Chapter 2, continued

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Chapter 2, continued
```continued
You can ride the rally track 4 times and the Grand Prix
track 4 times.
4. A relation involves two quantities that are related to
each other by some rule of correspondence. A function
f from set A to set B is a relation that assigns to each
element x in the set A exactly one element y in the
set B while a relation can have more than one element y
in the set B for each element x in the set A.
You can ride the rally track 2 times and the Grand Prix
track 5 times.
You can ride the rally track 7 times and the Grand Prix
track 2 times.
5. Input
Mixed Review for TEKS (p. 139)
22
21
2
3
1. C;
Vertex (h, k) 5 (12, 20)
So, y 5 a{x 2 h{ 1 k 5 a{x 2 12{ 1 20
22
0
6
8
The domain consists of all the x-coordinates:
22, 21, 2, 3.
The range consists of all the y-coordinates: 22, 0, 6, 8.
The relation is a function because each input is mapped
onto exactly one output.
Substitute (2, 0) into the equation and solve for a.
0 5 a{2 2 12{ 1 20
220 5 10a
22 5 a
So, y 5 22{x 2 12 {1 20.
6.
Input
21
1
3
2. G;
3
22 2 1
slope 5 } 5 2}2
0 2 (22)
The domain consists of all the x-coordinates: 21, 1, 3.
The range consists of all the y-coordinates:
27, 25, 2, 4.
The relation is not a function because the input 1 is
mapped onto both 27 and 2.
3
y a 2}2 x 2 2
3
y 1 }2 x a 22
3. C;
7.
32.85
5 10.95
cost per month 5 }
3
total cost 5 cost per month 3 months of service
5 10.95 3 12
5 131.40
A 12 month subscription costs \$131.40.
4. H;
The data show approximately no correlation.
5. B;
2m 1 3z a 30
6. s(t ) 5 215{t 2 5{ 1 180
Output
27
25
2
4
y-intercept 5 22
2y 1 3x a 24
Output
The function f is linear because it has the form
f (x) 5 mx 1 b.
f (x) 5 16 2 7x
f (25) 5 16 2 7(25) 5 51
y2 2 y1
3 2 (21)
4
2
8. m 5 } 5 } 5 } 5 }
x2 2 x1
6
3
4 2 (22)
y2 2 y1
2 2 (25)
7
9. m 5 } 5 } 5 } The slope is undefined.
x2 2 x1
121
0
y2 2 y1
7 2 (23)
10
5
10. m 5 } 5 } 5 } 5 2}
x2 2 x1
125
24
2
y2 2 y1
0
222
11. m 5 } 5 } 5 } 5 0
x2 2 x1
28 2 6
214
12. y 5 5 2 x
y
y 5 a{x 2 h{ 1 k
vertex (h, k ) 5 (5, 180)
The vertex is (5, 180), so the maximum amount of money
raised in one day was \$180.
Chapter 2 Review (pp. 141–144)
1. The linear equation 5x 2 4y 5 16 is written in
1
x
21
13. y 2 5x 5 24
1
y 5 5x 2 4
y
x
21
standard form.
2. A set of data pairs (x, y) shows a negative correlation if
y tends to decrease as x increases.
3. Two variables x and y show direct variation if y 5 ax
and a Þ 0.
14. x 5 4
y
1
21
94
Algebra 2
Worked-Out Solution Key
x
Chapter 2,
Chapter 2,
continued
15. 6x 2 4y 5 12
24y 5 12 2 6x
1
21
21. When x 5 23 and y 5 2.4:
y
y 5 ax
x
2.4 5 a(23)
3
y 5 }2 x 2 3
20.8 5 a
An equation that relates x and y is y 5 20.8x.
26 2 4
10
16. m 5 } 5 2} 5 22
5
2 2 (23)
When x 5 3:
y 5 20.8(3)
Choose (x1, y1) 5 (23, 4).
y 5 22.4
y 2 y1 5 m(x 2 x1)
22. When T 5 300 and V 5 4.8:
y 2 4 5 22(x 2 (23))
V 5 aT
y 5 22x 2 6 1 4
4.8 5 a(300)
y 5 22x 2 2
0.016 5 a
27 2 5
3
212
17. m 5 } 5 } 5 2}
16
4
12 2 (24)
An equation that gives V as a function of T is
V 5 0.016T.
