Your text almost certainly covers only the case where the degree of the numerator of a rational function is greater by 1 than is the degree of the denominator. But you should be aware that the graphs of rational functions behave in similar manners even when the degrees of a rational function's numerator and denominator are further apart.

In all cases, if a rational function's numerator is of a degree greater than that of the denominator — that is, if the rational function is an "improper" polynomial fraction — then the graph of the original rational function will approximate the polynomial part obtained by long division.

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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. Consider the following examples.

A degree-two polynomial function is a quadratic, and graphs as a parabola. It stands to reason that, should a rationl function's degree-difference be equal to two, then the function's graph ought to look roughly like a parabola, at least off to the sides.

the original rational function:

the long division of the numerator by the denominator:

the resulting form of the rational function:

the polynomial part of the rational function:

*y* = *x*^{2} + *x* + 4

a graph of the original rational function:

a graph of the polynomial part of the rational function:

Note the similarity between the two graphs above. Except for where the vertical asymptote caused a break in the middle, the two graphs are practically the same, as you can see from the overlay.

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If the degree-difference in favor of the numerator is 3, it stands to reason that the rational function's graph would approximate a cubic. And, below, you'll see that this reasoning is correct.

the original rational function:

the long division of the numerator by the denominator:

the resulting form of the rational function:

the polynomial part of the rational function:

*y* = –2*x*^{3} – 5*x*^{2} – 6*x* – 1

a graph of the original rational function:

a graph of the polynomial part of the rational function:

As the overlay in the second graph above displays, the rational function's graph is, except for the middle, very nearly equal to the graph of its polynomial portion.

You can probably guess what the rational function's graph will look like, off to the sides, when the degree-difference in favor of the numerator is 4. Yes, it'll look like a quartic.

the original rational function:

the long division of the numerator by the denominator:

the resulting form of the rational function:

the polynomial part of the rational function:

*y* = 5*x*^{4} + 5*x*^{3} + 8*x*^{2} + 8*x* + 12

a graph of the original rational function:

a graph of the polynomial part of the rational function:

Certainly, a given rational function's graph will frequently get a bit twitchy in the middle, especially if it has vertical asymptotes. But "at the sides" (or "on the ends"), you can expect the graph to be nearly the same as its associated polynomial function.

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