CPM Homework Help : PC3 Problem 6-140

Graph $r(x)=\frac{2x^2-4x-16}{x^2-9}$ . State the locations of any holes and/or asymptotes.

Step 1:

$r(x)=\frac{2\left(x^2-2x-8\right)}{\left(x+3\right)\left(x-3\right)}$

Step 2:

$r(x)=\frac{2\left(x-4\right)\left(x+2\right)}{\left(x+3\right)\left(x-3\right)}$

Step 3:

Since none of the factors create a Giant $1$ , there are no holes.
The vertical asymptotes are the values of $x$ that make the denominator $0$ .
The horizontal asymptote is $y = 2$ . (Why?)
The $x$ -intercepts are the values of $x$ that make the numerator $0$ .
The $y$ -intercept occurs when $x = 0$ .

Use this information to sketch a predicted graph of function. Then use a graphing calculator to check your work.
Be sure to identify and correct any mistakes.