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R E V I E W, p a g e s 6 9 2 – 6 9 7
8.1
1. Sketch a graph of each absolute function.
Identify the intercepts, domain, and range.
a) y = ƒ -4x + 2 ƒ
b) y = ƒ (x + 4)(x - 2) ƒ
y
8
y 4x 2
2
y
4
x
4
2
0
2
4
4
2
0
2
4
4
8
Draw the graph of
y ⴝ ⴚ4x ⴙ 2.
It has x-intercept 0.5.
Reflect, in the x-axis, the part
of the graph that is below the
x-axis.
From the graph, the
x-intercept is 0.5,
the y-intercept is 2, the
domain of y ⴝ 円ⴚ4x ⴙ 2円 is
x ç ⺢, and the range is
y » 0.
y (x 4)(x 2)
x
2
4
Draw the graph of y ⴝ (x ⴙ 4)(x ⴚ 2).
It has x-intercepts ⴚ 4 and 2. The axis
of symmetry is x ⴝ ⴚ1, so the vertex
is at (ⴚ1, ⴚ9).
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 2, the y-intercept is 8, the
domain of y ⴝ 円 (x ⴙ 4)(x ⴚ 2)円 is
x ç ⺢, and the range is y » 0.
2. Write each absolute value function in piecewise notation.
a) y = ƒ -x - 9 ƒ
b) y = ƒ 2x(x + 5) ƒ
y ⴝ ⴚx ⴚ 9 when
ⴚx ⴚ 9 » 0
ⴚx » 9
x ◊ ⴚ9
y ⴝ ⴚ (ⴚ x ⴚ 9),
or y ⴝ x ⴙ 9 when
ⴚx ⴚ 9<0
ⴚx<9
x>ⴚ9
So, using piecewise notation:
yⴝ e
44
ⴚ x ⴚ 9,
x ⴙ 9,
if x ◊ ⴚ9
if x>ⴚ9
The x-intercepts of the graph of
y ⴝ 2x(x ⴙ 5) are x ⴝ 0 and 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 円 2x(x ⴙ 5)円 :
For x ◊ ⴚ5 or x » 0, the value of
2x(x ⴙ 5) » 0
For ⴚ5<x<0, the value of
2x(x ⴙ 5)<0; that is, y ⴝ ⴚ2x(x ⴙ 5).
So, using piecewise notation:
yⴝ e
2x(x ⴙ 5),
ⴚ2x(x ⴙ 5),
if x ◊ ⴚ5 or x
if ⴚ5<x<0
»0
Chapter 8: Absolute Value and Reciprocal Functions—Review—Solutions
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8.2
3. Solve by graphing.
b) ƒ x2 - 5x ƒ = 6
a) 3 = ƒ -2x + 4 ƒ
4
y
y3
2
1
0
2
y |2x 4|
x
3
2
Enter y ⴝ 円x 2 ⴚ 5x円 and y ⴝ 6 in
the graphing calculator.
The line y ⴝ 6 appears to intersect
y ⴝ 円 x2 ⴚ 5x円 at 4 points:
(ⴚ1, 6), (2, 6), (3, 6), and (6, 6).
So, the equation has 4 solutions:
x ⴝ ⴚ1, x ⴝ 2, x ⴝ 3, and x ⴝ 6
y 2x 4
4
To graph y ⴝ 円ⴚ2x ⴙ 4円 , graph
y ⴝ ⴚ2x ⴙ 4, then reflect, in the
x-axis, the part of the graph that
is below the x-axis. The line y ⴝ 3
intersects y ⴝ 円ⴚ2x ⴙ 4円 at (0.5,
3) and (3.5, 3). So, the solutions
are x ⴝ 0.5 and x ⴝ 3.5.
4. Use algebra to solve each equation.
a) 2 = ƒ (x - 1)2 - 2 ƒ
When (x ⴚ 1)2 ⴚ 2
» 0: When (x ⴚ 1)2 ⴚ 2<0:
2 ⴝ (x ⴚ 1) ⴚ 2
2 ⴝ ⴚ (( x ⴚ 1)2 ⴚ 2)
2
4 ⴝ (x ⴚ 1)2
ⴚ2 ⴝ (x ⴚ 1) 2 ⴚ 2
x ⴝ 3 or x ⴝ ⴚ1
0 ⴝ (x ⴚ 1)2
xⴝ1
So, x ⴝ ⴚ1, x ⴝ 1, and x ⴝ 3 are the solutions.
