chapter solutions key 1 foundations for functions · 2014. 9. 2. · 1-2 properties of real...
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Solutions Key
Foundations for Functions1
CHAPTER
ARE YOU READY? PAGE 3
1. D 2. C
3. A 4. E
5. 3 ___ 10
= 0.3 6. 3 __ 5 = 0.6
7. - 4 _ 3 = -1.
− 3 8. 5 3 _
4 = 23 _
4 = 5.75
9–12.
13. 5 _ 6 > 2 _
3 14. 3 7 _
9 < 3 10 _
12
15. -0.38 < -0.3 16. - 15 ___
8 > -2
17. 14 ÷ 2(-3) + 1= 7(-3) + 1= -21 + 1= -20
18. 8 2 – (-12) + 15 ÷ 3= 64 + 12 + 5= 81
19. -2 (25 - 21) 2 + 11= -2 (4) 2 + 11= -2(16) + 11= -32 + 11= -21
20. 3 ( 21 - 9 ______ 6 - 1) ÷ 2
= 3 (2 - 1) ÷ 2 = 3(1) ÷ 2 = 3 ÷ 2
= 1.5 or 3 _ 2
21–24.
1-1 SETS OF NUMBERS, PAGES 6–13
CHECK IT OUT!
1a. π ≈ 3.14, 3 __ 2 = 1.5, - √ * 3 ≈ -1.73
The order is: -2, - √ * 3 , -0.321, 3 __ 2 , π
b. -2: ", #, %; - √ * 3 : ", irrational;
-0.321: ", #; 3 _ 2
: ", #;
π: ", irrational
2a. (-∞, -1] b. (-∞, 2] or (3, 11]
3a. even numbers between 1 and 9
b. {3, 4, 5, 6, 7} c. {x | x ≥ 99}
THINK AND DISCUSS
1. Possible answer: Interval notation is used to indicate infinite sets of real numbers over an interval. Roster notation is used to indicate finite or infinite sets that follow a pattern (such as multiples of 2). It is not possible to have a set represented by both methods.
2. Possible answer: No; any integer, n, can be expressed in the form n _
1 , which is a rational number.
3.
EXERCISES
GUIDED PRACTICE
1. Roster notation
2. 3 √ * 2 ≈ 4.24, √ * 7 ≈ 2.6, 4 3 _ 5 = 4.6
The order is: √ * 7 , 3 √ * 2 , 4 3 _ 5
, 4. − 6 , 5.125
√ * 7 : ", irrational; 3 √ * 2 : ", irrational;
4 3 _ 5 : ", #; 4.
− 6 : ", #;
5.125: ", #
3. - 100 ____
4 = -25,
√ * 4 = 2, 1 _
8 = 0.125, √ * 6 ≈ 2.4
The order is: - 100 ____
4 , -6.897, 1 _
8 , √ * 4 , √ * 6
- 100 ____
4 : ", #, %; -6.897: ", #;
1 _ 8 : ", #;
√ * 4 : ", #, %, ', );
√ * 6 : ", irrational
4. √ * 5 ≈ 2.2, π
__ 2 ≈ 1.57, -
√ * 3 ≈ -1.73, -1 1 __
3 = 1.
− 3
The order is: – √ * 3 , -1 1 __ 3 , 1.
− 3 , π
__ 2 , √ * 5
– √ * 3 : ", irrational; -1 1 __ 3
: ", #;
1. − 3 : ", #;
π
__ 2
: ", irrational;
√ * 5 : ", irrational
5. (-10, 10] 6. (-∞, -5)
7. [1, 20) or (30, ∞) 8. one
9. {x | -5 ≤ x < 3}
10. nonnegative integer multiples of 5
11. {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}
1 Holt McDougal Algebra 2
PRACTICE AND PROBLEM SOLVING
12. 2 √ # 5 ≈ 4.47, - 4 __ 5 = -0.8,
The order is: - 4 __ 5 , -0.75, 2.33, 2 √ # 5 , 5.
− 5
- 4 __ 5 : !, "; -0.75: !, ";
2.33: !, "; 2 √ # 5 : !, irrational;
5. − 5 : !, "
13. 1 __ 2 = 0.5, -
√ # 2 ≈ -1.4,
√ # 2 ____
3 ≈ 0.47
The order is: -2, - √ # 2 , -1. −− 25 ,
√ # 2 ___
3 , 1 __
2
-2: !, ", $; - √ # 2 : !, irrational;
-1.25: !, ";
√ # 2 ____
3 : !, irrational;
1 __ 2 : !, "
14. - √ # 9 = -3, 2π ≈ 6.28, - 7 __ 2 = -3.5
The order is: - 7 __ 2 , - √ # 9 , -1, 5.
−− 12 , 2π
- 7 __ 2 : !, "; - √ # 9 : !, ", $;
-1: !, ", $; 5. −− 12 : !, ";
2π: !, irrational
15. (-∞, 5) or (5, ∞) 16. (-15, 0)
17. [-3, 3]
18. less or equal to 3 or greater than 5 and less than or equal to 11
19. {11, 22, 33, 44, 55, 66, 77,…}
20. less than -3 or greater than 0
21. {x | -9 ≤ x ≤ -1 and x is odd}
22. Lithium, aluminum, sulfur, chlorine, calcium
23. " 24. $
25. Possible answer: Interval notation is used for ranges of numbers, but the set of atomic masses is a list of numbers.
26. negative even integers; cannot be expressed in interval notation; {x | x < 0 and x is even}
27. numbers greater than or equal to -4 and less than 8; cannot be expressed in roster notation;
{x | -4 ≤ x < 8}
28. {28, 30, 32, 34,36, 38}; cannot be expressed in interval notation; {x | 27 < x < 39 and x is even}
29. numbers greater than 0 and less than 1; cannot be expressed in roster notation; (0, 1)
30. (-∞, -4) or (4, ∞);{x | x < -4 or x > 4}
31. (-∞, 2) or (2, ∞);{x | x ≠ 2}
32. (∞, 2] or (3, 5);{x | x ≤ 2 or 3 < x < 5}
33. (1, 10); {x | 1 < x < 10}
34. (-∞, 6) or (10, ∞);{x | x < 6 or x > 10}
35. (-∞, 5) or (5, 10];{x | x < 5 or 5 < x ≤ 10}
36. true
37. False; possible answer: 3 is a real number but not irrational.
38. False; possible answer: -4 is an integer but not a whole number.
39. true
40. Soccer ball A: C = 2πr
= 2π (4.36) ≈ 27.38 in. Therefore, the size of ball A is 5. Soccer ball B: C = πd
= (7.54) π ≈ 23.68 in. Therefore, the size of ball B is 3.
Soccer ball C: V = 4 __ 3 π r 3
276.2 = 4 __ 3 π r 3
( 3 ___ 4π
) (276.2) = ( 3 ___ 4π
) ( 4π
___ 3
) r 3
65.94 ≈ r 3
3 √ ### 65.94 ≈
3
√ # r 3 4.04 ≈ r So, C = 2πr
= 2π (4.04) ≈ 25.38 in. Therefore, the size of ball C is 4.
41. size 3: {x | 11 ≤ x ≤ 12} size 4: {x | 12 ≤ x ≤ 13} size 5: {x | 14 ≤ x ≤ 16}
42. size 3: (0, 8) size 4: [8, 12] size 5: (12, ∞)
43. The circumference is always irrational because it is the product of an irrational number, π, and a rational number, the diameter d.
44a. !, "
b. 97 ____ 186
≈ 0.522, 117 ____ 310
≈ 0.377
The order is: Moon, Venus, Mercury, Mars.
c. The round-trip to Venus would take longer because twice the average distance between Earth and Venus is about 0.555 AU and the average distance between Earth and Mars is about 0.522 AU.
45. (∞, -1] or (3, 6) or [9, ∞)
46.
47.
48.
49.
50.
2 Holt McDougal Algebra 2
51.
52a. {talc, gypsum, calcite, fluorite, apatite}
b. 5; orthoclase to diamond are harder
c. Neither; quartz is harder than window glass but apatite is softer than window glass.
53. ! 54. "
55. #
56. No; possible answer: square roots of some numbers are not irrational. √ # 9 = 3, and 3 is rational.
57a. interior designer, police officer, pediatric nurse, marine biologist, astronaut
b. The order would not change, because each salary is increased by the same amount.
c. The order would not change, because the amount by which each salary is increased is relative to its original amount, and that does not allow one salary to increase so much or so little as to change the order.
d. {46,000, 52,900, 59,800, 79,350, 106,950}
58. Possible answer: rational: 5 __ 2
; irrational: √ # 2
___ 2 ; 5 is in
the set
59. Possible answer: rational: 6; irrational: 6 √ # 3 ; 5 is in the set.
60. Possible answer: rational: 11: irrational: 4 √ # 5 ; 5 is in the set.
61. Possible answer: rational: 3 3 __ 2 ; irrational: π; 5 is not
in the set.
62. Possible answer: Mathematical and everyday sets are similar because they are both made up of elements. They are different because mathematical sets can be infinite.
TEST PREP
63. D 2(-2) = -4
64. F
3 __ 7 ≈ 0.42,
√ # 3 ___
2 ≈ 0.866
65. B
- √ # 4 = -2, - 5 __ 3 = 1.
− 6 , 1 1 __
2 = 1.5
66. J
CHALLENGE AND EXTEND
67. finite; " 68. infinite; "
69. finite; ", $, #, ! 70. infinite; "
71a. Possible answer: 3.141
b. Possible answer: 3.142
SPIRAL REVIEW
72. Possible answer: −− AB and
−− BC
73. Possible answer: AEGD and BFHC
74. Possible answer: AEGD and ABFE
75. 1.065(21.49 + 11.59 + 12.95)= 1.065(46.03)= 49.02The cost is $49.02.Since she could only have 3 bills, then the only combination Debra could have is $20, $20, $10.
76. 3.7 ___ s = 1 ____
120
s = 3.7(120) s = 444 cm The square has area of (444)
2 = 197,136 cm
2 .
1-2 PROPERTIES OF REAL NUMBERS,
PAGES 14–19
CHECK IT OUT!
1a. additive inverse: -500; multiplicative inverse: 1 _ 500
b. additive inverse: 0.01; since -0.01 = - 1 ____
100 ,
multiplicative inverse: -100
2a. Comm. Prop. of Mult. b. Assoc. Prop. of Mult.
3. 20% = 2(10%) [2(10%)]15.60 = 2[(10%)15.60] = 2[(0.1)(15.60)] = 2(1.56) = $3.12
4a. always true by the Additive Inverse Property
b. sometimes true;true when a = 0, b = 1, c = 2false when a = 1, b = 2, c = 3
THINK AND DISCUSS
1. Possible answer: No, the Commutative Property does not apply to subtraction or division because the order in subtraction and division is essential.
2. Possible answer: The product of 0 and another number is always 0, so you cannot multiply 0 by any number and get a product of 1.
3.
EXERCISES
GUIDED PRACTICE
1. additive inverse: 36; multiplicative inverse: - 1 _ 36
2. additive inverse: 0.05; since -0.05 = 1 ____ -20
,
multiplicative inverse: -20
3. additive inverse: -2 √ # 2 ; multiplicative inverse: 1 _ 2 √ # 2
3 Holt McDougal Algebra 2
4. additive inverse: - 2 _ 5 ; multiplicative inverse: 5 _
2
5. additive inverse: 1 _ 500
; multiplicative inverse: -500
6. additive inverse: -0.25; since 0.25 = 1 _ 4 ,
multiplicative inverse: 4
7. Assoc. Prop. of Mult. 8. Comm. Prop. of Add.
9. Comm. Prop. of Mult.
10. 3(2.55) = 3(2 + 0.5 + 0.05) = 3(2) + 3(0.5) + 3(0.05) = 6 + 1.5 + 0.15 = $7.65
11. 33 1 __ 3
% = 1 __ 3
(33 1 __ 3
%) (21.99)
= ( 1 __ 3 ) (21 + 0.99)
= ( 1 __ 3 ) 21 + ( 1 __
3 ) 0.99
= 7 + 0.33 = $7.33
12. always true by the Distributive Property
13. sometimes true; true when a = b; false when a = 1 and b = 2
14. sometimes true; true when a = 0; false when a = 1, b = 2, and c = 3
PRACTICE AND PROBLEM SOLVING
15. additive inverse: 2.5; since -2.5 = - 5
__
2 ,
multiplicative inverse: - 2
__
5
16. additive inverse: -0.75; since 0.75 = 3 __
4 ,
multiplicative inverse: 4 __ 3
17. additive inverse: -2π; multiplicative inverse: 1 ___ 2π
18. additive inverse: 2 __ 3 ; multiplicative inverse: -
3
__
2
19. additive inverse: - 1 ___ 20
; multiplicative inverse: 20
20. additive inverse: -6231; multiplicative inverse: 1 _____ 6231
21. Distributive Property 22. Comm. Prop. of Mult.
23. Additive Identity Property
24. 9%(150) = 0.09(100 + 50) = 0.09(100) + 0.09(50) = 9 + 4.5 = $13.50
25. 5(2.00) = 10.00 5(0.04) = 0.2 10.00 - 0.20 = $9.80
26. always true by the Distributive Property
27. never true by the Multiplicative Inverse Property
28. 2(8.88) + 3(14.99) = 2(8 + 0.88) + 3(14 + 0.99) = 2(8) + 2(0.88) + 3(14) + 3(0.99) = 16 + 1.76 + 42 + 2.97 = $62.73
29. 4(11.99) - 2(8.88) = 4(11 + 0.99) - 2(8 + 0.88) = 4(11) + 4(0.99) - 2(8) - 2(0.88) = 44 + 3.96 - 16 - 1.76 = $30.20
30. 4(0.85)(14.99) = 3.4(15 - 0.01) = 51 - 0.034 = 50.966 ≈ $50.97
31. 3(0.9)(9.96) + 5(0.75)(11.99) = 2.7(9.96) + 3.75(11.99) = (2 + 0.7)(9 + 0.96) + (3 + 0.75)(11 + 0.99) = 71.8545 ≈ $71.85
32. time = distance _______
speed
= 11 + 32 + 38
___________ 40
= 81
___ 40
≈ 2 h
33. miles that can be driven on one tank: 24 × 8 = 192 mi length of one loop: 11 + 32 + 38 = 81 mi
number of loops per tank: miles per tank
____________ length of loop
= 192
____ 81
≈ 2 loops
34. length of new loop: 81 × 1.2 = 97.2 mi
number of miles driven in 10 h: x mi ____ 10 h
= 40 mi _____ 1 h
x = 400 mi
loops in 10 h day: miles in 10 h ___________ length of loop
= 400 ____ 97.2
≈ 4 loops
35. (10 + 5) + 23 = 10 + (5 + 23); Assoc. Prop. of Add.
