Chapter 2.2, Problem 99E

To determine

To identify any symmetry of function with respect to x- axis, y -axis or origin.

the numbers of $x$ intercepts.

Draw graph of function.

Answer to Problem 99E

Function $f\left(x\right)=x^2\left(x+5\right)$ , has not any symmetry.

Function $f\left(x\right)=x^2\left(x+5\right)$ has two $x$ intercepts.

Explanation of Solution

Given: Function is

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

Calculation:

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

i.e.

  $y=x^2\left(x+5\right)$

Function is symmetric about x axisif on replacing $x$ by $-x$ , function equation remain same.

On replacing, we get

  $\begin{array}{l}y={\left(-x\right)}^2\left(-x+5\right)\\ =-x^2\left(x-5\right)\end{array}$

Hence graph of function is not symmetric with respect to x axis.

Now function is symmetric about y axis if on replacing $y$ by $-y$ , functionequation remain same.

On replacing, we get

  $-y=x^2\left(x+5\right)$

Hence graph of function is not symmetric with respect to y axis.

Also function is symmetric about origin if on replacing $y$ by $-y$ and $x$ by $-x$ , function equation remain same.

On replacing, we get

  $\begin{array}{l}-y=-x^2\left(x-5\right)\\ y=x^2\left(x-5\right)\end{array}$

Hence graph of function is not symmetric with respect to origin.

For $x$ intercept, put

  $\begin{array}{l}f\left(x\right)=0\\ x^2\left(x+5\right)=0\\ \Rightarrow x=0,-5\end{array}$

Hence $x$ intercept are $\left(0,0\right)$ and $\left(-5,0\right)$ .

i.e. function has two $x$ intercepts.

Graph of function is