Since the directrix is y=-4, use the equation of a parabola that opens up or down.
[tex]\mleft(x-h\mright)^2=4p(y-k)[/tex]where vertex is at (h,k) and focus is at (h,k+p)
The vertex is halfway between the directrix and focus. Find the y-coordinate of the vertex using the following
[tex](-8,\frac{-1-4}{2})=(-8,-\frac{5}{2})[/tex]Find the distance from the focus to the vertex.
The distance from the focus to the vertex and from the vertex to the directrix is |p|. Subtract the y-coordinate of the vertex from the y-coordinate of the focus to find p.
[tex]p=-1+\frac{5}{2}=-\frac{2}{2}+\frac{5}{2}=\frac{-2+5}{2}=\frac{3}{2}[/tex]Substitute in the known values for the variables into the equation
[tex](x-h)^2=4p(y-k)[/tex][tex](x+8)^2=4\cdot\frac{3}{2}\cdot(y+\frac{5}{2})[/tex]Simplify
[tex](x+8)^2=6\cdot(y+\frac{5}{2})[/tex]in y=ax^2+bx+c form:
[tex]\begin{gathered} x^2+16x+64=6y+15 \\ x^2+16x+64-15=6y \\ x^2+16x+49=6y \\ y=\frac{1}{6}x^2+\frac{16}{6}x+\frac{49}{6} \end{gathered}[/tex]