36. 12 + 11 _ 15
x = 11
___
15 x + 12; Comm. Prop. of Add.
37. j + 0 = j; Additive Identity Property
38. 5 · 4 + 5 · 3 = 5 · (4 + 3); Distributive Property
39. 4 _ 5 ·
5
__
4 = 1; Multiplicative Inverse Prop.
40. ab = ba; Comm. Prop. of Mult.
41. Yes; by Distributive Property, both methods give the same results.
42. Find the ticket price, which is 60% of $185: 10% of 185 = 0.1(185) = 18.5 60% of 185 = 6(18.5) = $111.00 Add $16 + $12 = $28 for fees and surcharge: $111 + $28 = $139
4 Holt McDougal Algebra 2
43. Possible answer: The set of integers is made up of the set of natural numbers, their additive inverses, and the additive identity. The set of rational numbers is made up of the set of numbers that can be expressed as a ratio of two natural numbers, their additive inverses, and the additive identity.
44. Assoc. Prop. of Add.; Additive Inverse Prop.
45. Multiplicative Identity Prop.
46. Comm. Prop. of Add; Comm. Prop. of Mult.
47. Distrib. Prop.; Assoc. Prop. of Add.
48. Distrib. Prop.; Comm. Prop. of Add.
49. Distrib. Prop.
50a. (18 + 13) – 24 = 7
b. yes; possible answer: 9 + 19 = 4 mod 24, 19 + 9 = 4 mod 24
c. yes; possible answer: 5 + (12 + 20) = 13 mod 24, (5 + 12) + 20 = 13 mod 24
51. Possible answer: By the Distributive Property, the savings is 5% not 10%.5% (labor) + 5% (parts) = 5% (labor + parts) = 5% (total)
52. Possible answer: Opposites are used for addition. A number and its opposite have different signs. Reciprocals are used for multiplication. A number and its reciprocal have the same sign.
TEST PREP
53. D 54. J
55. C
56. 4(1 + 3) = 4(4) = 16 by the order of operations; 4(1 + 3) = 4(1) + 4(3) = 4 + 12 = 16 by the Distributive Property.
CHALLENGE AND EXTEND
57. n = 4 ( 1 __ n )
(n)n = 4 ( 1 __ n ) (n)
n 2 = 4
n = ± √ % 4 = ±2 n = 2 since n is positive
58a. 3 + 5 = 8, 5 + 3 = 8; 3 - 5 = -2, 5 - 3 = 2; 3 · 5 = 15, 5 · 3 = 15; 3 ÷ 5 = 0.6, 5 ÷ 3 = 1.
− 6 ; the pair 3 + 5 and 5 + 3, and the pair 3 · 5 and 5 · 3 b. a + b and b + a; a · b and b · a always represent
natural numbers.
c. Natural numbers are closed under addition and multiplication.
d. Integers are closed under addition, subtraction, and multiplication.
SPIRAL REVIEW
59. Area of garden last summer: 12 × 8 = 96 ft 2 Area of garden this summer: 16 × 10 = 160 ft 2 Let n represent the percent increase. 96 + 96n = 160 __________ - 96 ____ - 96 96n = 64
96n
____ 96
= 64 ___ 96
n = 0.66 − 6
n ≈ 66.7%
60. π (≈ 3.14) 61. -4 √ % 2 (≈ -5.66)
62. -4 √ % 2 and π 63. (-10, 0]
64. {x | -10 < x ≤ 0} 65. cannot be notated
1-3 SQUARE ROOTS, PAGES 21–26
CHECK IT OUT!
1. - √ %% 64 < - √ %% 55 < - √ %% 49 -8 < - √ %% 55 < -7 7.4 2 = 54.76 7.5 2 = 56.25- √ %% 55 ≈ -7.4
2a. √ %% 48 = √ %%% 16 · 3 = 4
√ % 3
b. √ %% 36 ___ 16
= 6 __ 4
= 3 __ 2
c. √ % 5 · √ %% 20
= √ %%% 5 · 20
= √ %% 100 = 10
d. √ %% 147
_____ √ % 3
= √ %% 147 ____ 3
= √ %% 49 = 7
3a. 3 √ % 5
____ √ % 7
= 3 √ % 5
____ √ % 7
· √ % 7 ___
√ % 7
= 3 √ %% 35
_____ 7
b. 5 ____ √ %% 10
= 5 ____ √ %% 10
· √ %% 10 ____
√ %% 10
= 5 √ %% 10
_____ 10
= √ %% 10
____ 2
4a. 3 √ % 5 + 10 √ % 5
= (3 + 10) √ % 5
= 13 √ % 5
b. √ %% 80 - 5 √ % 5
= √ %%% 16 · 5 - 5 √ % 5
= 4 √ % 5 - 5 √ % 5
= (4 - 5) √ % 5
= - √ % 5
THINK AND DISCUSS
1. Both are equal to 15 √ % 2 .
3 √ %% 50 = 3 √ %%% 25 · 2 = 15 √ % 2
5 √ %% 18 = 5 √ %% 9 · 2 = 15 √ % 2
2. √ %% 16 · √ % 4 = 4 · 2 = 8 or √ %% 16 · √ % 4 = √ %% 64 = 8
5 Holt McDougal Algebra 2
3.
EXERCISES
GUIDED PRACTICE
1. radicand
2. √ ## 64 < √ ## 75 < √ ## 81
8 < √ ## 75 < 9
8.6 2 = 73.96
8.7 2 = 75.69
√ ## 75 ≈ 8.7
3. √ ## 16 < √ ## 20 < √ ## 25
4 < √ ## 20 < 5
4.4 2 = 19.36
4.5 2 = 20.25
√ ## 20 ≈ 4.5
4. - √ ## 100 < - √ ## 93 < - √ ## 81
-10 < - √ ## 93 < -9
9.6 2 = 92.16
9.7 2 = 94.09
- √ ## 93 ≈ -9.6
5. √ # 9 < √ ## 13 < √ ## 16
3 < √ ## 13 < 4
3.6 2 = 12.96
3.7 2 = 13.69
√ ## 13 ≈ 3.6
6. - √ ## 300
= - √ ### 100 · 3
= -10 √ # 3
7. √ ## 24 · √ # 6
= √ ## 144
= 12
8. √ ## 72
____ √ # 2
= √ ## 36
= 6
9. √ ## 80
= √ ### 16 · 5
= 4 √ # 5
10. 1 ___ √ # 2
= 1 ___ √ # 2
· √ # 2 ___
√ # 2
= √ # 2
___ 2
11. 5 √ # 6
_____ - √ # 3
= 5 √ # 6
_____ - √ # 3
· - √ # 3 _____
- √ # 3
= -5 √ ## 18
_______ 3
= -5 √ ## 9 · 2
________ 3
-5(3) √ # 2
________ 3
= -5 √ # 2
12. √ ## 50
____ √ ## 12
= √ ## 50
____ √ ## 12
· √ ## 12 ____
√ ## 12
=
√ ## 600 ______
12
= √ ### 100 · 6
________ 12
= 10 √ # 6
_____ 12
= 5 √ # 6
____ 6
13. √ # 3 ______
- √ ## 21
= √ # 3 ______
- √ ## 21 · - √ ## 21
______ - √ ## 21
= - √ ## 63
______ 21
= - √ ## 9 · 7
_______ 21
= -3 √ # 7
______ 21
= - √ # 7
___
7
14. 6 √ # 7 + 7 √ # 7
= (6 + 7) √ # 7
= 13 √ # 7
15. 5 √ ## 32 - 15 √ # 2
= 5 √ ### 16 · 2 - 15 √ # 2
= 5 · 4 √ # 2 - 15 √ # 2
= 20 √ # 2 - 15 √ # 2
= (20 - 15) √ # 2
= 5 √ # 2
16. 4 √ # 5 + √ ## 245
= 4 √ # 5 + √ ### 49 · 5
= 4 √ # 5 + 7 √ # 5
= (4 + 7) √ # 5
= 11 √ # 5
17. - √ ## 50 + 6 √ # 2
= - √ ### 25 · 2 + 6 √ # 2
= -5 √ # 2 + 6 √ # 2
= (-5 + 6) √ # 2
= √ # 2
PRACTICE AND PROBLEM SOLVING
18. √ ## 49 < √ ## 60 < √ ## 64
7 < √ ## 60 < 8
7.7 2 = 59.29
7.8 2 = 60.84
√ ## 60 ≈ 7.7
19. - √ ## 16 < - √ ## 15 < - √ # 9
-4 < - √ ## 15 < -3
3.8 2 = 14.44
2.9 2 = 15.21
- √ ## 15 ≈ -3.9
20. √ ## 36 < √ ## 47 < √ ## 49
6 < √ ## 47 < 7
6.8 2 = 46.24
6.9 2 = 47.61
√ ## 47 ≈ 6.9
21. √ ## 81 < √ ## 99 < √ ## 100
9 < √ ## 99 < 10
9.8 2 = 96.04
9.9 2 = 98.01
√ ## 99 ≈ 9.9
22. √ ## 162
= √ ### 81 · 2
= 9 √ # 2
23. - √ ## 1 ____ 121
= - 1 ___
11
24. √ ## 50
___ 9
= √ ### 25 · 2
_______ 3
= 5 √ # 2
____ 3
25. -2 √ ## 10 · √ # 8
= -2 √ ## 80
= -2 √ ### 16 · 5
= -2(4) √ # 5
= -8 √ # 5
26. √ ## 288
_____ √ # 8
= √ ### 144 · 2
________ √ ## 4 · 2
= 12 √ # 2
_____ 2 √ # 2
= 6
27. √ ## 85 · √ # 5
= √ ## 425
= √ ### 25 · 17
= 5 √ ## 17
6 Holt McDougal Algebra 2
28. 2 √ ## 126
______ √ ## 14
= 2 √ ### 9 · 14
________ √ ## 14
= 2(3) √ ## 14
_______ √ ## 14
= 6
29. - √ ## 189
= - √ ### 9 · 21
= -3 √ ## 21
30. 2 ___ √ # 3
= 2 ___ √ # 3
· √ # 3 ___
√ # 3
= 2 √ # 3
____ 3
31. 3
√ ## 27 ______
2 √ # 6
= 3 √ ## 27
_____ 2 √ # 6
· √ # 6 ____
√ # 6
= 3
√ ## 162 _______
2 · 6
=
√ ### 81 · 2
_______ 2 · 2
= 9 √ # 2
____ 4
32. - 18 ____
√ # 6
= - 18 ____
√ # 6
· √ # 6 ___
√ # 6
= - 18 √ # 6 ______
6
= -3 √ # 6
33. √ # 11
______ 5 √ ## 132
= √ # 11
______ 5 √ ## 132
· √ ## 132 _____
√ ## 132
= √ ## 1452
______ 5(132)
= √ ### 484 · 3
________ 660
= 22 √ # 3
______ 22 · 30
= √ # 3
___ 30
34. 4 √ # 3 - 9 √ # 3
= (4 - 9) √ # 3
= -5 √ # 3
35. √ ## 112 + √ ## 63
= √ ### 16 · 7 + √ ## 9 · 7
= 4 √ # 7 + 3 √ # 7
= 7 √ # 7
36. √ # 8 - 15 √ # 2
= √ ## 4 · 2 - 15 √ # 2
= 2 √ # 2 - 15 √ # 2
= -13 √ # 2
37. √ ## 12 + 7 √ ## 27
= √ ## 4 · 3 + 7 √ ## 9 · 3
= 2 √ # 3 + 7 · 3 √ # 3
= 2 √ # 3 + 21 √ # 3
= 23 √ # 3
38. √ ## 45 + √ ## 20
= √ ## 9 · 5 + √ ## 4 · 5
= 3 √ # 5 + 2 √ # 5
= 5 √ # 5
39. 5 √ ## 28 - 2 √ # 7
= 5 √ ## 4 · 7 - 2 √ # 7
= 5 · 2 √ # 7 - 2 √ # 7
= 10 √ # 7 - 2 √ # 7
= 8 √ # 7
40. 2 √ ## 48 + 2 √ ## 12
= 2 √ ### 16 · 3 + 2 √ ## 4 · 3
= 2 · 4 √ # 3 + 2 · 2 √ # 3
= 8 √ # 3 + 4 √ # 3
= 12 √ # 3
41. √ ## 150 - 8 √ # 6
= √ ### 25 · 6 - 8 √ # 6
= 5 √ # 6 - 8 √ # 6
= -3 √ # 6
42. Let x represent the size of side of the square.
x · x = 4000
x 2 = 4000
√ # x 2 = √ ## 4000
x ≈ 63.25
The dimensions would be about 63.25 m by 63.25 m.
43. perimeter = 12 √ ## 40
___ 5
= 12 √ # 8
≈ 12(2.8284)
≈ 33.9 cm
44. perimeter = 14 √ ## 90
___ 6
= 14 √ ## 15
≈ 14(3.873)
≈ 54.2 ft
45. perimeter = 14 √ ## 300
____ 6
= 14 √ ## 50
≈ 14(7.07)
≈ 99.0 in.
