Chapter 4.1, Problem 76AYU

To determine

To analyze: The graph of a polynomial function.

Answer to Problem 76AYU

$\text{The graph of a polynomial function}f\left(x\right)=\left(x-1\right){\left(x+3\right)}^2\text{is analyzed}$.

Explanation of Solution

Given:

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

Square the binomial sum

$f\left(x\right)=\left(x-1\right)\left(x^2+6x+9\right)$

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

$f\left(x\right)=\left(x-1\right)\left(x^2+6x+9\right)$

Multiply

$f\left(x\right)=x^3+5x^2+3x-9$

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

The $y\text{-intercept }=f\left(0\right)=-9$

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

$f\left(x\right)=\left(x-1\right){\left(x+3\right)}^2=0$

$x=1 \text{or} \left(x+3\right)=0$

$x=1\text{or} x=-3$

The $x\text{-intercept }=x=1, -3$.

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 $f=1,-3$.

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

The zero $-3$ has even multiplicity therefore the graph of $f$ touches the $x\text{-axis}$.

Step 4: Using the graphing utility to graph the function

$y_1=\left(x-1\right){\left(x+3\right)}^2$

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 $=\left(-3,0\right)$.

Using the graph minimum turning point $=\left(-0.5,-9.5\right)$.

Step 6: Redraw the graph.

The graph passes through $\left(1,0\right)$ and $\left(0,-9\right)$.

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 $\left(-\infty ,-3\right]$ and $\left[0,\infty \right)$.

Based on the graph, $f$ is decreasing $\left[-3,0\right]$.