Chapter 3, Problem 80RE

To determine

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

Explanation of Solution

Given information:

The rational function is $r\left(x\right)=\frac{2x^2-6x-7}{x-4}$ .

Graph:

Factor: $r\left(x\right)=\frac{2x^2-6x-7}{x-4}$

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

  $\begin{array}{l}2x^2-6x-7=0\\ x=\frac{-\left(-6\right)\pm \sqrt{{\left(-6\right)}^2-4·2·\left(-7\right)}}{2·2}\\ x=\frac{6\pm \sqrt{36+56}}{4}\\ x=\frac{3\pm \sqrt{23}}{2}\end{array}$

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

  $\begin{array}{l}r\left(0\right)=\frac{2{\left(0\right)}^2-6\left(0\right)-7}{0-4}\\ r\left(0\right)=\frac{7}{4}\end{array}$

So, y-intercept is $\frac{7}{4}$ .

Horizontal asymptote: because here is degree of numerator is greater than denominator so horizontal asymptote is none.

Vertical asymptote: the vertical asymptote is occurs where denominator is zero,

$\begin{array}{l}x-4=0\\ x=4\end{array}$

So vertical asymptote is $x=4$ .

Slant asymptote: since the degree of the numerator is one more than the degree of the denominator, the function has a slant asymptote.

  $r\left(x\right)=2x+2+\frac{1}{x-4}$

  $y=2x+2$

So slant asymptote is $y=2x+2$ .

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

To sketch the graph,

x	y
-2	-2.16
-1	-0.20
1	3.66
2	-5.5

The graph is obtained as:

Interpretation:

From the above graph it can be observed that the x-intercept is $\frac{3\pm \sqrt{23}}{2}$ , y-intercept is $\frac{7}{4}$ , vertical asymptote is $x=4$ and slant asymptote is $y=2x+2$ .