46. length of diagonal = √ # 2 · length of side
= √ # 2 ·
√ ## 8100
= √ ### 16,200
≈ 127.28 ft
47. The dimensions are about 50 in. by 50 in.; four
canvases would cover a total of 2400 in 2 , which is
approximately 2500 in 2 ; √ ## 2500 = 50, so the sides
are about 50 in.
48. √ ## 900
_____ √ ## 20
= √ ## 45
= √ ## 9 · 5
= 3 √ # 5
49. 3 √ ## 50 · 3 √ # 8
= 9 √ ## 400
= 9(20)
= 180
50. -3 √ # x + √ # 9x
= -3 √ # x + √ #### 3 · 3 · x
= -3 √ # x + 3 √ # x
= 0
51. 2 √ # 5 - 5 √ # 2
cannot simplify further
52. √ ## 25x - 6 √ # x
= √ #### 5 · 5 · x - 6 √ # x
= 5 √ # x - 6 √ # x
= -1 √ # x , or - √ # x
53. 3 √ # 7 + 1
________ √ # 5
= 3 √ # 7 + 1
________ √ # 5
· √ # 5 ___
√ # 5
= √ # 5 (3 √ # 7 + 1)
____________ 5
= 3 √ ## 35 + √ # 5
__________ 5
7 Holt McDougal Algebra 2
54. 4 √ ## 10 - √ ## 90
___________ √ # 2
= 4 √ ## 10 - √ ## 90
___________ √ # 2
· √ # 2 ____
√ # 2
= √ # 2 (4 √ ## 10 - √ ## 90 )
_______________ 2
= 4 √ ## 20 -
√ ## 180 _____________
2
= 4
√ ## 4 · 5 -
√ ### 36 · 5 _________________
2
= 4 · 2 √ # 5 - 6 √ # 5
_____________ 2
= 8 √ # 5 - 6 √ # 5
__________ 2
= 2 √ # 5
____ 2
= √ # 5
55. 4 √ ## 32
_____ √ # 5
= 4 √ ## 32
_____ √ # 5
· √ # 5 ___
√ # 5
= 4 √ ## 160 _______
5
= 4 √ ### 16 · 10
_________ 5
= 4 · 4 √ ## 10
________ 5
= 16 √ ## 10
______ 5
56. √ ## 75x + √ ## 45x
= √ ##### 5 · 5 · 3 · x + √ ##### 5 · 3 · 3 · x
= 5 √ # 3x + 3 √ # 5x
57. side length of a ward =
√ ##### 8,640,000
_________ 24
= √ #### 360,000
= 600 ft
The dimensions of a ward are 600 ft by 600 ft.
58. 10 acres = 10(43,560)
= 435,600 ft 2
side length = √ #### 435,600
= 660 ft
59. 2 mi 2 = 2(27,880,000)
= 55,760,000 ft 2
side length = √ ##### 55,760,000
≈ 7467.3 ft
60. 5 hectare = 5(107,600)
= 538,000 ft 2
side length = √ #### 538,000
≈ 733.5 ft
61. 6.2 km 2 = 6.2(10,760,000)
= 66,712,000 ft 2
side length = √ ##### 66,712,000
≈ 8167.7 ft
62. sometimes true;
possible answer: true when both a and b equal 4,
false when a equals 5 and b equals 3
63. always true;
possible answer: when a = 4 and b = 9;
√ ## 4(9)
_____ √ # 4
= √ ## 36
____ √ # 4
= 6 __
2 = 3 = √ # 9
64. sometimes true;
possible answer: true for a = 2 and b = 1,
false for a = 2 and b = 4
65. no; for example, √ ### 3 + 3 = √ # 6 ≠ 3
66. Possible answer: no; √ # 2 is an irrational number
but is rounded by the calculator. The square of the
rounded number does not equal 2.
67a. t = √ ## h
____ 0.82
t = √ ## 50
____
0.82
t ≈ 7.81 s
b. t = √ ## h
____ 4.89
t = √ ## 50
____
4.89
t ≈ 3.20 s
TEST PREP
68. A
A) √ ## 20 = 2 √ # 5
B) √ # 8 · √ # 5 = 2 √ ## 10
C) 2 √ ## 10
D) 5 √ # 8
____ √ # 5
= 5 √ # 8 · √ # 5
_________ 5
= 2 √ ## 10
69. H
perimeter: 4 √ ## 30
≈ 4(5.48)
≈ 22 m
70. D
1 ___ √ # 2
, 1, √ # 2 , 2
0.707, 1, 1.414, 2
71. −− AC =
√ ##### −− AB
2 +
−− BC 2
−− AC = √ #### 8
2 + 4
2
−− AC = √ #### 64 + 16
−− AC = √ ## 80
−− AC ≈ 8.9 ft
CHALLENGE AND EXTEND
72. a √ # b - 3a √ ## 5ab
_____________ 3 √ # b
= 5 √ # 6 - 3(5) √ #### 5 · 5 · 6
__________________ 3 √ # 6
= 5 √ # 6 - 15 · 5 √ # 6
______________ 3 √ # 6
= 5 - 75
______ 3
= - 70 ___
3
73a. Small right triangle:
a 2 + b
2 = c
2
a 2 + 6
2 = (6 √ # 2 )
2
a 2 + 36 = (6 √ # 2 )(6 √ # 2 )
a 2 + 36 = (36 · 2)
a 2 + 36 = 72
_______ - 36 ____ - 36
a 2 = 36
√ # a 2 = √ ## 36
a = 6 in.
Large right triangle:
a 2 + b
2 = c
2
6 2 + 12
2 = c
2
36 + 144 = c 2
180 = c 2
√ ## 180 = √ # c 2
√ ### 36 · 5 = c 6 √ # 5 in. = c
b. A + = (b × h)
______ 2
= 18 × 6
______
2
= 108
____
2
= 54 in 2
c. perimeter: 18 + 6 √ # 2 + 6 √ # 5 in.
8 Holt McDougal Algebra 2
74. √ ## x
3 y
5 ________
x 2 √ ## 48 y
3
= √ ###### x
2 · x · y
4 · y _______________
x 2 √ ###### 16 · 3 · y
2 · y
= x y
2 √ # xy ________
4 x 2 y √ # 3y
= y √ # xy
______ 4x √ # 3y
= y √ # xy
______ 4x √ # 3y
· √ # 3y ____
√ # 3y
= y √ ## 3x y
2 _______
4x · 3y
= y · y √ # 3x
________ 4x · 3y
= y √ # 3x
_____ 12x
SPIRAL REVIEW, PAGE 26
75. tetrahedron or triangular pyramid
76. cylinder 77. triangular prism
78. -7 < x ≤ 1 79. 1.5 < x < 8
80. 2 ≤ x ≤ 12 81. 3 __ 4 < x < 5 __
2
82. Identity Property of Multiplication
83. Commutative Property of Addition
84. Assoc. Prop. of Mult.
85. Distributive Prop.
1-4 SIMPLIFYING ALGEBRAIC
EXPRESSIONS, PAGES 27–32
CHECK IT OUT!
1a. age = 18 + y
b. 60 s in 1 min; 60 min in 1 h60 × 60 s in 1 hseconds = 3600h
2. x 2 y - x y
2 + 3y
= (2) 2 (5) - (2)( 5)
2 + 3(5)
= 4(5) - 2(25) + 15= 20 - 50 + 15= -15
3. -3(2x - xy + 3y) - 11xy
= -6x + 3xy - 9y - 11xy
= -6x - 8xy - 9y
4a. Let h represent the number of Hawaii packages sold agent will make = 50h + 80(100 - h)
= 50h + 8000 - 80h
= -30h + 8000
b. agent will make = -30h + 8000 = -30(28) + 8000 = -840 + 8000 = $7160
THINK AND DISCUSS
1. Possible answer: An expression with 5 terms will have 4 addition or subtraction symbols, 1 separating each pair of terms.
2. Possible answer: When you add like terms such as 3x + 4x, you can use the Distributive Property to rewrite the sum. In this case, the sum becomes
(3 + 4)x. This expression can be simplified to 7x.
3.
EXERCISES
GUIDED PRACTICE
1. cost = 0.79c 2. area = 8
3. a 2 + b
2 - 2ab
= ( 5) 2 + (8)
2 - 2(5)(8)
= 25 + 64 - 80= 9
4. 3xy __________
x 2 - 9y + 2
= 3(2)(4) _____________
( 2) 2 - 9(4) + 2
= 24 __________ 4 - 36 + 2
= 24 ____ -30
= - 4 __ 5
5. -8a + 9 - 5a + a= -12a + 9
6. -2(2x + y) - 7x + 2y
= -4x - 2y - 7x + 2y
= -11x
7. 1 + (ab - 5a)5 - b 2
= 1 + 5ab - 25a - b 2
8a. total calories used = 9(r) + 7(60 - r) = 9r + 420 - 7r
= 2r + 420
b. calories used = 2r + 420 = 2(20) + 420 = 40 + 420 = 460 Calories
PRACTICE AND PROBLEM SOLVING
9. If angle measures x°, supplementry angle = (180 - x)° 10. bagels = d
____ 0.60
11. 6c - 3 c 2 + d
3
= 6(5) - 3 (5) 2 + 3
3
= 30 - 3(25) + 27= 30 - 75 + 27= -18
12. y 2 - 2x y
2 - x
= (3) 2 - 2(2)( 3)
2 - 2
= 9 - (4)(9) - 2= 9 - 36 -2= -29
13. 3 a 2 b - a b
3 + 5
= 3( 5) 2 (2) - (5) (2)
3 + 5
= 3(25)(2) - 5(8) + 5= 150 - 40 + 5= 115
9 Holt McDougal Algebra 2
14. 2s - t 2 ______
s t 2
= 2(5) - ( 3)
2 _________
(5)( 3) 2
= 10 - 9 ______ 5(9)
= 1 ___ 45
15. -x - 3y + 4x - 9y + 2= 3x - 12y + 2
16. -4(-a + 3b) - 3(a - 5b)= 4a - 12b - 3a + 15b
= a + 3b
17. 5 - (3m + 2n)= 5 - 3m - 2n
18. x(4 + y) - 2x(y + 7)= 4x + xy -2xy - 14x
= -10x -xy
19a. Let m represent the number of muffins. total time = 30(m) + 50(10 - m) = 30m + 500 - 50m
= -20m + 500
b. time = -20m + 500 = -20(2) + 500 = -40 + 500 = 460 min = 7 h 40 min
20. -a( a 2 + 2a - 1)
= - a 3 - 2 a
2 + a
= - (2) 3 - 2( 2)
2 + 2
= -(8) - 2(4) +2= -8 - 8 + 2= -14
21. ( 2g - 1) 2 - 2g + g
2
= (2g - 1)(2g - 1) - 2g + g 2
= 4 g 2 - 2g - 2g + 1 - 2g + g
2
= 5 g 2 - 6g + 1
= 5( 3) 2 - 6(3) + 1
= 5(9) - 18 + 1= 45 - 18 + 1= 28
22. u 2 - v
2 _______
uv
= u
2 ___
uv - v
2 ___
uv
= u __ v - v __
u
= 4 __ 2 - 2 __
4
= 2 - 1 __ 2
= 3 __ 2
23. a
2 - 2 ( b
2 - a) _____________
2 + a
= a
2 - 2 b
2 + 2a ____________
2 + a
= ( 3)
2 - 2(5 )
2 + 2(3) ________________
2 + 3
= 9 - 2(25) + 6
____________ 5
= 9 - 50 + 6
__________ 5
= - 35
___ 5
= -7
24. x (x + 3 ) 2 x
2 + 9 x
2 + 6x + 9
1 (1 + 3) 2 1
2 + 9 1
2 + 6(1) + 9
= (4) 2 = 1 + 9 = 1 + 6 + 9
= 16 = 10 = 16
2 (2 + 3) 2 2
2 + 9 2
2 + 6(2) + 9
= ( 5) 2 = 4 + 9 = 4 + 12 + 9
= 25 = 13 = 25
3 (3 + 3) 2 3
2 + 9 3
2 + 6(3) + 9
= ( 6) 2 = 9 + 9 = 9 + 18 + 9
= 36 = 18 = 36
4 (4 + 3) 2 4
2 + 9 4
2 + 6(4) + 9
= ( 7) 2 = 16 + 9 = 16 + 24 + 9
= 49 = 25 = 49
Therefore (x + 3) 2 = x
2 + 6x + 9.
25. x (x - 4) 2 x
2 + 16 x
2 - 8x + 16
1 (1 - 4) 2 1
2 + 16 1
2 - 8(1) + 16
= (- 3) 2 = 1 + 16 = 1 - 8 + 16
= 9 = 17 = 9
2 (2 - 4) 2 2
2 + 16 2
2 - 8(2) + 16
= (-2) 2 = 4 + 16 = 4 - 16 + 16
= 4 = 20 = 4
3 (3 - 4) 2 3
2 + 16 3
2 - 8(3) + 16
= (-1 ) 2 = 9 + 16 = 9 - 24 + 16
= 1 = 25 = 1
4 (4 - 4) 2 4
2 + 16 4
2 - 8(4) + 16
= 0 = 16 + 16 = 16 - 32 + 16
= 32 = 0
Therefore (x - 4) 2 = x
2 - 8x + 16.
26a. Let m represent an m-minute commercial first Super Bowl = 85,000m
Super Bowl XXXVIII = (2,300,000)(2)m = 4,600,000m
b. length of commercial: 85,000m = 170,000
85,000m
________ 85,000
= 170,000
_______ 85,000
m = 2 cost during Super Bowl XXXVIII: cost = 4,600,000(2) = $9,200,000 Super Bowl XXXVIII costs about 54 times as much.
10 Holt McDougal Algebra 2
c. first Super Bowl: 60,000,000 viewers
cost per 1000 viewers = $85,000
_______ 60,000
m ≈ 1.42m
Super Bowl XXXVIII: 800,000,000 viewers
cost per 1000 viewers = $4,600,000
__________ 800,000
m ≈ 5.75m
d. first Super Bowl cost = 1.42(2) = $2.84 Super Bowl XXXVIII cost = 5.75(2) = $11.50 Super Bowl XXXVIII costs about 4 times as much
per 1000 viewers.
