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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}
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.
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)
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.
− 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
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:
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.
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.
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.
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