Avon High School
ACE COLLEGE ALGEBRA II - NOTES
Rational Functions and Their Graphs
Section: 3.5
Mr. Record: Room ALC-129
Day 1 of 1
Rational Functions
Rational functions are quotients of polynomial function. This means that rational functions can be expressed
p ( x)
as f ( x) 
, where p and q are polynomial functions and q ( x)  0. The domain of a rational function is
q ( x)
the set of all real numbers except the x-values that make the denominator zero.
Finding the Domain of a Rational Function
Example 1
Find the domain of each rational function
x  25
x
b. f ( x)  2
f ( x) 
x  25
x 5
2
a.
c.
The most basic rational function is the reciprocal function, defined by f ( x ) 
and sketch a graph of f ( x) .
-4
-3
x
-2
-1
0
f ( x) 
x5
x 2  25
1
. Complete the charts below
x
1
2
3
4
f ( x)
x
-0.1
-0.01
-0.0001
0
0.001
0.01
0.1
f ( x)
y
x
1
10
100
1000

f ( x)


x
-1
-10
-100

-1000
x
f ( x)






As x  0 , f ( x)  _________ .
As x  0 , f ( x)  _________ .


As x  , f ( x)  __________ . As x  , f ( x)  _________ .






Arrow Notation
Symbol

xa
x  a
x 
x  
Meaning
x approaches a from the right
x approaches a from the left
x approaches infinity; x increases without bound
x approaches negative infinity; x decreases without bound
Vertical Asymptotes of Rational Functions
Definition of 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.
Finding vertical asymptotes is actually quite easy.
Locating Vertical Asymptotes
p ( x)
is a rational function in which p ( x ) and q( x) have no common factors and a is a zero of
q ( x)
q( x), the denominator, then x  a is a vertical asymptote of the graph of f .
If f ( x) 
Example 2
b.
Finding the Vertical Asymptotes of a Rational Function
Find the vertical asymptotes, if any, of the graph of each rational function.
x
x 1
x 1
b. f ( x)  2
c. f ( x)  2
f ( x)  2
x 1
x 1
x 1
Horizontal Asymptotes of Rational Functions
Definition of Horizontal Asymptote
Let f be the rational function given by
an x n  an 1 x n 1   a1 x  a0
f ( x) 
, an  0, bm  0.
bm x m  bm1 x m1   b1 x  b0
The degree of the numerator is n. The degree of the denominator is m.
1. If n  m , the x-axis, or y  0 , is the horizontal asymptote of the graph of f.
a
2. If n  m , the line y  n is the horizontal asymptote of the graph of f.
bm
3. If n  m , the graph of f has no horizontal asymptotes.
Example 3
c.
Finding the Horizontal Asymptotes of a Rational Function
Find the horizontal asymptotes, if any, of the graph of each rational function.
9 x2
9 x3
9x
b. f ( x)  2
c. f ( x)  2
f ( x)  2
3x  1
3x  1
3x  1
Using Transformations to Graph Rational Functions
Graphs of Two Common Rational Functions
Example 4
Using Transformations to Graph a Rational Function
Use the graph of f ( x) 
f ( x) 
1
to graph
x2
y

1
1 .
( x  2)2



x












Graphing Rational Functions
Strategies for Graphing Rational Functions
p ( x)
,
q ( x)
where p and q are polynomial functions with no common factors.
The following steps can be used to graph f ( x) 
1. Determine whether the graph of f has symmetry.
a. f ( x)  f ( x) : y-axis symmetry
b. f ( x)   f ( x) : origin symmetry
2. Find the y-intercepts (if there are any) by evaluating f (0) .
3. Find the x-intercepts (if there are any) by solving the equation f ( x)  0.
4. Find the vertical asymptote(s) by solving the equation q ( x)  0.
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.
7. Use the information obtained previously to graph the function between and beyond the vertical
asymptotes.


Example 5
Graphing a Rational Function
Graph f ( x) 
3x  3
.
x2
y




x














Example 6
Graphing a Rational Function
Graph f ( x) 
2 x2
.
x2  9
y




x














Example 7
Graphing a Rational Function
Graph f ( x) 
x4
.
x2  2
y




x














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. The equation of the slant asymptote is found by long division of the rational functions
numerator and denominator and using the non-remainder portion of the quotient.
Example 8
Finding the Slant Asymptote
2 x2  5x  7
Find the slant asymptote of f ( x) 
. Use a graphing calculator to verify your
x2
result.
Applications
There are numerous examples of asymptotic behavior in functions that model real-world phenomena. Let’s
consider an example from the business world. The cost function, C, for a business is the sum of its fixed and
variable costs:
C ( x)  (fixed cost) + cx
Cost per unit times the
number of units produced, x
The average cost per unit for a company to produce x units is the sum of its fixed and variable costs divided by
the number of units produced. The average cost function is a rational function that is denoted by C . Thus,
Cost of producing x units
C ( x) 
(fixed cost) + cx
x
Example 9
Number of units produced
Average Cost for a Business
A Japanese company manufactures a robotic exoskeleton to assist elder people in their day-to-day activities.
(Think Robo-Grandma!). The fixed monthly cost to produce such a device is $1,000,000. It costs $5000 to
produce each robotic system.
a. Write the cost function, C, of producing x robotic systems.
b. Write the average cost function, C , of producing x robotic systems.
c. Find and interpret C (1000), C (10,000), and C (100,000).
d. What is the horizontal asymptote for the graph of the average cost function, C ?
Describe what this represents for the company.