Chapter 3, Problem 81RE

To determine

To graph the rational function and show the x-andy-intercepts and asymptotes.

Explanation of Solution

Given information:

The rational function is $r\left(x\right)=\frac{x^2-9}{2x^2+1}$ .

Graph:

Factor: $r\left(x\right)=\frac{(x-3)(x+3)}{2x^2+1}$

x-intercepts: the x-intercept are the zeros of the numerator,

  $\begin{array}{l}x+3=0\\ x=-3\\ x-3=0\\ x=3\end{array}$

So, x-intercept is $-3,3$ .

y-intercepts: to find y-intercept, substitute $x=0$ in the original form of the function,

  $\begin{array}{l}r\left(0\right)=\frac{{\left(0\right)}^2-9}{2{\left(0\right)}^2+1}\\ r\left(0\right)=-9\end{array}$

So, y-intercept is $-9$ .

Horizontal asymptote: because here is degree of numerator same denominator so horizontal asymptote

  $y=\frac{1}{2}$

Vertical asymptote: the vertical asymptote is occurs where denominator is zero, but here the denominator has imaginary roots so vertical asymptote does not occur.

Slant asymptote: the slant asymptote is occurs when the numerator degree is one more than denominator,

  $\begin{array}{l}r\left(x\right)=\frac{x^2-9}{2x^2+1}\\ r\left(x\right)=\frac{1}{2}-\frac{19}{2\left(2x^2+1\right)}\end{array}$

So vertical asymptote is $y=\frac{1}{2}$

Use the above information together with some additional values which is show in table below

To sketch the graph,

x	y
-2	-0.55
-1	-2.67
1	-2.67
2	-0.55

The graph is obtained as:

Interpretation:

From the above graph it can be observed that the x-intercept is $-3,3$ , y-intercept is $\frac{1}{2}$ , horizontal asymptote $y=\frac{1}{2}$ and slant asymptote is $y=\frac{1}{2}$ .