Asymptotes: Comparing Graphs

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.

But what if the difference isn't just 1?

What happens if the degree difference in a rational function is more than one (in the numerator's favor)?

When the numerator of a rational function has a degree that is greater than the denominator's degree by more than 1, then the shape of the rational function's graph will be approximated by the polynomial found by doing the long division. The rational graph will veer away from the polynomial graph when you're near any vertical asymptotes, but will generally otherwise be very close. In effect, you will have a curvy asymptote (using "asymptote" to mean "the curve that the graph gets very, very close to, at least off to the sides").

You've already seen how the asymptote works when the numerator's degree is exactly 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.

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What if the numerator's degree is 2 more than the denomintor's degree?

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:

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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What if the numerator's degree is 3 more than the denomintor's degree?

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:

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.

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What if the numerator's degree is 4 more than the denomintor's degree?

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:

a graph of the original rational function:

a graph of the polynomial part of the rational function:

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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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