Rational Functions

Rational Functions are functions of the form

where P(x) and Q(x) are polynomials. The behaviour of these functions can be of considerable complexity, so we will treat here only the case where the dgree of the polynomials is small.  One of the main characteristics of rational functions is the existence of asymptotes. An asymptote is a straight line to which the graph  of the function gets arbitrarily close. Typically one can classify the asymptotes into two types.

• Vertical: These are the equations of the straight lines (if any) the one gets by setting the linear factors of Q(x) to 0. If Q(x) is a prime polynomial such as {x^2 +1), there are no vertical asymptotes.
• Oblique: When the degree of P(x) is greater than or equal to the degree of Q(x), the division alogorithm tells as that the ratio is equal to a quotient D(x) plus a remainder R(x) whose degree is smaller than that of Q(x).                       

The remainder goes to 0 as x goes to infinity, so for large x, the ratio behaves as the quotient. The equation obtained by setting the quotient D(x)=0  represents the oblique asymptote. Thus, for example, f(x) = (x^2)/(x+2) = (x-2) + 4/(x+2).  So the asymptotes of this function are  y = x-2  and x = -2.

In the table below we list most of the rational functions that appear in elementary calculus textbooks. We also provide a graphical animations of these functions as the parameter  c  is varied.

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Gabriel G. Lugo, lugo@uncwil.edu
Last updated November 29, 1998