27. 2a + a + 2b + (a + b) + a + (2a + b)= 3a + 2b + a + b + a + 2a + b= 7a + 4b
28. (2 x 2 - 5) + (9 - x) + (x + 2) + (3x - 6)
+ ( x 2 - 4) + (x + 3) + x + x
= 2 x 2 - 5 + 9 - x + x + 2 + 3x - 6 + x
2 - 4 + x
+ 3 + 2x
= 3 x 2 + 6x - 1
29a. total budgeted cost = 100d + 275(15 - d) = 100d + 4125 - 275d
= -175d + 4125
b. budgeted cost = -175(5) + 4125 = -875 + 4125 = $3250
c. Each additional day they stay with relatives saves 275 - 100 = $175 per day.
30a. time in minutes = 119n
b. time in hours = 119n _____
60
c. time = 119n _____
60
= 119(30)
_______ 60
= 3570
_____ 60
= 59.5 hr
d. time for 1 orbit = 119(1)
______ 60
= 1.98 − 3 h
hours in 1 week = 24 × 7 = 168 h
number of orbits in a week = 168
_____ 1.98
− 3
≈ 84.7 So, almost 85 orbits would be made.
31. x y = -2 x 2 + 5x - 7
-3 -2 (-3) 2 + 5(-3) - 7 = -40
-2 -2 (-2) 2 + 5(-2) - 7 = -25
0 -2 (0) 2 + 5(0) - 7 = -7
2 -2 (2) 2 + 5(2) - 7 = -5
3 -2 (3) 2 + 5(3) - 7 = -10
32. x y = -
3x + 9 ______
x 2 - 1
-3 - 3(-3) + 9
_________ (-3)
2 - 1
= 0
-2 - 3(-2) + 9
_________ (-2)
2 - 1
= -1
0 - 3(0) + 9
_______ (0)
2 - 1
= 9
2 - 3(2) + 9
_______ (2)
2 - 1
= -5
3 - 3(3) + 9
_______ (3)
2 - 1
= - 9 __ 4
33. x y = x 3 - 11x + 1
-3 (-3) 3 - 11(-3) + 1 = 7
-2 (-2) 3 - 11(-2) + 1 = 15
0 (0) 3 - 11(0) + 1 = 1
2 (2) 3 - 11(2) + 1 = -13
3 (3) 3 - 11(3) + 1 = -5
34. A; the minus sign was not distributed in the second step.
35. Distributive Property; possible answer: the Distributive Property says that multiplying by a sum is the same as adding products.
TEST PREP
36. DA) -2x(1 - 3x) = -2x + 6 x
2
B) 2(3x - 1)x = (6x - 2)x = 6 x 2 - 2x
C) (3x - 1)2x = 6 x 2 - 2x
D) 6x 2 + 2x
37. GF) 12 in. = 1 ftG) 60 min = 1 hH) 7 days = 1 wkJ) 36 in. = 1 yd
38. C3x( y - 1)
2
= 3(4) (3 - 1) 2
= 12 (2) 2
= 12(4)= 48
CHALLENGE AND EXTEND
39. 2a - 5 = 11 ______ + 5 ___ + 5 2a = 16
2a ___
2 = 16 ___
2
a = 8
40. 2a - 5 = -5 ______ + 5 ___ + 5 2a = 0
2a
___ 2 = 0 __
2
a = 0
41. 2a - 5 = 39 ______ + 5 ___ + 5 2a = 44
2a
___ 2 = 44 ___
2
a = 22
42. 2a - 5 = 225 ______ + 5 ___ + 5 2a = 230
2a
___ 2 = 230 ____
2
a = 115
11 Holt McDougal Algebra 2
43a. x y =
3(x + 2) 2 ____________
(x - 1)(x - 3)
0 3(0 + 2)
2 ____________
(0- 1)(0 - 3) = 4
1 3(1 + 2)
2 ____________
(1 - 1)(1 - 3) = undefined
2 3(2 + 2)
2 ____________
(2 - 1)(2 - 3) = -48
3 3(3 + 2)
2 ____________
(3 - 1)(3 - 3) = undefined
4 3(4 + 2)
2 ____________
(4 - 1)(4 - 3) = 36
5 3(5 + 2)
2 ____________
(5 - 1)(5 - 3) = 147 ____
8
b. The expression cannot be evaluated for x = 1, x = 3.
c. { x | x ≠ 1 and x ≠ 3 }
SPIRAL REVIEW
44. triangular prism 45. square pyramid
46. !, #, $ 47. !
48. ! 49. irrational
50. √ && 52 ___
25
= √ &&& 4 · 13 _____
25 · 1
= 2 __ 5 √ && 13
51. √ && 24 + √ & 6
= √ && 4 · 6 + √ & 6
= 2 √ & 6 + √ & 6
= 3 √ & 6
52. 4 √ && 27
_____ 18
= 4 √ && 9 · 3
_______ 18
= 4 · 3 √ & 3
_______ 18
= 12 √ & 3
_____ 18
= 2 √ & 3
____ 3
53. √ && 28 · √ & 7
= √ &&& 4 × 7 · √ & 7
= 2 √ & 7 √ & 7 = 2 · 7= 14
1-5 PROPERTIES OF EXPONENTS,
PAGES 34–41
CHECK IT OUT!
1a. (2a) 5 = (2a)(2a)(2a)(2a)(2a)
b. 3b 4 = 3 · b · b · b · b
c. -(2x - 1) 3 y 2
= -(2x - 1)(2x -1)(2x - 1) · y · y
2a. ( 1 __ 3 ) -2
= ( 3 __ 1
) 2
= (3) 2 = 9
b. (-5) -5
= 1 _____ (-5) 5
= 1 ___________________ (-5)(-5)(-5)(-5)(-5)
= - 1 _____ 3125
3a. ( 5x 6 ) 3
= 5 3 x (6)(3)
= 125 x 18
b. (-2 a 3 b) -3
= 1 ________ (-2 a 3 b)
3
= 1 ___________ (-2) 3 a (3)(3) b 3
= - 1 _____ 8 a 9 b 3
4a. 2.325 × 10 6 __________
9.3 × 10 9
= 0.25 × 10 -3
= 2.5 × 10 -4
b. (4 × 10 -6 ) (3.1 × 10 -4 ) = 12.4 × 10 -10 = 1.24 × 10 -9
5. speed of light: 3 × 10 5 km ___ s
= 3 × 10 5 km ___
s ( 10
3 m _____ 1 km
) ( 60 s _____ 1 min
) = 1.8 × 10 10 m ____
min
time = distance _______ speed
= 1.5 × 10 11 m ____________
1.8 × 10 10 m ___ min
= 0.8 − 3 × 10 min ≈ 8.33 min
THINK AND DISCUSS
1. Possible answer: The Product and Quotient of Powers Properties both require the same base.
2. Possible answer: Move the decimal point so that there is one nonzero digit in front of it. Use the number of places moved for the exponent of 10. If you moved the decimal point left, use a positive exponent. If you moved the decimal point right, use a negative exponent.
3.
EXERCISES
GUIDED PRACTICE
1. Possible answer: a number between 1 and 10 multiplied by an integer power of 10.
2. 4 (a - b) 2 = 4(a - b)(a - b)
12 Holt McDougal Algebra 2
3. (12xy) 4
= (12xy)(12xy)(12xy)(12xy)
4. - s 3 (-2t)
5
= -s · s · s(-2t)(-2t)(-2t)(-2t)(-2t)
5. (- 1 __ 2 d)
3
= (- 1 __ 2
d) (- 1 __ 2 d) (- 1 __
2 d)
6. (- 3 __ 5 ) -2
= (- 5
__ 3 )
2
= (- 5
__ 3 ) (-
5 __
3 )
= 25
___ 9
7. 5 0
= 1
8. ( 2 __ 3 ) -3
= ( 3 __ 2
) 3
= ( 3 __ 2
) ( 3 __ 2 ) ( 3 __
2 )
= 27 ___ 8
9. 10 -1
= 1 ___ 10
10. (- 3 a 2 b
3 ) 2
= (-3) 2 a
(2)(2) b
(3)(2)
= 9 a 4 b
6
11. c 3 d
2 ( c -2
d 4 )
= c d 6
12. 5u v
6 ____
u 2 v
2
= 5 u -1
v 4
= 5v
4 ___
u
13. 10 ( y 5 __
x 2 )
2
= 10 ( y (5)(2)
_____
x (2)(2)
)
= 10y
10 _____
x 4
14. - 2s -3
t(7 s -8
t 5 )
= -14 s -11
t 6
= - 14 t
6 ____
s 11
15. -4m (m n 2 ) 3
= -4m( m 3 n
(2)(3) )
= -4 m 4 n
6
16. (4b)
2 _____
2b
= 4
2 b
2 ____
2b
= 16 b
2 ____
2b
= 8b
17. x -1
y -2
______
x 3 y -5
= x -4
y 3
= y
3 __
x 4
18. (2.2 × 10 5 ) (4.5 × 10
11 )
= 9.9 × 10 16
19. 7.8 × 10
8 __________
2.6 × 10 -3
= 3 x 10 11
20. 16 × 10
-3 _________
4.0 × 10 4
= 4 × 10 -7
21. width of hair = 80 microns or 8.0 × 10 -5
m
width of hair
_____________________ width of nanoguitar string
= 8.0 × 10
-5 __________
2.0 × 10 -7
= 4.0 × 10 2
= 4.0 × 100
= 400
Therefore, 400 nanoguitar strings would have the
same width as a human hair
PRACTICE AND PROBLEM SOLVING
22. (m + 2n) 3
= (m + 2n)(m + 2n)(m + 2n)
23. 5 x 3
= 5 · x · x · x
24. (-9fg) 3 h
4
= (-9fg)(-9fg)(-9fg) · h · h · h · h
25. 2a (- b 2 - a)
2
= 2a ( -b 2 - a) (- b
2 - a)
26. (-4) -2
= 1 _____ (-4)
2
= 1 ___ 16
27. (- 3 __
4 ) -1
= (- 4 __
3 )
1
= - 4 __ 3
28. (- 5 __
2 ) -3
= (- 2 __
5 )
3
= (- 2 __
5 ) (-
2 __
5 ) (-
2 __
5 )
= - 8 ____
125
29. - 6 0
= -(1)
= -1
30. -100 s
3 t -5
_________
25 s -2
t 6
= -4 s 5 t -11
= - 4 s
5 ___
t 11
31. (- x 4 y
2 ) 5
= - x (4)(5)
y (2)(5)
= - x 20
y 10
32. (16 u 4 v
6 ) -2
= 1 _________ (16 u
4 v
6 )
2
= 1 ________ 256 u
8 v
12
33. 8 a 2 b
5 (-2 a
3 b
2 )
= -16 a 5 b
7
34. (3.2 × 10 6 ) (1.7 × 10
-4 )
= 5.44 × 10 2
35. 5.1 × 10
4 __________
3.4 × 10 -5
= 1.5 × 10 9
36. (6.8 × 10 3 ) (9.5 × 10
5 )
= 64.6 × 10 8
= 6.46 × 10 9
37. 5.02 × 10
11 __________
5.4 × 10 9
≈ 0.930 × 10 2
= 93 s or 1.55 min
13 Holt McDougal Algebra 2
38. 0.00173 g ___
kg = 1.73 × 10
-3
0.02 kg = 2 × 10 -2
Smallest amount of venom that will be fatal to the
mouse is
(1.73 × 10 -3
) (2 × 10 -2
) = 3.46 × 10
-5 g
39. 8 2 = ( 2
3 )
2 = 2
6 ; 4
1 = 2
2 ;
2 5 ; 16
-2 = ( 2
4 ) -2
= 2 -8
The order is: 16 -2
, 4 1 , 2
5 , 8
2
40. 2 -1
; -4 3 = - ( 2
2 )
3 = - 2
6 ;
4 2 = ( 2
2 )
2 = 2
4 ; 8
-2 = ( 2
3 ) -2
= 2 -6
The order is: - 4 3 , 8
-2 , 2
-1 , 4
2
41. - 8 2 = - ( 2
3 ) 2 = - 2
6 ; 4
0 = ( 2
2 ) 0 = 2
0
16 1 = 2
4 = 2
4 ; 2
-2
The order is: - 8 2 , 2
-2 , 4
0 , 16
1
42. 1.3 × 10 15
× 128
= 166.4 × 10 15
= 1.664 × 10 17
oz of water in Lake Michigan
The faucet’s leaking rate per year is
1.5 oz
____
min × 60 min = 90
oz ___
h
90 oz
___ h × 24 h = 2160
oz
____
day
2160 oz
____
day × 365 days = 788,400 or 7.884 × 10
5 oz
___
yr
The number of years it will take for the amount of
water that is leaking to be equal to the amount of
water in Lake Michigan is
1.664 × 10
17
___________
7.884 × 10 5
≈ 0.211 × 10 12
= 2.11 × 10 11
yr
43. V = × w × h
= m n 2 · m
3 n · 3mn
= 3 m 5 n
4
44. V = π r 2 h
= π ( a 2 b)
2 (abc)
= π ( a 4 b
2 ) (abc)
= π a 5 b
3 c
45. 27 x
3 y ______
18 x 2 y
4
= 3x y
-3 _____
2
= 3x
___ 2 y
3
46. ( 3 a 3 b ______
2 a -1
b 2 )
2
= 3
2 a
(3)(2) b
2 ____________
2 2 a
(-1)(2) b
(2)(2)
= 9 a
6 b
2 ______
4 a -2
b 4
= 9 a
8 ___
4 b 2
47. 12 a 0 b
5 (-2 a
3 b
2 )
= -24 a 3 b
7
48. 72 a
2 b
3 ________
-24 a 2 b
5
= -3 a 0 b -2
= - 3 __
b 2
49. ( 5mn _____
-3 m 2 ) -2
= ( -3 m 2 _____
5mn )
2
= (-3)
2 m
(2)(2) __________
5 2 m
2 n
2
= 9 m
4 _______
25 m 2 n
2
= 9 m
2 ____
25 n 2
50. 6 x 5 y
3 (-3 x
2 y -1
)
= -18 x 7 y
2
51. 1 yd = 36 in.
