We can start with the equation of the parabola in vertex form, as we know the coordinates of the vector:
[tex]y=a(x-h)^2+k[/tex]
The vertex is (h,k) = (-5,2), so we get:
[tex]y=a(x-(-5))^2+2=a(x+5)^2+2[/tex]
We can use another known point, as (-6,4) to calculate the parameter a:
[tex]\begin{gathered} y=a(x+5)^2+2 \\ 4=a(-6+5)^2+2 \\ 4=a\cdot(-1)^2+2 \\ 4=a+2 \\ a=4-2 \\ a=2 \end{gathered}[/tex]
Now that we have no more unknown parameters, we can expland the equation to the standard form:
[tex]\begin{gathered} y=2(x+5)^2+2=2(x^2+10x+25)+2=2x^2+20x+50+2 \\ y=2x^2+20x+52 \end{gathered}[/tex]
The equation is y=2x^2+20x+52.
