# 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.
• 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.
• 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.
• 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 the graph might cross somewhere along the way).
• Don't forget the vertical intercept (unless there's a hole there), which works the same as always.
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.
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