2.6 Rational Functions and Asymptotes

Rational Functions can be written in the form:

Where N(x) and D(x) are polynomials and D(x) is not the zero polynomial.

Graphs of rational functions look like this...

Domain of a Rational Function: The domain of rational functions of x includes all real numbers except x-values, that will make the denominator zero.

(Eg. the function below has a domain of (-∞,3) U (3,∞))

Vertical and Horizontal Asymptotes:

Vertical asymptotes:

are found by setting the denominator equal to zero and solving D(x)=0.

(Eg. the function below has a vertical asymptote at x=3)

Horizontal Asymptotes:

There are three different scenarios for finding the horizontal asymptote. However, the first consideration that should be taken is that only the leading term on both the numerator and denominator when finding horizontal asymptotes. This is because asymptotes really only have an effect at the ends of the graph and due to the magnitude of the numbers at the end, the numbers following the leading term are insignificant.

Scenarios:

1. Leading terms are to an equal degree. When this occurs the answer is the constant in the leading term of the numerator over the constant of the leading term of the denominator.

(Eg. In the function below ignore everything after the leading term, the horizontal asymptote is y=3/4)

2. The leading term of the numerator is higher than the leading term of the denominator. In this x keeps growing and the function never levels off. There is no horizontal asymptote in this situation.

(Eg. In the function below there is no horizontal asymptote)

3. The leading term of the denominator is higher than the leading term of the numerator. In this x approaches zero and gets very tiny. The horizontal asymptote is y=0

(Eg. In the function below the horizontal asymptote is y=0)

Intercepts:

X-Intercepts:

To find the x intercepts of a rational function you must set y equal to zero, and therefore N(x)=0.

(Eg. the x-int for this function=0)

Y-Intercepts:

To find the y intercepts of a rational function you must set x equal to zero.

(Eg. the x-int for the function is 0.)
Things to Consider:

• Asymptotes are lines and therefore should be written in proper notation, such as x=a and y=b


• It is possible for a rational function to have multiple x intercepts and asymptotes


• It is possible for a rational function to lack both vertical and horizontal asymptotes


• It is possible for a rational function to lack both x and y intercepts