Final answer:
To write the equation of a parabola with a focus at (-4, 8) and a directrix at y = 4, find the vertex halfway between the focus and directrix, calculate the distance p, and use the general form of the equation for a vertical parabola: (x - h)² = 4p(y - k). The resulting equation is (x + 4)² = 8(y - 6).
Step-by-step explanation:
The question asks for the equation of a parabola given a focus at (-4, 8) and a directrix at y = 4. The general equation for a parabola with a vertical axis of symmetry is (x - h)² = 4p(y - k), where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus or directrix.
To find the equation, we first determine the vertex. Since the directrix is y = 4 and the focus is at y = 8, the vertex is halfway between the two, which means it's at y = 6. The x-coordinate of the vertex is the same as the x-coordinate of the focus, so the vertex is (-4, 6). The value of p is then the distance from the vertex to the focus, which is 2 (from y = 6 to y = 8).
The equation of the parabola is therefore (x + 4)² = 4*2*(y - 6). Simplifying this gives us (x + 4)² = 8(y - 6), which can further be expanded or rearranged into any required form.