no work is shown.
0
Score
Explanation
冢
冣
4
x⫹4
1
, x ⫽ ⫺2, ⫺ , 0, 7 and appropriate work is shown.
x
2
3
An appropriate method is used but one factoring or computational error is made.
2
An appropriate method is used but two computational errors are made.
1
Multiple factoring or mathematical errors are made or
1
x⫹4
but no work is shown.
x
0
Part IV
Use the specific criteria to award a maximum of 6 credits.
20
76
Score
Explanation
6
x ⫽ ⫺1, 4 and appropriate work is shown.
5
Appropriate work is shown, but one computational error is made.
4
Appropriate work is shown, but one conceptual error is made.
3
Only one solution is given with appropriate work shown.
2
Multiple errors are made in the algebraic solution.
1
x ⫽ ⫺1, 4 but no work is shown
1
Student shows minimal understanding/correct algebra.
0
Cumulative Reviews
or
Chapters 1–3
(pages 595–596)
Part I
1
2
3
4
5
6 (1) 1
7 (4) 3a4b4 兹2b
8 (1) ⫺7 ⬍ x ⬍ 3
9 (3) 7兹2
10 (2) 7 only
(3) 3 ⫺ 兹5
1
(1)
x⫹y
(3) {⫺1, 4}
(4) ⫺1
(4) 4
11
12
13
2兹5 ⫹ 5兹2
10
(1) 0 ⬍ x ⬍ 6
(3) 3
(4)
Part II
14
Score
2
5x
x⫹1
1
Appropriate work is shown, but one computational error is made or
1
5x
x⫹1
0
15
16
Explanation
Score
Explanation
2
y ⫽ ⫺1 and appropriate work is shown.
1
1
Appropriate work is shown, but the solution y ⫽ ⫺
1
y ⫽ ⫺1 but no work is shown.
0
Score
5
is not rejected or
2
Explanation
2
1
1
Appropriate work is shown, but the negative solution is not rejected or
1
0
Chapters 1–3
77
17
Score
Explanation
2
冤⫺ 32 , 5冥 or ⫺ 32 ⱕ x ⱕ 5 and appropriate work is shown.
1
1
Student correctly solves for the related equality
1
冤⫺ 32 , 5冥 or ⫺ 32 ⱕ x ⱕ 5 but no work is shown.
0
or
Part III
18
78
Score
Explanation
4
x ⫽ 2 and appropriate work is shown, such as correctly solving the equation
兹x ⫹ 14 ⫽ x ⫹ 2 algebraically or graphically.
3
2
Appropriate work is shown, but two or more computational errors are made or
2
Appropriate work is shown, but one conceptual error is made, such as incorrectly
squaring both sides.
1
Appropriate work is shown, but one conceptual error and one computational error
are made or
1
0
Cumulative Reviews
19
Score
Explanation
4
a 兩 t ⫺ 110 兩 ⬍ 20
b 90 ⬍ t ⬍ 130 and appropriate work is shown, such as correctly solving the
inequality 兩 t ⫺ 110 兩 ⬍ 20 algebraically or graphically.
3
2
2
Appropriate work is shown, but one conceptual error is made when solving the
inequality or
2
An incorrect, but similar, inequality is given and is solved correctly.
1
are made or
1
兩 t ⫺ 110 兩 ⬍ 20 and 90 ⬍ t ⬍ 130 but no work is shown.
0
Part IV
20
Score
Explanation
6
5
4
Appropriate work is shown, but student does not eliminate extraneous root.
3
A correct equation is solved but only one solution is given.
2
1
x ⫽ 3 but no work is shown or
1
Student shows minimal understanding of quadratics/correct algebra.
0
Chapters 1–3
79
Chapters 1–4
(pages 597–598)
Part I
1 (2) 6兹5
2 (3) {0, 2}
3 (4) y2 ⫺ 3y ⫺ 2 ⫽ x
4 (2) 兩 140 ⫺ t 兩 ⬍ 5
5 (3) $210
6 (4) all real numbers except
⫾3
7 (1) 1
8
(3) f ⫺1(x) ⫽
冪
x⫹1
3
13
(4)
9 (4) 10 ⫺ 5兹3
10 (1) 120 ⫹ 31x兹3 ⫹ 6x 2
11 (2) (x ⫺ 1)(x ⫹ 1)(2x ⫹ 7)
12 (4) {0, 3}
Part II
14
15
16
80
Score
Explanation
2
7兹6 and appropriate simplification of radicals is shown.
1
One term is correctly simplified; others are not.
0
Score
Explanation
2
Explanations will vary but should mention the use of the conjugate of the
denominator. The result of the process should be ⫺10 ⫺ 5兹7.
1
A correct explanation or a correct solution is given but not both.
0
Score
Explanation
2
(x ⫹ 3)2 ⫹ (y ⫺ 8)2 ⫽ 25
1
One term in the equation is incorrect.
0
Cumulative Reviews
17
Score
Explanation
2
x ⬍ ⫺2 or x ⬎ 2; these values cause the denominator of the fraction to be a real,
nonzero number. Values between ⫺2 and 2 would produce a fraction that is
undefined.
1
Correct domain but no explanation given or
1
Correct explanation but incorrect domain is given.
0
Part III
18
Score
冢
冣 and appropriate work is shown.
4
x⫺4
3 1
, x ⫽ ⫺4, ⫺ ,
2
2 3
3
An appropriate method is used but one factoring error is made.
2
Appropriate work is shown with two factoring or algebraic errors or
2
Solution is not simplified.
1
0
19
Explanation
Score
x⫺4
2
Explanation
4
9x2 ⫺ 18x ⫺ 3 and appropriate work is shown.
3
An appropriate method is used but one computational error is made.
2
An appropriate method is used but two computational errors are made or
2
An appropriate method is used but a conceptual error, such as the incorrect order of
composition, is made.
1
1
9x2 ⫺ 18x ⫺ 3 but no work is shown.
0
Chapters 1–4
81
Part IV
20
Score
Explanation
冢
冣
6
3 ⫺ 2x
5 3
, x ⫽ ⫺4, ⫺ , , 4 and appropriate work is shown.
2
3 2
5
Appropriate work is shown, but one factoring/algebraic error is made.
4
Appropriate work is shown, but two factoring/algebraic errors are made.
3
Some appropriate work is shown, but the factor of ⫺1 is not used.
2
1
Student shows minimal understanding/correct algebra.
0
Chapters 1–5
(pages 599–600)
Part I
1
2
3
4
5
6
(3)
(3)
(4)
(1)
(1)
(2)
4 ⫹ 兹15
⫺5 ⫹ 12i
7
11
{x : ⫺1 ⬍ x ⬍ 5}
⫺1
a
a⫹1
8 (4) ⫺3 ⬍ x ⬍ 3
9 (4) (2, 7)
2
1
x⫺
10 (4)
3
3
7
(4)
11
12
13
(2) ⫺5 ⫺ 5i
(3) real, irrational, and
unequal
(2) y ⫽ 3
Part II
14
82
Score
Explanation
2
34 ⫺ 42i and appropriate work is shown.
