Chapter 4.1, Problem 70AYU

To determine

To analyze: the given polynomial function

Answer to Problem 70AYU

  

  

Explanation of Solution

Given:

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

Formula Used:

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$

Calculation:

Given: $f\left(x\right)=x{\left(x+2\right)}^2$

Step1: Determine the end behavior of the graph of the function

Expand the polynomial to write it in the form

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$

  $\begin{array}{l}f\left(x\right)=x{\left(x+2\right)}^2\\ =x\left(x^2+4x+4\right) Multiply\\ =x^3+4x^2+4x\end{array}$

The polynomial function $f$ is of degree $3$ .The graph of $f$ behaves like $y=x^3$ for large values of $\left|x\right|$ .

Step2: find the $x - and y-$ intercepts of the graph of the function

The $y-intercept$ is $f\left(0\right)=0$ .To find the $x-intercepts$ , we solve $f\left(x\right)=0$ .

  $\begin{array}{l}f\left(x\right)=0\\ x{\left(x+2\right)}^2=0\\ x=0 or {\left(x+2\right)}^2=0\\ x=0 or x+2=0\\ x=-2\end{array}$

They $x-intercepts$ are $-2 and 0$ .

Steps3: Determine the zeros of the function and their multiplicity.Use this information whether the graph crosses or touches the $x-axis$ ateach $x-intercept$ .

The zeros of $f$ are $-2 and 0$ . The zero $-2$ is a zero of multiplicity $2$ , so the graph of $f$ touches the $x-axis$ at $y=-2$ . The zero $0$ is a zero of multiplicity $1$ , so the graph of $f$ crosses the $x-axis$ at $x=0$ .

Step4: Determine the maximum number of turning points on the graph of the function. Because the polynomial function is of degree $3$ (step1), the graph of the function will have at most $3-1=2$ points.

Steps5: Determine the behavior of the graph of $f$ neareach $x-intercept$

The two $x-intercepts$ are $-2and0$ .

  $\begin{array}{l}Near -2 f\left(x\right)=x{\left(x+2\right)}^2\\ \approx -2{\left(x+2\right)}^2 \text{A parabola that opens down}\\ \text{Near} 0: f\left(x\right)=x{\left(x+2\right)}^2\\ \approx x{\left(0+2\right)}^2\\ \text{A line with slope 4}\\ =4x\end{array}$

Step6: Put all the information from step 1through5together to obtain the graph of $f$ Figure $1\left(a\right)$ illustrates the information obtained from Steps 1 through 5. We evaluate $f$ at $-3,-1,1$ to help establish the scale on the $y-axis$ . The graph of $f$ is given in Figure $1\left(b\right)$ .

Figure1

  

  

Conclusion:

Thus, the graph of $f$ is obtained which illustrates the information obtained from Steps 1 through 5.