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Checkpoint: Assess Your Understanding, pages 648–651
8.1
1. Multiple Choice Which statement about y = -4x + 8 and
y = ƒ -4x + 8 ƒ is false?
A. Both functions have the same x-intercept.
B. Both functions have the same y-intercept.
C. Both functions have the same domain.
D. Both functions have the same range.
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Chapter 8: Absolute Value and Reciprocal Functions—Checkpoint—Solutions
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2. Sketch a graph of each absolute function.
Identify the intercepts, domain, and range.
b) y = ƒ x2 + 6x + 8 ƒ
a) y = ƒ -5x + 10 ƒ
4
y
y
6
y |5x 10|
2
4
x
0
4
6
8
2
y |x 2 6x 8|
2
6
4
Draw the graph of
y ⴝ ⴚ5x ⴙ 10.
It has x-intercept 2 and
y-intercept 10.
Reflect, in the x-axis,
the part of the graph that is
below the x-axis.
The x-intercept is 2 and the
y-intercept is 10. The domain
of y ⴝ 円 ⴚ5x ⴙ 10円 is x ç ⺢,
and the range is y » 0.
x
4
0
Draw the graph of: y ⴝ x2 ⴙ 6x ⴙ 8
y ⴝ (x ⴙ 2)(x ⴙ 4)
The graph opens up and has x-intercepts
ⴚ4 and ⴚ2. The axis of symmetry is
ⴚ4 ⴚ 2
xⴝ
, or ⴚ3 and the vertex
2
is at (ⴚ3, ⴚ1). Reflect, in the x-axis,
the part of the graph that is below the
x-axis.
From the graph, the x-intercepts are
ⴚ4 and ⴚ2, and the y-intercept is 8.
The domain of y ⴝ 円 x2 ⴙ 6x ⴙ 8円 is
x ç ⺢ and the range is y » 0.
3. Write each absolute value function in piecewise notation.
b) y = ƒ (x + 4)2 - 1 ƒ
a) y = ƒ 2x - 7 ƒ
y ⴝ 2x ⴚ 7 when
2x ⴚ 7 » 0
x
»
7
2
y ⴝ ⴚ(2x ⴚ 7),
or y ⴝ ⴚ2x ⴙ 7 when
2x ⴚ 7<0
x<
7
2
So, using piecewise notation:
yⴝ c
2x ⴚ 7, if x
»
7
2
7
2
ⴚ2x ⴙ 7, if x<
Determine the x-intercepts of
the graph of y ⴝ (x ⴙ 4)2 ⴚ 1.
0 ⴝ (x ⴙ 4)2 ⴚ 1
(x ⴙ 4)2 ⴝ 1
x ⴝ ⴚ3 or x ⴝ ⴚ5
The graph opens up, so between
the x-intercepts, the graph is below
the x-axis.
For the graph of y ⴝ (x ⴙ 4)2 ⴚ 1:
For x ◊ ⴚ5 or x » ⴚ3, the value of
(x ⴙ 4)2 ⴚ 1 » 0
For ⴚ5<x<ⴚ3, the value of
(x ⴙ 4)2 ⴚ 1<0;
that is, y ⴝ ⴚ ((x ⴙ 4)2 ⴚ 1), or
y ⴝ ⴚ(x ⴙ 4)2 ⴙ 1
So, using piecewise notation:
yⴝ e
20
(x ⴙ 4)2 ⴚ 1, if x ◊ ⴚ5 or x » ⴚ3
ⴚ(x ⴙ 4)2 ⴙ 1, if ⴚ5<x<ⴚ3
Chapter 8: Absolute Value and Reciprocal Functions—Checkpoint—Solutions
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8.2
4. Multiple Choice How many solutions does the equation
ƒ x2 + x - 9 ƒ = 6 have?
