Chapter 5, Problem 8T

Solution

To graph: The function $y=\frac{x^2-4}{x^2+8x+15}$.

Given information:

The function $y=\frac{x^2-4}{x^2+8x+15}$.

Graph:

The $x-$ intercepts of the graph of a rational function are the real zeros of its numerator.

Consider the function $y=\frac{x^2-4}{x^2+8x+15}$ . To find $x-$ intercept, equate numerator to zero.

  $\begin{array}{c}{x}^2-4=0\\ x^2=4\\ x=\pm 2\end{array}$

So, the $x-$ intercept is $2$ and $-2$.

To find the vertical asymptote, equate the denominator to zero.

  $\begin{array}{c}{x}^2+8x+15=0\\ x^2+3x+5x+15=0\\ \left(x+3\right)\left(x+5\right)=0\end{array}$

So the vertical asymptote is $x=-3$ and $x=-5$.

The degree of the numerator is same as that of the denominator, so the horizontal asymptote is,

  $y=1$.

Draw the graph of the given function.

  

Interpretation:

The $x-$ intercept is $2$ and $-2$ , the vertical asymptote is $x=-3$ and $x=-5$ , and the horizontal asymptote is $y=1$.