Chapter 5.6, Problem 5E

Solution

To sketch: the graph of the function $f\left(x\right)=x^4-8x^2+16$ and to check the answer with the help of a graphing calculator.

Given:

The function $f\left(x\right)=x^4-8x^2+16$ .

Graph:

Identify the zeroes of the function $f\left(x\right)=x^4-8x^2+16$ :

Observe that the zeroes of the function $f\left(x\right)=x^4-8x^2+16$ are those pairs $\left(x,0\right)$ for which $f\left(x\right)=0$ .

Substitute $f\left(x\right)=0$ in $f\left(x\right)=x^4-8x^2+16$ :

  $0=x^4-8x^2+16$ .

Solve the equation $0=x^4-8x^2+16$ for $x$ :

  $\begin{array}{l}0=x^4-8x^2+16\\ 0={\left(x^2-4\right)}^2\\ 0={\left(x^2-2^2\right)}^2\\ 0={\left[\left(x-2\right)\left(x+2\right)\right]}^2\\ 0={\left(x-2\right)}^2{\left(x+2\right)}^2\\ 0={\left(x-2\right)}^2\text{or}0={\left(x+2\right)}^2\\ 0=\left(x-2\right)\text{or }0={\left(x+2\right)}^2\\ 2=x\text{or}-2=x\end{array}$ .

Thus, the zeroes of the function $f\left(x\right)=x^4-8x^2+16$ are $\left(-2,0\right)$ and $\left(2,0\right)$ .

Plot the zeroes $\left(-2,0\right)$ and $\left(2,0\right)$ of the function $f\left(x\right)=x^4-8x^2+16$

on a coordinate grid:

  

Determine whether $f\left(x\right)=x^4-8x^2+16$ is positive or negative on each interval determined by the zeroes $\left(-2,0\right)$ and $\left(2,0\right)$ :

Observe that the intervals determined by the zeroes $\left(-2,0\right)$ and $\left(2,0\right)$ are $x<-2,-2<x<2$ and $x>2$ .

To determine whether $f\left(x\right)=x^4-8x^2+16$ is positive or negative over the interval $x<-2$ , evaluate $f\left(x\right)=x^4-8x^2+16$ for some value of $x$ that lies in $x<-2$ , say $x=-3$ , for its sign.

That is, substitute $x=-3$ in $f\left(x\right)=x^4-8x^2+16$ and evaluate $f\left(-3\right)$ :

  $\begin{array}{l}f\left(-3\right)={\left(-3\right)}^4-8\cdot {\left(-3\right)}^2+16\\ f\left(-3\right)=25\end{array}$ .

Observe that $f\left(-3\right)=25$ is positive.

Thus, $f\left(x\right)=x^4-8x^2+16$ is positive over $x<-2$ .

Similarly, it may be observed that $f\left(0\right)=16$ and $f\left(3\right)=25$ .

That is, $f\left(x\right)=x^4-8x^2+16$ is positive over both the intervals $-2<x<2$ and $x>2$ .

Sketch the graph of the function $f\left(x\right)=x^4-8x^2+16$ incorporating all the information gained:

  

Verify the sketch with the help of a graphing calculator:

Graph the function $f\left(x\right)=x^4-8x^2+16$

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.