A. 1 solution
B. 2 solutions
C. 3 solutions
D. 4 solutions
5. Use the graphs to determine the solutions of each equation.
b) 5 = ƒ -2(x - 1)2 + 3 ƒ
a) ƒ 2x - 4 ƒ = 6
y 2(x 1)2 3
y
8
y 2x 4
y
8
y6
6
y5
4
4
2
x
2
2
0
x
4
4
The line y ⴝ 6 intersects
y ⴝ 円2x ⴚ 4 円 at 2 points:
(ⴚ1, 6) and (5, 6). So, the
solutions are x ⴝ ⴚ1 and
x ⴝ 5.
2
0
2
The line y ⴝ 5 intersects
y ⴝ 円ⴚ2(x ⴚ 1)2 ⴙ 3円 at 2
points: (ⴚ 1, 5) and (3, 5).
So, the solutions are
x ⴝ ⴚ1 and x ⴝ 3.
6. Solve by graphing.
b) ƒ (x - 1)2 - 4 ƒ = 5
a) 7 = ƒ 2x - 7 ƒ
y7
y
6
y |2x 7|
4
2
x
0
2
4
6
y 2x 7
To graph y ⴝ 円2x ⴚ 7円 ,
graph y ⴝ 2x ⴚ 7, then
reflect, in the x-axis, the part
of the graph that is below
the x-axis. The line y ⴝ 7
intersects y ⴝ 円2x ⴚ 7円
at (0, 7) and (7, 7). So, the
solutions are x ⴝ 0 and
x ⴝ 7.
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Enter y ⴝ 円(x ⴚ 1)2 ⴚ 4円 and
y ⴝ 5 in the graphing calculator.
The line y ⴝ 5 intersects
y ⴝ 円 (x ⴚ 1)2 ⴚ 4円 at 2 points:
(ⴚ2, 5) and (4, 5). So, the equation
has 2 solutions: x ⴝ ⴚ2
and x ⴝ 4.
Chapter 8: Absolute Value and Reciprocal Functions—Checkpoint—Solutions
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7. Use algebra to solve each equation.
a) 9 = ƒ -2x + 6 ƒ
ⴚ2x ⴙ 6 ⴝ 9
if ⴚ2x ⴙ 6 » 0
that is, if x ◊ 3
When x ◊ 3:
ⴚ 2x ⴙ 6 ⴝ 9
ⴚ2x ⴝ 3
ⴚ (ⴚ2x ⴙ 6) ⴝ 9
if ⴚ2x ⴙ 6<0
that is, if x>3
When x>3:
ⴚ (ⴚ 2x ⴙ 6) ⴝ 9
ⴚ2x ⴙ 6 ⴝ ⴚ9
3
2
x ⴝ ⴚ , or ⴚ 1.5
ⴚ2x ⴝ ⴚ15
xⴝ
ⴚ1.5 ◊ 3, so this root
is a solution.
15
, or 7.5
2
7.5>3, so this root
is a solution.
The solutions are x ⴝ ⴚ1.5 and x ⴝ 7.5.
b) ƒ x2 - 4x - 5 ƒ = 7
When x2 ⴚ 4x ⴚ 5 » 0:
x2 ⴚ 4x ⴚ 5 ⴝ 7
When x2 ⴚ 4x ⴚ 5<0:
ⴚ (x2 ⴚ 4x ⴚ 5) ⴝ 7
ⴚx2 ⴙ 4x ⴙ 5 ⴝ 7
ⴚx2 ⴙ 4x ⴚ 2 ⴝ 0
x2 ⴚ 4x ⴙ 2 ⴝ 0
x2 ⴚ 4x ⴚ 12 ⴝ 0
(x ⴚ 6)(x ⴙ 2) ⴝ 0
x ⴝ 6 or x ⴝ ⴚ2
√
xⴝ
4_ ( ⴚ 4)2 ⴚ 4(1)(2)
√
xⴝ
2(1)
4_ 8
2
√
4_2 2
xⴝ
2
√
So, x ⴝ 6, x ⴝ ⴚ2, x ⴝ 2 ⴙ
22
√
x ⴝ 2_ 2
2, and x ⴝ 2 ⴚ
√
2 are the solutions.
Chapter 8: Absolute Value and Reciprocal Functions—Checkpoint—Solutions
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