- 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).

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