1 yd 2 = 36 in. × 36 in.
1 yd 2 = 1296 in
2
52. 1 m = 100 cm
1 m 2 = 100 cm × 100 cm
1 m 2 = 10,000 cm
2
53. 1 ft = 12 in.
1 ft 3 = 12 in. × 12 in. × 12 in.
1 ft 3 = 1728 in
3
54. 1 km = 1000 m
1 km 3 = 1000 m × 1000 m × 1000 m
1 km 3 = 10
9 m
3
55a. speed = distance
_______ time
= 384,500
_______ 102.75
≈ 3742 km ___ h
b. 3742 km
________ 1 h
= 3742 km
________ 60 min
≈ 62.367 km
_________ 1 min
= 62.367 km
_________ 60 s
= 1.03945 km ___ s is the speed of Apollo 11.
Future spaceships will travel
3 × 10
5 ________
1.03945 = 288,608 times as fast as Apollo 11.
c. time = distance
_______ speed
= 384,500
_______ 3 × 10
5
≈ 1.28 s
56. -9 a 2 b
6 (-7a b
-4 )
= 63 a 3 b
2
57. 14 x
-2 y
3 ________
-8 x -5
y 5
= - 7 x
3 y -2
______
4
= - 7 x
3 ___
4 y 2
58. - ( 20 x 6 ____
2 x 2 )
3
= - 20
3 x
(6)(3) ________
2 3 x
(2)(3)
= - 8000 x
18 _______
8 x 6
= - 1000x 12
14 Holt McDougal Algebra 2
59. (10 x -2
y 0 z -3
) 2
= 100 x -4
z -6
= 100
____ x
4 z
6
60. (-3 a 2 b -1
) -3
= 1 __________
(-3 a 2 b -1
) 3
= 1 _______________ (-3)
3 a
(2)(3) b
(-1)(3)
= 1 _________ -27 a
6 b -3
= - b
3 ____
27 a 6
61. (8 m 4 n -2
) (- 3m -2
n) 0
= (8 m 4 n -2
)
= 8 m
4 ____
n 2
62. China;
1.25 × 10
9
_________
9.60 × 10 6
= 130.2 people
______ mi
2
63. Laos;
6.07 × 10
6 _________
2.37 × 10 5
= 25.6 people
______ mi
2
64. Thailand;
6.49 × 10
7 _________
5.14 × 10 5
= 126.3 people
______ mi
2
65. Vietnam;
7.88 × 10
7 _________
1.28 × 10 5
= 615.6 people
______ mi
2
66. Cambodia;
1.34 × 10
7 _________
1.81 × 10 5
= 74.0 people
______ mi
2
67. 1.2 beats
_____ s × 60 s = 72
beats _____
min
72 beats
_____
min
× 60 min = 4320 beats
_____
h
4320 beats
_____
h × 24 h = 103,680
beats
_____
day
103,680 beats
_____
day
× 365 day = 37,843,200 beats
_____
yr
37,843,200 beats
_____
yr
× 75 yr = 2,838,240,000 beats
______ lifetime
or ≈ 2.84 × 10 9 beats in 75 years.
68. 16 breaths
_______ min
× 60 min = 960 breaths
_______
h
960 breaths
_______
h × 24 h = 23,040
breaths _______
day
23,040 breaths
_______ day
× 365 day = 8,409,600 breaths
_______ yr
8,409,600 breaths
_______ yr
× 75 yr = 630,720,000 breaths
_______ lifetime
or ≈ 6.3 × 10 8 breaths in a lifespan of 75 years.
69. 254 hairs
_____ cm
2 × 500 cm
2 = 127,000
hairs _____
head
or 1.27 × 10 5 hairs on a human head.
70. Power of a Power Property
71. Power of a Product Property or Power of a Power
Property
72. Quotient of Powers Property
73. Power of a Quotient Property or Power of a Power
Property
74. 1 million = 10 6 , so 3.8 million = 3.8 × 10
6 .
The word million can be represented by the
expression 10 6 .
75. Possbile answer: 0 0 = 0
(2 - 2) =
0 2 __
0 2 =
0 __
0
but division by zero undefined.
76. (3.7 × 10 -3
) (8.1 × 10 -5
) = 2.997 × 10
-7
77. 2.05 × 10
-8 ___________
3.0 × 10 6
= 6.5 × 10 -15
78. (4.75 × 10 2 ) (4.2 × 10
-7 )
= 1.995 × 10 -4
79. 8.4 × 10
9 __________
2.4 × 10 -5
= 3.5 × 10 14
80. 17.068 × 10
-4 _____________
6.8 × 10 3
= 2.51 × 10 -7
81. (1.83 × 10 13
) (6.2 × 10 10
) = 1.1346 × 10
24
82. Possible answer: First compare exponents.
Since 9 > 8, 1.23 × 10 9 is greater than 4.56 × 10
8 .
If the exponents are equal, compare the initial
factors. Since 1.23 < 4.56, 1.23 × 10 7 is less than
4.56 × 10 7 .
TEST PREP
83. C 84. J
85. C 86. J
a
4 b -3
_____
a 2 c
0 = a
2 b -3
= a
2
__
b 3
CHALLENGE AND EXTEND
87. ( 7.82 × 10 6 _________
5.48 × 10 8 )
2
≈ (1.427 × 10 -2
) 2
= 1.427 2 × 10
(-2)(2)
= 2.0363 × 10 -4
88. (6.18 × 10
7 ) (2.05 × 10
8 )
2
= ( 12.669 × 10 15
) 2
= (1.2669 × 10 16
) 2
= 1.2669 2 × 10
(16)(2)
≈ 1.605 × 10 32
89. Possible answer: ( 1 __ 2 ) -2
, (0.7) -2
, (- 2 __ 5 ) -2
;
numbers between -1 and 1, excluding 0, are
greater than 1 when raised to the exponent -2.
90. Possible answer: 2 3 < 3
2 , 1
3 < 3
1 , 0
2 < 2
0 ;
3 4 > 4
3 , 2
5 > 5
2 , 4
5 > 5
4
15 Holt McDougal Algebra 2
SPIRAL REVIEW
91. Probability of rock, paper, or scissors is the same:
P (r) = P (p) = P (s) = 1 __ 3
;
P(same choice) = P (r, r) + P (p, p) + P (s, s)
= 1 __ 3 · 1 __
3 + 1 __
3 · 1 __
3 + 1 __
3 · 1 __
3
= 3 __ 9 = 1 __
3
92. 1 __ 3 · 3 = 1 93. 4(-3 + 8) = -12 + 32
94. 0 = √ & 7 + (- √ & 7 )
95. 2mn ____________
n 2 - 2n + 5m
= 2(3)(-1)
__________________ (-1) 2 - 2(-1) + 5(3)
= -6 __________ 1 + 2 + 15
= -6 ___ 18
= - 1 __ 3
96. 2x (9y - x 2 ) = 2(-3) 9(10) - (-3) 2 = -6(90 - 9)= -6(81)= -486
READY TO GO ON? PAGE 43
1. -3 1 __ 3 = -3.3
− 3 , √ & 5 ≈ 2.23, - 4 __
5 = -0.8
The order is: -3 1 __ 3
, - 4 __ 5 , 0.
−− 75 , √ & 5 , 2.5
-3 1 __ 3 : ', ); - 4 __
5 : ', );
0. −− 75 : ', ); √ & 5 : ', irrational;
2.5: ',)
2. √ & 3 ≈ 1.732, - π __ 2
≈ -1.57, 5 __
6 = 0.8
− 3
The order is: -2, - π __ 2 , -1.
−− 15 , 5 __
6 , √ & 3
-2: ', ), *; - π __ 2
: ', irrational;
-1. −− 15 : ', );
5 __
6 : ', );
√ & 3 : ', irrational
3. [-4, 2) 4. {x | x < -2 or x > 0}
5. Distributive Property 6. Additive Identity Prop.
7. Assoc. Prop. of Mult.
8. 12% of $250 = (0.12)(250) = (0.1 + 0.02)(250) = (0.1)(250) + (0.02)(250) = 25 + 2(2.5) = 25 + 5 = $30
9. 75 ft 2 dance floor: side size = √ && 75 ≈ 8.7 ft 125 ft 2 dance floor: side size = √ && 125 ≈ 11.2 ft 150 ft 2 dance floor: side size = √ && 150 ≈ 12.2 ft The 75 ft 2 dance floor is the largest that would fit in an 11 ft by 13 ft room.
10. - √ && 72
= - √ &&& 36 · 2
= -6 √ & 2
11. 5 √ && 12 + 9 √ & 3
= 5 √ && 4 · 3 + 9 √ & 3
= 5 · 2 √ & 3 + 9 √ & 3
= 10 √ & 3 + 9 √ & 3
= 19 √ & 3
12. -4 √ && 10
_______ √ & 2
= -4 √ && 10 ___ 2
= -4 √ & 5
13. √ && 32 · √ & 6
= √ && 192
= √ &&& 64 · 3
= 8 √ & 3
14. a 2 __
3 + ab
___ 4
= (3)
2 ____
3 +
3(-4) _____
4
= 9 __ 3
- 12 ___ 4
= 3 - 3= 0
15. d 2 ____
2cd
= (2)
2 _______
2(-1)(2)
= 4 ___ -4
= -1
16. 2 x 2 - 3y + 5x - x 2
= 6 x 2 - 3y
17. 3(x + 2y) - 5x + y= 3x + 6y - 5x + y= -2x + 7y
18. ( x 11 y -2 ) 4
= x (11)(4) y (-2)(4)
= x 44 y -8
= x 44 ____
y 8
19. -3 s 3 t 2 ______
s -2 t 8
= -3 s 5 t -6
= -3 s 5 _____
t 6
20. 4 ( a 2 b 6 ) -3
= 4 _______ ( a 2 b 6 ) 3
= 4 _________ a (2)(3) b (6)(3)
= 4 _____ a 6 b 18
21. ( m 4 ________
-5 m -2 n 3 )
2
= m (4)(2) _______________
(-5) 2 m (-2)(2) n (3)(2)
= m 8 ________
25 m -4 n 6
= m 12 ____
25 n 6
22. 4.515 × 10 26 ___________
6.02 × 10 23
= 0.75 × 10 3 = 750 moles
1-6 RELATIONS AND FUNCTIONS,
PAGES 44–50
CHECK IT OUT!
1. D: {-2, -1, 0, 1, 2, 3}R: {-3, -2, -1, 0, 1, 2}
2a. function; There is only one price for each size.
b. not a function; Two different carts with the same number of items could cost different amounts.
3a. function
b. not a function; A vertical line can pass through (1, 2) and (1, -2).
16 Holt McDougal Algebra 2
THINK AND DISCUSS
1. Possible answer: ordered pairs, table, mapping
diagram, and graph
2. Possible answer: any point that is above another
on a vertical line has the same input value but a
different output value, such as (2, 4) and (2, 1).
3.
EXERCISES
GUIDED PRACTICE
1. range 2. D: {0, 1, 2}
R: {-2, -1, 0, 1, 2}
3. D: {2000, 2001, 2002, 2003}
R: {5.39, 5.65, 5.80, 6.03}
4. function; Each value in the domain is mapped to
only one value in the range.
5. not a function; Each value for car model can be
mapped to multiple car colors.
6. not a function; Possible answer: a vertical line can
be drawn through (2, 1) and (2, 0).
7. function
8. not a function; Possible answer: a vertical line can
pass through (2, 2) and (2, -2).
PRACTICE AND PROBLEM SOLVING
9. D: {Irene, Anna, Lea, Kate}
R: {12, 16, 22}
10. D: {-3, 2, 3, 4}
R: {-2, 1, 3, 4}
11. function; Each value in the domain is mapped to
only one value in the range.
12. not a function; The value 3 is mapped onto two
values, 1 and 0.
13. not a function; Possible answer: (1, 1) and (1, -1).
14. function 15. function
16. D: {-5, 0, 5}
R: {-5, 0, 5}
17. D: {-2, -1, 0, 1, 2}
R: {-2, 0, 2}
18. D: {-2, -1, 1, 3}
R: {-3, 0, 3}
19. D: {jumbo, extra large, large, medium}
R: {1.75, 2, 2.25, 2.5}
20. D: {e, n, s, v}
R: {5, 14, 19, 22}
21a. function; Each year has exactly 5 states for which
new quarters are released.
b. function; Each state’s quarter is produced during
one year.
c. not a function; Each year releases quarters for
5 different states.
d. function; Each year has exactly 5 states for which
new quarters are released.
e. not a function; The number of quarters stays the
same each year, while the year changes.
22. D: {-1, 0, 1, 2, 3}
R: {-1, 1, 3}
function; For every x-value there is only one y-value.
23. D: {a, b, c, d}
R: {1, 2, 4}
function; For each letter there is only one
corresponding number
24. D: {7}
R: {1, 2, 3, 4, 6}
not a function; The domain value 7 is mapped onto
5 range values.
25. D: {1, 3, 5, 7, 9}
R: {3}
function; For every x-value there is only one y-value.
26. D: {-3, -1, 0, 3}
R: {-4, -3, -2, -1, 0}
not a function; The value 0 is mapped onto 2 range
values.
27. D: {3, 4, 5, 6, 7}
R: {-1, 2, 3}
function; For every x-value there is only one y-value.
17 Holt McDougal Algebra 2
28. D: {January, February, March, April, May, June,
July, August, September, October, November,
December}
R: {28, 30, 31}
function; Each month in the domain is mapped to
one value in the range.
29. D: {Monday, Tuesday, Wednesday, Thursday,
Friday, Saturday, Sunday}
R: {24}
function; Each day has only one number of hours.
