Q. 64 - Calculus

Question

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f\text{'}, and f\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f. 

$f\left(x\right)=(x-2)(x+2)$

Step-by-Step Solution

Answer

The sign chart is

The sketch of the graph is

 

Step 1. Given Information.

The given function is $f\left(x\right)=\left(x-2\right)\left(x+2\right).$

Step 2. Finding the roots.

To find the roots we will put the given function equal to zero.

So,

$$\begin{gathered} f\left(x\right)=(x-2)(x+2) \\ x-2=0 \mathrm{and} x+2=0 \\ x=2 \mathrm{and} x=-2 \end{gathered}$$

Therefore, the given function have roots at $x=2 \mathrm{and} -2.$

Step 3. Testing the signs of f.

To sketch the sign chart, let's test the signs on both sides.

For f

$$\begin{gathered} f(-3)=\left(-3-2\right)\left(-3+2\right) \\ f(-3)=5 \\ \mathrm{Now}, f(-1)=\left(-1-2\right)\left(-1+2\right) \\ f(-1)=-3 \\ \mathrm{Now}, f\left(4\right)=\left(4-2\right)\left(4+2\right) \\ f\left(4\right)=12 \end{gathered}$$

Step 4. Testing the signs.

Now, let's test the sign for $f\text{'} \mathrm{and} f\text{'}\text{'}.$

Let's differentiate the equation to find $f\text{'}.$

So, 

 $$\begin{gathered} f\left(x\right)=(x-2)(x+2) \\ f\left(x\right)=x^2-4 \\ f\text{'}\left(x\right)=2x \\ 0=2x \\ 0=x \end{gathered}$$

Testing the signs on both sides,

$$\begin{gathered} f\text{'}(-2)=2(-2) \\ f\text{'}(-2)=-4 \\ \mathrm{And} \\ f\text{'}\left(2\right)=2\left(2\right) \\ f\text{'}\left(2\right)=4 \end{gathered}$$

Thus, $f\text{'}$ is negative on the interval $\left(-\infty ,0\right)$ and positive on the interval $\left(0,\infty \right).$ Hence the graph of f will be increasing on the positive intervals and decreasing on the negative intervals.

Let's differentiate again.

So, $f\text{'}\text{'}=2$

Thus, $f\text{'}\text{'}$ is positive. Hence f will be concave up everyplace.

Step 5. Sketch the sign chart.

The sign chart is

Step 6. Examine the relevant limit.

Let's examine the limits of $f\left(x\right)=(x-2)(x+2) \mathrm{as} x\to \pm \infty .$

$$\begin{gathered} \underset{x\to \infty }{\lim }f\left(x\right)=\infty \\ \underset{x\to -\infty }{\lim }f\left(x\right)=\infty \end{gathered}$$

Step 7. Sketch the graph of function f.

The graph of the function is