Section 3.5 - Rational Functions and Asymptotes

Section Objectives

  1. Determine the vertical, horizontal, and/or slant asymptotes of the graph of a rational function.
  2. Sketch the graph of a rational function.

Rational Functions

A rational function is a ratio of two polynomials. That is, a rational function is a function of the form

where and are polynomials and .

The domain of a rational function is the set of all real numbers for which the denominator is nonzero.

Here are some examples of rational functions...

Reciprocal function

One of the simplest rational functions is the reciprocal function .


The vertical line is a vertical asymptote of the graph of if the -values of grow without bound (positively or negatively) as the -values get closer and closer to (from either side).

IMPORTANT IDEA: The zeros of the denominator of a rational function give possible vertical asymptotes...


The line is a horizontal asymptote of the graph of if the -values of get closer and closer to the number as the -values grow without bound (positively or negatively).

IMPORTANT IDEA: Horizontal asymptotes can be found by examining the degrees of the numerator and denominator polynomials.

While the graph of a rational function can have any number of vertical asymptotes, it can have at most one horizontal asymptote. (In general, functions can have at most two horizontal asymptotes---but rational functions can have at most one!)


A non-horizontal or non-vertical line that a graph approaches as the -values grow without bound is called a slant (or oblique) asymptote.


Graphing Rational Functions

To graph the rational function :

  1. Completely factor the denominator and specify all real numbers excluded from the domain of (restricted values).
  2. Completely factor the numerator and cancel any factors common to both the numerator and the denominator. (If any factors cancel, your graph will have holes at the corresponding -values.)
  3. Examine the factors that remain in the denominator. Then find and sketch the vertical asymptotes. (Sketch with a dashed line.)
  4. Find and sketch (dashed) any horizontal or slant asymptotes.
  5. Determine the - and -intercepts of the graph.
  6. To determine when the graph is above or below the axis, construct a sign chart: Away from your graph, sketch a number line separated (in order) by the -values associated with the vertical asymptotes, -intercepts, and other points excluded from the domain. Mark which is which. Then determine and mark the signs ( or ) of on each interval of your number line.
  7. Determine -values (if any) where the graph crosses the horizontal or slant asymptotes.
  8. Plot some points.
  9. Determine if there is symmetry in the graph.
  10. Use all the information above to sketch the graph. Use your graphing calculator as an aid.