Answer
$ F(x)=-3(x-5)^{2}-7$
Work Step by Step
$ F(x)=a(x-h)^{2}+k, \quad$ where the vertex is $(h,k)$
has the same shape as $y=ax^{2}$, as
the graph is obtained by translating (shifting left/right and up/down)
To keep the same shape, a remains the same.
The graph of F(x) is a parabola.
If $a < 0,$ the parabola opens down, the vertex is its maximum point.
If $a > 0$, the parabola opens up, the vertex is its minimum point.
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To have the same shape as f(x) or g(x), a is either $-3$ or $3.$
Since our function has a maximum, its graph opens down, so $a=-3.$
The vertex$,\qquad (h,k)=(5,-7)$
so
$ F(x)=a(x-h)^{2}+k$
$ F(x)=-3(x-5)^{2}-7$
