Chapter 4.2, Problem 86AYU

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

$f\left(x\right)=20x^4+73x^3+46x^2-52x-24$

To determine

To analyze: The polynomial function $f\left(x\right)=20x^4+73x^3+46x^2-52x-24$.

Answer to Problem 86AYU

$\text{The polynomial function }f\left(x\right)=20x^4+73x^3+46x^2-52x-24\text{ is analyzed}$.

Explanation of Solution

Given:

$f\left(x\right)=20x^4+73x^3+46x^2-52x-24$

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 $f\left(0\right)$

$f\left(x\right)=20x^4+73x^3+46x^2-52x-24$

$f\left(0\right)=-24$

$y\text{-intercept }=-24$

To find the $x\text{-intercept}$, solving $f\left(x\right)=20x^4+73x^3+46x^2-52x-24=\text{ 0}$

Using synthetic division

$\left(x+2\right)\left(20x^3+33x^2-20x-12\right)=\left(x+2\right)\left(4x+3\right)\left(5x+2\right)\left(x+2\right)$

$x\text{-intercept }=-2,-\frac{3}{4},-\frac{2}{5}$

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 $-\frac{3}{4}$, $-\frac{2}{5}$ each have multiplicity 1. Therefore at $-\frac{3}{4}$, $-\frac{2}{5}$ the graph crosses the $x\text{-axis}$

The zero $-2$ has multiplicity 2 therefore the graph touches 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 $=\left(-1.1,\text{ 21.5}\right)$ rounded to two decimal places

Using the graph minimum turning point $=\left(0.15,-33.55\right)$ rounded to two decimal places

Step 6: Redraw the graph

The graph passes through $\left(-2,\text{ 0}\right)$ and $\left(0.8,0\right)$

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 $\left[-2,-1\right]$ and $\left[0.3,\infty \right)$

Based on the graph, $f$ is decreasing $\left(\infty ,-2\right]\left[-1,0.3\right]$