Warm Up

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Preview
Warm Up
California Standards
Lesson Presentation
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Warm Up
The inverse variation xy 8 relates the constant
speed x in mi/h to the time y in hours that it
takes to travel 8 miles. Graph this inverse
variation.
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Vocabulary
rational function excluded value discontinuous
function asymptote
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A rational function is a function whose rule is a
quotient of polynomials. The inverse variations
you studied in the previous lesson are a special
type of rational function.
Rational functions
For any function involving x and y, an excluded
value is any x-value that makes the function
value y undefined. For a rational function, an
excluded value is any value that makes the
denominator equal to 0.
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Additional Example 1 Identifying Excluded Values
Identify any excluded values for each rational
function.
A.
x 0
Set the denominator equal to 0.
The excluded value is 0.
B.
x 2 0
Set the denominator equal to 0.
Solve for x.
x 2
The excluded value is 2.
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Check It Out! Example 1
Identify any excluded values for each rational
function.
a.
x 0
Set the denominator equal to 0.
The excluded value is 0.
b.
x 1 0
Set the denominator equal to 0.
x 1
Solve for x.
The excluded value is 1.
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Check It Out! Example 1
Identify any excluded values for each rational
function.
c.
x 4 0
Set the denominator equal to 0.
Solve for x.
x 4
The excluded value is 4.
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Many rational functions are discontinuous
functions, meaning their graphs contain one or
more jumps, breaks, or holes. This occurs at an
excluded value.
One place that a graph of a rational function may
be discontinuous is at an asymptote. An asymptote
is a line that a graph gets closer to as the
absolute value of a variable increases. In the
graph shown, both the x- and y-axes are
asymptotes. A graph will
get closer and closer to but
never touch its asymptotes.
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Additional Example 2A Identifying Asymptotes
Identify the asymptotes.
Step 2 Identify the asymptotes.
vertical x 7
horizontal y 0
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Additional Example 2B Identifying Asymptotes
Identify the asymptotes.
Step 1 Identify the vertical asymptote.
2x 3 0
Find the excluded value. Set the denominator
equal to 0.
Add 3 to both sides.
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Additional Example 2B Continued
Identify the asymptotes.
Step 2 Identify the horizontal asymptote.
c 8
y 8
y c
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Check It Out! Example 2a
Identify the asymptotes.
Step 1 Identify the vertical asymptote.
x 5 0
Find the excluded value. Set the denominator
equal to 0.
Add 5 to both sides.
x 5
Solve for x. 5 is an excluded value.
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Check It Out! Example 2a Continued
Identify the asymptotes.
Step 2 Identify the horizontal asymptote.
c 0
y 0
y c
Vertical asymptote x 5 horizontal asymptote
y 0
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Check It Out! Example 2b
Identify the asymptotes.
Step 1 Identify the vertical asymptote.
4x 16 0
Find the excluded value. Set the denominator
equal to 0.
Subtract 16 from both sides.
x 4
Solve for x. 4 is an excluded value.
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Check It Out! Example 2b Continued
Identify the asymptotes.
Step 2 Identify the horizontal asymptote.
c 5
y 5
y c
Vertical asymptote x 4 horizontal asymptote
y 5
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Check It Out! Example 2c
Identify the asymptotes.
Step 1 Identify the vertical asymptote.
Find the excluded value. Set the denominator
equal to 0.
x 77 0
Subtract 77 from both sides.
Solve for x. 77 is an excluded value.
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Check It Out! Example 2c Continued
Identify the asymptotes.
Step 2 Identify the horizontal asymptote.
c 15
y 15
y c
Vertical asymptote x 77 horizontal
asymptote y 15
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To graph a rational function in the form y
, you can use the asymptotes and
a table of values.
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Additional Example 3A Graphing Rational
Functions Using Asymptotes
Graph the function.
Step 1 Identify the asymptotes.
Use x b. x 3 0, so b 3.
vertical x 3
horizontal y 0
Use y c. c 0
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Additional Example 3A Continued
Step 2 Graph the asymptotes using dashed lines.
