Review Exercises for Chapter 2 Review Exercises for Chapter 2 237 1. (a) (b) Vertical stretch Vertical stretch and a reflection in the x-axis (c) (d) Vertical shift two units upward Horizontal shift two units to the left (a) y x - 4 - 3 - 2 - 4 - 3 - 2 - 1 1 4 - 1 1 2 3 4 y x - 4 - 3 - 2 - 4 - 3 - 2 - 1 1 3 4 - 1 1 2 3 4 y = x + 22 y = x 2 + 2 y x - 4 - 3 - 2 - 4 - 3 2 1 3 4 - 1 1 2 3 4 y x - 4 - 3 - 2 - 4 - 3 - 2 - 1 2 3 4 - 1 1 2 3 4 y =-2x 2 y = 2x 2 2. (a) Vertical shift four units downward y x - 4 - 3 - 1 - 5 - 2 - 1 2 3 1 3 4 y = x 2 - 4 (b) Reflection in the x-axis and a vertical shift four units upward y x - 4 - 3 - 1 - 3 - 2 - 1 1 2 3 5 1 3 4 y = 4 - x 2 (c) Horizontal shift three units to the right y x - 3 - 2 - 1 - 3 - 2 - 1 1 2 3 4 5 2 1 3 4 5 y = x - 32 (d) Vertical shrink (each y-value is multiplied by and a vertical shift one unit downward y x - 4 - 3 - 2 - 4 - 3 -2 1 2 3 4 2 3 4 1 2 , y = 1 2 x 2 - 1
22
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Chapter 2 Review Solution Key - Montville Township Public ...€¦ · Review Exercises for Chapter 2 Review Exercises for Chapter 2 237 1. (a) (b) Vertical stretch Vertical stretch
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Review Exercises for Chapter 2
Review Exercises for Chapter 2 237
1. (a) (b)
Vertical stretch Vertical stretch and a reflection in the x-axis
(c) (d)
Vertical shift two units upward Horizontal shift two units to the left
(a) y
x−4 −3 −2
−4
−3
−2
−1
1
4
−1 1 2 3 4
y
x−4 −3 −2
−4
−3
−2
−1
1
3
4
−1 1 2 3 4
y 5 sx 1 2d2y 5 x2 1 2
y
x−4 −3 −2
−4
−3
2
1
3
4
−1 1 2 3 4
y
x−4 −3 −2
−4
−3
−2
−1
2
3
4
−1 1 2 3 4
y 5 22x2y 5 2x2
2. (a)
Vertical shift four units downwardy
x−4 −3 −1
−5
−2
−1
2
3
1 3 4
y 5 x2 2 4 (b)
Reflection in the x-axis and a vertical shift
four units upwardy
x−4 −3 −1
−3
−2
−1
1
2
3
5
1 3 4
y 5 4 2 x2
(c)
Horizontal shift three units to the righty
x−3 −2 −1
−3
−2
−1
1
2
3
4
5
21 3 4 5
y 5 sx 2 3d2(d)
Vertical shrink (each y-value is multiplied by
and a vertical shift one unit downward
y
x−4 −3 −2
−4
−3
−2
1
2
3
4
2 3 4
12d,
y 512 x2 2 1
238 Chapter 2 Polynomial and Rational Functions
3.
Vertex:
Axis of symmetry:
x-intercepts: s0, 0d, s2, 0d
0 5 x2 2 2x 5 xsx 2 2d
x 5 1
s1, 21d
5 sx 2 1d2 2 1
5 x2 2 2x 1 1 2 1
x1
−1
−2
−1−2−3
3
4
5
6
7
2 3 4 5 6
ygsxd 5 x2 2 2x 4.
Vertex:
Axis of symmetry:
x-intercepts: s0, 0d, s6, 0d
0 5 6x 2 x2 5 xs6 2 xd
x 5 3
s3, 9d
5 2sx 2 3d2 1 9
5 2sx2 2 6x 1 9 2 9d
x4
−2
8−2
2
4
8
6
10
2 10
y fsxd 5 6x 2 x2
5.
Vertex:
Axis of symmetry:
x-intercepts: s24 ± !6, 0d x 5 24 ± !6
x 1 4 5 ±!6
sx 1 4d2 5 6
0 5 sx 1 4d2 2 6
x 5 24
s24, 26d
x
2
2
−8 −4
−2
−4
−6
y 5 sx 1 4d2 2 6
5 x2 1 8x 1 16 2 16 1 10
f sxd 5 x2 1 8x 1 10 6.