1
1
Appropriate work is shown, but one simplification error is made.
0
Cumulative Reviews
15
16
17
Score
Explanation
2
1
1
Appropriate work is shown, but one conceptual error is made, such as not rejecting
x ⫽ ⫺1 or
1
x ⫽ 2, but no work is shown.
0
Score
Explanation
2
1
0
Score
Explanation
2
Mary Lu is 4 years old and Jane is 8 years old, and appropriate work is shown.
1
1
Appropriate work is shown, but student answers Mary Lu is 8 years old and Jane is 4
years old or
1
Mary Lu is 4 years old and Jane is 8 years old, but no work is shown.
0
Chapters 1–5
83
Part III
18
Score
4
Explanation
a, b
yi
9
7 + 8i
8
7
6
5
2 + 5i
4
3
5 + 3i
2
1
–3 –2 –1 O 1
–1
–2
–3
c
19
84
2
3
4
5
6
7
8
9
x
7 ⫹ 8i, and appropriate work is shown.
3
Appropriate work is shown, but one computational or graphing error is made.
2
Appropriate work is shown, but two or more computational or graphing errors are
made or
2
1
Appropriate work is shown, but one conceptual error and one computational or
graphing error are made or
1
7 ⫹ 8i but no work is shown.
0
Score
Explanation
4
x ⫽ ⫺3, x ⫽ 6 and appropriate work is shown.
3
3
Appropriate work is shown, but only one answer is given.
2
2
Appropriate work is shown, but one conceptual error is made, such as not finding
an appropriate common denominator.
1
are made or
1
x ⫽ ⫺3, x ⫽ 6 but no work is shown.
0
Cumulative Reviews
Part IV
20
Score
Explanation
6
{(⫺7, ⫺25), (5, 11)} and appropriate work is shown.
5
4
3
A correct equation is solved, but only one solution is given or
3
{(⫺7, ⫺25), (5, 11)}, but one solution is achieved without using algebra.
2
1
Student shows minimal understanding/correct algebra
1
{(⫺7, ⫺25), (5, 11)} but no work is shown.
0
or
Chapters 1–6
(pages 601–603)
Part I
1
2
3
4
5
6
7
(2) ⫺14
(1)
8
9
(3)
(1)
(4)
(4)
(1)
113.6
10
36
82
The product is a real,
rational number.
1
(4)n
(2)
2
(4) 4
10
11
12
13
3
4
3 ⫹ 24i
(2)
13
(1) 1 ⫾ 2i
(2) 2
(3) ⫾
Part II
14
Score
Explanation
2
second root: 3 ⫺ 2i; equation: x2 ⫺ 6x ⫹ 13 ⫽ 0 and appropriate work is shown.
1
The second root or the equation is correct and appropriate work is shown, but the
other is not or
1
3 ⫺ 2i; x2 ⫺ 6x ⫹ 13 ⫽ 0 but no work is shown.
0
Chapters 1–6
85
15
16
17
Score
Explanation
2
Two real, unequal, irrational roots; explanations will vary but should include
mention of the discriminant, 40.
1
Correct description of roots or appropriate explanation, but not both.
0
Score
Explanation
2
1
One or more errors are made in the solution of the equation or
1
The equation is solved correctly, but the extraneous root is not discarded.
0
Score
Explanation
1
⫾ 2i and appropriate work is shown.
2
2
x⫽
1
Equation is correctly solved but solution is not simplified
1
Computational errors are made in the solution of the quadratic equation or
1
x⫽
0
or
1
⫾ 2i, but no work is shown.
2
Part III
18
86
Score
Explanation
4
sum ⫽ 399; 5th term: ⫺486; 6th term: 729, and appropriate work is shown.
3
Two of the three answers are correct and appropriate work is shown.
2
Appropriate work is shown with two formula or computational errors.
1
Only one of the three solutions is correct
1
sum ⫽ 399; 5th term: ⫺486; 6th term: 729 but no work is shown.
0
Cumulative Reviews
or
19
Score
Explanation
4
$2,800,000 in 2009, and appropriate work or graph is shown.
3
2009 or $2,800,000 is correct and appropriate work or graph is shown.
2
An appropriate method is used, but two computational errors are made.
1
Multiple errors are made
1
$2,800,000 in 2009, but no work is shown.
0
or
Part IV
20
Score
Explanation
6
a an ⫽ 96(0.6)n⫺1 and appropriate work is shown.
b 6 swings and appropriate work is shown.
5
The equation is correct and appropriate work is shown, but the number of swings is
incorrect.
4
An appropriate formula is used, but one computational error is made.
3
Student uses arithmetic sequence formulas but does so incorrectly.
2
1
Student shows minimal understanding of geometric sequences/correct algebra
1
an ⫽ 96(0.6)n⫺1 and 6 swings but no work is shown.
0
or
Chapters 1–7
(pages 604–606)
Part I
(1)
(2)
(3)
(4)
(2)
(4)
I
2
760
16
2a2b⫺2
(⫺4, ⫺1)
7
(3)
8
n
(3)
9 (4)
10 (1)
11 (2)
Game Owners
1
2
3
4
5
6
t
12
(4)
13
(4)
1
3
(x ⫺ 3)2 ⫹ (y ⫹ 6)2 ⫽ 25
I and II
x 2 ⫹ 2x ⫺ 24 ⱕ 0
1
2
{8}
Time
Chapters 1–7
87
Part II
14
15
16
17
88
Score
Explanation
2
29 and appropriate work is shown.
1
1
Student correctly multiplies by an incorrect conjugate with appropriate work shown or
1
29, but no work is shown.
0
Score
Explanation
2
domain: All real numbers or (⫺⬁, ⬁); range: {y : y ⱖ 3} or [3, ⬁), and appropriate
work is shown.
1
1
Appropriate work is shown, but only one correct answer is given.
0
Score
Explanation
2
1
1
Student sets the exponents 2x and x ⫺ 1 equal to each other and solves correctly
with appropriate work shown or
1
0
Score
Explanation
2
1
x ⫽ ⫺ , x ⫽ 3 and appropriate work is shown.
3
1
1
1
x ⫽ ⫺ , x ⫽ 3 but no work is shown or
3
1
0
Cumulative Reviews
Part III
18
19
Score
Explanation
4
x ⫽ 0, x ⫽ ⫺3 ⫹ 4i, x ⫽ ⫺3 ⫺ 4i, and appropriate work is shown, such as solving
the cubic equation by factoring and using the quadratic formula or completing the
square.
3
2
2
Appropriate work is shown, but one conceptual error is made, such as not including
the complex roots in the solution.