30a. a diamond or a square
b. No; (0, 20) and (0, -20) are on the same vertical
line.
c. D: {-20, 0, 20} d. R: {-20, 0, 20}
31. B to A is a function; Each person has one date of
birth, but a date may have several people born on it.
32. both are functions; Each person has a unique
thumbprint and each thumbprint belongs to a unique
person.
33. A to B is a function; Each area code belongs to a
single state, while a state can have several area
codes.
34. both are functions; the amount of sales tax is unique
to the purchase total.
35. B to A is a function; The percentage of tax remains
the same while the total of the purchase changes.
36. B to A is a function; Each player has a single jersey
number, but the same jersey number may belong to
different players.
37. Both are functions; Each player has a unique
number, and each number belongs to one player.
38. Statement A is incorrect; Possible answer: the input
value 0 is paired with 2 output values, which violates
the definition of a function.
39. No; the relation is not a function. One input gauge
can produce 2 output values.
40. Lengths increase in increments of 1 __ 4 inch;
the relation is a function
41a. Yes, the relation is a function.
b. It is a function; As size increases, number per 1 lb
decreases, so average weight increases.
c. size average weight
2d 16
____ 876
≈ 0.0183 oz
3d 16 ____ 568
≈ 0.0282 oz
4d 16 ____ 316
≈ 0.0506 oz
5d 16 ____ 271
≈ 0.0590 oz
6d 16 ____ 181
≈ 0.0884 oz
42. No; Possible answer: switching the domain and
range of the function {(0, 2), (1, 2)} results in
{(2, 0), (2, 1)}. There is more than one output for the
input 2. The resulting relation is not a function.
43. Possible answer: Set of ordered pairs: Look for a
duplicate x-coordinate; Mapping diagram: Look for
2 arrows starting at one domain value; Graph:
Use the vertical-line test.
TEST PREP
44. B
-2 in the domain is mapped twice.
45. F 46. D
CHALLENGE AND EXTEND
47. for the set to be a function, the elements in the
domain cannot be equal, D: {a, -a, 2a, a 2 }, so
a ≠ - a ___ + a ___ + a
2a ≠ 0
a ≠ 0
a ≠ 2a
____ - 2a ____ - 2a
-a ≠ 0
a ≠ 0
a ≠ a 2
____ - a 2 ____ - a
2
a(1 - a) ≠ 0
a ≠ 0 or 1 - a ≠ 0
a ≠ 1
-a ≠ 2a
____ - 2a ____ - 2a
-3a ≠ 0
a ≠ 0
-a ≠ a 2
____ - a 2 ____ - a
2
-a(1 + a) ≠ 0
-a ≠ 0 or 1 + a ≠ 0
a ≠ 0 a ≠ 1
2a ≠ a 2
____ - a 2 ____ - a
2
a(2 - a) ≠ 0
a ≠ 0 or 2 - a ≠ 0
a ≠ 2
a ≠ {-1, 0, 1, 2} and b ∈ !
48. Not one to one; each y-value may correspond to
more than one x-value.
49. One to one; each length in feet corresponds to only
one length in inches.
50. For the set to be a one-to-one function, the elements
in the range cannot be equal, R: b, ab,
ab ___
2 , so
b ≠ ab
___ - b ___ - b
0 ≠ b(a - 1)
b ≠ 0 or a - 1 ≠ 0
a ≠ 1
b ≠ ab
___ 2
2b ≠ 2 ( ab ___
2 )
2b ≠ ab
____ - 2b ____ - 2b
0 ≠ b(a - 2)
b ≠ 0 or a - 2 ≠ 0
a ≠ 2
ab ≠ ab
___ 2
2(ab) ≠ 2 ( ab ___
2 )
2ab ≠ ab
____ - ab ____ - ab
ab ≠ 0
a ≠ 0 or b ≠ 0
a ≠ {0, 1, 2} and b ≠ {0}
18 Holt McDougal Algebra 2
SPIRAL REVIEW
51. P = 2 + 2w
= 2(50) + 2(94) = 100 + 188 = 288 ft
52. A = × w = 50 × 94 = 4700 ft
2
53. A = π r 2
= π (6) 2
= 36π ≈ 113.1 ft
2
54. √ && 36 < √ && 42 < √ && 49
6 < √ && 42 < 7
6.4 2 = 40.96
6.5 2 = 42.25
√ && 42 ≈ 6.5
55. √ && 16 < √ && 22 < √ && 25
4 < √ && 22 < 5
4.6 2 = 21.16
4.6 2 = 22.09
√ && 22 ≈ 4.7
56. - √ & 9 < - √ & 8 < - √ & 4
-3 < - √ & 8 < -2
2.8 2 = 7.84
2.9 2 = 8.41
- √ & 8 ≈ -2.8
57. √ && 81 < √ && 90 < √ && 100
9 < √ && 90 < 10
9.4 2 = 88.36
9.5 2 = 90.25
√ && 90 ≈ 9.5
58. (-3 y 4 ) 3
= (-3) 3 y
(4)(3)
= -27 y 12
59. (10 w
2 ) 2 _______
5 w 5
= 10 2 w
(2)(2) ________
5 w 5
= 100 w
4 ______
5 w 5
= 20
___ w
60. (4 c 6 d
2 ) 2
= 4 2 c
(6)(2) d
(2)(2)
= 16 c 12
d 4
61. ( x 3 __
z )
7
= x (3)(7)
_____ z
7
= x 21
___ z
7
1-7 FUNCTION NOTATION, PAGES 51–57
CHECK IT OUT!
1a. f(0) = 0 2 - 4(0)
= 0
f ( 1 __ 2 ) = ( 1 __
2 )
2 - 4 ( 1 __
2 )
= - 7 __ 4
f(-2) = (-2) 2 - 4(-2)
= 12
b. f(0) = -2(0) + 1 = 1
f ( 1 __ 2 ) = -2 ( 1 __
2 ) + 1
= 0 f(-2) = -2(-2) + 1 = 5
2a. b.
3a. Let x represent the number of pictures, and let f represent the cost of photo processing.
f(x) = 0.27x
b. f(24) = 0.27(24) = $6.48 $6.48 represents cost of processing 24 prints.
THINK AND DISCUSS
1. Possible answer: A reasonable domain is 0, 380 ____
156 ' h
because time cannot be negative and the train
completes the trip in 380 ____ 156
h.
2. Possible answer: The name of the function is g, the independent variable is t, and the dependent variable is g(t).
3.
EXERCISES
GUIDED PRACTICE
1. independent
2. f(0) = 3(0) - 4 = -4f(1.5) = 3(1.5) - 4 = 0.5 f(-4) = 3(-4) - 4 = -16
3. f(0) = 0 2 + 9
= 9
f(1.5) = (1.5) 2 + 9
= 11.25
f(-4) = (-4) 2 + 9
= 25
4. f(0) = 3 (0) 2 - 0 + 2
= 2f(1.5) = 3 (1.5)
2 - 1.5 + 2
= 7.25 f(-4) = 3 (-4)
2 - (-4) + 2
= 54
5. f(0) = 3 f(1.5) = 4 f(-4) = 4
6. f(0) = 1 f(1.5) = 3 f(-4) = 1
7. f(0) = -5 f(1.5) = 1 f(-4) = 1
8. 9.
19 Holt McDougal Algebra 2
10.
11. Let x represent the number of living room sets sold. f(x) = 125x
f(50) = 125(50) = 6250This represents the loss, in dollars, if 50 customers purchase the living room set.
PRACTICE AND PROBLEM SOLVING
12. f(0) = 7(0) - 4 = -4
f ( 3 __ 2 ) = 7 ( 3 __
2 ) - 4
= 13
___ 2
f(-1) = 7(-1) - 4 = -11
13. f(0) = - (0) 2 + 0
= 0
f ( 3 __ 2 ) = - ( 3 __
2 )
2 +
3 __
2
= - 3 __
4
f(-1) = - (-1) 2 + (-1)
= -2
14. f(0) = -2 (0) 2 + 1 = 1
f ( 3 __ 2 ) = -2 ( 3 __
2 )
2 + 1
= - 7 __
2
f(-1) = -2 (-1) 2 + 1
= -1
15. f(0) = 2
f ( 3 __ 2 ) = 5
f(-1) = 0
16. f(0) = 4
f ( 3 __ 2 ) = 4
f(-1) = -1
17. f(0) = 0
f ( 3 __ 2 ) = 3
f(-1) = 1 __ 2
18. 19.
20.
21. Let m represent the number of miles over the speed limit.
f(m) = 160 + 4m
f(8) = 160 + 4(8) = 192 A fine of $192 for driving 8 mi/h over the speed limit.
22. Let d represnt the depth in feet. P(d) = 14.7 + 0.445d
P(50) = 14.7 + 0.445(50) = 36.9536.95 is the pressure in psi at a depth of 50 ft.
23. f(-3.5) = 3(-3.5) - 6 = -16.5 f(-1) = 3(-1) - 6 = -9
f ( 1 __ 4 ) = 3 ( 1 __
4 ) - 6
= -5 1 __ 4
f(2) = 3(2) - 6 = 0 f(11) = 3(11) - 6 = 27
24. f(-8) = -8[1 - 2(-8)] = -136
f ( 2 __ 3
) = ( 2 __ 3 ) 1 - 2 ( 2 __
3 )
= - 2 __
9
f(1) = 1[1 - 2(1)] = -1 f(9) = 9[1 - 2(9)] = -153 f(4) = 4[1 - 2(4)] = -28
25. f(-4) = 2(-4) - 1
_________ 3
= -3
f(0) = 2(0) - 1
_______ 3
= - 1 __ 3
f ( 1 __ 2 ) =
2 ( 1 __ 2 ) - 1
________ 3
= 0
f(5) = 2(5) - 1
_______ 3
= 3
26. f(-6) = (-6 - 1) 2 + 4 = 53
f (- 3 __ 2 ) = (- 3 __
2 - 1)
2 + 4
= 10 1 __ 4
f(1) = (1 - 1) 2 + 4
= 4 f(4) = (4 - 1) 2 + 4 = 13
27. f(-2) = -1 f(-1) = 2 f(1) = 2 f(2) = -1
28. f (- 3 __ 2 ) = 2
f(-1) = 1 f(0) = -1
f ( 1 __ 2
) = -2
29. D: {A | A ≥ 0}R: {y | y ∈ )}Possible answer: For every area, there is only 1 appropriate number of boxes of tile.
30. D: {h | h ∈ )}R: {y | y ≥ 0 and y is a multiple of 4}Possible answer: The situation represents a function because each horse needs exactly 4 shoes.
31. D: {t | t ≥ 0}R: {y | -16 < y ≤ 32.8}Possible answer: For every time, the diver can be in only one place.
32. Possible answer: D: {h | h ≥ 0} R: {y | -130 < y < 60}For every time, there is only one temperature reading for a given thermometer.
33. t = 35; the number of years it takes for plan h to reach a value of $7500.
34. h(25) ≈ 4400; g(25) ≈ 3500
35. t = 40; the time when plan g is worth half the value of plan h.
20 Holt McDougal Algebra 2
36. h: 12 years; g: 10 years
37. h(40) - g(40) = 5000; the difference in the plan after 40 years.
38a. c(125) = 175 + 3.5(125) = 175 + 437.5 = 612.5 or $612.50
b. 450 = 175 + 3.5p
_____ - 175 ___________ - 175 275 = 3.5p
275
____ 3.5
= 3.5p
____ 3.5
78.6 ≈ p 78 pots can be produced for $450.
c. a line starting at (0, 175) and rising to the right
39. When x = 3, f(x) = 1 _____ x - 3
= 1 __ 0 , but division by 0 is
undefined.
40. For -5 < x < 1, g(x) is the square root of a negative number, which is not defined for real numbers.
41. For -5 < x < 0, x represents negative hours, and distance traveled would be negative.
42. (2, 8) and (3, 11)
43. independent: number of shirts;dependent: total cost;D: { x | x ≥ 15}
44. independent: hospital charges;dependent: the amount Belinda pays;D: { x | x ≥ 0}
45. f(x) = 2.37x
46. f(x) = 7.5x
47. f(x) = 0.8x
48. f(x) = 250 + 0.05x
49. Possible answer: A domain and range are reasonable if they make sense for the problem. For example, a domain that includes negative values is not reasonable for a problem involving the number of boxes of kitchen tile required to cover a floor with area A.
TEST PREP
50. C f(1) = 2; g(1) = 14; so f(1) < g(1)
51. H h(3) = 15; h(1) = 15; so h(3) = h(1)
52. D f(1) = -3 (1) 2 + 12 = 9 f(3) = -3 (3) 2 + 12 = -15 f(4) = -3 (4) 2 + 12 = -36 f(9) = -3 (9) 2 + 12 = -231 f(10) = -3 (10) 2 + 12 = -288 f(x) = 9 when x = 1.
53. f(-1) = 3 (-1 - 2) 2 + 4 = 3 (-3) 2 + 4 = 3(9) + 4 = 27 + 4 = 31
CHALLENGE AND EXTEND
54. f(2c) = √ )) (2c) 3
= √ ))))) (2c)(2c)(2c)
= √ )))) 4 c 2 · 2c
= 2c √ ) 2c
55. g (- h __ 4 ) =
6 (- h __ 4
) + h
_________ 2 (- h
__ 4
)
= - 6h
___ 4
+ h ________
- 2h
___ 4
= - 2h
___ 4
÷ - 2h
___ 4
= - 2h
___ 4
× - 4 ___ 2h
= 1
21 Holt McDougal Algebra 2
56. h ( t 2 + 3t) = 4 ( t 2 + 3t) + 7t
= 4 t 2 + 12t + 7t
= 4 t 2 + 19t
57. r ( t 4 ) = √ $$$$$$
( t 4 ) 2 + ( 2 __
t 4 )
2
= √ $$$ t
8 + 4 __
t 8
= √ $$$$
t 16
___
t 8 + 4 __
t 8
= √ $$$
t 16
+ 4 ______
t 8
=
√ $$$ t 16
+ 4 ________
√ $ t 8
=
√ $$$ t
16 + 4 ________
t 4
58a. Yes; for each value of h, there is only one value of
A.
b. function; possible answer: any combination of
values for b and h gives only one possible value
of A.
