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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. In the column 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 in the same table, 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:
y = [-3x^2 + 2] / [x - 1] graph of y = [-3x^2 + 2] / [x - 1]
long division:
long division
rearranged function:
y = -3x - 3 + [-1] / [-3x^2 + 2]
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.

graph of y = -3x - 3

The graphs show that, if the degree of the numerator is exactly one more 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 be nearly equal to this slanted straight-line 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 2003-2011 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.)

  • Find the slant asymptote of the following function:
  •  

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      y = [x^2 + 3x + 2] / [x - 2]

    To find the slant asymptote, I'll do the long division:

      long division: y = x + 5 + 12/(x - 2]

    The slant asymptote is the polynomial part of the answer, not the remainder.

      slant asymptote:  y = x + 5

  • Find the slant asymptote of the following function:
    • y = [2x^3 + 4x^2 - 9] / [3 - x^2]

    I'll need to be careful of the missing linear term in the numerator, and of the signs when I reverse the terms in the denominator for the long division.

      long division: -2x - 4, with remainder 6x + 3

    The slant asymptote is the polynomial part of the answer, so:

      slant asymptote: y = 2x 4


A note for the curious regarding the horizontal and slant asymptote rules.

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Cite this article as:

Stapel, Elizabeth. "Slant, or Oblique, Asymptotes." Purplemath. Available from
    http://www.purplemath.com/modules/asymtote3.htm. Accessed
 

 



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