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Lesson 8.3 Exercises, pages 657–665
A
3. For each function, write the equation of the corresponding
reciprocal function.
a) y = 5x - 2
yⴝ
b) y = 3x
1
5x ⴚ 2
yⴝ
1
c) y = -4
yⴝ ⴚ
1
3x
d) y = 2
1
4
yⴝ
1
1
2
yⴝ2
4. Sketch broken lines to represent the vertical and horizontal
asymptotes of each graph.
a)
b)
y
6
y
4
4
2
x
4
2
0
2
4
4
6
1
y
f (x)
0
y
1
f (x)
2
x
4
2
4
6
The graph approaches the
x-axis, so the line y ⴝ 0 is a
horizontal asymptote. The
graph approaches the y-axis,
so the line x ⴝ 0 is a vertical
asymptote.
The graph approaches the x-axis,
so the line y ⴝ 0 is a horizontal
asymptote. The graph approaches
the line x ⴝ ⴚ2, so x ⴝ ⴚ2 is a
vertical asymptote.
5. Identify the equation of the vertical asymptote of the graph of each
reciprocal function.
1
a) y = 2x
b) y =
Let denominator equal 0.
2x ⴝ 0
xⴝ0
So, graph of y ⴝ
1
has a
2x
vertical asymptote at x ⴝ 0.
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1
- 3x + 12
Let denominator equal 0.
ⴚ3x ⴙ 12 ⴝ 0
ⴚ3x ⴝ ⴚ12
xⴝ4
So, graph of y ⴝ
1
has a
ⴚ3x ⴙ 12
vertical asymptote at x ⴝ 4.
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c) y =
1
5x + 15
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1
d) y = 6x - 3
Let denominator equal 0.
5x ⴙ 15 ⴝ 0
5x ⴝ ⴚ15
Let denominator equal 0.
6x ⴚ 3 ⴝ 0
6x ⴝ 3
x ⴝ ⴚ3
xⴝ
So, graph of y ⴝ
1
has a
5x ⴙ 15
vertical asymptote at x ⴝ ⴚ3.
1
2
1
has a
6x ⴚ 3
1
vertical asymptote at x ⴝ .
2
So, graph of y ⴝ
B
6. Use each graph of y = f (x) to sketch a graph of y =
1
.
f (x)
Identify the asymptotes of the graph of the reciprocal function.
a)
y
y
4
1
f (x)
2
x
2
7
0
2
4
y f (x)
Horizontal asymptote: y ⴝ 0
x-intercept is ⴚ4, so vertical asymptote is x ⴝ ⴚ 4.
Points (ⴚ3, 1) and (ⴚ5, ⴚ1) are common to both graphs.
Some points on y ⴝ f(x) are: (ⴚ2, 2), (0, 4), (ⴚ8, ⴚ4), and
(ⴚ6, ⴚ2). So, points on y ⴝ
1
are (ⴚ2, 0.5), (0, 0.25), (ⴚ8, ⴚ0.25),
f(x)
and (ⴚ6, ⴚ0.5).
b)
6
y f (x)
y
4
y
1
f (x)
2
x
4
2
0
4
6
2
4
6
Horizontal asymptote: y ⴝ 0
x-intercept is 1, so vertical asymptote is x ⴝ 1.
Points (0.5, 1) and (1.5, ⴚ1) are common to both graphs.
Some points on y ⴝ f(x) are: (2, ⴚ2), (3, ⴚ4), (0, 2), and (ⴚ1, 4).
So, points on y ⴝ
24
1
are (2, ⴚ0.5), (3, ⴚ0.25), (0, 0.5), and (ⴚ1, 0.25).
f(x)
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7. For each pair of functions, use a graph of the linear function to
sketch a graph of the reciprocal function.
State the domain and range of each reciprocal function.
1
a) y = -x and y = - x
y x
y
3
x
4
2
0
4
2
4
1
y x
The graph of y ⴝ ⴚx has slope ⴚ1 and y-intercept 0.
1
The graph of y ⴝ ⴚ x has horizontal asymptote y ⴝ 0 and vertical
asymptote x ⴝ 0.
Points (ⴚ1, 1) and (1, ⴚ1) are common to both graphs.
Some points on y ⴝ ⴚx are (ⴚ2, 2), (ⴚ4, 4), (2, ⴚ2), and (4, ⴚ4).
1
So, points on y ⴝ ⴚ x are (ⴚ2, 0.5), (ⴚ4, 0.25), (2, ⴚ0.5), and (4, ⴚ0.25).
1
From the graph, y ⴝ ⴚ x has domain x ç ⺢, x ⴝ 0 and range:
y ç ⺢, y ⴝ 0.
b) y = x + 2 and y =
4
y
1
x2
6
1
x + 2
y
2
x
0
2
2
yx2
4
The graph of y ⴝ x ⴙ 2 has slope 1 and y-intercept 2.
The graph of y ⴝ
1
has horizontal asymptote y ⴝ 0 and vertical
xⴙ2
asymptote x ⴝ ⴚ2.
Points (ⴚ1, 1) and (ⴚ3, ⴚ1) are common to both graphs.
