We have been given that the graph of [tex]F(x)[/tex] has the same shape as the graph of [tex]G(x)=x^2[/tex], but it is flipped over the x-axis and shifted down 1 unit. We are asked to find the equation of [tex]F(x)[/tex].
We will use transformation rules to solve our given problem.
The rule of reflection of a graph about x-axis is [tex]f(x)\Rightarrow -f(x)[/tex].
Let us find [tex]-G(x)[/tex] as:
[tex]-G(x)=-(x^2)=-x^2[/tex]
Now we will shift our graph 1 unit down.
[tex]f(x)-a\Rightarrow\text{Graph shifted downwards by a units}[/tex], where a is a positive number.
This is same as subtracting 1 from [tex]-x^2[/tex] as:
[tex]F(x)=-x^2-1[/tex]
Therefore, our required function would be [tex]F(x)=-x^2-1[/tex].
