Section 3.5Rational Functions
and Their Graphs
Rational Functions
Rational Functions are quotients of polynomial
functions. This means that rational functions can
( )be expressed as f(x)= where p and q are
( )
polynomial functions and q(x) 0. The domain
of a rat
p x
q x
ional function is the set of all real numbers
except the x-values that make the denominator zero.
Example
Find the domain of the rational function.
2 16( )
4
xf x
x
Example
Find the domain of the rational function.
2( )
36
xf x
x
Vertical Asymptotes of
Rational Functions
x
y1The equation f(x)=
xVertical Asymptote on
the y-axis.
2
1The equation f(x)=
Vertical Asymptote on
the y-axis.
x
Two Graphs with Vertical Asymptotes, one without
Graphing Calculator2
f(x)=9
x
x Input the equation as you see at left.
The first graph is Connected Mode. In connected
mode, the graphing calculator plots many points
and connects the points with "curves."
In dot mode, the graphing calculator plots the
same points but does not connect them. To
change the mode on the calculator press the
MODE key then scroll down to the line that says
Connected Dot and choose the one that
you want to use.
Connected mode
Dot Mode
Example
Find the vertical asymptote, if any, of the graph of the rational function.
2( )
36
xf x
x
Example
Find the vertical asymptote, if any, of the graph of the rational function.
2( )
36
xf x
x
Example
Find the vertical asymptote, if any, of the graph of the rational function.
2
6( )
36
xf x
x
2 4Consider the function f(x)= . Because the denominator is zero when
2x=2, the function's domain is all real numbers except 2. However, there is
a reduced form of the equation in which 2 does not
x
x
cause the denominator
to be zero.
A graph with a hole corresponding to the denominator’s zero. Your calculator will not show the hole.
Horizontal Asymptotes of Rational Functions
Two Graphs with Horizontal Asymptotes, one without
Notice how the horizontal asymptote intersects the graph.
Example
Find the horizontal asymptote, if any, of the graph of the rational function.
2
3( )
1
xf x
x
Example
Find the horizontal asymptote, if any, of the graph of the rational function.
2
2
6( )
1
xf x
x
Using Transformations to Graph Rational Functions
Graphs of Common Rational Functions
Transformations of Rational Functions
Example
1 1Use the graph of f(x)= to graph g(x)= 4
3x x
x
y
Example
2 2
1 1Use the graph of f(x)= to graph g(x)= 2
4x x
x
y
Graphing Rational Functions
Example5
Graph f(x)= using the 7 step strategy from 2
the previous slide.
x
x
x
y
Example2
2
2Graph f(x)= using the 7 step strategy.
25
x
x
x
y
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 denominator. The equation of the slant
asymptote can be found by division. It is the equation
of the dividend with the term containing the remainder
dropped.
Example
2 6 2Find the slant asymptote of the function f(x)= .
x x
x
Example
3
2
1Find the slant asymptote of the function f(x)= .
2 2
x
x x
Applications
The cost function, C, for a business is the sum
of its fixed and variable costs.
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 . ThusC
Example
The Fort Myers Fishing Company discovered a better material for making fishing reels. The fixed monthly cost is $10,000 for the cost of rental of space, manufacturing equipment, as well as wages and benefits for it’s employees. It costs $10 for materials to make each fishing reel.
a.Write the cost function, C, of producing x fishing poles.
b. Write the average cost function, ,of producing x fishing poles.
c. Find and interpret (1,000), (100,000)
d. What is the horizontal asympt
C
C C
ote for the graph of the average cost
function . Describe what this represents for the company.C
(a)
(b)
(c)
(d)
2
2
Find the vertical asymptote(s) for the graph of
4the rational function f(x)= .
16
x
x 4
4
6, 6
4, 4
y
x
x
y
(a)
(b)
(c)
(d)
2
2
Find the horizontal asymptote(s) for the graph of
4the rational function f(x)= .
16
x
x 4
4
6, 6
4, 4
y
x
x
y
(a)
(b)
(c)
(d)
3
2
3Find the horizontal asymptote for f(x)= .
36
x
x
3
6, 6
6, 6
y
y
x
none