Vertex:
Axis of symmetry:
x-intercepts: s2 ± !7, 0d
54 ± !28
25 2 ± !7
x 52s24d ± !s24d2 2 4s1ds23d
2s1d
0 5 x2 2 4x 2 3
0 5 3 1 4x 2 x2
x 5 2
s2, 7d
5 2sx 2 2d2 1 7
x4−2
2
4
8
6
10
2 106 8
y
5 2fsx 2 2d2 2 7g
5 2sx2 2 4x 1 4 2 4 2 3d
5 2sx2 2 4x 2 3d
hsxd 5 3 1 4x 2 x2
7.
Vertex:
Axis of symmetry:
t-intercepts: 11 ±!6
2, 02
t 5 1 ±!6
2
t 2 1 5 ±!3
2
2st 2 1d2 5 3
0 5 22st 2 1d2 1 3
t 5 1
s1, 3d
t1 2−1−2−3 3 4 5 6
2
4
5
6
3
1
y 5 22st 2 1d2 1 3
5 22fst 2 1d2 2 1g 1 1
5 22st2 2 2t 1 1 2 1d 1 1
f std 5 22t2 1 4t 1 1 8.
Vertex:
Axis of symmetry:
x-intercepts: s2, 0d, s6, 0d
0 5 sx 2 2dsx 2 6d
0 5 x2 2 8x 1 12
x 5 4
x4 8−2
2
6
4
8
10
−2
−4
ys4, 24d
5 sx 2 4d2 2 4
5 x2 2 8x 1 16 2 16 1 12
f sxd 5 x2 2 8x 1 12
Review Exercises for Chapter 2 239
9.
Vertex:
Axis of symmetry:
No real zeros
x-intercepts: none
1x 11
222
5 23
0 5 41x 11
222
1 12
x 5 21
2
121
2, 122
x
−1−2−3 1 2 3
5
10
15
20
y 5 41x 11
222
1 12
5 41x2 1 x 11
42 2 1 1 13
5 41x2 1 x 11
42
1
42 1 13
5 4sx2 1 xd 1 13
hsxd 5 4x2 1 4x 1 13 10.
Vertex:
Axis of symmetry:
x-intercepts: s3 ± 2!2, 0dx
4 8
2
102
−2
−2
−4
−6
−8
y 56 ± !32
25 3 ± 2!2
x 52s26d ± !s26d2 2 4s1ds1d
2s1d
0 5 x2 2 6x 1 1
x 5 3
s3, 28d
5 sx 2 3d2 2 8
5 x2 2 6x 1 9 2 9 1 1
f sxd 5 x2 2 6x 1 1
11.
Vertex:
Axis of symmetry:
By the Quadratic Formula,
x-intercepts: 125 ± !41
2, 02
x 525 ± !41
2.
0 5 x2 1 5x 2 4
x 5 25
2
125
2, 2
41
4 2 5 1x 1
5
222
241
4
x
−2
−2 2−4−6−8
−4
−10
y 5 1x 15
222
225
42
16
4
5 x2 1 5x 125
42
25
42 4
hsxd 5 x2 1 5x 2 4 12.
Vertex:
Axis of symmetry:
By the Quadratic Formula,
The equation has no real zeros.
intercepts: Nonex-
x 524 ± 8i
85 2
1
2± i.
0 5 4x2 1 4x 1 5
x 5 21
2
121
2, 42
5 41x 11
222
1 4
x
8
10
12
2
−2−2
2 4 6−4−6−8
y 5 431x 11
222
1 14
5 41x2 1 x 11
42
1
41
5
42 fsxd 5 4x2 1 4x 1 5
13.
Vertex:
Axis of symmetry: x 5 25
2
125
2, 2
41
122
51
31x 15
222
241
12
51
331x 15
222
241
4 4 5
1
31x2 1 5x 125
42
25
42 42
x
4
−4
−2 2−4−6−8
−6
2
y
f sxd 51
3sx2 1 5x 2 4d
By the Quadratic Formula,
x-intercepts: 125 ± !41
2, 02
x 525 ± !41
2.
0 5 x2 1 5x 2 4
240 Chapter 2 Polynomial and Rational Functions
14.
