Chapter 8.4, Problem 54E

To determine

To graph function $g(x)=-f(x),\text{where}f(x)=-2{\left(x-4\right)}^2-8$

Explanation of Solution

Given information :

Given functions $g(x)=-f(x),\text{where}f(x)=-2{\left(x-4\right)}^2-8$

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 $f(x)=-2{\left(x-4\right)}^2-8$ with $f(x)=m{(x-h)}^2+k$ , we get $h=4,k=-8$

Thus, the vertex of $f(x)$ is $(4,-8)$ and the axis of symmetry $x=4$

Given function $g(x)=-f(x)$ is in the form of $y=-f(x)$ , where $k=-1$ .

So, the graph of $g(x)$ is a reflection of the graph of $f(x)$ along the $x-$ axis.

To graph $g(x)$ , multiply $-$ with $y-$ coordinates of the points on the graph of $f(x)$ .

Thus, the vertex of $g(x)$ is $(4,8)$ and axis of symmetry remains same $x=4$ .

Graph: The graph of $g(x)=-f(x)$ is

  

Interpretation: From above graph, we observe that vertex of $g(x)$ is $(4,8)$ and axis of symmetry remains same $x=4$ .