Use the information provided to write the vertex form equation of the parabola:
Focus: (-8, -6)
The vertex form equation of a parabola with focus (h, k) is given by:
(x-h)^2 = 4p(y-k)
In this case, the focus is (-8, -6), so we have h = -8 and k = -6. Since the vertex form equation requires the coordinate of the vertex, we first need to find the vertex using the formula for the vertex of a parabola with focus (h, k) which is (h, k-p).
We know that the focal length is the distance between the focus and the vertex and is given by the equation p = 1/4a, where a is the distance between the vertex and the directrix (a is also the distance between the vertex and the focus in the case of a parabola that opens upward or downward).
Since the focus is (-8, -6), and the parabola opens upwards or downwards, its directrix will be a horizontal line p units below the focus, so the directrix is y = -6 + 1/4p = -6 -1 = -7. This means that the vertex will be (h, k-p) = (-8, -7).
Now that we have the coordinates of the vertex, we can plug them into the vertex form equation:
(x+8)^2 = 4p(y+7)
So the vertex form of the equation of the parabola is:
(x+8)^2 = -28(y+7)