### Home > PC3 > Chapter 4 > Lesson 4.2.3 > Problem4-100

4-100.

State the coordinates of any holes and the equations of any asymptotes for the graph of the function $f(x)=\frac{2x^2+3x-2}{x^2-4}$. Then sketch a graph of the function.

Factor completely:

$f(x)=\frac{(2x-1)(x+2)}{(x-2)(x+2)}$

HOLES or VERTICAL ASYMPTOTES occur when the denominator $=0$. In this case $x=−2$ is where a hole occurs because the factor divides evenly into the numerator. $x=2$ is a vertical asymptote because the factor does not divide evenly into the numerator.

If you were to rewrite this equation using polynomial division, what would the "whole" term(s) be? This will tell you the horizontal or slant asymptote.
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