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
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 2018 by Toby Bartels,
last edited on 2018 May 20.
Toby reserves no legal rights to it.
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