Some points on y ⴝ x ⴙ 2 are (0, 2), (2, 4), (ⴚ4, ⴚ2), and (ⴚ6, ⴚ4).
1
are (0, 0.5), (2, 0.25), (ⴚ4, ⴚ0.5), and (ⴚ6, ⴚ0.25).
xⴙ2
1
From the graph, y ⴝ
has domain x ç ⺢, x ⴝ ⴚ2 and range:
xⴙ2
So, points on y ⴝ
y ç ⺢, y ⴝ 0.
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1
2x + 1
c) y = 2x + 1 and y =
y
1
2x 1
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y
x
4
0
2
y 2x 1
The graph of y ⴝ 2x ⴙ 1 has slope 2 and y-intercept 1.
The graph of y ⴝ
1
has horizontal asymptote y ⴝ 0 and
2x ⴙ 1
vertical asymptote x ⴝ ⴚ 0.5.
Points (0, 1) and (ⴚ1, ⴚ1) are common to both graphs.
Some points on y ⴝ 2x ⴙ 1 are (0.5, 2), (2, 5), (ⴚ1.5, ⴚ2), and (ⴚ3, ⴚ5).
1
are (0.5, 0.5), (2, 0.2), (ⴚ1.5, ⴚ0.5), and
2x ⴙ 1
1
(ⴚ3, ⴚ0.2). From the graph, y ⴝ
has domain x ç ⺢, x ⴝ ⴚ0.5
2x ⴙ 1
So, points on y ⴝ
and range: y ç ⺢, y ⴝ 0.
d) y = -x + 4 and y =
4
y
y
1
-x + 4
1
x 4
2
x
0
2
2
y x 4
4
The graph of y ⴝ ⴚx ⴙ 4 has slope ⴚ1 and y-intercept 4.
The graph of y ⴝ
1
has horizontal asymptote y ⴝ 0 and
ⴚx ⴙ 4
vertical asymptote x ⴝ 4.
Points (3, 1) and (5, ⴚ1) are common to both graphs.
Some points on y ⴝ ⴚx ⴙ 4 are (2, 2), (0, 4), (6, ⴚ2), and (8, ⴚ4).
1
are (2, 0.5), (0, 0.25), (6, ⴚ0.5), and
ⴚx ⴙ 4
1
(8, ⴚ0.25). From the graph, y ⴝ
has domain x ç ⺢, x ⴝ 4
ⴚx ⴙ 4
So, points on y ⴝ
and range: y ç ⺢, y ⴝ 0.
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8. Use each graph of y =
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1
to graph the linear function y = f (x).
f (x)
Describe the strategy you used.
a)
6
y
y
1
f (x)
4
2
x
0
2
4
6
2
4
x5
y f (x )
Vertical asymptote is x ⴝ 5, so graph of y ⴝ f(x) has x-intercept 5.
Mark points at y ⴝ 1 and y ⴝ ⴚ1 on graph of y ⴝ
through these points for the graph of y ⴝ f(x).
b)
6
1
, then draw a line
f(x)
y
y
4
2
1
f (x)
x3
x
0
2
6
8
2
4
y f (x)
Vertical asymptote is x ⴝ 3, so graph of y ⴝ f(x) has x-intercept 3.
Mark points at y ⴝ 1 and y ⴝ ⴚ 1 on graph of y ⴝ
1
, then draw a line
f(x)
through these points for the graph of y ⴝ f(x).
9. Use graphing technology. Graph each pair of functions on the same
screen. State the domain and range of each reciprocal function.
a) y = 3x + 5 and y =
1
3x + 5
5
3
The graph of y ⴝ 3x ⴙ 5 has x-intercept ⴚ , so the graph of
1
5
has vertical asymptote x ⴝ ⴚ . The reciprocal function
3x ⴙ 5
3
5
has domain x ç ⺢, x ⴝ ⴚ and range: y ç ⺢, y ⴝ 0.
3
yⴝ
b) y = -5x + 0.5 and y =
1
- 5x + 0.5
Using the Intersect feature, the graph of y ⴝ ⴚ5x ⴙ 0.5 has
x-intercept 0.1, so the graph of y ⴝ
1
has vertical
ⴚ 5x ⴙ 0.5
asymptote x ⴝ 0.1. The reciprocal function has domain x ç ⺢, x ⴝ 0.1
and range: y ç ⺢, y ⴝ 0.
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10. a) The function y = f (x) is linear. Is it possible for y =
1
to be a
f (x)
horizontal line? If so, how do the domain and range of the reciprocal
function compare to the domain and range of f(x)?
Yes, if the linear function is a horizontal line of the form
1
y ⴝ c, c ⴝ 0, then the graph of the reciprocal function y ⴝ c
is also a horizontal line. The domain of both functions is x ç ⺢.
The range of the linear function is y ⴝ c and the range of the
1
reciprocal function is y ⴝ c .
b) The function y = f (x) is linear. Is it possible for y =
1
to be
f (x)
undefined for all real values of x? Explain your thinking.
