Chapter 9.4, Problem 11PPS

To determine

To graph: The given function.

Explanation of Solution

Given information:

The following function:

  $f\left(x\right)=\frac{x^3+64}{16x-24}$

Calculation:

Step 1:

Find the zeroes.

  $\begin{array}{l}{x}^3+64=0[\text{Set }a(x)=0]\\ x^3=-64[\text{Subtract 64 from each side]}\\ x=-4[\text{Take the cube root of each side]}\end{array}$

There is a zero at $x=-4$ .

Step 2:

Find the asymptotes.

  $\begin{array}{l}16x-24=0[\text{Set }b(x)=0]\\ 16x=24[\text{Add 24 to each side]}\\ x=1.5[\text{Divide both sides by 16]}\end{array}$

There is a vertical asymptote at $x=1.5$ .

The degree of numerator is greater than the degree of the denominator.

Therefore, there is no horizontal asymptote.

Step 3:

Draw the graph

Use a table to find ordered pairs on the graph. Then connect the points.

The table of values:

  

The graph: