Slant, or Oblique, Asymptotes


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Slant, or Oblique, Asymptotes (page 3 of 4)

Sections: Vertical asymptotes, Horizontal asymptotes, Slant asymptotes, Examples

What happens if the degree is greater in the numerator than in the denominator? Reasonably, the numerator, being "stronger", ought to "pull" the graph away from the x-axis (y = 0) or any other fixed y-value.

To investigate, look at the following table. To the left of the first graph, you can see the original function, the original function after rearrangement by long division, and the polynomial-only part of the original function. Then look at the second graph, and note the similarity between the graph of the original function and the graph of the polynomial part of the function.

original function:	graph of the original function:

long division:

result function:

polynomial part:	graph of polynomial part:
y = –3x – 3 Note the similarity between the two graphs. Except for where the vertical asymptote causes a break in the middle, the two graphs are practically the same, as you can see from the overlay.

As you can see, if the degree of the numerator is one greater than the degree of the denominator (so that the polynomial fraction is "improper"), then the graph of the rational function will be, roughly, a slanty straight line with some fiddly bits in the middle. Because the graph will approximate this "slanty" polynomial equivalent, the asymptote for this sort of rational function is called a "slant" (or "oblique") asymptote. The equation for the slant asymptote is the polynomial part of the rational that you get after doing the long division. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

(By the way, this relationship -- between an improper rational function, its associated polynomial, and their graphs -- holds true regardless of the difference in the degrees of the numerator and denominator.)