Chapter 4.1, Problem 81AYU

To determine

To analyze: The graph of a polynomial function.

Answer to Problem 81AYU

$\text{The graph of a polynomial function }f\left(x\right)=x^2\left(x-2\right)\left(x+2\right)\text{is analyzed}$.

Explanation of Solution

Given:

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

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

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

Multiply

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

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: Find $x\text{and}y\text{intercepts}$ of the graph of the function.

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

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

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

$x=0\text{or} x=2 \text{or} x= -2$

The $x\text{-intercept }=x=0, 2, -2$.

Step 3: Determine the zeros of the function and their multiplicity. Using this information determine whether the graph crosses or touches the $x\text{-axis}$ at each $x\text{-intercept}$.

The zeros of $f=0,2,-2$.

The zero 0 has even multiplicity therefore the graph of $f$ touches the $x\text{-axis}$.

The zero 2 has odd multiplicity therefore the graph of $f$ crosses the $x\text{-axis}$.

The zero $-2$ has odd multiplicity therefore the graph of $f$ crosses the $x\text{-axis}$.

Step 4: Using the graphing utility to graph the function

$y_1=x^2\left(x-2\right)\left(x+2\right)$

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 $=\left(0,0\right)$.

Using the graph minimum turning point $=\left(-1.45,-4\right)$ and $=\left(1.45,-4\right)$.

Step 6: Redraw the graph.

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

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 $\left[-\mathrm{1.45},0\right]$ and $\left[1.45,\infty \right)$.

Based on the graph, $f$ is decreasing $\left(\infty ,-1.45\right]$ and $\left[0,1.45\right]$.