Chapter 5.6, Problem 3E

Solution

To sketch: the graph of the function $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ and to check the answer with the help of a graphing calculator.

Given:

The function $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ .

Graph:

Identify the zeroes of the function $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ :

Observe that the zeroes of the function $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ are those pairs $\left(x,0\right)$ for which $p\left(x\right)=0$ .

Substitute $p\left(x\right)=0$ in $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ :

  $0=x\left(x+4\right)\left(x-4\right)$ .

Solve the equation $0=x\left(x+4\right)\left(x-4\right)$ for $x$ :

  $\begin{array}{c}0=x\text{or }0=\left(x+4\right)\text{or }0=\left(x-4\right)\\ 0=x\text{ or -4}=x\text{or}4=x\end{array}$ .

Thus, the zeroes of the function $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ are $\left(0,0\right),\left(-4,0\right)$ and $\left(4,0\right)$ .

Plot the zeroes $\left(0,0\right),\left(-4,0\right)$ and $\left(4,0\right)$ of the function $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$

on a coordinate grid:

  

Determine whether $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ is positive or negative on each interval determined by the zeroes $\left(0,0\right),\left(-4,0\right)$ and $\left(4,0\right)$ :

Observe that the intervals determined by the zeroes $\left(0,0\right),\left(-4,0\right)$ and $\left(4,0\right)$ are $x<-4,-4<x<0,0<x<4$ and $x>4$ .

To determine whether $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ is positive or negative over the interval $x<-4$ , evaluate $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ for some value of $x$ that lies in $x<-4$ , say $x=-5$ , for its sign.

That is, substitute $x=-5$ in $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ and evaluate $p\left(-5\right)$ :

  $\begin{array}{l}p\left(-5\right)=-5\left(-5+4\right)\left(-5-4\right)\\ p\left(-5\right)=-45\end{array}$ .

Observe that $p\left(-5\right)=-45$ is negative.

Thus, $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ is negative over $x<-4$ .

Similarly, it may be observed that $g\left(-2\right)=24,g\left(2\right)=-24$ and $g\left(5\right)=45$ .

That is, $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ is positive over $-4<x<0$ , negative over $0<x<4$ and finally positive over $x>4$ .

Sketch the graph of the function $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$ incorporating all the information gained:

  

Verify the sketch with the help of a graphing calculator:

Graph the function $p\left(x\right)=x\left(x+4\right)\left(x-4\right)$

with the help of a graphing calculator on a similar grid:

  

Observe that the graph so obtained looks exactly like the one sketched previously.

Thus, it is verified that the sketch is correct.