Choose (x1, y1) 5 (24, 5).
When T 5 420:
y 2 y1 5 m(x 2 x1)
V 5 0.016T
3
y 2 5 5 2}4 (x 2 (24))
V 5 0.016(420)
V 5 6.72 liters
3
y 5 2}4 x 2 3 1 5
23. A scatter plot and possible line of best fit are shown.
The line appears to pass through (x1, y1)5 (21, 3.3) and
(x2, y2) 5 (1, 1.3).
3
y 5 2}4 x 1 2
26 2 1
27
18. m 5 } 5 } 5 21
7
3 2 (24)
1.3 2 3.3
Choose (x1, y1) 5 (24, 1).
An equation is y 2 1.3 5 21(x 2 1), or y 5 2x 1 2.3.
y
y 2 y1 5 m(x 2 x1)
y 2 1 5 21(x 2 (24))
22
m5}
5}
5 21
2
1 2 (21)
(1, 1.3)
(21, 3.3)
y 5 2x 2 4 1 1
1
y 5 2x 2 3
x
1
19. When x 5 6 and y 5 248:
y 5 ax
248 5 a(6)
24. The graph of y 5 {x 2 3{ 1 2 is the graph of y 5 {x{
28 5 a
An equation that relates x and y is y 5 28x.
translated right 3 units and up 2 units.
y
When x 5 3:
y 5 28(3)
y 5 224
20. When x 5 29 and y 5 15:
1
y 5 ax
x
21
15 5 a(29)
5
2}3 5 a
5
An equation that relates x and y is y 5 2}3 x.
3
25. The graph of y 5 }{x{ is the graph of f (x) 5 {x{
4
3
shrunk vertically by a factor of }4.
y
When x 5 3:
5
y 5 2}3 (3)
1
21
x
y 5 25
Algebra 2
Worked-Out Solution Key
95
Chapter 2,
continued
26. The graph of f(x) 5 24{x 1 2{ 1 3 is the graph of
34. Let x represent the number of windmills producing
1.5 megawatts and y represent the number of windmills
producing 2.5 megawatts. And inequality describing
how many of each size of windmill it takes to supply the
electric company is 1.5x 1 2.5yq180.
f (x) 5 {x{reflected in the x-axis, stretched vertically by
a factor of 4, and translated left 2 units and up 3 units.
y
y
80
1
x
22
60
40
When d 5 0.25:
y
20
d 5 {a 2 1.50{
0
1
5 a 2 1.50
x
21
a 2 1.50 5 0.25
a 5 1.75 or a 2 1.50 5 20.25
0
30
60
90
x
120
Chapter 2 Test (p. 145)
1. The relation is not a function because the input 1 is
mapped onto both 25 and 2, and the input 2 is mapped
onto both 3 and 7.
a 5 1.25
d will be \$.25 when a is \$1.75 or \$1.25.
28. (0, 1):
2. The relation is a function because each input is mapped
onto exactly one output.
2ya5x
5(0)
21a
3.
f (x) 5 3x2 2 2x 1 11
f(26) 5 3(26)2 2 2(26) 1 11 5 108 1 12 1 11 5 131
21a0 4. Let (x1, y1) 5 (3, 22) and (x2, y2) 5 (5, 4).
y2 2 y1
4 2 (22)
6
m5}
5}
5 }2 5 3
x2 2 x1
523
(0, 1) is a solution.
29. (24, 6):
y > 23x 2 7
6
> 23(24) 2 7
The line rises from left to right.
5. Let (x1, y1) 5 (6, 27) and (x2, y2) 5 (13, 27).
y2 2 y1
27 2 (27)
0
m5}
5}
5 }7 5 0
x2 2 x1
13 2 6
6>5
(24, 6) is a solution.
30. (22, 0):
The line is horizontal.
3x 2 4y < 28
3(22) 2 4(0) < 28
6. Let (x1, y1) 5 (22, 1) and (x2, y2) 5 (1, 24).
y2 2 y1
5
24 2 1
m5}
5}
5 2}3
x2 2 x1
1 2 (22)
26 ñ 28
The line falls from left to right.