Yes, if the linear function is the horizontal line y ⴝ 0,
then the graph of the reciprocal function y ⴝ
1
is undefined.
0
3
11. The reciprocal of a linear function has a vertical asymptote x = 4 .
What is an equation for the reciprocal function?
1
.
mx ⴙ b
3
3
The vertical asymptote is x ⴝ , so mx ⴙ b ⴝ 0 when x ⴝ .
4
4
3
ma b ⴙ b ⴝ 0
4
The equation of the reciprocal function has the form y ⴝ
Choose a value for m.
When m ⴝ 4, 3 ⴙ b ⴝ 0, or b ⴝ ⴚ3.
So, an equation for the function is: y ⴝ
1
4x ⴚ 3
12. Two linear functions with
6
opposite slopes were used to
create graphs of the
reciprocal functions
1
1
y =
and y =
.
f(x)
g(x)
a) Which linear function has
a positive slope?
y
4
y
6
4
1
g(x)
y
0
1
f(x)
4
x
6
4
6
The graph of y ⴝ
1
goes through the points (1, 1) and (ⴚ1, ⴚ1). A line
f(x)
through these points goes up to the right, so y ⴝ f(x) has a positive slope.
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b) Which linear function has a negative slope?
Explain your reasoning.
The graph of y ⴝ
1
goes through the points (1, ⴚ1) and (ⴚ1, 1). A
g(x)
line through these points goes down to the right, so y ⴝ g(x) has a
negative slope.
13. a) Write a reciprocal function that describes the length, l metres, of
a rectangle with area 1 m2, as a function of its width, w metres.
Use the formula for the area, A, of a rectangle with length l and width w:
A ⴝ lw Substitute: A ⴝ 1
1 ⴝ lw
1
lⴝw
b) What are the domain and range of the reciprocal function in part a?
Both length and width are positive. The reciprocal function has vertical
asymptote w ⴝ 0 and horizontal asymptote l ⴝ 0. So, the domain is
w ç ⺢, w>0 and the range is l ç ⺢, l>0.
c) Graph the reciprocal function in part a. Describe the graph. How
does it differ from the graphs of other reciprocal functions you
have seen? Explain.
6
w
4
2
1
l w
l
2
0
2
4
6
2
1
Both l>0 and w>0, so the graph of l ⴝ w
is in Quadrant 1. The arm of the graph that
would be in Quadrant 3 is not included because
both length and width must be positive.
14. A linear function has the form y = ax + b, a Z 0. Why does the
graph of its reciprocal function always have a vertical and a
horizontal asymptote?
b
A linear function of the form y ⴝ ax ⴙ b, a ⴝ 0, has an x-intercept of ⴚ a
b
when y ⴝ 0, so the graph of its reciprocal has a vertical asymptote at x ⴝ ⴚ a .
The numerator of a reciprocal function is 1. So, the value of the function cannot
1
be 0 for any value of x. When |x| is very large,
is close to 0. So, the x-axis
ax ⴙ b
is a horizontal asymptote.
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C
15. The graphs of two distinct linear functions y = f (x) and y = g(x) are
parallel. Do the graphs of their reciprocal functions intersect? How
do you know?
The functions are parallel so y ⴝ f(x) and y ⴝ g(x) have the same slope.
Let the equations of the lines be y ⴝ ax ⴙ b and y ⴝ ax ⴙ c.
By graphing some examples on my calculator, it seems that the graphs
never intersect.
1
1
ⴝ ax ⴙ c .
I have to show that there is no value of x for which
ax ⴙ b
1
1
ⴝ
Assume there is a value of x, x ⴝ d, where
a(d) ⴙ b
a(d) ⴙ c
Then, ad ⴙ c ⴝ ad ⴙ b
cⴝb
But, b ⴝ c because the linear functions are distinct. So, the graphs of
the reciprocal functions do not intersect.
16. a) How can you tell without graphing whether the graphs of
1
1
intersect?
y = x - 2 and y =
-x + 4
I can equate the two reciprocals to determine if there is a value of x
for which the y-values are the same. If there is, then the graphs
intersect.
b) Determine the coordinates of any points of intersection.
The y-values are equal when:
1
1
ⴝ
xⴚ2
ⴚx ⴙ 4
x ⴚ 2 ⴝ ⴚx ⴙ 4
2x ⴝ 6
xⴝ3
1
:
xⴚ2
1
yⴝ
3ⴚ2
Substitute x ⴝ 3 in y ⴝ
yⴝ1
The graphs of the reciprocal functions intersect at (3, 1).
c) Use graphing technology. Graph the functions to verify your
answer.
The graphs of the reciprocal functions intersect at (3, 1).
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17. Determine the equation of the linear function y = f (x) you
graphed in question 8, part a.
The linear function has x-intercept 5 and y-intercept ⴚ5.
So, the equation has the form y ⴝ mx ⴚ 5.
The line passes through (6, 1) so substitute x ⴝ 6 and y ⴝ 1.
1 ⴝ m(6) ⴚ 5
6 ⴝ 6m
mⴝ1
So, the equation on the linear function is: y ⴝ x ⴚ 5
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