Vertex:
Axis of symmetry:
x-intercepts: 12 ±!3
3, 02
512 ± !12
65 2 ±
!3
3 x 5
2s212d ± !s212d2 2 4s3ds11d2s3d
0 5 3x2 2 12x 1 11
x 5 2
s2, 21d
5 3sx 2 2d2 2 1
5 3sx 2 2d2 1 3s24d 1 11
5 3sx2 2 4x 1 4 2 4d 1 11
5 3x2 2 12x 1 11
–6 –4 –2 4 6 8 10
2
4
6
8
10
12
14
x
y
f sxd 51
2s6x2 2 24x 1 22d
15. Vertex:
Point:
Thus, f sxd 5 212sx 2 4d2 1 1.
212 5 a
22 5 4a
s2, 21d ⇒ 21 5 as2 2 4d2 1 1
s4, 1d ⇒ f sxd 5 asx 2 4d2 1 1 16. Vertex:
fsxd 514sx 2 2d2 1 2
14 5 a
1 5 4a
3 5 4a 1 2
Point: s0, 3d ⇒ 3 5 as0 2 2d2 1 2
s2, 2d ⇒ fsxd 5 asx 2 2d2 1 2
17. Vertex:
Point:
Thus, f sxd 5 sx 2 1d2 2 4.
1 5 a
s2, 23d ⇒ 23 5 as2 2 1d2 2 4
s1, 24d ⇒ f sxd 5 asx 2 1d2 2 4 18. Vertex:
fsxd 513sx 2 2d2 1 3
13 5 a
3 5 9a
6 5 9a 1 3
Point: s21, 6d ⇒ 6 5 as21 2 2d2 1 3
s2, 3d ⇒ fsxd 5 asx 2 2d2 1 3
19. (a)
x
y
(c)
The maximum area occurs at the vertex when
and The dimensions with the
maximum area are meters and meters.y 5 50x 5 50
y 5 100 2 50 5 50.
x 5 50
5 2sx 2 50d2 1 2500
5 2fsx 2 50d2 2 2500g
5 2sx2 2 100x 1 2500 2 2500d
Area 5 100x 2 x2(b)
5 100x 2 x2
5 xs100 2 xd
Area 5 xy
y 5 100 2 x
x 1 y 5 100
2x 1 2y 5 200
20.
(a)
Rs30d 5 $15,000
Rs25d 5 $13,750
Rs20d 5 $12,000
R 5 210p2 1 800p
(b) The maximum revenue occurs at the vertex of the parabola.
The revenue is maximum when the price is $40 per unit.
The maximum revenue is $16,000.
Rs40d 5 $16,000
2b
2a5
2800
2s210d 5 $40
Review Exercises for Chapter 2 241
21.
The minimum cost occurs at the vertex of the parabola.
Approximately 1091 units should be produced each day to
yield a minimum cost.
Vertex: 2b
2a5 2
2120
2s0.055d< 1091 units
C 5 70,000 2 120x 1 0.055x2 22.
The age of the bride is
approximately 24 years
when the age of the groom
is 26 years.
y
x20 21 22 23 24 25
22
23
24
25
26
27
Age of bride
Age
of
gro
om
x < 23.7, 29.4
x 525.68 ± !s5.68d2 2 4s20.107ds274.5d
2s20.107d
0 5 20.107x2 1 5.68x 2 74.5
26 5 20.107x2 1 5.68x 2 48.5
23.
Transformation: Reflection in
the x-axis and a horizontal shift
four units to the right
x
2
1
1 2 7−2
5
−3
−4
3
4
3 4 6
y
y 5 x3, f sxd 5 2sx 2 4d3 24.
is a reflection in the x-axis
and a vertical stretch of the graph
of y 5 x3.
f sxd
x−1−2−3
2
1
1 2 3
3
−1
−2
−3
y
f sxd 5 24x3y 5 x3, 25.
Transformation: Reflection in
the x-axis and a vertical shift
two units upward
x
1
1 2 3−2−3
3
−1
−2
−3
y
y 5 x4, f sxd 5 2 2 x4
26.
is a shift to the right two
units and a vertical stretch of the
graph of y 5 x4.
f sxd
x−1−2−3 1 2 3 4 5 6
2
−2
−3
4
5
6
3
1
y
f sxd 5 2sx 2 2d4y 5 x4, 27.
