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Asymptotes: Comparing Graphs

While your text almost certainly covers only the case where the numerator's degree is 1 greater than the denominator's, you should be aware that the graphs of rational functions behave in similar manners even when the degrees are further apart. If the numerator's degree is greater than the denominator's  — if the rational function is an "improper" polynomial fraction — then the graph will approximate the polynomial part obtained by long division.

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. Compare:

original function: graph of original function:
y = [x^3 + 3x - 2] / [x - 1] graph of y = [x^3 + 3x - 2] / [x - 1]
long division:
long division
result function:
y = x^2 + x + 4 + [2] / [x - 1]
polynomial part: graph of
polynomial part:
y = x2 + x + 4

  

 graph of y = x^2 + x + 4

Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
Note the similarity between the two graphs above. 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.

The table above displayed the graphs of a rational function in which the degree of the numerator was two more than the degree of the denominator. In the following tables, this relationship is demonstrated when the degrees are three apart and four apart.

original function: graph of original function:
y = [-2x^4 - 3x^3 - x^2 + 5x - 1] / [x - 1] graph of y = [-2x^4 - 3x^3 - x^2 + 5x - 1] / [x - 1]
long division:
long division
result function:
y = -2x^3 - 5x^2 - 6x - 1 + [-2] / [x - 1]
polynomial part: graph of
polynomial part:
y = –2x3 – 5x2 – 6x – 1

 

 y = -2x^3 - 5x^2 - 6x - 1

    Copyright © Elizabeth Stapel 2003-2011 All Rights Reserved

original function: graph of original function:
y = [5x^5 + 3x^3 + 4x] / [x - 1] graph of y = [5x^5 + 3x^3 + 4x] / [x - 1]
long division:
long division
result function:
y = 5x^4 + 5x^3 + 8x^2 + 8x + 12 + [12]/[x - 1]
polynomial part: graph of
polynomial part:
y = 5x4 + 5x3 + 8x2 + 8x + 12

 

 graph of y = 5x^4 + 5x^3 + 8x^2 + 8x + 12

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

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Cite this article as:

Stapel, Elizabeth. "Asymptotes: Comparing Graphs." Purplemath. Available from
    http://www.purplemath.com/modules/asymnote2.htm. Accessed
 

 

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