1
are made or
1
0
Score
Explanation
4
a f(x) ⫽ 3x ⫹ 2
b f(x) ⫽ 3x⫺3
3
One of the equations f(x) ⫽ 3x ⫹ 2, f(x) ⫽ 3x⫺3 is correct, but the other one has an
error, such as an error in the sign of the constant.
2
One of the equations f(x) ⫽ 3x ⫹ 2, f(x) ⫽ 3x⫺3 is correct, but the other equation is
completely wrong or
2
Both of the equations f(x) ⫽ 3x ⫹ 2, f(x) ⫽ 3x⫺3 have errors in the signs of the
constant.
1
One of the equations f(x) ⫽ 3x ⫹ 2, f(x) ⫽ 3x⫺3 is wrong, and the other one has an
error, such as an error in the sign of the constant.
0
Chapters 1–7
89
Part IV
20
Score
Explanation
6
a
5
Appropriate work is shown, but one computational or rounding error is made.
4
Appropriate work is shown, but only one of the explanations given for the meaning
of N(4) or N(t) ⫽ 4 is correct or
4
3
Only one of the evaluations of N(4) or N(t) ⫽ 4 is correct, along with its explanation.
2
1
Student shows minimal understanding of exponents/logarithms.
0
N(4) ⫽ 5,247,000; four months after being introduced, the show has
approximately 5,247,000 viewers.
b t ⫽ 2.058; the show can be expected to have 4,000,000 viewers slightly more than
2 months after first airing.
Chapters 1–8
(pages 607–608)
Part I
1
2
3
4
(4) imaginary
3
(1)
2
c⫺2
(3)
c
(3) are the same graph
5
6
7
8
9
(2)
(4)
(2)
(3)
2
It is a complex number.
{5}
m ⫹ 2p
1
(1) 2x 2 ⫺ 5x ⫽
2
10
(3) 3
11
(3) ⫺
12
13
(4) 4
(4) y ⫽ 1,000(1.04)18
64
81
Part II
14
90
Score
Explanation
2
4x2 ⫹ 4x ⫺ 3 and appropriate work is shown.
1
4x2 ⫹ 4x ⫺ 3, but no work is shown
1
4x2 ⫹ 8x ⫹ 9; student performed composition in wrong order or
1
One computational error is made in performing the composition.
0
Cumulative Reviews
or
15
16
17
Score
Explanation
2
The population is decreasing by (1 ⫺ 0.7882)% or 21.18% yearly. 17,432 is the initial
population in 2009, and 0.7882 represents the decay in population.
1
One of the two explanations is incorrect.
0
Score
Explanation
冢
冣
1
, 2 , and appropriate work is shown.
2
2
x, x ⫽ 0,
1
Appropriate work is shown but one factoring or computational error is made or
1
x, but no work is shown.
0
Score
Explanation
2
1
1
⫾ i and appropriate work is shown.
3
3
1
1
1
⫾ i, but no work is shown or
3
3
1
Equation is solved correctly but solution is not simplified.
0
Chapters 1–8
91
Part III
18
Score
4
Explanation
a, b
f –1(x) = 2x
f(x) = log2 x
19
92
3
Function and inverse are graphed over the wrong domain or
3
The function is graphed correctly but there is one error on the inverse.
2
Only the original function is graphed correctly or
2
The graph of the function is correct but the student does not graph the inverse or
2
The graph of the function is incorrect but the correct inverse of the student’s graph
is shown.
1
Student demonstrates minimal understanding of functions and their graphs.
0
Score
Explanation
4
a 2.1425 and appropriate work is shown.
b 0.643 and appropriate work is shown.
3
One solution is correct but one computational error is made in the other.
2
Appropriate use of logarithms is shown, but two computational errors are made.
1
Multiple computational errors are made
1
Minimal understanding of logarithms is displayed
1
2.1425 and 0.643, but no work is shown.
0
Cumulative Reviews
or
or
Part IV
20
Score
Explanation
6
6.3 hours and appropriate work is shown.
5
Appropriate work is shown, but a rounding or computational error is made.
4
3
Student expresses 6.3 without a unit of measure or with an incorrect unit of
measure, such as milligrams.
2
1
Student shows minimal understanding of appropriate methodology or
1
6.3 hours, but no work is shown.
0
Chapters 1–9
(pages 609–610)
Part I
1 (1) 0°
2 (3) 73
3 (3) 28i
4 (4) imaginary
5 (4) 4,882,812
6
7
8
9
10
(4)
(2)
(3)
(1)
(3)
15
330°
8
⫺6
210°
11 (1) {1}
12 (4) y ⫽ log2 x
13 (3) 8
Part II
14
Score
Explanation
2
10x ⫹ 26
x2 ⫺ 9
1
1
Student adds length and width correctly, and appropriate work is shown or
1
10x ⫹ 26
x2 ⫺ 9
0
Chapters 1–9
93
15
16
17
94
Score
Explanation
2
k ⫽ 3 and appropriate work is shown.
1
1
k ⫽ 3 but no work is shown.
0
Score
Explanation
xy 2
z
2
log
1
1
Appropriate work is shown, but one incorrect logarithmic rule is used or
1
log
0
Score
xy 2
z
Explanation
2
1
1
0
Cumulative Reviews
Part III
18
19
Score
Explanation
4
15 hours and appropriate work is shown.
3
2
2
Student uses an incorrect equation using the given information, but solves it
correctly.
1
are made or
1
15 hours, but no work is shown.
0
Score
Explanation
4
⫺1 and appropriate work is shown.
3
Appropriate work is shown, but one computational or factoring error is made.
2
Appropriate work is shown, but two or more computational or factoring errors are
made or
2
1
are made or
1
⫺1 but no work is shown.
0
Chapters 1–9
95
Part IV
20
Score
Explanation
6
a A(t) ⫽ 500(0.98)t
b A(50) ⫽ 182.08 grams and appropriate work is shown.
c 34.31 years and appropriate work is shown.
5
Appropriate work is shown, but one or more rounding error is made or
5
The correct equation is solved incorrectly, and that answer is used correctly for the
other parts of the problem.
4
An incorrect equation is solved correctly, and that answer is used correctly for the
other parts of the problem or
4
3
The initial equation is written correctly, but no other work is correct.
2
Only one of the three required answers is correct, and no work is shown for the
other two parts.
1
A(t) ⫽ 500(0.98)t; A(50) ⫽ 182.08; approximately 34.31 years, and no work is
shown or
1
Student shows minimal understanding of exponential functions.
0
Chapters 1–10
(pages 611–612)
Part I
1
2
3
4
5
96
(3) y ⫽ 兹1 ⫺ x 2
125
(2)
16
(1) 1
(4) $120.00
(4) 315°
Cumulative Reviews
(2) 兩 58 ⫺ h 兩 ⬍ 4
11p
7 (3)
6
8 (4) 12
9 (2) a ⫹ 3b
10 (1) 4x2 ⫺ 4x ⫹ 17 ⫽ 0
6
11
12
13
(1) 42°, 48°
(2) 3.04
(2) 2
Part II
14
15
16
17
Score
Explanation
2
An appropriate explanation is given, such as the sine function is represented by the
y-coordinate on the unit circle and in the y-coordinate is positive in Quadrants I and II.