1
b) 2x = 2 ƒ 3x - 5 ƒ
4x ⴝ 円3x ⴚ 5円
4x ⴝ 3x ⴚ 5
if 3x ⴚ 5 » 0
that is, if x
When x
»
4x ⴝ ⴚ (3x ⴚ 5)
if 3x ⴚ 5<0
5
3
» 53 :
4x ⴝ 3x ⴚ 5
x ⴝ ⴚ5
5
3
that is, if x<
5
3
When x< :
4x ⴝ ⴚ(3x ⴚ 5)
4x ⴝ ⴚ3x ⴙ 5
7x ⴝ 5
xⴝ
5
7
5
3
ⴚ5 is not greater than or equal to , so ⴚ5 is not a solution.
5
5
5
< so is a root.
7
3
7
5
The solution is x ⴝ .
7
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8.3
5. Identify the equation of the vertical asymptote of the graph of
1
, then graph the function.
y =
- 2x + 5
5
2
The graph of y ⴝ ⴚ2x ⴙ 5 has slope ⴚ2, x-intercept , and y-intercept 5.
1
The graph of y ⴝ
has a horizontal asymptote y ⴝ 0 and a
ⴚ 2x ⴙ 5
5
vertical asymptote x ⴝ . Points (3, ⴚ1) and (2, 1) are common to both
2
graphs. Some points on y ⴝ ⴚ2x ⴙ 5 are (1, 3), (0, 5), (4, ⴚ3), and (5, ⴚ5).
So, points on y ⴝ
1
1
1
are a1, b , (0, 0.2), a4, ⴚ b , and (5, ⴚ0.2).
ⴚ2x ⴙ 5
3
3
6. Use the graph of y = f (x) to sketch a graph of y =
4
y
y
1
2x 5
4
6
2
x
2
0
2
2
4
1
.
f (x)
Identify the equations of the asymptotes of the graph of each
reciprocal function.
a)
4
y
( 12
2
0
y
b)
1
f(x )
y f(x)
2
x
, 0)
x
0
4
2
y f(x)
4
2
4
y
6
Horizontal asymptote: y ⴝ 0
1
x-intercept is , so vertical
2
1
asymptote is x ⴝ . Points
2
(0, 1) and (1, ⴚ1) are common
to both graphs. Some points
on y ⴝ f(x) are: (ⴚ1, 3) and
(2, ⴚ3). So, points on y ⴝ
1
f(x)
2
y
8
1
f(x )
Horizontal asymptote: y ⴝ 0
x-intercept is 6, so vertical
asymptote is x ⴝ 6.
Points (7, 1) and (5, ⴚ1) are
common to both graphs.
Some points on y ⴝ f(x) are:
(8, 2) and (4, ⴚ2). So, points on
yⴝ
1
are (8, 0.5) and (4, ⴚ0.5).
f(x)
are aⴚ1, b and a2, ⴚ b .
1
3
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1
3
Chapter 8: Absolute Value and Reciprocal Functions—Review—Solutions
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7. Use the graph of y =
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1
to graph the linear function y = f (x).
f (x)
Describe your strategy.
4
y
y f (x)
2
4
2
0
y
2
1
f(x)
4
x
6
2
4
Vertical asymptote is x ⴝ 1.5,
so graph of y ⴝ f(x) has
x-intercept 1.5. Mark points at
y ⴝ 1 and y ⴝ ⴚ1 on graph of
yⴝ
1
, then draw a line
f(x)
through these points for
the graph of y ⴝ f(x).
8.4
8. Use a graphing calculator or graphing software.
For which values of q does the graph of
1
y =
have:
2
- (x - 3) + q
a) no vertical asymptotes?
