To determine the equation of the quadratic equation, which is a parabola, we substitute the coordinates of the vertex of the parabola (h,k) into the general equation.
[tex]y=a(x-h)^2+k[/tex]
From the given, notice that the only value of f(x) that does not repeat is 8. This means that the vertex is at (7,8).
[tex](h,k)=(7,8)[/tex]
Thus, we only need to obtain the value of a.
Substitute the coordinate of a point (x,y) into the equation and the vertex as well. In this case, let us use the first given point, (5,-4).
[tex]\begin{gathered} y=a(x-h)^2+k \\ \\ (5,-4)\rightarrow-4=a(5-7)^2+8 \end{gathered}[/tex]
Simplify the obtained equation.
[tex]\begin{gathered} (5,-4) \\ -4=a\mleft(5-7\mright)^2+8 \\ -4=a\mleft(-2\mright)^2+8 \\ -4=a(4)+8 \\ -4-8=4a \\ -12=4a \\ a=\frac{-12}{4} \\ a=-3 \end{gathered}[/tex]
Now that we have the value of a, substitute the coordinates of the obtained vertex and the value of a into the equation of the quadratic equation.
[tex]\begin{gathered} y=a(x-h)^2+k \\ y=-3(x-7)^2+8 \end{gathered}[/tex]
To check, the graph of the given function is as follows:
Therefore, the equation of the quadratic equation is y=-3(x-7)²+8.