1
An incomplete explanation is given.
0
Score
Explanation
2
16.1 years, and appropriate work is shown.
1
16.1 years and no work is shown
1
0
Score
or
Explanation
2
S(30) ⫽ 47,531.44, which will be Kelly Ann’s salary after having worked at the job
for three years.
1
1
S(30) ⫽ 47,531.44, but no work is shown or no explanation is provided.
0
Score
Explanation
2
1
Graph of the parabola is correct but no shading or incorrect shading is drawn or
1
Shading is appropriate, but there is an error on the parabola.
0
Chapters 1–10
97
Part III
18
19
Score
Explanation
4
a1 ⫽ 20; d ⫽ 7, and appropriate work is shown.
3
2
Only the common difference or the first term is correct.
1
a1 ⫽ 20; d ⫽ 7, but no work is shown.
0
Score
Explanation
1
2
4
x⫽⫺
3
An appropriate method is used, but one computational error is made.
2
1
Only minimal understanding of logarithmic equations is displayed
1
x⫽⫺
0
or
1
2
Part IV
20
98
Score
Explanation
6
a $12,845.16 and appropriate work is shown.
b $20,911.84 and appropriate work is shown.
5
A correct exponential equation is shown, but one computational or rounding error is
made.
4
Appropriate work is shown, but one conceptual error is made or
4
Appropriate work is shown, but two computational or rounding errors are made.
3
An incorrect equation is solved correctly.
2
1
Student shows minimal understanding of exponential functions.
0
Cumulative Reviews
Chapters 1–11
(pages 613–615)
Part I
2
3
4
(3) 5 ⫹ 6i
兹3 , 1
(3)
2 2
(4) 兩 32 ⫺ t 兩 ⬍ 2
(2) ⫺1
冢
冣
(1)
5
6
d
Distance
1
t
Time
(2)
兹2
2
7 (4) ⫺2 ⬍ y ⬍ 4
8 (1) 20.5
3
9 (1) ⫺
2
10 (1) a
11 (1) {⫺5, 5}
5
12 (2) ⫺
4
13 (2) y ⫽ sec x
Part II
14
Score
2
x 2 ⫺ 3x ⫹ 2
x2 ⫺ x ⫺ 6
1
1
x 2 ⫺ 3x ⫹ 2
x2 ⫺ x ⫺ 6
0
15
Explanation
Score
Explanation
2
150 people, and appropriate work is shown.
1
1
150 people but no work is shown.
0
Chapters 1–11
99
16
17
Score
Explanation
2
x ⫽ ⫺3, x ⫽ 6, and appropriate work is shown.
1
1
x ⫽ ⫺3, x ⫽ 6, but no work is shown
1
0
Score
or
Explanation
2
50° and appropriate work is shown.
1
1
Appropriate work is shown, but the coterminal angle given is not the smallest
positive acute angle or
1
50°, but no work is shown.
0
Part III
18
100
Score
Explanation
4
5.7 feet and appropriate work is shown, either algebraic or graphical.
3
2
Appropriate work is shown, but two or more computational or rounding errors are
made or
2
1
rounding error are made or
1
5.7 feet, but no work is shown.
0
Cumulative Reviews
19
Score
Explanation
4
25 indicates that Mathland had a population of 25,000 when it was founded; 1.03
indicates that the population is growing at a rate of 3% per year; 46.9 years, and
appropriate work is shown.
3
Appropriate work is shown, but one computational or rounding error is made or
3
Appropriate work is shown, but one of the interpretations of 25 and 1.03 is incorrect.
2
made or
2
Appropriate work is shown, but both of the interpretations of the 25 and 1.03 are
incorrect.
1
Appropriate work is shown, but one conceptual and one computational or rounding
error are made or
1
46.9 years, but no work is shown.
0
Part IV
20
Score
Explanation
6
h(t) ⫽ 1.5cos
冢 p6 t冣 ⫹ 2.1 or h(t) ⫽ ⫺1.5sin 冢 p6 (x ⫺ 3)冣 ⫹ 2.1; 4.6 hours and
appropriate work/graph is shown.
5
The amplitude, frequency, or midline is incorrect, but the resulting hour is correct
for the student’s equation or
5
The equation is incorrectly rounded, but the resulting hour is correct for the
incorrect equation.
4
The equation is correct, but no other work is done
4
Two of the values for amplitude, frequency, or midline are incorrect, and the
resulting hour is correct for the student’s equation.
3
Two of the values of amplitude, frequency, or midline are incorrect, and the resulting
hour is also incorrect for the student’s equation.
2
None of the values for amplitude, frequency, or midline are correct; however, the
resulting hour is correct for the student’s equation.
1
h(t) ⫽ 1.5cos
or
冢 p6 t冣 ⫹ 2.1 or h(t) ⫽ ⫺1.5sin 冢 p6 (x ⫺ 3)冣 ⫹ 2.1; 4.6 hours, but no
work/graph is shown or
1
Student shows minimal understanding of trigonometric graphs.
0
Chapters 1–11
101
Chapters 1–12
(pages 616–618)
Part I
1
(2) y ⫽ 3cos
2
(3)
3
4
1
x⫺1
2
7p
6
(2) x ⫽ 3, y ⫽ 6
(3) 14
5
6
7
8
9
(3) 14.2 hundred thousand
dollars
(2) 2
(4) 6.8%
(1) 1
(2) {6}
10
11
12
13
(1)
(3)
(3)
(1)
1
500
52
⫺30°
Part II
14
15
16
102
Score
Explanation
2
82 and 100 and appropriate work is shown.
1
82 and 100, but no work is shown
1
Only one of the two arithmetic means is correct.
0
Score
or
Explanation
2
y ⫽ 2(x ⫺ 2)2 ⫹ 5 and appropriate work is shown.
1
y ⫽ 2(x ⫺ 2)2 ⫹ 5, but no work is shown
1
Appropriate work is shown, but errors are made in completing the square.
0
Score
or
Explanation
2
July 1; 28 blankets sold, and appropriate work is shown.
1
Only one of the two required answers is correct
1
Appropriate work is shown, but computational errors are made or
1
July 1; 28 blankets sold, but no work is shown.
0
Cumulative Reviews
or
17
Score
Explanation
2
2
3
1
Appropriate work is shown, but computational errors are made or
1
2
3
0
Part III
18
19
Score
Explanation
4
a 3.25 seconds and appropriate work is shown.
b 181 feet and appropriate work is shown.
3
Appropriate work is shown, but one rounding or computational error is made.
2
Appropriate work is shown, but two computational or rounding errors are made or
2
Only time is correct or distance is correct but not both.
