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
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
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.