2-6 Rational Functions and Asymptotes

Rational Function is a function that can be written as

F(x)= N(x)/D(x)

Where N(x) and D(x) are polynomials.

Asymptotes of Rational Functions

1) The graph of F has the vertical asymptotes at the zeros of D(x).

2) The graph of F has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x).

a) If n < m, the line y=0(x axis) is a horizontal asymptote at the zeros of D(x)

b) If n=m, the line y=a/b is a horizontal asymptote.

c) If n>m the graph of f has no horizontal asymptotes.

As or the distance between the horizontal asymptotes and the points on the graph must approach zero.

Example 1

Graph of 1/x In the graph 1/x the Horizontal Asymptote is the X axis and the Vertical Asymptote is the Y axis. Because the denominator is zero when x=o, the domain is of F is all real numbers except x=0.

Example 2:

Graph of 1/(x-3)

Finding the Vertical asymptote:

D(x)= x-3

0=x-3

x=3

3 is a zero of the function, so the vertical asymptote on the graph

Example 3

Graph of 1/(x-3) +2

Vertical Asymptote

D(x)= x-3

X=3

Vertical Asymptote= 3

Horizontal Asymptote

Horizontal Asymptote= 2

If the exponent is greater in the denominator than in the numerator, the horizontal asymptote will be x=o as the denominator approaches infinity. Therefore in this example, as 1/(x-3) approaches zero, the horizontal asymptote is two, f(x) approaches 2.