Final answer:
The equation for the parabola with a vertex at (-5,1) and a directrix at y = -3 is y = 16(x+5)^2 + 1 or y = -16(x+5)^2 + 1.
Step-by-step explanation:
The equation for a parabola with a vertex at (-5,1) and a directrix at y = -3 can be written in the form y = a(x-h)^2 + k, where (h,k) represents the vertex. In this case, the vertex is (-5,1), so the equation becomes y = a(x+5)^2 + 1. To find the value of a, we need to use the distance formula between the vertex and the directrix. The distance between the vertex and the directrix is the same as the distance between the vertex and the focus, which is equal to |a|/4. In this case, the distance is |-3-1| = 4, so |a|/4 = 4. Solving for a, we find that a = ±16. Therefore, the equation of the parabola is y = 16(x+5)^2 + 1 or y = -16(x+5)^2 + 1.