1
3.25 seconds and 181 feet, but no work is shown
1
Minimal understanding of quadratic functions is shown.
0
Score
or
Explanation
4
x ⫽ 16.034 and appropriate work is shown.
3
Logarithmic equation is correctly displayed, but one computational error is made.
2
Logarithms are used, but two computational or rounding errors are made or
2
x ⫽ 16.034, but the problem is not solved using logarithms.
1
Multiple computational errors are made
1
x ⫽ 16.034, but no work is shown.
0
or
Chapters 1–12
103
Part IV
Score
20
6
Explanation
a
The sequence is geometric because it has a constant ratio, r ⫽
b a n ⫽ 324
c
冢冣
2
3
2
.
3
n⫺1
2,660
2
or 866 and appropriate work is shown.
3
3
5
5
Appropriate work is shown, but no explanation is provided for the type of sequence.
4
Appropriate work is shown, but only two of the solutions are correct.
3
Student solves for an arithmetic sequences but does so incorrectly or
3
Student identifies the sequence as geometric, but no explanation is provided, and
solves only one of the other parts of the question.
2
a n ⫽ 324
2
1
Student shows minimal understanding of sequences.
0
冢冣
2
3
n⫺1
and
2
2,660
or 866 , but no work is shown or
3
3
Chapters 1–13
(pages 619–620)
Part I
1
2
3
4
(2) 2
(2) ⫺7 ⫹ 24i
p
(2)
2
a2 ⫹ 2a ⫹ 2
(2)
a⫹2
104
Cumulative Reviews
(4) 2cos u
(1) 32
3p
7 (3)
4
8 (3) N(t) ⫽ 5(1.08)t
9 (2) 44.444
5
6
(1) 512
3p
11 (4)
2
12 (2) P(t) ⫽ 250e1.3t
13 (3) 3
10
Part II
14
15
16
17
Score
Explanation
2
1 ⫾ 3i and appropriate work is shown.
1
1
1 ⫾ 3i, but no work is shown.
0
Score
Explanation
2
⫺2 ⬍ n ⬍ 6 and appropriate work is shown.
1
1
Student solves related equality correctly and appropriate work is shown or
1
Appropriate work is shown, but only one portion of the correct inequality is given.
0
Score
Explanation
2
Moisha is correct. The discriminant, 25, is a perfect square, which indicates that the
roots are real, rational, and unequal.
1
1
1
Stating that the discriminant is 25 or that the roots are real, rational, and unequal,
0
Score
Explanation
2
No, and an appropriate explanation is given, such as the graph fails the vertical line
test.
1
An answer of “no” without an explanation or
1
An answer of “no” with an incorrect explanation.
0
Chapters 1–13
105
Part III
18
19
Score
Explanation
4
a 2 seconds and appropriate work is shown.
b 1.9 seconds and appropriate work is shown.
3
2
made or
2
2
2 seconds and 1.9 seconds, but no work is shown.
1
or rounding error are made or
1
2 seconds or 1.9 seconds, but no work is shown.
0
Score
4
Explanation
a
y
3
2
y = 2sin 3x
1
p
3
–1
2p
3
x
–2
–3
y = –2sin 3x
b y ⫽ ⫺2sin 3x
c Reflecting in the y-axis would produce the curve y ⫽ 2sin (⫺3x) and, since
sin (⫺x) ⫽ ⫺sin x, we see that y ⫽ 2sin (⫺3x) ⫽ ⫺2sin 3x.
106
3
Appropriate work is shown, but either the graph or its equation or the answer or
explanation about the reflection in the y-axis is incorrect.
2
Appropriate work is shown, but either the graph and its equation or the answer and
explanation about the reflection in the y-axis are incorrect.
1
Appropriate work is shown, but only one of the following is correct: the graph or its
equation or the answer or explanation about the reflection in the y-axis.
0
Cumulative Reviews
Part IV
20
Score
Explanation
6
u ⫽ 90°, 210°, 330° and appropriate work is shown.
5
5
Appropriate work is shown, but either 210° or 330° is not given.
4
Appropriate work is shown, but two computational errors are made or
4
Appropriate work is shown, but either 90° or both 210° and 330° are not given or
4
Appropriate work is shown, and all three answers are correct but are given in radian
measure.
3
3
Appropriate work is shown, but three computational errors are made.
2
Appropriate work is shown, but one conceptual and one computational error are
made or
2
90° or 210° or 330° and appropriate work is shown.
1
90°, 210°, 330°, but no work is shown
1
Student shows minimal understanding of trigonometric equations.
0
or
Chapters 1–14
(pages 621–623)
Part I
1
2
3
4
5
2
(1) ⫺45°
3
(1) ⫺
4
(2) Its axis of symmetry is
x ⫽ ⫺2.
(3) ⫺
5
6
7
8
9
10
(4) 5.4
(3) ⫺cos 32°
x⫺1
(2)
2
(1) 25
(3) 1,158
(4) 36
(4) 2.963
(4) The solution is a portion
of the number line
between 0 and 2 but
excluding 0 and 2.
13 (4) {270°}
11
12
Chapters 1–14
107
Part II
14
15
16
17
108
Score
Explanation
2
⫺6,465 and appropriate work is shown.
1
1
⫺6,465, but no work is shown.
0
Score
Explanation
2
4 ⫾ 2i and appropriate work is shown.
1
Appropriate work is shown, but solution is not simplified or
1
Quadratic is set up correctly, but algebraic errors are made or
1
4 ⫾ 2i, but no work is shown.
0
Score
Explanation
2
Decreasing exponential function, and appropriate explanation is given.
1
Decreasing exponential function, and no appropriate explanation is given.
0
Score
Explanation
24
25
2
⫺
1
Appropriate work is shown, but student solves for b only, not sin 2b or
1
⫺
0
24
, but no work is shown.
25
Cumulative Reviews
Part III
18
Score
4
冦(⫺2, 0), 冢 72 , ⫺ 774 冣冧 and appropriate work is shown.
3
Appropriate work is shown, but student finds only the x-values.
2
Appropriate work is shown, but student finds only one solution.
1
1
冦(⫺2, 0), 冢 72 , ⫺ 774 冣冧, but student uses a graphic solution
1
冦(⫺2, 0), 冢 72 , ⫺ 774 冣冧, but no work is shown.
0
19
Explanation
Score
or
Explanation
4
Brandon is correct; the measurements are not accurate. Appropriate justification
should include the Law of Cosines.
3
An appropriate method is used to justify that Brandon is correct, but one
computational error is made.
2
Student correctly solves a problem using Law of Sines.
1
Student demonstrates minimal understanding of trigonometry
1
Brandon is correct, but no appropriate justification is given.
0
or
Chapters 1–14
109
Part IV
20
Score
Explanation
6
a 0.924 ⫽ 0.999879t
b Approximately 653 years old, and appropriate work is shown.
c Picasso did not paint the picture because he was not alive 653 years ago.