(22, 0) is not a solution.
31. 24y < 16
7. Let (x1, y1) 5 (24, 9) and (x2, y2) 5 (24, 8).
y2 2 y1
829
21
m5}
5}
5}
x2 2 x1
0
24 2 (24)
32. y 2 2x > 8
y > 24
y > 2x 1 8
1
y
y
The slope is undefined, so the line is vertical.
x
21
8.
9.
y
y
1
x
21
1
x
21
33. 12x 2 8ya
28ya212x 1 24
1
21
10.
y
y
x
yq}3 x 2 3
2
96
Algebra 2
Worked-Out Solution Key
1
21
1
21
x
x
27.
Chapter 2,
11.
continued
16. When x 5 8 and y 5 18:
3y 5 2x 2 12
22x 1 3y 5 212
y 5 ax
x-intercept: 22x 1 3(0) 5 212
18 5 a(8)
x56
9
4
}5a
y-intercept: 22(0) 1 3y 5 212
9
An equation that relates x and y is y 5 }4 x.
y 5 24
1
When y 5 6:
y
9
6 5 }4 x
x
21
8
3
}5x
17. When x 5 16 and y 5 26:
1
12. The slope of a line parallel to y 5 2} x 2 8 is
3
1
m2 5 m1 5 2}3 .
Let (x1, y1) 5 (9, 21).
y 5 ax
26 5 a(16)
3
2}8 5 a
3
An equation that relates x and y is y 5 2}8 x.
y 2 y1 5 m2(x 2 x1)
When y 5 6:
1
y 2 (21) 5 2}3 (x 2 9)
3
6 5 2}8 x
1
y 5 2}3 x 1 2
216 5 x
13. The slope of a line perpendicular to y 5 25x 1 7 is
1
1
1
5 2}
5 }5 .
m2 5 2}
25
m1
18. a.
(5, 91)
80
Let (x1, y1) 5 (10, 2).
60
(3, 55)
y 2 y1 5 m2(x 2 x1)
40
1
y
y 2 2 5 }5 (x 2 10)
20
1
0
y 5 }5 x
0
2
4
6 x
b. Sample answer: On the scatter plot, draw a line that
14. When x 5 4 and y 5 28:
appears to best fit the data. The line appears to pass
through (3, 55) and (5, 91).
y 5 ax
28 5 a(4)
91 2 55
36
22 5 a
m5}
5}
5 18
523
2
An equation that relates x and y is y 5 22x.
y 2 y1 5 m(x 2 x1)
When y 5 6:
y 2 55 5 18(x 2 3)
6 5 22x
y 5 18x 1 1
23 5 x
An approximation of the best-fitting line is
y 5 18x 1 1.
15. When x 5 22 and y 5 21:
c. When x 5 10:
y 5 ax
y 5 18(10) 1 1
21 5 a(22)
y 5 181
1
}5a
2
1
An equation that relates x and y is y 5 }2 x.
19. a.
y
100
When y 5 6:
80
1
6 5 }2 x
60
12 5 x
(1, 97)
(5, 7
5)
0
0
2
4
6 x
Algebra 2
Worked-Out Solution Key
97
Chapter 2,
continued
b. Sample answer: On the scatter plot, draw a line that
26. When p 5 0.5 and t 5 2.5:
appers to best fit the data. The line appears to pass
through (1, 97) and (5, 75).
75 2 97
p 5 at
0.5 5 a(2.5)
222
m5}
5}
5 25.5
521
4
0.2 5 a
y 2 y1 5 m(x 2 x1)
An equation for the situation is p 5 0.2t.
When t 5 20:
y 2 97 5 25.5(x 2 1)
p 5 0.2(20)
y 5 25.5x 1 102.5
p54
An approximation of the best-fitting line is
y 5 25.5x 1 102.5.
If the sprayer is operated for 20 minutes, 4 gallons of
paint is applied.
c. When x 5 10:
27. A scatter plot and possible line of best fit are shown. The
y 5 25.5(10) 1 102.5
line appears to pass through (x1, y1) 5 (40, 1.25) and
(x2, y2) = (108, 3).
y 5 47.5
20.
