College Algebra
Notes: 3.3 Properties of Rational Functions
Name: _______________________
Exploring Rational Functions with your Graphing Calculator
A. Vertical Asymptotes and Holes Examine the behavior of the graph at the zeros of the denominator.
( )
and
( )
Notation:
a) What is the domain of each function?
b) Use a graphing calculator to graph
( ) and
What differences do you observe between the graph of
( )
and the graph of ?
𝑥
: x approaches 2 from the left
𝑥
: x approaches 2 from the right
c) Examine the TABLE, notice that ( )
and ( )
. Why is this an error at x=2? _________________
We will look at the -values in the
to see what happens as gets close to 2.
Set
in
and
As
, ( )
____________
( )
__________
As
, ( )
____________
( )
__________
Set
and
As
, ( )
____________
( )
__________
As
, ( )
____________
( )
__________
As
gets infinitely close to 2, the values of ( ) go to _______ and the values of ( ) get close to ___________.
d) Does ( ) have a vertical asymptote? What is it?
Does ( ) have a vertical asymptote?
e) Let ( ) = numerator; ( ) = denominator of each rational expression.
What happens to numerator and denominator of ( ) at
? ( ) = ______ ( ) = ___________
What happens to numerator and denominator of ( ) at
? ( ) = ______
( ) = ____________
f) Reduce both and to lowest terms, what happens?
The graph of 𝑔(𝑥) demonstrates a rational function with a hole at a zero of the denominator.
The graph of 𝑓(𝑥) demonstrates a rational function with a vertical asymptote at a zero of the denominator.
B. Horizontal and Oblique Asymptotes
1. Use a graphing calculator to graph
Examine the End-Behavior. What happens at the end of the graphs?
( )
)?
What do the y- values get close to as x gets large (as
2. Use a graphing calculator to graph
as x gets small (
)?
( )
What do the y- values get close to as x gets large (
)?
as x gets small (
?
The graph of 𝑓(𝑥) demonstrates a rational function with a horizontal asymptote at 𝑦
The graph of 𝑔(𝑥) demonstrates a rational function with a horizontal asymptote at 𝑦
0
A function will have only ONE horizontal asymptote or none at all.
3. Use a graphing calculator to graph
( )
This function does not have a horizontal asymptote, but it does have an oblique asymptote at
.
Graph
. Zoom out to see that the graph of ( ) follows the graph of
for large values of
The graph of (𝑥) demonstrates a rational function with an oblique (slant) asymptote
A rational function that does not have a horizontal asymptote will have an oblique asymptote.
Horizontal and Oblique asymptotes describe the End-Behavior of the graph of a rational function!
C. Find
the rational function that fits the given information
(Do you recall our discussion of end-behavior for polynomials? Same idea, different way to show it)
College Algebra
a) Write a rational function ( )
Notes: 3.3 Properties of Rational Functions
that meets the given criteria.
Name: _______________________
( )
( )
Adjust the degree and coefficients according to the Horizontal Asymptote.
b) Graph using a graphing calculator to verify the required properties
Example: If
is a zero and
and
is a horizontal asymptote, then ( )
is a Vertical Asymptote and
1.
has zeros at
, vertical asymptotes at
2.
has no x-intercepts, has a vertical asymptote given by
3.
has zeros at x={-2, 0,2}, vertical asymptotes given by
and
, and a horizontal asymptote at
, a horizontal asymptote
(
)
(
)
.
.
0.
, a horizontal asymptote
.
For each graph below, determine
a) x intercepts are given
b) horizontal asymptotes (look for a horizontal line that graph follows at ends)
c) vertical asymptotes (look for vertical line that graph follows, may be more than one)
d) An equation for the function represented by this graph
4.
5.
6.
a)
a)
a)
b)
b)
b)
c)
c)
c)
d)
( )
d)
( )
d) ( )
You may need to adjust the (multiplicity) of a linear term to assure that the function lies on the appropriate side of the x-axis.
 If the y-values are the same sign on each side of a vertical asymptote, then the degree of the linear term is even (2).
 If the graph touches (does not cross) at an x-intercept, then the degree of the linear term is even (2).