SPIRAL REVIEW
59. 4(x + 2) - x(y - 8)
= 4x + 8 - xy + 8x
= 12x - xy + 8
60. (2a) 2 + 6 a
2
= 4 a 2 + 6 a
2
= 10 a 2
61. 3c - 10 + 2c
___________ 5c
= 5c - 10
_______ 5c
= 5(c - 2)
_______ 5c
= c - 2
_____ c
62. s(s + 7) - 4s
= s 2 + 7s - 4s
= s 2 + 3s
63. b is any value. 64. b ≠ -3, 0, or 5
65. function
66. not a function; All x-value inputs are the same.
1-8 EXPLORING TRANSFORMATIONS,
PAGES 59–66
CHECK IT OUT!
1a. (3, 3) b. (-2, 1)
2a. x + 3 x y
-2 + 3 = 1 -2 4
-1 + 3 = 2 -1 0
0 + 3 = 3 0 2
2 + 3 = 5 2 2
b. x y -y
-2 4 -1(4) = -4
-1 0 -1(0) = 0
0 2 -1(2) = -2
2 2 -1(2) = -2
3. x y 2y
-1 3 2(3) = 6
0 0 2(0) = 0
2 2 2(2) = 4
4 2 2(2) = 4
4. The transformation will be a vertical compression by
a factor of 3 __
4 .
THINK AND DISCUSS
1. Possible answer: translation 2 units left or horizontal
compression by a factor of 1 __ 2
2. Possible answer: both squeeze the graph toward
the y-axis. In a vertical stretch, the y-coordinates
change. In a horizontal compression, the
x-coordinates change.
3.
EXERCISES
GUIDED PRACTICE
1. compression 2. (-1, 2)
3. (4, -1) 4. (5, 8)
22 Holt McDougal Algebra 2
5. x y y + 2
-2 1 1 + 2 = 3
0 1 1 + 2 = 3
1.5 0 0 + 2 = 2
3 -2 -2 + 2 = 0
5 0 0 + 2 = 2
6. -x x y
-1(-2) = 2 -2 1
-1(0) = 0 0 1
-1(1.5) = -1.5 1.5 0
-1(3) = -3 3 -2
-1(5) = -5 5 0
7. x y -y
-2 1 -1(1) = -1
0 1 -1(1) = -1
1.5 0 -1(0) = 0
3 -2 -1(-2) = 2
5 0 -1(0) = 0
8. 3x x y
3(-4) = -12 -4 1
3(-3) = -9 -3 0
3(-1) = -3 -1 2
3(0) = 0 0 1
3(1) = 3 1 2
3(3) = 9 3 0
3(4) = 12 4 1
9. x y 3y
-4 1 3(1) = 3
-3 0 3(0) = 0
-1 2 3(2) = 6
0 1 3(1) = 3
1 2 3(2) = 6
3 0 3(0) = 0
4 1 3(1) = 3
10. x y 1 __ 3 y
-4 1 1 __ 3 (1) = 1 __
3
-3 0 1 __ 3 (0) = 0
-1 2 1 __ 3 (2) = 2 __
3
0 1 1 __ 3 (1) = 1 __
3
1 2 1 __ 3 (2) = 2 __
3
3 0 1 __ 3 (0) = 0
4 1 1 __ 3 (1) = 1 __
3
11. vertical compression by a factor of 1 __ 2
23 Holt McDougal Algebra 2
12. vertical shift up 1.5 units
13. horizontal shift right 5 units
PRACTICE AND PROBLEM SOLVING
14. (5, 1) 15. (3, 5)
16. (-2, -3)
17. x y y - 2
-3 2 2 - 2 = 0
-1 0 0 - 2 = -2
0 1 1 - 2 = -1
1 0 0 - 2 = -2
3 2 2 - 2 = 0
18. x y -y
-3 2 -1(2) = -2
-1 0 -1(0) = 0
0 1 -1(1) = -1
1 0 -1(0) = 0
3 2 -1(2) = -2
19. x + 3 x y
-3 + 3 = 0 -3 2
-1 + 3 = 2 -1 0
0 + 3 = 3 0 1
1 + 3 = 4 1 0
3 + 3 = 6 3 2
20. -x x y
-1(-3) = 3 -3 2
-1(-1) = 1 -1 0
-1(0) = 0 0 1
-1(1) = -1 1 0
-1(3) = -3 3 2
21. x y 2 __ 3 y
-3 2 2 __ 3 (2) = 4 __
3
-1 0 2 __ 3 (0) = 0
0 1 2 __ 3 (1) = 2 __
3
1 0 2 __ 3 (0) = 0
3 2 2 __ 3
(2) = 4 __ 3
24 Holt McDougal Algebra 2
22. 1 __ 2 x x y
1 __ 2 (-3) = -
3 __
2 -3 2
1 __ 2 (-1) = -
1 __
2 -1 0
1 __ 2 (0) = 0 0 1
1 __ 2 (1) =
1 __
2 1 0
1 __ 2 (3) =
3 __
2 3 2
23. 3 __ 2 x x y
3 __ 2 (-3) = -
9 __
2 -3 2
3 __ 2 (-1) = -
3 __
2 -1 0
3 __ 2 (0) = 0 0 1
3 __ 2 (1) =
3 __
2 1 0
3 __ 2 (3) =
9 __
2 3 2
24. x y 2y
-3 2 2(2) = 4
-1 0 2(0) = 0
0 1 2(1) = 2
1 0 2(0) = 0
3 2 2(2) = 4
25. vertical shift down 5 units
26. vertical compression by a factor of 3 __ 4
27. horizontal stretch by a factor of 2
28. 10 square units; the same as the original
29. 10 square units; the same as the original
30. 20 square units; larger than the original
31. 7 square units; smaller than the original
32. 7 square units; smaller than the original
33. 10 square units; the same as the original
34. 10 square units; the same as the original
35. 30 square units; larger than the original
36a. a horizontal shift 10 units right or a vertical shift 30 units down
b. possible answers: f(x) = 3(x - 10)
37a. vertical translation
b. horizontal compression
c. the increase in the per-hour labor rate 60 + 65(3) = $255 50 + 75(3) = $275
25 Holt McDougal Algebra 2
38a. They are both linear graphs.
b. The graphs are parallel lines.
c. Add 10 to f or subtract 10 from g.
d. The graph is 10 units above until x = 150,then 10 units below.
39. 40.
41.
42. Roberta started half an hour later.
43. The library is half as far from Roberta’s house.
44. Possible answer: Order is important in these transformations: horizontal translation and reflection across the y-axis; vertical translation and reflection across the x-axis. Order is not important in these transformations: horizontal translation and reflection across the x-axis; vertical translation and reflection across the y-axis.
45. Possible answer: You might not need to make a table of values to graph a transformation of a function. For example, if the graph of a function is translated 2 units right, you can graph the transformation by shifting each point on the graph of the original function 2 units right.
TEST PREP
46. D(x, y) → (x, ay).
47. H (x, y) → (-x, y)
48. D(x, y) → (bx, y)
49. H (x, y) → (-x, y)
50. D
51. Possible answer: Translate down 6 units, or reflect across the x-axis.
CHALLENGE AND EXTEND
52. 2x = 22
x __ 2 = 22 ___
2
x = 11
y - 3 = 7 _____ + 3 ___ + 3 y = 10
The original point was (11, 10).
53a. c(n) = 0.37n b. Vertical stretch
c. 15 in 1999 and 13 in 2002.
d. The number of letters that can be mailed for $5.00 must be rounded down to the nearest whole number.
54. for (x, -y) = (-x, y) x = -x
2x = 0 x = 0
y = -y
2y = 0 y = 0
(0, 0) is the only point that satisfies this condition.
SPIRAL REVIEW
55. 172 + 150 + x
____________ 3 = 144
322 + x
_______ 3
= 144
3 ( 322 + x _______
3 ) = 3(144)
322 + x = 432 _________ - 322 _____ - 322 x = 110
56. function 57. function
26 Holt McDougal Algebra 2
58. not a function 59. f(1) = 4(1) - 5
_______ 2
= - 1 __ 2
f(-3) = 4(-3) - 5
_________ 2
= - 17 ___
2
f ( 1 __ 4 ) =
4 ( 1 __ 4 ) - 5
________ 2
= -2
60. f(1) = 2 (1) 3
= 2
f(-3) = 2 (-3) 3
= -54
f ( 1 __ 4 ) = 2 ( 1 __
4 )
3
= 1 ___ 32
61. f(1) = [1 - (1) 2 ] 2
= 0
f(-3) = [1 - (-3) 2 ] 2
= 64
f ( 1 __ 4 ) =
1 - ( 1 __
4 )
2 2
= 225
____ 256
1-9 INTRODUCTION TO PARENT
FUNCTIONS, PAGES 67–73
CHECK IT OUT!
1a. g(x) = x 3 + 2 is cubic.
g(x) = x 3 + 2 represents a vertical translation of the
cubic parent function 2 units up.
b. g(x) = (-x) 2 is quadratic.
g(x) = (-x) 2 represents a reflection of the quadratic
parent function across the y-axis.
2. The data points resemble a linear function. The data set is a vertical stretch of the linear parent
function by a factor of 3.
126
12
6
-12 -6
3. The graph resembles a linear function. The cost for 5 months of online services is about
$72.
03 6 9 12
30
60
90
120
Time (mo)
Co
st (
$)
THINK AND DISCUSS
1. Possible answer: Look at its function rule or sketch the graph of the function to see the shape.
2. Possible answer: Recognizing the parent function can help you predict what the graph will look like and help you fill in the missing parts.
3.
EXERCISES
GUIDED PRACTICE
1. Possible answer: Within a family of functions, each function is a transformation of the parent function.
2. g(x) = (x - 1) 3 is cubic.
g(x) = (x - 1) 3 represents a translation of the cubic
parent function 1 unit right.
3. g(x) = (x + 1) 2 is quadratic.
g(x) = (x + 1) 2 represents a translation of the
quadratic parent function 1 unit left.
4. g(x) = -x is linear. g(x) = -x represents a reflection of the linear parent
function across the y-axis.
5. g(x) = √ %%% x + 3 is a square root.
g(x) = √ %%% x + 3 represents a translation of the
square root parent function 3 units left.
6. g(x) = x 2 + 4 is quadratic.
g(x) = x 2 + 4 represents a translation of the
quadratic parent function 4 units up.
7. g(x) = x - √ % 2 is linear.
g(x) = x - √ % 2 represents a translation of the linear
parent function √ % 2 units down.
8. The data points resemble a linear function. The data set is a vertical stretch or horizontal
compression of the linear parent function by a factor
of 5 or 1 __ 5 , respectively.
27 Holt McDougal Algebra 2
9. The data points resemble a cubic function. The data set is a vertical compression or horizontal
stretch of the cubic parent function by a factor of 1 ___ 27
or 27, respectively.
10a.
b. The graph resembles the shape of a square root parent function.
c. The string length must be about 5 m to have a complete swing of 4.5 s.
d. It takes about 7.5 s to complete a swing if the string length is 14 m.
PRACTICE AND PROBLEM SOLVING
11. g(x) = x 2 - 1 is quadratic. g(x) = x 2 - 1 represents a translation of the
quadratic parent function 1 unit down.
12. g(x) = √ $$$ x - 2 is a square root.
g(x) = √ $$$ x - 2 represents a translation of the
square root parent function 2 units right.
13. g(x) = x 3 + 3 is cubic. g(x) = x 3 + 3 represents a translation of the cubic
parent function 3 units up.
14. The data points resemble a quadratic function. The data set is a vertical compression or horizontal
stretch of the quadratic parent function by a factor
of 1 __ 3
or 3, respectively.
15. The data points resemble a square root function. The data set is a vertical stretch or horizontal
compression of the quadratic parent function by a
factor of 2 or 1 __ 2 , respectively.
16a.
b. quadratic parent function
c. 10 points d. 21 segments
17. D: {x | x ≥ 0}; R: {y | y ≥ 0}; vertical stretch by a factor of 3
18. D: {x | x ∈ (}; R: {y | y ∈ (}; vertical compression by a factor of 2 __
3
19. D: {x | x ≥ 0}; R: {y | y ≤ 0}; reflection across the x-axis
20. D: {x | x ∈ (}; R: {y | y ≤ 0}; horizontal shift right 2 units and then reflection
across the x-axis
21. D: {x | x ∈ (}; R: {y | y ≤ 1}; reflection across the x-axis and then a vertical shift
up 1 unit
28 Holt McDougal Algebra 2
22. D: {x | x ∈ #}; R: {y | y ∈ #}; reflection across the x-axis and a vertical
compression by a factor of 1 __ 2
23. The total cost of 15 tickets is $195; possible answer: cost could be determined by estimating from a graph of the data in the table.
24. The data set is a reflection of the cubic parent function across the x-axis or reflection across the y-axis.
25. The data set is a translation of the quadratic parent function by 7 units right.
26. The data set is a reflection of the square root parent function across the y-axis.
27. The data set is a reflection of the linear parent function across y-axis and a vertical shift down by 1 unit.
28a. linear function b. quadratic function
c. square root function
29. linear function; The width of a photo 1000 pixels high is about 1500 pixels.
30. linear function; The height of a photo 500 pixels wide is about 334 pixels.
31. quadratic function; The width of a photo with a file size of 1000 KB is about 1417 pixels.
32. linear function; D: {h | h ≥ 0}; R: {y | y ≥ 0}; Unlike the linear parent function, the domain and
range of this situation do not include the negative values in #.
29 Holt McDougal Algebra 2
33. cubic function; D: { | ≥ 0}; R: {y | y ≥ 0}; Unlike the cubic parent function, the domain this
situation does not include the negative values in ".
34. linear function; D: {w | w ≥ 0}; R: {y | y ≥ 0}; Unlike the linear parent function, the domain and
range of this situation do not include the negative values in ".
