Chapter 4.1, Problem 92AYU

In Problems 99-106, analyze each polynomial function $f$ by following Steps 1 through 8 on page 195.

$f\left(x\right)\text{ = }{x}^3-\text{ 0}\text{.8}{x}^2-\text{ 4}\text{.6656}x\text{ + 3}\text{.73248}$

To determine

To analyze: The polynomial function

Answer to Problem 92AYU

$\text{The polynomial function }f\left(x\right)=x^3-0.8x^2-4.6656x+3.73248\text{ is analyzed}$.

Explanation of Solution

Given:

$f\left(x\right)=x^3-0.8x^2-4.6656x+3.73248$

Step 1: Determine the end behavior of the graph of the function.

Analyze the graph of the polynomial function

$f\left(x\right)=x^3-0.8x^2-4.6656x+3.73248$

The polynomial function $f$ is of degree 3. The graph of $f$ behaves like $y=x^3$ for large values of $\left|x\right|$.

Step 2: Graph the function using a graphing utility.

$Y_1=x^3-0.8x^2-4.6656x+3.73248$

Step 3: Use a graphic utility to approximate the $x$ and $y\text{-intercepts}$ of the graph.

The $y\text{-intercept }=f\left(0\right)=3.73248$

Using graphic utility find the lone $x\text{-intercept }=-2.09,2.09,0.85$

Step 4: Using the graphic utility to find the points on the graph around each $x\text{-intercept}$.

The points $\left(0,3.8\right)$ and $\left(0.9,0\right)$ are on the graph

Step 5: Approximate the turning point of the graph.

From the graph of $f$, we see $f$ has 2 turning points

Using the graph maximum turning point $=\left(-1,6.4\right)$ rounded to two decimal places

Using the graph minimum turning point $=\left(1.8,-1.8\right)$ rounded to two decimal places

Step 6: Redraw the graph.

The graph passes through $\left(0,3.8\right)$ and $\left(0.9,0\right)$

The end behavior of the graph $=y=x^3$

Step 7: Find the domain and range of the function.

The domain and range is the set of all real numbers

Step 8: Use the graph to determine where the function is increasing or where it is decreasing.

Based on the graph, $f$ is increasing on the intervals $\left(-\infty ,-1\right]$ and $\left[1.8,\infty \right)$

Based on the graph, $f$ is decreasing $\left[-1,1.8\right]$.