Chapter 3
Polynomial and
Rational Functions
3.5 Rational Functions and
Their Graphs
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Objectives:
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Find the domains of rational functions.
Use arrow notation.
Identify vertical asymptotes.
Identify horizontal asymptotes.
Use transformations to graph rational functions.
Graph rational functions.
Identify slant asymptotes.
Solve applied problems involving rational functions.
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Rational Functions
Rational functions are quotients of polynomial
functions. This means that rational functions can be
expressed as
p( x)
f ( x) 
q( x)
where p and q are polynomial functions and q( x)  0.
The domain of a rational function is the set of all real
numbers except the x-values that make the denominator
zero.
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Example: Finding the Domain of a Rational Function
Find the domain of the rational function:
x 2  25
f ( x) 
x5
Rational functions contain division. Because division
by 0 is undefined, we must exclude from the domain of
each function values of x that cause the polynomial
function in the denominator to be 0.
x 5  0
x5
The domain of f consists of all real numbers except 5.
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Example: Finding the Domain of a Rational Function
(continued)
Find the domain of the rational function:
x 2  25
f ( x) 
x5
The domain of f consists of all real numbers except 5.
We can express the domain in set-builder or interval
notation.
Domain of f =  x x  5
Domain of f = (,5) (5, )
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Example: Finding the Domain of a Rational Function
Find the domain of the rational function:
g ( x) 
x
x 2  25
Rational functions contain division. Because division
by 0 is undefined, we must exclude from the domain of
each function values of x that cause the polynomial
function
in the denominator to be 0.
2
x  25  0
The domain of g consists of
2
x  25
all real numbers except –5 or 5.
x  5
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Example: Finding the Domain of a Rational Function
(continued)
Find the domain of the rational function:
g ( x) 
x
x 2  25
The domain of g consists of all real numbers except
–5 or 5.
We can express the domain in set-builder or interval
notation.
Domain of g =  x x  5, x  5
Domain of g =
(, 5)
(5,5)
(5, )
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Example: Finding the Domain of a Rational Function
Find the domain of the rational function:
x5
h( x )  2
x  25
Rational functions contain division. Because division
by 0 is undefined, we must exclude from the domain of
each function values of x that cause the polynomial
function in the denominator to be 0.
No real numbers cause the denominator
x 2  25  0
of h(x) to equal 0. The domain of h
x 2  25
x  5i
consists of all real numbers.
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Example: Finding the Domain of a Rational Function
(continued)
Find the domain of the rational function:
h( x ) 
x5
x 2  25
The domain of h consists of all real numbers. We can
write the domain in interval notation:
Domain of h = (, )
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Arrow Notation
We use arrow notation to describe the behavior of some
functions.
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Definition of a Vertical Asymptote
The line x = a is a vertical asymptote of the graph of a
function f if f(x) increases or decreases without bound as
x approaches a.
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Definition of a Vertical Asymptote (continued)
The line x = a is a vertical asymptote of the graph of a
function f if f(x) increases or decreases without bound as
x approaches a.
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Locating Vertical Asymptotes
If
f ( x) 
p( x)
q( x)
is a rational function in which p(x)
and q(x) have no common factors and a is a zero of q(x),
the denominator, then x = a is a vertical asymptote of
the graph of f.
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Example: Finding the Vertical Asymptotes of a Rational
Function
Find the vertical asymptotes, if any, of the graph of the
x
rational function:
f ( x)  2
x 1
There are no common factors in the numerator and the
denominator.
x2  1  0
x2  1
x  1
The zeros of the denominator are –1 and 1.
Thus, the lines x = –1 and x = 1 are the vertical
asymptotes for the graph of f.
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Example: Finding the Vertical Asymptotes of a Rational
Function (continued)
Find the vertical asymptotes, if any, of the graph of the
rational function:
x
f ( x)  2
The zeros of the
x 1
denominator are
x  1
–1 and 1. Thus,
the lines x = –1
x 1
and x = 1 are the vertical
asymptotes for the graph of f.
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Example: Finding the Vertical Asymptotes of a Rational
Function
Find the vertical asymptotes, if any, of the graph of the
rational function: h( x)  x2  1
x 1
We cannot factor the
denominator of h(x) over
the real numbers. The
denominator has no real
zeros. Thus, the graph of h
has no vertical asymptotes.
