Chapter 3.7, Problem 69E

To determine

To find: the slant asymptote, the vertical asymptotes, and sketch a graph of the function.

Answer to Problem 69E

The quotient $y=x+8$ is the slant asymptote.

Vertical asymptote is $x=3$ .

Explanation of Solution

Given information:

Given function,

  $r\left(x\right)=\frac{x^2+5x+4}{x-3}$

Calculation:

Consider the function,

  $r\left(x\right)=\frac{x^2+5x+4}{x-3}$

Vertical asymptote of $r$ are the lines $x=a$ , where $a$ is a zero of the denominator.

Al slant asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator. To find the slant asymptote must divide the numerator by the denominator using either long division or synthetic division.

Here, denominator is $x-3$ .

So,

  $x-3=0$

This implies that,

  $x=3$

Therefore, vertical asymptote is $x=3$ .

Using long division, we get

  $\begin{array}{c}x-3\begin{array}{c}\hfill x+8\\ \hfill \overline{)x^2+5x+4}\end{array}\\ \underset{\_}{x^2-3x}\\ 8x+4\\ \underset{\_}{8x-24}\\ 28\end{array}$

This implies that,

  $r\left(x\right)=x+8+\frac{28}{x-3}$

Therefore, the quotient $y=x+8$ is the slant asymptote.

Now to draw the graph of the function $r\left(x\right)=\frac{x^2+5x+4}{x-3}$ , slant asymptote $y=x+8$ and vertical asymptote $x=3$ .

Thus, the required graph is given below,