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Asymptotes: Comparing Graphs While your text almost certainly covers only the case where the numerator's degree is 1 greater than the denominator's, you should be aware that the graphs of rational functions behave in similar manners even when the degrees are further apart. If the numerator's degree is greater than the denominator's — if the rational function is an "improper" polynomial fraction — then the graph will approximate the polynomial part obtained by long division. You've already seen how this works when the numerator's degree is one greater than the denominator's. But the relationship holds whatever the difference in degrees. As long as the rational function is "improper", its graph will approximate the polynomial found by doing the long division. Compare:
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© Elizabeth Stapel 19992009 All Rights Reserved The table above displayed the graphs of a rational function in which the degree of the numerator was two more than the degree of the denominator. In the following tables, this relationship is demonstrated when the degrees are three apart and four apart.
Copyright © Elizabeth Stapel 20032011 All Rights Reserved
Certainly, the rational function's graph will frequently get a bit twitchy in the middle (around its vertical asymptotes). But "at the sides" or "on the ends", if you will, the graph will be nearly the same as the associated polynomial.



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