Transformation: Horizontal shift
three units to the right
x
2
1
1−2
5
−5
3
4
3 4 5 6 7
y
y 5 x5, f sxd 5 sx 2 3d5 28.
is a vertical shrink and a
vertical shift three units upward
of the graph of y 5 x5.
f sxd
x6−2−4−6 2 4
4
8
6
y
f sxd 512x5 1 3y 5 x5,
29.
The degree is even and the leading coefficient is negative.
The graph falls to the left and falls to the right.
fsxd 5 2x2 1 6x 1 9 30.
The degree is odd and the leading coefficient is positive.
The graph falls to the left and rises to the right.
fsxd 512x3 1 2x
31.
The degree is even and the leading coefficient is positive.
The graph rises to the left and rises to the right.
g sxd 534sx4 1 3x2 1 2d 32.
The degree is odd and the leading coefficient is negative.
The graph rises to the left and falls to the right.
hsxd 5 2x5 2 7x2 1 10x
242 Chapter 2 Polynomial and Rational Functions
33.
Zeros: all of
multiplicity 1 (odd multiplicity)
Turning points: 1
x 532, 27,
5 s2x 2 3dsx 1 7d
0 5 2x2 1 11x 2 21 −9
−40
9
20 f sxd 5 2x2 1 11x 2 21 34.
Zeros: of multiplicity 1
(odd multiplicity)
of multiplicity 2
(even multiplicity)
Turning points: 2
x 5 23
x 5 0
0 5 xsx 1 3d2−6
−5
6
3f sxd 5 xsx 1 3d 2
35.
Zeros: all of
multiplicity 1 (odd multiplicity)
Turning points: 2
t 5 0, ±!3
0 5 tst2 2 3d
0 5 t3 2 3t−5
−3
4
3 f std 5 t3 2 3t 36.
Zeros: of multiplicity 2
(even multiplicity)
of multiplicity 1
(odd multiplicity)
Turning points: 2
x 5 8
x 5 0
0 5 x2sx 2 8d
0 5 x3 2 8x2−10
−80
10
10 f sxd 5 x3 2 8x2
37.
Zeros: of multiplicity 2
(even multiplicity)
of multiplicity 1
(odd multiplicity)
Turning points: 2
x 5 53
x 5 0
0 5 24x2s3x 2 5d
0 5 212x3 1 20x2
−5
−5
5
10 f sxd 5 212x3 1 20x2 38.
Zeros: of multiplicity 2 (even multiplicity)
of multiplicity 1 (odd multiplicity)
of multiplicity 1 (odd multiplicity)
Turning points: 3
x 5 2
x 5 21
x 5 0
5 x2sx 1 1dsx 2 2d
0 5 x2sx2 2 x 2 2d
0 5 x4 2 x3 2 2x2
−4
−3
5
3gsxd 5 x4 2 x3 2 2x2
39.
(a) The degree is odd and the leading coefficient is
negative. The graph rises to the left and falls to
the right.
(b) Zero:
(c)
(d) y
x−4 −3 −2
−4
−3
1
2
3
4
1 2 3 4
(−1, 0)
x 5 21
f sxd 5 2x3 1 x2 2 2 40.
(a) The degree is odd and the leading coefficient, 2, is
positive. The graph rises to the right and falls to the left.
(b)
The zeros are 0 and
(c)
(d) y
x−4 −3 −1
−4
−3
−2
−1
2
3
4
21 3 4
(−2, 0) (0, 0)
22.
0 5 x2sx 1 2d
0 5 2x2sx 1 2d
0 5 2x3 1 4x2
gsxd 5 2x3 1 4x2
gsxd 5 2x3 1 4x2
x 0 1 2
34 10 0 262222f sxd
212223
x 0 1
0 2 0 6218gsxd
212223
Review Exercises for Chapter 2 243
48.
4x 1 7
3x 2 25
4
31
29
3s3x 2 2d
293
4x 283
3x 2 2 ) 4x 1 7
43
41.
(a) The degree is even and the leading coefficient is
positive. The graph rises to the left and rises to
the right.
(b) Zeros:
(c)
(d) y
x−4 −1−2
3
21 3 4
−15
−18
−21
(−3, 0)
(0, 0)
(1, 0)
x 5 0, 1, 23
f sxd 5 xsx3 1 x2 2 5x 1 3d
x 0 1 2 3
100 0 0 0 10 7228218f sxd
21222324
42.
(a) The degree is even and the leading coefficient, , is
negative. The graph falls to the left and falls to the right.