We have a quadratic function with unknown parameters b and c and we have to define them in order for the function to have its vertex at (6,-2).
The formula for the x-coordinate of the vertex is:
[tex]x_v=-\frac{b}{2a}[/tex]
As xv = 6 and a=20, we can find b as:
[tex]\begin{gathered} x_v=-\frac{b}{2a} \\ 6=\frac{-b}{2\cdot20} \\ 6\cdot40=-b \\ 240=-b \\ b=-240 \end{gathered}[/tex]
As the y-coordinate of the vertex is:
[tex]y_v=f(x_v)[/tex]
then we can replace with the known information and calculate c as:
[tex]\begin{gathered} y_v=20x^2_v-240x_v+c=-2 \\ 20(6)^2-240\cdot6+c=-2 \\ 20\cdot36-1440+c=-2 \\ 720-1440+c=-2 \\ c=-2+1440-720 \\ c=718 \end{gathered}[/tex]
Then, the quadratic equation can be written as:
[tex]y=20x^2-240x+718[/tex]
We can check it with a graph:
