Chapter 4.1, Problem 94AYU

To determine

To analyze: The polynomial function

Answer to Problem 94AYU

$\text{The polynomial function }f\left(x\right)=x^3-2.91x^2-7.668x-3.8151\text{ is analyzed}$.

Explanation of Solution

Given:

$f\left(x\right)=x^3-2.91x^2-7.668x-3.8151$

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

Analyze the graph of the polynomial function

$f\left(x\right)=x^3-2.91x^2-7.668x-3.8151$

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.

$Y_1=x^3-2.91x^2-7.668x-3.8151$

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

The $y\text{-intercept }=f\left(0\right)=-3.8151$

Using graphic utility find the lone $x\text{-intercept }=-0.89,5$

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

The points $\left(0,-2\right)$ and $\left(5,0\right)$ 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 $=\left(-0.89,0\right)$ rounded to two decimal places

Using the graph minimum turning point $=\left(2.5,-27\right)$ rounded to two decimal places

Step 6: Redraw the graph.

The graph passes through $\left(0,-2\right)$ and $\left(5,0\right)$

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 $\left(-\infty ,-0.89\right]$ and $\left[2.5,\infty \right)$

Based on the graph, $f$ is decreasing $\left[-0.89,2.5\right]$.