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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 xaxis (y = 0) or any other fixed yvalue. 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 polynomialonly 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.
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 straightline 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 20032011 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.)
To find the slant asymptote, I'll do the long division: The slant asymptote is
the polynomial part of the answer, not the remainder. slant asymptote: y = x + 5
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. 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. << Previous Top  1  2  3  4  Return to Index Next >>



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