Rational functions (§§4.3&4.4)
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;
these are vertical lines (which should be dashed).
- The roots of the factors that you cancelled
give you holes
(except where you already have asymptotes);
plug the roots into the reduced expression to get the holes' second coordinates
(and mark them on the graph with hollow circles).
- The roots of the reduced numerator
give you horizontal intercepts
(except where you already have holes);
these are points on the horizontal axis (which should be solid dots).
- 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 or curve)
using methods for graphing linear and other polynomial functions.
(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 the solutions into the polynomial to get the points' second coordinates,
and mark them with solid dots except where you already have holes.)
- Finally, plug 0 into the reduced expression
to find the vertical intercept
(unless you already have an asymptote or 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 vertical asymptotes
and the horizontal intercepts (and holes that would be intercepts),
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 and would-be intercepts,
you can use the same optional technique as for polynomial functions.)
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