35. linear function; D: {n | n ∈ &}; R: {y | y ∈ &}; Unlike the linear parent function, the domain and
range include only values in &.
36. linear function; D: {p | p ≥ 0}; R: {y | y ≥ 0}; Unlike the linear parent function, the domain and
range in this situation do not include the negative values in ".
37. square root function; D: {a | a ≥ 0}; R: {y | y ≥ 0}; Same domain and range as parent function.
38. The volume of 1 g of aerogel is about 333 cm 3 .
39a. linear function b. cubic function
c. quadratic function d. square root function
e. linear function; horizontal stretch by a factor of 2 and a vertical shift up 3 units
40. Possible answer: A horizontal translation results from a constant being added to x before squaring,
such as (x + a) 2 . A vertical translation results from a constant being added to x 2 , such as x 2 + a. A reflection across the x-axis results from negating x 2 , such as -x 2 .
41. Constant, square root, linear, quadratic, cubic; the constant function does not increase at all; the
square root function increases slowly; the linear function increases 1 to 1 as x increases; the quadratic and cubic functions increase quickly, with cubic being the faster of the 2.
TEST PREP
42. D 43. H
44. B x ≠ 0; x 2 > 0; - x 2 < 0
45. G
46. D
CHALLENGE AND EXTEND
47. quadratic function, since highest power of x is 2
48. constant function, since h(x) = 1 + 2 = 3 and the highest power of x is 0
49. linear function, since highest power of x is 1
50a. b. D: {x | x ∈ "}; R: {y | y > 0}
30 Holt McDougal Algebra 2
c. f(0) = 2 0 = 1; So, function crosses y-axis at (0, 1).
d. (0, 1); Possible answer: 3 0 = 1, so f(x) = 3 x will
have the same y-intercept as f(x) = 2 x .
SPIRAL REVIEW
51. (1.5 × 10 -4 ) (5.0 × 10 13 ) = 7.5 × 10 9
52. (8.1 × 10 3 ) 2
= 65.61 × 10 6
= 6.561 × 10 7
53. 1.9 × 10 -6 __________
9.5 × 10 18
= 0.2 × 10 -24
= 2.0 × 10 -25
54. f(-3) = 1 __ 2 (-3) + 3
= 3 __ 2
f(0) = 1 __ 2 (0) + 3
= 3
f ( 1 __ 3 ) = 1 __
2 ( 1 __
3 ) + 3
= 19 ___ 6
f(6) = 1 __ 2 (6) + 3
= 6
55. f(-5) = (-5)(-5 + 2) = 15
f (- 2 __ 3
) = (- 2 __ 3 ) (- 2 __
3 + 2)
= - 8 __ 9
f(1.6) = (1.6)(1.6 + 2) = 5.76 f(4) = (4)(4 + 2) = 24
56. (1, 1) 57. (4, -10)
58. (-3, -5)
READY TO GO ON? PAGE 75
1. D: {x | -3 ≤ x ≤ 3} R: {y | -1 ≤ y ≤ 2} not a function
2. D: {0, 2, 4, 6} R: {5, 8, 10, 12, 20} not a function
3. D: {x | -1 ≤ x ≤ 3} R: {y | 0 ≤ y ≤ 4} function
4. f(0) = 12 - 3(0) = 12 f(1) = 12 - 3(1) = 9 f(-2) = 12 - 3(-2) = 18
5. f(0) = 3 (0) 3 + 1 = 1
f(1) = 3 (1) 3 + 1 = 4
f(-2) = 3 (-2) 3 + 1 = -23
6. f(0) = 4 - (0) 2 = 4
f(1) = 4 - (1) 2 = 3
f(-2) = 4 - (-2) 2 = 0
7a. Let m represent the distance driven in miles and let c represent the cost per mile in dollars, c(m) = 1.75 + 0.25(4m) = 1.75 + m
b.
c. c(5.5) = 1.75 + 5.5 = 7.25 It represents the cost in dollars for a taxi ride of
5.5 miles.
8. vertical translation up by 15 units
9. vertical compression by a factor of 0.6
10. g(x) = - x 2 is quadratic. g(x) = - x 2 represents a reflection of the quadratic
parent function across the x-axis.
11. g(x) = √ ''' x - 3 is a square root.
g(x) = √ ''' x - 3 represents a translation of the
square root parent function 3 units right.
12. g(x) = 1.5x is linear. g(x) = 1.5x represents a vertical stretch of the linear
parent function by a factor of 1.5.
31 Holt McDougal Algebra 2
13. quadratic; A stand with a maximum load of 7920 kg has a diameter of about 17.5 mm.
STUDY GUIDE: REVIEW, PAGES 76–79
1. domain; range
LESSON 1-1
2. {x | x ≥ -5} 3. (1, 5]
4. {4, 5, 6, 7, …} 5. {x | x < -2 or x > 5}
6. integers greater than -4 and less than or equal to 5
7. [5.5, 5.6]
LESSON 1-2
8. Comm. Prop. of Mult. 9. Distributive Property
10. -0.55; 1 ____ 0.55
11. 7 __ 8 ; - 8 __
7
12. -1. − 2 ; 1 ___
1. − 2 or 9 ___
11
LESSON 1-3
13. √ ( 9 <
√ (( 12 <
√ (( 16
3 < √ (( 12 < 4
3.4 2 = 11.56
3.5 2 = 12.25
√ (( 12 ≈ 3.5
14. √ (( 49 <
√ (( 55 <
√ (( 64
7 < √ (( 55 < 8
7.4 2 = 54.76
7.5 2 = 56.25
√ (( 55 ≈ 7.4
15. √ (( 64 <
√ (( 74 <
√ (( 81
8 < √ (( 74 < 9
8.6 2 = 73.96
8.7 2 = 75.69
√ (( 74 ≈ 8.6
16. √ (( 25 <
√ (( 29 <
√ (( 36
5 < √ (( 29 < 6
5.3 2 = 28.09
5.4 2 = 29.16
√ (( 29 ≈ 5.4
17. √ (( 32
= √ ((( 16 · 2
= 4 √ ( 2
18. √ (( 64
____ √ ( 4
= 8 __ 2
= 4
19. 2 √ ( 2 - √ (( 72
= 2 √ ( 2 - √ ((( 36 · 2
= 2 √ ( 2 - 6 √ ( 2
= -4 √ ( 2
20. √ ( 3 · √ (( 21
= √ (( 63
= √ (( 9 · 7
= 3 √ ( 7
21. 7 ___ √ ( 2
= 7 ___ √ ( 2
· √ ( 2
___ √ ( 2
= 7 √ ( 2
____ √ ( 4
= 7 √ ( 2
____ 2
22. 2 √ (( 20
_____ 5 √ ( 8
= 2 √ (( 4 · 5
_______ 5 √ (( 4 · 2
= 2 · 2 √ ( 5
_______ 5 · 2 √ ( 2
= 2 √ ( 5
____ 5 √ ( 2
= 2 √ ( 5
____ 5 √ ( 2
· √ ( 2
___ √ ( 2
= 2 √ (( 10
_____ 5 √ ( 4
= 2 √ (( 10
_____ 5 · 2
= √ (( 10
____ 5
LESSON 1-4
23. x 2 y - x y 2 = (6) 2 (-2) - (6) (-2) 2 = 36(-2) - 6(4)= -72 - 24= -96
24. - x 2 __ 2 + 5xy - 9y
= - (4)
2 ____
2 + 5(4)(2) - 9(2)
= - 16 ___
2 + 40 - 18
= -8 + 40 - 18 = 14
25. n
2 + mn - 1 ___________
4 m 2 n
= (-1)
2 + (2)(-1) - 1 _________________
4 (2) 2 (-1)
= 1 - 2 - 1 _________ 4(4)(-1)
= -2 ____ -16
= 1 __ 8
26. -x - 2y + 9x - y + 3x
= -x + 9x + 3x - 2y - y= 11x - 3y
27. 7 - (5a - b) + 11= 7 - 5a + b + 11= 7 + 11 - 5a + b= 18 - 5a + b
28. -4(2x + 3y) + 5x
= -8x - 12y + 5x
= -8x + 5x - 12y
= -3x - 12y
29. c ( a 2 - b) + 3bc
= a 2 c - bc + 3bc
= a 2 c + 2bc
32 Holt McDougal Algebra 2
LESSON 1-5
30. (-2 x 5 y -3
) 3
= -8 x 15
y -9
= -8 x
15
______
y 9
31. -24 x
4 y -6
_________
14x -3
y 3
= -12 x
7 y -9
_________
7
= -12 x
7
______
7 y 9
32. ( r 2 s ___
s 3 )
2
= r 4 s
2
____
s 6
= r 4
__
s 4
33. 4mn ( m 5 n -5
) = 4 m
6 n -4
= 4 m
6
____
n 4
34. 7.7 × 10
5 __________
1.1 × 10 -2
= 7 × 10 7
35. (4.5 × 10 -2
) (1.2 × 10 3 )
= 5.4 × 10 1
LESSON 1-6
36. D: {3, 5, 7}
R: {-1, 0, 9}
not a function
37. D: [-2, ∞)
R: [-4, ∞)
not a function
38. D: {-2, 0, 3, 4}
R: {3, 4}
function
39. D: {5, 10, 15, 20, 25}
R: {-5, -4, -3, -2, -1}
function
40. D: {a, b, c}
R: {Alabama, Alaska, Arizona, Arkansas, California,
Colorado, Connecticut}
not a function
LESSON 1-7
41. f(2) = - (2) 2 + 2
= -2
f ( 1 __
2 ) = - ( 1
__
2 )
2 + 2
= 7
__
4
f(-2) = - (-2) 2 + 2
= -2
42. f(2) = -5(2) - 6
= -16
f ( 1 __
2 ) = -5 ( 1
__
2 ) - 6
= - 17
___
2
f(-2) = -5(-2) - 6
= 4
43. f(2) = -1
f ( 1 __
2 ) = 1
f(-2) = 2
44. f(2) = 1 __ 2
f ( 1 __
2 ) = 2
f(-2) = - 1
__
2
45. 46.
47. A(s) = 6 s 2 , where A is the surface area in square
units and s is the side length in linear units.
A(10) = 600; the surface area of a cube of side
length 10 cm is 600 cm 2 .
LESSON 1-8
48. (0, -5) 49. (5, 1)
50. vertical compression by a factor of 1 __ 2
Parking Fees
01 2 3
4
8
12
Time (h)
Fee
($)
4 5
51. vertical stretch by a factor of 1.1
Parking Fees
01 2 3
4
8
12
Time (h)
Fee
($)
4 5
52. translation 1 unit up
Parking Fees
01 2 3
4
8
12
Time (h)
Fee
($)
4 5
LESSON 1-9
53. quadratic function;
translation 1 unit down
54. square root function;
reflection across the
x-axis
55. linear function;
For a 95 lb rider, the tire pressure is about 90 psi.
33 Holt McDougal Algebra 2
CHAPTER TEST, PAGE 80
1. - √ $ 3 ≈ -1.73
The order is: -2, - √ $ 3 , 0.95, 1, 1. − 5
-2: !, ', (; - √ $ 3 : !, irrational; 0.95: !, '; 1: !, ', (, ), *;
1. − 5 : !, '
2. (-∞, -2) and (1, 3] 3. {x | x ≤ 12}
4. Comm. Prop. of Add. 5. Distributive Property
6. Multiplicative Identity Property
7. √ $ 6 ≈ 2.4 ft; √ $ 8 ≈ 2.8 ft; √ $$ 15 ≈ 3.9 ft;The 8 ft 2 window is the largest that could fit in the wall.
8. -2 √ $ 3 + √ $$ 75
= -2 √ $ 3 + √ $$$ 25 · 3
= -2 √ $ 3 + 5 √ $ 3
= 3 √ $ 3
9. √ $$ 24 - √ $$ 54
= √ $$ 4 · 6 - √ $$ 9 · 6
= 2 √ $ 6 - 3 √ $ 6
= - √ $ 6
10. √ $$ 22 · √ $$ 55
= √ $$ 1210
= √ $$$$ 121 · 10
= 11 √ $$ 10
11. 2(x + 1) + 9x
= 2x + 2 + 9x
= 11x + 2
12. 5x - 5y - 7x + y= -2x - 4y
13. 12x + 4(x + y) - 6y
= 12x + 4x + 4y - 6y
= 16x - 2y
14. 8 a 2 b 5 (-2 a 3 b 2 ) = -16 a 5 b 7
15. 28 u -2 v 3 _______
4 u 2 v 2
= 7 u -4 v
= 7v ___ u 4
16. (5 x 4 y -3 ) -2
= 5 -2 x (4)(-2) y (-3)(-2)
= 5 -2 x -8 y 6
= y 6 ____
25 x 8
17. ( 3 x 2 y ____
x y 2 ) -1
= x y
2 ____
3 x 2 y
= y ___
3x
18. 2.25 × 10 8 _________ 5 × 10 6
= 0.45 × 10 2
= 45
19. D: {8, 9, 10}R: {2, 4, 6, 8, 10}not a function
20. D: [-5, 5]R: [-2, 2]function
21. f(-2) = -4(-2) = 8
f ( 1 __ 2 ) = -4 ( 1 __
2 )
= -2 f(0) = -4(0) = 0
22. f(-2) = -3 (-2) 2 + (-2) = -14
f ( 1 __ 2 ) = -3 ( 1 __
2 )
2 + 1 __
2
= - 1 __ 4
f(0) = -3 (0) 2 + (0)
= 0
23. f(-2) = √ $$$$ (-2) + 3
= 1
f ( 1 __ 2 ) = √ $$$$
( 1 __ 2 ) + 3
= √ $$ 3.5 ≈ 1.87
f(0) = √ $$$ 0 + 3
= √ $ 3 ≈ 1.73
24. square root function; For a building with height of 80 m, the distance to
the horizon is about 32 km.
020 40 60 80
8
16
24
32
Height of building (m)
Dis
tance
to
ho
rizo
n (
km
)
100
34 Holt McDougal Algebra 2