Chapter 8.4, Problem 43E

To determine

To graph function $g(x)=-\frac{1}{3}{(x+3)}^2-2$ and compare with the graph of $f(x)=x^2$

Answer to Problem 43E

The graph of $g(x)$ is a horizontal translation of $3$ units left, a vertical shrink by a factor of $\frac{1}{3}\text{units}$ , a vertical translation of $2\text{units}$ down and reflection along the $x-$ axis of the graph of $f(x)=x^2$ .

Explanation of Solution

Given information :

Given functions $g(x)=-\frac{1}{3}{(x+3)}^2-2,andf(x)=x^2$

As we know that the vertex of the graph of $f(x)=m{(x-h)}^2+k\left(\text{where}m\text{is}\text{constant}\right)$ is $(h,k)$ and the axis of symmetry is $x=h$ .

Now compare given function $f(x)=x^2$ with $f(x)=m{(x-h)}^2+k$ , we get $h=0,k=0$

Thus, the vertex of $f(x)$ is $(0,0)$ and the axis of symmetry $x=0$

Given function $g(x)=-\frac{1}{3}f(x+3)-2,\text{where}\text{f(x)}\text{=}{\text{x}}^2$ is in the form of $y=af(x-h)+k$ , where $h=-3,a=\frac{1}{3}<1andk=-2$ .

So, the graph of $g(x)$ is a horizontal translation of $3$ units left, a vertical shrink by a factor of $\frac{1}{3}\text{units}$ , a vertical translation of $2\text{units}$ down and reflection along the $x-$ axis of the graph of $f(x)=x^2$ .

Graph: The graph of $g(x)=-\frac{1}{3}{(x+3)}^2-2,andf(x)=x^2$ is

  

Interpretation: From above graph, we observe that the graph of $g(x)$ is a horizontal translation of $3$ units left, a vertical shrink by a factor of $\frac{1}{3}\text{units}$ , a vertical translation of $2\text{units}$ down and reflection along the $x-$ axis of the graph of $f(x)=x^2$ .