Chapter 4, Problem 14RE

To determine

To analyze: the given polynomial function

Answer to Problem 14RE

  

Explanation of Solution

Given:

  $f\left(x\right)=\left(x-2\right){\left(x+4\right)}^2$

Calculation:

Let us consider the following polynomial function,

  $f\left(x\right)=\left(x-2\right){\left(x+4\right)}^2$

Analyse the graph of the given function.

For this follow the following steps:

Step 1:Determination of the end behaviour of the graph of the function

The end behaviour of the graph of a function is similar to the leading term of the function. So, first of all determine the degree of the function.

So,

  $\begin{array}{l}f\left(x\right)=\left(x-2\right){\left(x+4\right)}^2\hfill \\ \begin{array}{l}=\left(x-2\right)\left(x^2+8x+16\right)\\ =x^3+8x^2+16x-2x^2-16x-32\end{array}\hfill \\ =x^3+6x^2-32\hfill \end{array}$

Here, the degree of the polynomial is 3. So, the graph off behaves like $y=x^3$ for large values of $\left|x\right|$ .

Step 2:

Graph the function using a graphing utility.

Plot the given function.

  

Step 3: Next, approximate the x and the y intercepts of the graph.

From the graph, the graph passes through y -axis at a single point, so the y -intercept is at $y=-32$ .

Also, the graph intersects the x -axis at 1 points, $x=2$ and touched the axis at $x=-4$ .

Step 4:

Create a table to find points on the graph around each x intercept.

  

Here, for $x=-4$ , the multiplicity is 2, as the f touches the axis at this point.

Step 5:

Approximate the turning points of the graph.

From the graph off , f has two turning points.

Using MAXIMUM, one turning point is at $\left(-4,\text{0}\right)$ , using MINIMUM, the other turning point is at $\left(0,-16\right)$ .

Step 6:

Complete the graph using all the above information.

  

Step 7:

Determine the range and domain of the function.

Domain and range of the function are the set of all real numbers. So,

  $\begin{array}{l}\text{Domain}=\left(-\infty ,\infty \right)\hfill \\ \text{Range}=\left(-\infty ,\infty \right)\hfill \end{array}$

Step 8:

Determine where the function is increasing and where it is decreasing.

From the graph, the function is increasing in the interval $\left(-\infty ,-4\right)$ and $\left(0,\infty \right)$ . The function is decreasing in the interval $\left(-4,0\right)$ .

Conclusion:

Therefore, the polynomial function $f\left(x\right)=\left(x-2\right){\left(x+4\right)}^2$ is analysed with graph.