section 2.6 rational functions and their graphs · rational functions are quotients of poly nomial...
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Section 2.6
Rational 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)=
x
Vertical Asymptote on
the y-axis.
2
1The equation f(x)=
Vertical Asymptote on
the y-axis.
x
Two Graphs with Vertical Asymptotes, one without
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
2 4Consider the function f(x)= . Because the denominator is zero when
2
x=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
Example 5
Graph f(x)= using the 7 step strategy from 2
the previous slide.
x
x
x
y
Example
2
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
(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