Question

What is the equation of a parabola with a focus (-2,4) and directrix y = 0?

Answer 1

The general form of a parabola  when using the focus and directrix is:
(x - h)² = 4p(y - k) where (h, k) is the vertex of the parabola and 'p' is distance between vertex and the focus.  We use this form due to the fact we can see the parabola will open up based on the directrix being below the focus.  Remember that the parabola will hug the focus and run away from the directrix.  The formula would be slightly different if the parabola was opening either left or right.

Given a focus of (-2,4) and a directrix of y = 0, we can assume the vertex of the parabola is exactly half way in between the focus and the directrix.  The focus and vertex with be stacked one above the other, therefore the vertex will be (-2, 2) and the value of 'p' will be 2.  We can now write the equation of the parabola:
(x + 2)² = 4(2)(y - 2)
(x + 2)² = 8(y - 2)  Now you can solve this equation for y if you prefer solving for 'y' in terms of 'x'