Chapter 4.2, Problem 82AYU

In Problems 91-98, analyze each polynomial function using Steps 1 through 8 on page 193 in Section 4.1.

$f\left(x\right)=x^4-x^3-6x^2+4x+8$

[Hint: See Problem 60.]

To determine

To analyze: The polynomial function $f\left(x\right)=x^4-x^3-6x^2+4x+8$.

Answer to Problem 82AYU

$\text{The polynomial function }f\left(x\right)=x^4-x^3-6x^2+4x+8\text{ is analyzed}$.

Explanation of Solution

Given:

$f\left(x\right)=x^4-x^3-6x^2+4x+8$

Step 1: Determine the end behavior of the function

The polynomial function $f$ is of degree 4. The graph of $f$ behaves like $y=x^4$ for large values of $\left|x\right|$.

Step 2: Finding the $x$ and $y\text{-intercepts}$ of the graph of the function

To find the $y\text{-intercept}$, finding $f\left(0\right)$

$f\left(x\right)=x^4-x^3-6x^2+4x+8$

$f\left(0\right)=8$

$y\text{-intercept }=8$

To find the $x\text{-intercept}$, solving $f\left(x\right)=x^4-x^3-6x^2+4x+8=\text{ 0}$

Using synthetic division

$\left(x-2\right)\left(x^3+x^2-4x-4\right)=\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x-2\right)$

$x\text{-intercept }=1,-1,-2,\text{ 2}$

Step 3: Determining the zeros of the function and their multiplicity to find the graph crosses or touches the $x\text{-axis}$ at each intercept.

The zeros 1, 2, $-1$ and $-2$ each have multiplicity 1. Therefore at 1,2, $-1$, $-2$ graph crosses the $x\text{-axis}$

Step 4: Using the graphing utility to graph the function.

Step 5: Approximate the turning point of the graph

From the graph of $f$, we see $f$ has 3 turning points

Using the graph maximum turning point $=\left(0.15,\text{ 9.5}\right)$ rounded to two decimal places

Using the graph minimum turning point $=\left(-1.85,-2.65\right)$ rounded to two decimal places

Step 6: Redraw the graph

The graph passes through $\left(-2,\text{ 0}\right)$ and $\left(2,0\right)$

The end behavior of the graph $=y=x^4$

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 $\left[-\mathrm{1.85},0.15\right]$ and $\left[2,\infty \right)$

Based on the graph, $f$ is decreasing $(\infty ,-1.85]\left[0.15,2\right]$