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Definition of a Horizontal Asymptote
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Locating Horizontal Asymptotes
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Example: Finding the Horizontal Asymptote of a Rational
Function
Find the horizontal asymptote, if any, of the graph of
the rational function: f ( x)  9 x 2
3x 2  1
The degree of the numerator, 2, is equal to the degree of
the denominator, 2. The leading coefficients of the
numerator and the denominator, 9 and 3, are used to
obtain the equation of the horizontal asymptote.
The equation of the horizontal asymptote is y  9
3
or y = 3.
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Example: Finding the Horizontal Asymptote of a Rational
Function
Find the horizontal asymptote, if any, of the graph of
2
9
x
the rational function:
f ( x) 
2
3x  1
y3
The equation of the
horizontal asymptote
is y = 3.
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Example: Finding the Horizontal Asymptote of a Rational
Function
Find the horizontal asymptote, if any, of the graph of
9x
the rational function:
g ( x) 
3x 2  1
The degree of the numerator, 1, is less than the degree
of the denominator, 2. Thus, the graph of g has the
x-axis as a horizontal asymptote.
The equation of the horizontal asymptote is y = 0.
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Example: Finding the Horizontal Asymptote of a Rational
Function (continued)
Find the horizontal asymptote, if any, of the graph of
9x
the rational function:
g ( x) 
3x 2  1
y0
The equation of the
horizontal asymptote
is y = 0.
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Example: Finding the Horizontal Asymptote of a Rational
Function
Find the horizontal asymptote, if any, of the graph of
3
the rational function: h( x)  9 x
3x 2  1
The degree of the numerator, 3, is greater than the
degree of the denominator, 2. Thus the graph of h has
no horizontal asymptote.
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Example: Finding the Horizontal Asymptote of a Rational
Function (continued)
Find the horizontal asymptote, if any, of the graph of
the rational function: h( x)  9 x3
3x 2  1
The graph has
no horizontal
asymptote.
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Basic Reciprocal Functions
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Example: Using Transformations to Graph a Rational
Function
Use the graph of
Begin with
f ( x) 
f ( x) 
1
x
We’ve identified two
points and the
asymptotes.
1
x
to graph
g ( x) 
Vertical
asymptote
x=0
( 1,1)
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1
1
x2
(1,1)
Horizontal
asymptote
y=0
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Example: Using Transformations to Graph a Rational
Function (continued)
Use the graph of f ( x)  1 to graph g ( x)  1  1
x
The graph will shift
Vertical
2 units to the left.
asymptote
Subtract 2 from each x = –2
x-coordinate. The
( 3, 1)
vertical asymptote
is now x = –2
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x2
( 1,1)
Horizontal
asymptote
y=0
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Example: Using Transformations to Graph a Rational
Function (continued)
Use the graph of f ( x)  1 to graph g ( x)  1  1
x
The graph will shift
1 unit down.
Subtract 1 from each
y-coordinate. The
horizontal asymptote
is now y = –1.
Vertical
asymptote
x = –2
( 3, 2)
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x2
( 1, 0)
Horizontal
asymptote
y = –1
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Strategy for Graphing a Rational Function
The following strategy can be used to graph f ( x)  p ( x)
q( x)
where p and q are polynomial functions with no
common factors.
1. Determine whether the graph of f has symmetry.
f (  x )  f ( x ) y-axis symmetry
f (  x )   f ( x ) origin symmetry
2. Find the y-intercept (if there is one) by evaluating
f(0).
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Strategy for Graphing a Rational Function (continued)
f ( x) 
p( x)
q( x)
The following strategy can be used to graph
where p and q are polynomial functions with
no common factors.
3. Find the x-intercepts (if there are any) by solving the
equation p(x) = 0.
4. Find any vertical asymptote(s) by solving the
equation q(x) = 0.
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Strategy for Graphing a Rational Function (continued)
f ( x) 
p( x)
q( x)
The following strategy can be used to graph
where p and q are polynomial functions with no
common factors.
5. Find the horizontal asymptote (if there is one) using
the rule for determining the horizontal asymptote of a
rational function.
6. Plot at least one point between and beyond each
x-intercept and vertical asymptote.
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Strategy for Graphing a Rational Function (continued)
f ( x) 
p( x)
q( x)
The following strategy can be used to graph
where p and q are polynomial functions with no
common factors.
7. Use the information obtained previously to graph the
function between and beyond the vertical asymptotes.
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Example: Graphing a Rational Function
f ( x) 
3x  3
x2
Graph:
Step 1 Determine symmetry.
3(  x )  3
3 x  3
f ( x) 

