Graphing Rational Functions
We are now ready to use the information and techniques we have learned in the preceding pages to be able to sketch the graph of a rational function. Here we've summarized the steps used in graphing a rational function based on all the information gathered. As usual, we include locating the y intercept, if it exists.

Factor the numerator and denominator (into linear factors if possible). Simplify.
Use the factored form of the function (including any factors you have canceled from the denominator) to determine the
  • x-intercepts (i.e. the zeros of the numerator that are not zeros of the denominator),
  • the vertical asymptotes (i.e. the zeros of the denominator that are not zeros of the numerator, or zeros of the denominator that have larger multiplicity than they do in the numerator), and
  • the "holes" (the zeros of the numerator that have the same or larger multiplicity than they do in the denominator).
Determine where the function is positive and where it is negative.
Determine the behaviour of the function near the vertical asymptotes.
Find the horizontal asymptote if there is one, and the behaviour of the function as x and as x.
If f is defined at 0, find the y-intercept, f (0).

With all the information you have accumulated about your function, you should now be ready to sketch the graph. The next exercise will lead you through the above steps.

At each stage, answer the questions in the box, then press NEXT. If you are asked for an intercept, asymptote or hole when there is none, press NEXT immediately. When you have completed all the questions, the software will tell you which are correct and which need to be fixed.

If something needs to be fixed, return to the appropriate page (by clicking the appropriate button), undo your previous answers by pressing UNDO to cancel the entry on that screen, and reenter the information.

Once all your information is correct, sketch the graph on paper. Then click on "Next" to see the graph on the computer and compare it with your sketch.

Graphing Rational Functions

Note: The rules that we have described here are helpful for all rational functions for which the degree of the denominator is not smaller than the degree of the numerator. However, these rules are not completely sufficient for functions such as f (x) = (x2 - 1) / (x + 2), or f (x) = (x3 + x + 2) / (x - 1)(2x + 3).

In cases such as these, there may be other kinds of asymptotes: oblique straight lines, or even parabolas or higher degree polynomials! These are beyond the scope of the existing course, therefore we have not gone into detail regarding such graphs.