Look at the quadratic function y ⴝ ⴚ(x ⴚ 3)2 ⴙ q. The graph opens
down. For the graph of its reciprocal function to have no vertical
asymptotes, the quadratic function must have no x-intercepts. So, the
vertex of the quadratic function must be below the x-axis; that is, q<0.
b) one vertical asymptote?
Look at the quadratic function y ⴝ ⴚ(x ⴚ 3)2 ⴙ q. The graph opens
down. For the graph of its reciprocal function to have one vertical
asymptote, the quadratic function must have one x-intercept. So, the
vertex of the quadratic function must be on the x-axis; that is, q ⴝ 0.
c) two vertical asymptotes?
Look at the quadratic function y ⴝ ⴚ(x ⴚ 3)2 ⴙ q. The graph opens
down. For the graph of its reciprocal function to have two vertical
asymptotes, the quadratic function must have two x-intercepts. So, the
vertex of the quadratic function must be above the x-axis; that is, q>0.
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9. Determine the equations of the vertical asymptotes of the graph of
each reciprocal function.
Graph to check the equations.
a) y =
1
(x - 2)2 - 9
b) y =
(x ⴚ 2)2 ⴚ 9 ⴝ 0 when
(x ⴚ 2)2 ⴝ 9; that is, when
x ⴝ 5 or x ⴝ ⴚ1. So, the
graph of y ⴝ
1
- (x - 2)2 - 9
ⴚ(x ⴚ 2)2 ⴚ 9 ⴝ 0 when
(x ⴚ 2)2 ⴝ ⴚ9. Since the square
of a number is never negative,
the graph of y ⴝ ⴚ(x ⴚ 2)2 ⴚ 9
has no x-intercepts, and the
1
has
(x ⴚ 2)2 ⴚ 9
2 vertical asymptotes, x ⴝ 5
and x ⴝ ⴚ1. I used my graphing
calculator to show that my
equations are correct.
graph of y ⴝ
1
has
ⴚ(x ⴚ 2)2 ⴚ 9
no vertical asymptotes. I used my
graphing calculator to show that
my equations are correct.
8.5
10. On the graph of each quadratic function y = f (x), sketch a graph
1
of the reciprocal function y =
. Identify the vertical asymptotes,
f (x)
if they exist.
a)
b)
y
x
0
2
2
4 1
y
f(x)
6
y
y f (x)
4
y
2
4
x
6
y f(x)
The graph of y ⴝ f(x) has no
x-intercepts, so the graph of
yⴝ
1
has no vertical
f(x)
asymptotes and has Shape 1.
Horizontal asymptote: y ⴝ 0
Points (2, ⴚ4), (3, ⴚ 2), and
(4, ⴚ4) lie on y ⴝ f(x), so
points (2, ⴚ0.25), (3, ⴚ 0.5),
and (4, ⴚ 0.25) lie on
yⴝ
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1
f(x)
0
2
4
6
8
The graph of y ⴝ f(x) has 1
x-intercept, so the graph of
yⴝ
1
has 1 vertical asymptote,
f(x)
x ⴝ 5, and has Shape 2.
Horizontal asymptote: y ⴝ 0
Plot points where the line y ⴝ 1
intersects the graph of y ⴝ f(x).
These points are common to both
graphs.
1
.
f(x)
Chapter 8: Absolute Value and Reciprocal Functions—Review—Solutions
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11. On the graph of each reciprocal function y =
the quadratic function y = f (x).
a)
y
0
y f (x)
b)
1
, sketch a graph of
f (x)
4
x
2
y
y
1
f(x)
2
y f (x)
0
2
2
x
4
4
6
y
DO NOT COPY.
4
1
f(x)
The graph has one vertical
asymptote, x ⴝ 5, so the graph
of y ⴝ f(x) has vertex (5, 0).
The line y ⴝ ⴚ1 intersects the
graph at 2 points that are
common to both graphs.
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2
The graph has 2 vertical
asymptotes, so the graph of
y ⴝ f(x) has 2 x-intercepts.
Plot points where the asymptotes
intersect the x-axis. The point
(0, ⴚ1) is on the line of
symmetry, so (0, ⴚ1) is the
vertex of y ⴝ f(x).
Chapter 8: Absolute Value and Reciprocal Functions—Review—Solutions
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