Answer:
f(x) = - (x - 5)² - 5
Explanation:
The vertex form of the quadratic function is:
[tex]\begin{gathered} f\mleft(x\mright)=a\mleft(x-h\mright)^2+k \\ \text{where (h,k) is the vertex} \end{gathered}[/tex]Given that it has a vertex at (5, -5),
(h,k)=(5,-5)
[tex]\begin{gathered} f(x)=a(x-5)^2+(-5) \\ f(x)=a(x-5)^2-5 \end{gathered}[/tex]Since it passes through the point (7,-9)
When x=7, f(x)=-9
[tex]\begin{gathered} -9=a(7-5)^2-5 \\ -9=a\times2^2-5 \\ -9+5=4a \\ -4=4a \\ a=\frac{-4}{4}=-1 \end{gathered}[/tex]Therefore, the function is:
f(x) = - (x - 5)² - 5