3
y
3 2 1.25
1.75
m5}
5}
ø 0.026
108 2 40
68
y 2 y1 5 m(x 2 x1)
x
21
y 2 1.25 5 0.026(x 2 40)
y 5 0.026x 1 0.21
An approximation for the best-fitting line is
y 5 0.026x 1 0.21.
The graph of y 5 23{x 1 1{ 1 3 is the grpah of
y 5 {x{reflected in the x-axis, stretched vertically by
a factor of 3, and translated left 1 unit and up 3 units.
1
y
1
x
1
21
23.
24.
y
1
1
(40, 1.25)
0
40
80
120 x
Number of transistors
(millions)
y
21
1
TAKS Practice (pp. 148–149)
x
1. A;
1.80 2 1.53
1.80
} 5 0.15 5 15%
x
21
25. Let x represent the number of miles driven and y
2. G;
3 defects
130 batteries
0.12 millimeter per 1000 miles, the slope is
20.12
1000
} 5 20.00012.
y 2 y1 5 m(x 2 x1)
y 2 8 5 20.00012(x 2 0)
y 5 20.00012x 1 8
So, an equation for the situation is y 5 20.00012x 1 8.
When y 5 2:
2 5 20.00012x 1 8
26 5 0.00012x
50,000 5 x
50,000 miles.
Algebra 2
Worked-Out Solution Key
x defects
6500 batteries
} 5 }}
130x 5 6500 + 3
represent the tread depth (in millimeters). When
x 5 0, y 5 8. Because the tread depth decreases
98
2
0
x
(108, 3)
3
x 5 150
3. C;
first enlargement: 4 3 2.5 5 10
6 3 2.5 5 15
second enlargement: 10 3 2.5 5 25
15 3 2.5 5 37.5
4. J;
2 freshmen/sophomores
5 members
x freshmen/sophomores
20 members
}} a }}
2 + 20 a 5x
8ax
22.
y
Speed (gigahertz)
21.
y
Chapter 2,
continued
5. C;
14. F;
women in 2005 2 women in 1985
65 2 22
}}} 5 } 5 4.3
10
number of Congresses
6. H;
Total pay 5 hourly wage + hours worked 1
commission + sales
115 5 7 + 10 1 0.05s
45 5 0.05s
new y-intercept 5 32.6 2 65.2 5 232.6
y 5 2.7x 2 32.6
The slopes are the same, so the lines are parallel.
15. C;
1
V 5 }3:r 2h
1
900 5 s
original cone: 16 5 }3:r 2h
7. A;
2(m 2 3) 1 3m 5 9m 1 12
2m 2 6 1 3m 5 9m 1 12
24m 5 18
9
m 5 2}2
8. J;
y 5 2.7x 1 32.6
1
new cone: V 5 }3 :(2r)2(2h)
V 5 81 }3:r 2h 2
1
V 5 8(16) 5 128 cm3
16. horizontal distance 5 388 inches
number of steps 5 5
x 1 y 5 10
width of each step 5 x 1 8
y 5 2x 1 10
5(x 1 8) 5 388
So, m 5 21.
5x 1 40 5 388
The perpendicular line through (1, 23) will have the
negative reciprocal slope, m 5 1.
y 5 mx 1 b
5x 5 348
x 5 69.6
The distance x between timbers is 69.6 inches.
23 5 1(1) 1 b
24 5 b
y5x24lx2y54
9. C;
1
1
V 5 }2 + * + w + h 5 }2 (7)(5)(4) 5 70 in.3
10. G;
3.5c 1 4.5w a 30
11. B;
mA 5 mB
5x 1 4 5 3x 1 10
2x 5 6
x53
12. F;
2y 5 10 1 5x
2(0) 5 10 1 5x
25x 5 10
x 5 22
The coordinate of the x-intercept is (22, 0).
13. A;
3y 5 26(x 1 2)
3y 5 26x 2 12
y 5 22x 2 4
The slope is 22.
Algebra 2
Worked-Out Solution Key
99
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