Rational functions (§§5.2&5.3)

To graph a rational function:

First factor both the numerator and the denominator.
Cancel any common factors to reduce the fraction.
The roots of the reduced denominator give you vertical asymptotes; each one is a vertical line.
The roots of the factors that you cancelled give you holes (unless you already have a vertical asymptote there); plug each one into the reduced expression to get its second coordinate.
The roots of the reduced numerator give you horizontal intercepts (unless you already have a hole there); each one is a point on the horizontal axis.
If you perform long division (or a shortcut) and throw out the remainder, then you get a polynomial; this is the formula for the other asymptote (which the graph might cross somewhere along the way).
Don't forget the vertical intercept (unless there's a hole there), which works the same as always.

You should definitely mark all intercepts, asymptotes, and holes; if the graph crosses the non-vertical asymptote, then you can mark that too. You may want to plug in some more numbers to find more points; on the other hand, using multiplicity as a guide, you should have enough information for a rough graph already.

Go back to the course homepage.

This web page was written in 2011 and 2014 by Toby Bartels, last edited on 2014 December 10. Toby reserves no legal rights to it.

The permanent URI of this web page is http://tobybartels.name/MATH-1150/2014FA/rational/.