5
A correct equation is solved with appropriate work shown, but one computational
error is made or
5
0.924 ⫽ 0.999879t and approximately 653 years old, but student omits discussion of
Picasso’s artistry.
4
A correct exponential equation is solved correctly but using a method other than
logarithms.
3
An incorrect exponential equation is solved correctly using logarithms.
2
0.924 ⫽ 0.999879t, but no solution or discussion of Picasso’s artistry is given or
2
Multiple errors are made in the solution of the question.
1
Student shows minimal understanding of exponential functions and logarithms.
0
Chapters 1–15
(pages 624–626)
Part I
1 (1) 15
2 (2) a ⫽ 1, b ⫽ 7
3 (2) linear
4 (1) 1
5 (3) 27
110
Cumulative Reviews
6
7
8
9
(1) 10
(3) 3
(4) ⫺0.9
63
(2)
65
10
11
12
13
(2)
(2)
(3)
(2)
8
II
75–95
2
Part II
14
Score
16
冢
冣 and appropriate work is shown.
2
3x ⫹ 1
1
, x ⫽ 0,
3x
3
1
1
Appropriate work is shown, but one simplification error is made or
1
3x ⫹ 1
1
, but no work is shown.
, x ⫽ 0,
3x
3
0
15
Explanation
Score
冢
冣
Explanation
2
length: 10.5 feet; width: 8.5 feet, and appropriate work is shown.
1
1
Appropriate work is shown, but only one dimension is given or
1
length: 10.5 feet; width: 8.5 feet, but no work is shown.
0
Score
Explanation
2
No, and an appropriate explanation is given, such as two people could have the
same height but different weights.
1
An answer of “no” without an explanation or
1
An answer of “no” with an incorrect explanation.
0
Chapters 1–15
111
17
Score
Explanation
2
u ⫽ 60°, 90° and appropriate work is shown.
1
1
Appropriate work is shown, but only one angle is given or
1
u ⫽ 60°, 90°, but no work is shown.
0
Part III
18
19
Score
4
69.51° and appropriate work is shown.
3
2
made or
2
Appropriate work is shown, but one conceptual error is made, such as solving for
the wrong angle.
1
1
69.51°, but no work is shown.
0
Score
Explanation
p
, D ⫽ 300, and appropriate work is shown.
6
4
A ⫽ ⫺200, B ⫽
3
2
made or
2
1
p
A ⫽ ⫺200, B ⫽ , D ⫽ 300, but no work is shown.
6
1
0
112
Explanation
Cumulative Reviews
Part IV
20
Score
6
Explanation
a
y ⫽ 20.125x ⫹ 199.536
b Approximately 501 students, and appropriate work is shown.
c 2028; students’ explanations will vary but should mention the probable
inaccuracy due to extrapolation; appropriate work is shown.
5
Appropriate work is shown, but a rounding error is made or
5
Appropriate work is shown, but no discussion of the accuracy of 2028 is given.
4
Appropriate work is shown, but no scatter plot is drawn or
4
Only two of the three parts of the problem are completed or
4
Appropriate work is shown for all parts, but one conceptual error is made.
3
The scatter plot and regression equation are correct, but no other work is done.
2
Only one of the three parts of the problem is completed with work shown.
1
Student shows minimal understanding of regression equations
1
y ⫽ 20.125x ⫹ 199.536; approximately 501 students; 2028, but no work is shown.
0
or
Chapters 1–16
(pages 627–629)
Part I
1
(2)
冦⫺ 12 , 3冧
6
7
2
3
4
5
(2)
(2)
(4)
(1)
11.4
30
195°
⫺26 ⫹ 18i
8
9
(1) 0 ⬍ r ⱕ 1
(3) The rate of change is
83 percent.
(3) 12
(4) every fifth person who
enters the local mall
10
11
12
13
(2) 3.72
(1) 0
(3) 60
4
p
(2)
3
Chapters 1–16
113
Part II
14
Score
Explanation
2
yi
5
4
1 + 3i
3
2
1
O
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
15
16
114
1
2
3
4
x
5
5 – 2i
4 – 5i
1
Graph is correctly labeled, but sum is incorrect.
0
Score
Explanation
2
x ⫽ 3.6; s ⫽ 4.695, and appropriate work is shown.
1
Appropriate work is shown, but rounding errors are made or
1
Appropriate work is shown, but only one of two values is correct or
1
x ⫽ 3.6; s ⫽ 4.695, but no work is shown.
0
Score
Explanation
2
y ⫽ 4.2sin 2x ⫺ 3, and appropriate work is shown.
1
An error is made in the amplitude, period, or midline.
0
Cumulative Reviews
17
Score
Explanation
2
720a3b2 and appropriate work is shown.
1
Binomial expansion is attempted, but an error is made or
1
720a3b2, but no work is shown.
0
Part III
18
19
Score
Explanation
4
u ⫽ {63.4°, 126.9°, 243.4°, 306.9°} and appropriate work is shown.
3
Equation is solved for u, but additional angle measures in interval are not found or
3
Appropriate work is shown, but rounding errors are made.
2
Appropriate work is shown, but student finds only two solutions.
1
u ⫽ {63.4°, 126.9°, 243.4°, 306.9°}, but no work is shown.
0
Score
Explanation
4
a 58° and appropriate work is shown.
b 58.5 inches and appropriate work is shown.
3
One or both solutions are incorrectly rounded or
3
An appropriate method is used, but one computational error is made.
2
Student uses an incorrect formula, but solves it correctly.
1
Minimal understanding of trigonometry is demonstrated
1
58° and 58.5 inches, but no work is shown.
0
or
Chapters 1–16
115
Part IV
20
Score
Explanation
6
a y ⫽ 0.474t ⫹ 11.391 and appropriate work is shown.
b 30.35% and appropriate work is shown.
c approximately 81.5 years; explanation is given, such as it is unreasonable
because it is unlikely that the model would be valid for that period of time.
5
Appropriate work is shown, but there is a rounding error or
5
Appropriate work is shown, but there is no discussion of the reliability of the
extrapolated value.
4
Only two of the three parts of the question are answered.
3
Student finds a nonlinear regression equation and bases answers on that.
2
y ⫽ 0.474t ⫹ 11.391, but no other solutions or discussions are given or
2
Only one of the three parts of the question is answered.
1
Student shows minimal understanding of linear regression.