Step 3 Make a table of values. Choose x-values on
both sides of the vertical asymptote.
Step 4 Plot the points and connect them with
smooth curves. The curves should not touch the
asymptotes.
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Additional Example 3B Graphing Rational
Functions Using Asymptotes
Graph the function.
Step 1 Identify the asymptotes.
vertical x 4
Use x b. b 4
Use y c. c 2
horizontal y 2
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Additional Example 3B Continued
Step 2 Graph the asymptotes using dashed lines.
Step 3 Make a table of values. Choose x-values on
both sides of the vertical asymptote.
Step 4 Plot the points and connect them with
smooth curves. The curves should not touch the
asymptotes.
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Check It Out! Example 3a
Graph each function.
Step 1 Identify the asymptotes.
vertical x 7
Use x b. b 7
Use y c. c 3
horizontal y 3
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Check It Out! Example 3a Continued
Graph each function.
Step 2 Graph the asymptotes using dashed lines.
Step 3 Make a table of values. Choose x-values on
both sides of the vertical asymptote.
Step 4 Draw smooth curves to show the translation.
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Check It Out! Example 3b
Graph each function.
Step 1 Identify the asymptotes.
vertical x 3
Use x b. x 3 0, so b 3.
horizontal y 2
Use y c. c 2
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Check It Out! Example 3b
Step 2 Graph the asymptotes using dashed lines.
Step 3 Make a table of values. Choose x-values on
both sides of the vertical asymptote.
Step 4 Plot the points and connect them with
smooth curves. The curves will get very close to
the asymptotes, but will not touch them.
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Additional Example 4 Application
Your club has 75 with which to purchase snacks
to sell at an afterschool game. The number of
snacks y that you can buy, if the average price
of the snacks is x-dollars, is given by y
a. Describe the reasonable domain and range
values.
Both the number of snacks purchased and their
cost will be nonnegative values so nonnegative
values are reasonable for both domain and range.
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Additional Example 4 Continued
b. Graph the function.
Step 1 Identify the vertical and horizontal
asymptotes.
vertical x 0 horizontal y 0
Use x b. b 0
Use y c. c 0
Step 2 Graph the asymptotes using dashed lines.
The asymptotes will be the x- and y-axes.
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Additional Example 4 Continued
Step 3 Since the domain is restricted to
nonnegative values, only choose x-values on the
right side of the vertical asymptote.
Number of snacks 2 4 6 8
Cost of snacks() 37.5 18.75 12.5 9.38
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Additional Example 4 Continued
Step 4 Plot the points and connect them with a
smooth curve.
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Check It Out! Example 4
a. Describe the reasonable domain and range
values.
The domain would be all values greater than 0 up
to 500 dollars and the range would be all
natural numbers greater than 10.
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Check It Out! Example 4 Continued
b. Graph the function.
Step 1 Identify the vertical and horizontal
asymptotes.
Use x b. b 0
vertical x 0 horizontal y 10
Use y c. c 10
Step 2 Graph the asymptotes using dashed lines.
The asymptotes will be the x- and y-axes.
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Check It Out! Example 4 Continued
Step 3 Since the domain is restricted to
nonnegative values, only choose x-values on the
right side of the vertical asymptote.
Number of copies 20 40 60 80
Price () 35 22.5 18.3 16.25
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Check It Out! Example 4 Continued
Step 4 Plot the points and connect them with a
smooth curve.
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The table shows some of the properties of the
three types of functions you have studied and
their graphs.
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Lesson Quiz Part I
Identify the exceeded value for each rational
function.
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1.
2.
0
3. Identify the asymptotes of and
then graph the function.
x 4 y 0
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Lesson Quiz Part II
4. You have 100 to spend on CDs. A CD club
advertises 6 free CDs for anyone who becomes a
member. The number of CDs y that you can
receive is given by y ,
where x is the average price per CD. a.
Describe the reasonable domain and range values.
D x gt 0
R natural numbers gt 6
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Lesson Quiz Part III
b. Graph the function.