Chapter 4.1, Problem 100AYU

In Problems 107-114, analyze each polynomial function by following Steps 1 through 8 on page 193.

$f\left(x\right)\text{ = }x-x^3$

To determine

To analyze: The polynomial function.

Answer to Problem 100AYU

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

Explanation of Solution

Given:

$f\left(x\right)=x-x^3$

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

$f\left(x\right)=x-x^3$

Factor

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

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

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

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

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

$x=0\text{ or} x=-1,+1$

The $x\text{-intercept }=x=-1,0,1$.

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 $f=-1,0,1$.

The zero $-1$ has odd multiplicity therefore the graph of $f$ crosses the $X\text{-axis}$.

The zero $0$ has odd multiplicity therefore the graph of $f$ crosses the $X\text{-axis}$.

The zero $1$ has odd multiplicity therefore the graph of $f$ crosses the $X\text{-axis}$.

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

$y_1=x-x^3$

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 $=\left(0.6,0.4\right)$.

Using the graph minimum turning point $=\left(-0.6,-0.4\right)$.

Step 6: Redraw the graph.

The graph passes through $\left(0,0\right)\left(-1,0\right)$ and $\left(1,0\right)$.

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 $\left[-0.6,0.6\right]$.

Based on the graph, $f$ is decreasing $\left(\infty ,-0.6\right]$ and $\left[0.6,-\infty \right)$.