( x)  2
x  2
Because f(–x) does not equal either f(x) or –f(x), the
graph has neither y-axis symmetry nor origin symmetry.
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Example: Graphing a Rational Function (continued)
f ( x) 
3x  3
x2
Graph:
Step 2 Find the y-intercept. Evaluate f(0).
3(0)  3 3
f (0) 

02
2
Step 3 Find x-intercepts.
3x  3  0
3x  3
x 1
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Example: Graphing a Rational Function (continued)
Graph:
f ( x) 
3x  3
x2
Step 4 Find the vertical asymptote(s).
x20
x2
Step 5 Find the horizontal asymptote.
The numerator and denominator of f have the same
degree, 1. The horizontal asymptote is y  3  3
1
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Example: Graphing a Rational Function (continued)
Graph:
3x  3
f ( x) 
x2
We will evaluate f at
Step 6 Plot points between
and beyond each
x  1
x-intercept and
vertical asymptotes.
The x-intercept is (1,0)
 0, 3 
The y-intercept is 


x
x
1
2
2
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3
2
x

5
2
x3

vertical
asymptote
x=2
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Example: Graphing a Rational Function (continued)
3x  3
f ( x) 
x2
Graph:
Step 6 (continued)
We evaluate f at the selected points.
x
f ( x) 
3x  3
x2
1
1
2
3
2
5
2
2
1
3
9
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6
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Example: Graphing a Rational Function (continued)
Graph:
f ( x) 
3x  3
x2
Step 7
Graph the
function
vertical
asymptote
x=2
horizontal
asymptote
y=3
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Slant Asymptotes
The graph of a rational function has a slant
asymptote if the degree of the numerator is one more
than the degree of the denominator.
p( x)
f
(
x
)

, p and q have no common
In general, if
q( x)
factors, and the degree of p is one greater than the
degree of q, find the slant asymptotes by dividing q(x)
into p(x).
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Example: Finding the Slant Asymptotes of a Rational
Function
2 x2  5x  7
f ( x) 
x2
Find the slant asymptote of
The degree of the numerator, 2, is exactly one more
than the degree of the denominator, 1.
To find the equation of the slant asymptote, divide the
numerator by the denominator.
2 2
2
5
4
1
7
2
5
5
(2 x  5 x  7)  ( x  2)  2 x  1 
x2
2
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Example: Finding the Slant Asymptotes of a Rational
Function
2
2
x
 5x  7
Find the slant asymptote of f ( x) 
x2
5
(2 x 2  5 x  7)  ( x  2)  2 x  1 
x2
The equation of the
slant asymptote is
y = 2x – 1.
The graph of f(x)
is shown.
vertical
asymptote
x=2
slant asymptote
y = 2x – 1
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Example: Application
A company is planning to manufacture wheelchairs that
are light, fast, and beautiful. The fixed monthly cost will
be $500,000 and it will cost $400 to produce each
radically innovative chair.
Write the cost function, C, of producing x wheelchairs.
C ( x)  500, 000  400 x
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Example: Application
(continued)
A company is planning to manufacture wheelchairs that
are light, fast, and beautiful. The fixed monthly cost
will be $500,000 and it will cost $400 to produce each
radically innovative chair.
Write the average cost function, C , of producing x
wheelchairs.
C ( x) 
500, 000  400 x
x
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Example: Application
A company is planning to manufacture wheelchairs that
are light, fast, and beautiful. The fixed monthly cost
will be $500,000 and it will cost $400 to produce each
radically innovative chair.
The average cost per
Find and interpret C (1000).
wheelchair of producing
1000 wheelchairs
500, 000  400 x
C ( x) 
month is $900.
x
500, 000  400(1000)
 900
C (1000) 
1000
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per
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Example: Application (continued)
A company is planning to manufacture wheelchairs that
are light, fast, and beautiful. The fixed monthly cost
will be $500,000 and it will cost $400 to produce each
radically innovative chair.
The average cost per
Find and interpret C (10,000).
wheelchair of producing
500, 000  400 x
10,000 wheelchairs per
C ( x) 
x
month is $450.
C (10, 000) 
500, 000  400(10, 000)
10, 000
 450
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Example: Application
(continued)
A company is planning to manufacture wheelchairs that
are light, fast, and beautiful. The fixed monthly cost
will be $500,000 and it will cost $400 to produce each
radically innovative chair.
The average cost per
Find and interpret C (100,000). wheelchair of producing
C ( x) 
500, 000  400 x
x
C (100, 000) 
100,000 wheelchairs
per month is $405.
500, 000  400(100, 000)  405
100, 000
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Example: Application
(continued)
A company is planning to manufacture wheelchairs that are
light, fast, and beautiful. The fixed monthly cost will be
$500,000 and it will cost $400 to produce each radically
innovative chair.
Find the horizontal asymptote for the average cost function,
C ( x) 
500, 000  400 x
.
x
C ( x) 
500, 000  400 x
400 x  500, 000

x
x
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The horizontal
asymptote
is y = 400
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Example: Application (continued)
A company is planning to manufacture wheelchairs that are
light, fast, and beautiful. The fixed monthly cost will be
$500,000 and it will cost $400 to produce each radically
innovative chair. The average cost function is
C ( x) 
500, 000  400 x
x
The horizontal asymptote for this function is y = 400.
Describe what this represents for the company.
The cost per wheelchair approaches 400 as more
wheelchairs are produced.
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