0
Practice Algebra 2
and Trigonometry
Regents Examinations
Practice Regents Examination One
(pages 633–638)
Part I
1 (4) 4
2 (4) x ⫽ 0
3 (3) 3 inches
116
4
5
(1) 77
1 2
(3) 4x 3y 3
Practice Algebra 2 and Trigonometry Regents Examination
6
7
5p
6
(3) ⫺1 ⫹ 2i
(2)
8
9
10
11
12
13
14
15
16
(2) 64
(3) sec x
y⫺x
(3)
y
(4) trigonometric
(4) 16
(2) ⫺432a3b
(2) 5,700
3
(3)
z⫺3
(1) (⫺1, 8)
(2) {(1, 2), (1, 3), (5, 4),
(7, 5), (9, 6)}
18 (1) {⫺2, 5}
19 (4) 3兹5 ⫺ 6
20 (2) (y ⫺ 3)2 ⫽ 19
21 (1) ⫺1.25
22 (2) 2
p 3p
,
23 (3)
4 4
119
24 (2) ⫺
169
17
25
26
27
(2) (9, 10)
(4) y ⫽ 3sin px
y
(3)
x
Part II
28
29
Score
Explanation
2
x ⫺ 3, (x ⫽ 0, ⫺3) and appropriate work is shown.
1
1
x ⫺ 3 but no work is shown.
0
Score
Explanation
1
2
b 2兹a
a 2b 2
log
or log
c
c
1
b 2兹a
a 2b 2
or
but no work is shown or
c
c
1
b 2兹a
a 2b 2
log
or log
c
c
1
1
0
30
Score
Explanation
2
f ⫺1(x) ⫽ log2 x and appropriate work is shown.
1
f (x) ⫽ log2 x and appropriate work is shown
1
f ⫺1(x) ⫽ log2 x but no work is shown.
0
or
117
31
32
Score
Explanation
2
1
冦⫺ 32 , 9冧 and appropriate work is shown
1
0
Score
Explanation
d
Distance
2
or
t
Time
33
1
Time and distance are transposed or
1
One error on graph but some work is correct.
0
Score
2
3
兹15 i and appropriate work is shown.
⫾
4
4
1
The quadratic formula is used correctly, but one computational error is made or
1
15
3
⫾ 兹
i but no work is shown.
4
4
0
118
Explanation
34
35
Score
Explanation
2
55.8% and appropriate work is shown.
1
28.3%, the probability of exactly two, is found and appropriate work is shown or
1
1
55.8% but no work is shown.
0
Score
2
Explanation
a
yi
8
7
Z1
6
5
4
Z2
3
2
Z1 – Z2
1
–3 –2 –1 O 1
–1
–2
2
3
4
5
6
7
x
b 4 ⫹ 3i
1
4 ⫹ 3i without a graph or
1
Correct graph but no 4 ⫹ 3i or
1
8 ⫹ 13i and a graph depicting addition of vectors.
0
119
Part III
36
37
120
Score
Explanation
4
No, it would take approximately 14.2 years. Appropriate work is shown.
Explanations will vary, but should make use of the equation 2 ⫽ (1.05)t.
3
Appropriate work and explanation are shown, but one computational or rounding
error is made or
3
14.2 years with appropriate work but no explanation.
2
Appropriate work and explanation are shown, but two computational or rounding
errors are made.
1
14.2 years but no work is shown.
0
Score
Explanation
4
240.6 pounds and appropriate work is shown.
3
An appropriate method is used, such as the Law of Cosines, but one computational
or rounding error is made.
2
1
Multiple mathematical errors are made
1
240.6 pounds, but no work is shown.
0
or
38
Score
4
Explanation
a
y ⫽ ⫺2cos
冢 2p5 x冣 ⫹ 2 and appropriate work is shown.
b 1.7 seconds and appropriate work is shown.
3
One of the values of a, b, or c in the equation y ⫽ acos bx ⫹ c is incorrect
3
The equation is correct, but the time is incorrect.
2
Two of the values of a, b, or c in the equation y ⫽ acos bx ⫹ c are incorrect or
2
Time is incorrect, and one of the values of a, b, or c in the equation y ⫽
acos bx ⫹ c is incorrect.
1
Only one value of a, b, or c in the equation y ⫽ acos bx ⫹ c is correct or
1
Only 1.7 seconds is correct.
0
or
Part IV
39
Score
Explanation
6
a y ⫽ 29.08x ⫹ 492.62 and appropriate work is shown.
b 842 and appropriate work is shown.
c 1994 and appropriate work is shown.
5
4
y ⫽ 29.08x ⫹ 492.62 and 842 and appropriate work is shown
4
y ⫽ 29.08x ⫹ 492.62 and 1994 and appropriate work is shown.
3
A correct equation is shown, but parts b and c are missing or incorrect
3
The equation is incorrect, but all further work is appropriate.
2
The equation is incorrect, but some further work is appropriate.
1
An incorrect equation is shown, but part b or c is correct.
0
or
or
121
Practice Regents Examination Two
(pages 639–645)
Part I
1 (4) {4}
2 (4) x2 ⫹ 4x ⫺ 12 ⱖ 0
3 (1) ⫺7
2
4 (1)
2⫺a
5 (2) 2
6 (2)
y = 2x2 – 5
7
8
9
10
(2)
(3)
(3)
(3)
11
(4)
12
(4)
13
(2)
14
(3)
15
(1)
16
(2)
$1,333.94
8,943.9
III
y ⫽ log3 x
3
2
interquartile range
p
,1
2
3
3
t⫽⫺
2
increases, then
decreases
冢
冣
17
(1)
18
(2)
19
20
21
(3)
(1)
(4)
22
(2)
23
24
25
26
27
(1)
(2)
(2)
(4)
(3)
p
4
1
16
19
I and II
330°
3
⫺
4
35.66°
48.65
1.29
120
54c2h2
Part II
28
29
Score
2
4兹x 3y 5 and appropriate work is shown.
1
1
4兹x 3y 5 but no work is shown.
0
Score
Explanation
冢
冣
2
1
2 2
, x ⫽ ⫺3, ⫺ , , 3 and appropriate work is shown.
3x ⫺ 2
3 3
1
Student multiplied instead of divided but factored correctly or
1
Factoring errors are present, which produce an incorrect answer
1
1
3x ⫺ 2
0
122
Explanation
or
30
31
32
33
Score
Explanation
2
i and appropriate work is shown.
3
2
3⫾
1
Fractions are not reduced or in a ⫹ bi form
1
The quadratic formula is used correctly, but one computational error is made.
0
Score
or
Explanation
2
5, ⫺1 and appropriate work is shown.
1
Appropriate work is shown with one computational error or
1
5, ⫺1 but no work is shown.
0
Score
Explanation
2
a 7.9 and appropriate work is shown.
b 2 and appropriate work is shown.
1
Student uses an appropriate method, but makes one computational or rounding
error or
1
Student uses population standard deviation correctly or
1
7.9 and 2 but no work is shown.
0
Score
Explanation
77
85
2
⫺
1
Identity is used correctly but one computational error is made or
1
Incorrect identity is used with correct trigonometric values or
1
⫺
0
77
85
123
34
35
Score
Explanation
2
1
1
0
Score
Explanation
2
{210°, 330°} and appropriate work is shown.
