Graphing Rational Functions

A rational function is a function that looks like a fraction and has a variable in the denominator. The following are examples of rational functions:

Note that a function such as

is not considered a rational function. Even though it is in the form of a fraction, the denominator does not contain a variable.

Rational functions sometimes have limitations on what values can be put in for the variable. In order to graph a rational function, you will need to know how to find the domain. For more information on finding the domain of a rational function, click here to go to the rational domain lesson.

In addition to the domain, we will need to know if the rational function has any vertical or horizontal asymptotes. Asymptotes are lines that the graph approaches, but does not touch.

Once we know the domain and asymptotes, we will need to plot two or three points to get an idea of what the graph will look like. We will also be able to use the graphing calculator to graph rational functions.

The first step is to completely factor both the numerator and demoninator of the rational function if possible.

Rule for Domain

Set each factor of the denominator equal to zero. This is where the function will be undefined. The domain for the function will be all real numbers except those that make the denominator zero.

Rule for Vertical Asymptotes

Simplify the factored rational expression. Set any remaining factors of the denominator equal to zero. A vertical asymptote will occur at each of these x-valus.

Rules for Horizontal Asymptotes

The rules for finding a horizontal asymptote depend on the largest power in the numerator and denominator.
If the degree (highest power) of the numerator is larger than the degree of the denominator, then there is no horizontal asymptote.
If the degree of the numerator is smaller than the degree of the denominator, then the horizontal asymptote is at (the x–axis)
If the degree of the numerator is equal to the degree of the denominator, then you must compare the coefficients in front of the terms with the highest power. The horizontal asymptote is the coefficient of the highest power of the numerator divided by the coefficient of the highest power of the denominator.

Examples

#1: Graph

Determine the domain by setting the denominator equal to zero.

The domain is all real numbers except x = -2.

Determine vertical asymptote(s). Since the rational function is already in simplest form, the vertical asymptote(s) will occur at the domain restriction(s).

Determine the horizontal asymptote. Note that the degree of the numerator (0) is smaller than the degree of the denominator (1).

Plot other points.

x y
-4 - 1/2
-3 -1
-1 1
0 1/2

Describe and graph the function.

What is your answer?

#2: Graph

Determine the domain.

1 + x = 0
x = -1
All real numbers except

Determine the vertical asymptote(s).

Determine the horizontal asymptote. Note that the degree of the numerator (1) is the same as the degree of the denominator (1).

Plot other points.

x y
-4 1
-3 1/2
0 5
2 3

Describe and graph the function.

What is your answer?

#3: Graph

Determine the domain.

Domain is all real numbers except

Determine the vertical asymptote(s).

Determine the horizontal asymptote. Note that the degree of the numerator (2) is larger than the degree of the denominator (1).

There is no horizontal asymptote.

Plot other points.

x y
-1 1/8
0 0
2 4
3 9/4

Describe and graph the function.

What is your answer?

#4: Graph

Determine the domain.

and
All real numbers except and and

Determine the vertical asymptote(s).

and

Determine the horizontal asymptote. The degree of the numerator (1) is the same as the degree of the denominator (1).

Plot other points.

x y
-5 25/16
-2 -4/5
0 0
2 4/5
5 25/16

Describe and graph the function.

What is your answer?

#5: Graph

Determine the domain.

and
Domain is all real numbers except and

Determine the vertical asymptote(s).

and

Determine the horizontal asymptote. Note that the degree of the numerator (1) is smaller than the degree of the denominator (2).

Plot other points.

x y
-7 -1/2
-4 -11/10
-2 1/8
0 -1/2
2 -95/100
4 -17/4
7 22/5
9 3/2

Describe and graph the function.

What is your answer?

You may have noticed a couple of interesting features of the graphs in Example 3 and Example 5.

In Example 3, while there is no horizontal asymptote, there is a diagonal line that the graph approaches but does not touch. This is called a slant or oblique asymptote. Finding this type of asymptote requires long division of a polynomial.

In Example 5, there was a horizontal asymptote along the x–axis. However, close inspection of the graph will show that the graph does cross the x–axis. This occasionally happens with horizontal asymptotes.

For further information about these special situations, consult your algebra text or your instructor. Either can provide more details about what to do with these types of rational functions.

One other situation you need to be aware of is when you set the denominator equal to zero to find a vertical asymptote, yet when you graph the function, there is not a vertical asymptote where you expected. Do not automatically assume that you have done something wrong. If the rational function will simplify there will be a hole in the graph rather than an asymptote. Whether there is a hole or an asymptote, the value you find is still excluded from the domain of the rational function. As with the other special cases, you should consult your book or your instructor for more information about these situations.

Example

#6: Graph

First, factor the numerator and the denominator.

Determine the domain.

x = -3 and x = 2
Domain is all real numbers except x = -3 and x = 2
Determine the vertical asymptotes(s). Since the (x+3) factor can be cancel from the nuerator and the denominator, there is only a sole in the graph at x = -3. The only remining factor in the denominator after simplifation is (x-2). This is where the vertical asymptote occurs.
x = 2
Determine the horizontal asymptote. The degree of the numerator (1) is smaller than the degree of the denominator (2).
y = 0
Plot other points.
x y
-3 hole
-1 -4/3
0 -2
3 4
4 2
Describe and graph the function.

What is your answer?