- 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).

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2019 May 22. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1150/2020SP/rational/`

.