Rational functions (§§5.2&5.3)
To graph a rational function:
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.
- 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.
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This web page was written in 2011 and 2014 by Toby Bartels,
last edited on 2014 December 10.
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