The vertex form of the equation of a parabola is given to be:
[tex]y=a(x-h)^2+k[/tex]where
[tex](h,k)\Rightarrow\text{ Vertex coordinates}[/tex]The question provides the vertex coordinates to be:
[tex](h,k)=(7,4)[/tex]Substituting this value into the vertex form equation, we get:
[tex]y=a(x-7)^2+4[/tex]We are given the coordinates of a point on the parabola to be:
[tex](x,y)=(5,16)[/tex]We can use this point to get the value of a in the vertex formula by substituting into the equation:
[tex]\begin{gathered} 16=a(5-7)+4 \\ 16-4=a(-2)^2 \\ 12=4a \\ a=\frac{12}{4} \\ a=3 \end{gathered}[/tex]Therefore, the vertex form of the equation of the parabola is given to be:
[tex]y=3\mleft(x-7\mright)^2+4[/tex]