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 (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,
you should have enough information for a rough graph already.
Go back to the course homepage.
This web page was written between 2011 and 2016 by Toby Bartels,
last edited on 2016 August 24.
Toby reserves no legal rights to it.
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http://tobybartels.name/MATH-1150/2017SP/rational/.
