Rational functions

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 (which should be dashed).
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 (and mark it on the graph with a hollow circle).
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 (which should be a solid dot).
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 you can graph (with a dashed line) using the methods for graphing polynomials. (The graph of your rational function might cross the graph of this polynomial function; set the remainder equal to zero to see when this happens, plug this into the polynomial to get the second coordinate, and mark it with a solid dot unless you already have a hole there.)
Finally, plug 0 into the reduced expression to find the vertical intercept (unless you already have a hole there).

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 near the horizontal intercepts and vertical asymptotes, you should have enough information for a rough graph already. (For a more precise graph, at least near the vertical asymptotes and the horizontal intercepts, you can use the same optional technique as for polynomial functions.)

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This web page was written between 2011 and 2016 by Toby Bartels, last edited on 2018 May 20. Toby reserves no legal rights to it.

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