1
Equation is solved with one computational error or
1
Equation is solved correctly, but only one angle is given or
1
{210°, 330°} but no work is shown.
0
Part III
36
124
Score
Explanation
4
a Price of $8.99: $14,240; price of $10.50: $16,065, and appropriate work is shown.
b $12.90 and appropriate work or graph is shown.
c $16,641, and appropriate work or graph is shown.
3
Appropriate work or graph is shown with one incorrect solution or
3
$14,240, $16,065, $12.90, and $16,641 with a rough sketch but no labels or
explanation.
2
Appropriate work or graph is shown, but two computational or rounding errors are
made.
1
$14,240, $16,065, $12.90, and $16,641, but no work is shown.
0
37
Score
4
25
729
3
25
and appropriate method is indicated, but insufficient work is shown or
729
3
2
24
, student finds exactly two rather than at least two.
729
1
An appropriate method is indicated, but multiple mathematical errors are made or
1
25
729
0
38
Explanation
Score
Explanation
4
66.4 feet and appropriate work is shown.
3
An appropriate method is used, such as the Law of Sines, but one computational or
rounding error is made.
2
The length of the hypotenuse is found, but no other correct work is shown.
1
The initial setup for the problem is correct, but multiple errors are made
1
Student uses the Pythagorean theorem to find the height of the flagpole or
1
66.4 feet but no work is shown.
0
or
125
Part IV
Score
39
6
Explanation
a, c
g(x) = 2cos x – p
(
2
)
f (x) = 2cos x
冢
b
g(x) ⫽ 2cos x ⫺
d
g(x) ⫽ 2sin x
x
h(x) ⫽ 4cos
3
e
p
2
冣
5
Appropriate graph is shown, but one equation is incorrect.
4
Appropriate graph is shown, but two equations are incorrect.
3
Graphs of f(x) and g(x) are correct, but no other work is correct.
2
The graph of f(x) or g(x) is correct, but all further work is incorrect or
2
One equation is incorrect, and graphs are incorrect or not shown.
1
An incorrect graph is correctly shifted, but no other work is correct or
1
Two equations are incorrect, and graphs are incorrect or not shown.
0
Practice Regents Examination Three
(pages 646–652)
Part I
1
2
3
4
5
6
7
8
(4) 164
(3) real, irrational, and
unequal
(3) 95
(2) 2
p
(2)
2
(2) 29
(3) 0.6%
(1) 2sin2 u ⫹ sin u ⫺ 1 ⫽ 0
126
9
10
(2) {12}
(4)
11
(4)
12
(1)
13
(3)
14
(4)
兹2
⫺
2
exponential
4
⫺
5
1⬍y⬍6
21 ⫹ 8兹5
11
15 (3) ⫺3c ⫺ 2
16 (1) f ⫺1(x) ⫽ log3 x
3
17
(2)
兺 (3t 2 ⫺ 4t ⫹ 12)
t⫽1
18
19
20
21
(1) 780
1
a
(2)
2
(3) III
(4) acute or obtuse
22
23
10
p
3
(2) ⫺527 ⫹ 336i
(1)
24
25
(4) ⫺
26
27
兹3
2
(3) {⫺2, ⫺1, 1, 2}
(2) 0 and 4
(4) $36
Part II
28
29
30
31
Score
Explanation
2
x2 ⫺ 6x ⫹ 10 ⫽ 0 and appropriate work is shown.
1
1
x2 ⫺ 6x ⫹ 10 and appropriate work is shown or
1
x2 ⫺ 6x ⫹ 10 but no work is shown.
0
Score
Explanation
2
$1,074.50 and appropriate work is shown.
1
1
$1,074 but no work is shown.
0
Score
Explanation
2
g(x) ⫽ 兩 x ⫹ 3 兩 ⫺ 4 and appropriate work is shown.
1
Student has either the vertical shift or horizontal shift correct but not both or
1
g(x) ⫽ 兩 x ⫹ 3 兩 ⫺ 4 but no work is shown.
0
Score
Explanation
2
{0, 5} and appropriate work is shown.
1
x ⫽ 0 or x ⫽ 5 but not both
1
{0, 5} but no work is shown.
0
or
127
32
33
34
Score
Explanation
2
.
3
b arithmetic; there is a common difference of ⫺1.25.
2
a
1
Student correctly identifies type of sequence but does not give explanations or
1
One sequence is correctly identified and explained, but the other is not.
0
Score
Explanation
2
Katie is correct. The cosine function is symmetric about the y-axis.
1
Katie is correct, but not explanation is provided.
0
Score
2
Explanation
a
b
128
28
55
26
33
28
26
or
and appropriate work is shown or
55
33
1
Either
1
28
26
and
55
33
0
35
geometric; there is a common ratio of
Score
Explanation
2
1.7 seconds and appropriate work or graph is shown.
1
Student sets up a correct quadratic equation, but one computational error is
made or
1
1.7 seconds but no work is shown.
0
Part III
36
37
Score
Explanation
4
a 15,145 people and appropriate work is shown.
b 36.68 years and appropriate work is shown.
3
3
15,145 people with appropriate work or diagram, and 36.68 years without an
algebraic solution.
2
15,145 people or 36.68 years, but not both, and appropriate work is shown
2
1
15,145 people and 36.68 years, but no work is shown.
0
Score
or
Explanation
4
u ⫽ {0°, 45°, 153.4°, 180°, 225°, 333.4°} and appropriate work is shown.
3
An appropriate method is used, but one computational error is made or
3
An appropriate method is used, but one or two angles are missing.
2
An appropriate method is used, but two computational errors are made or
2
An appropriate method is used, but three or four angles are missing.
1
Multiple computational/formula errors are made
1
u ⫽ {0°, 45°, 153.4°, 180°, 225°, 333.4°} but no work is shown.
0
or
129
38
Score
Explanation
4
a 6.3 million and appropriate work is shown.
b 0.7 million and appropriate work is shown.
c Range ⫽ 1.9 million, Interquartile range ⫽ 1.4 million and appropriate work is
shown.
d Answers will vary, but student should indicate some knowledge of how
statistics, although correct, can lead people to come to false conclusion.
Alternatively, student may provide a regression analysis.
3
One of the parts a, b, c, or d is incorrect.
2
Two of the parts a, b, c, or d are incorrect.
1
Only one of the parts a, b, c, or d is correct.
0
Part IV
39
130
Score
Explanation
6
CT ⫽ 2.3 miles, TH ⫽ 2.6 miles and appropriate work is shown. Police cars from
point C will arrive first.
5
4
CT ⫽ 2.3 miles and TH ⫽ 2.6 miles and appropriate work is shown, but student omits
point C.
3
2
Appropriate work is shown, but only point C is correct.
1
Student shows minimal understanding of trigonometric relationships
1
CT ⫽ 2.3, TH ⫽ 2.6 miles, point C, but no work is shown.
0
or

algebra 2 trigonometry - Sewanhaka Central High School

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