Question

Analyze each polynomial function.

$f\left(x\right)=2x^4+12x^3-8x^2-48x$

Short answer

The graph of the function is:

Step-by-step solution

Step 1. Given Information

The given function is:

$f\left(x\right)=2x^4+12x^3-8x^2-48x$

Step 2. The polynomial function

$f\left(x\right)=2x^4+12x^3-8x^2-48x=2x(x+6)(x+2)(x-2)$

The polynomial function is of degree $4.$The graph of $f$behaves like $y=2x^4.$

Find x intercept

The $y$-intercept is localid="1648640322447" $f\left(0\right)=0.$$x-$intercepts:

$2x(x+6)(x+2)(x-2)=0$

$$\begin{gathered} 2x=0 \\ x=0 \end{gathered}$$

or

$$\begin{gathered} x+6=0 \\ x=-6 \end{gathered}$$

or

$$\begin{gathered} x+2=0 \\ x=-2 \end{gathered}$$

or

$$\begin{gathered} x-2=0 \\ x=2 \end{gathered}$$

The zero $0$is a zero of multiplicity $1,$so the graph of $f$crosses the $x$-axis at $x=0.$

The zero $-6$is a zero of multiplicity $1,$so the graph of $f$crosses the $x$-axis at $x=-6.$

The zero $-2$is a zero of multiplicity $1,$so the graph of $f$crosses the $x$-axis at $x=-2.$

The zero $2$is a zero of multiplicity $1,$ so the graph of $f$crosses the $x$-axis at $x=2.$

Since the degree of the function is $4,$ the graph of the function has $4-1=3$ turning points.

Step 4. Find a line with a slope  

Near $0:$

$$\begin{gathered} f\left(x\right)=2x(x+6)(x+2)(x-2) \\ =2x(0+6)(0+2)(0-2) \\ =-48x \end{gathered}$$

a line with slope -48

Near $-6:$

$$\begin{gathered} f\left(x\right)=2x(x+6)(x+2)(x-2) \\ =2·(-6)(x+6)(-6+2)(-6-2) \\ =576x+3456 \end{gathered}$$

a line with slope $576.$

Near $3:$

$$\begin{gathered} f\left(x\right)=2x(x+6)(x+2)(x-2) \\ =2·(-2)(-2+6)(x+2)(-2-2) \\ =64x+128 \end{gathered}$$

a line with slope $64.$

Near$2:$

$$\begin{gathered} f\left(x\right)=2x(x+6)(x+2)(x-2) \\ =2·2(2+6)(2+2)(x-2) \\ =128x-256 \end{gathered}$$

a line with slope$128.$

Step 5. Interval of increase and decrease. 

The function$f\left(x\right)=2x^4+12x^3-8x^2-48x=2x(x+6)(x+2)(x-2)$

increases in the interval$\left(-\infty ,-6\right)\cup \left(-2,0\right)\cup \left(2,\infty \right)$and interval of decrease is$\left(-6,-2\right)\cup \left(0,2\right)$

Step 6.The graph

The graph of the given function is: