Write the equation of a parabola with a vertex at (-5, 2) and a directrix y = -1.

To find the equation of a parabola with a given vertex and directrix, we can use the following steps:

Step 1: Identify the axis of symmetry.
The axis of symmetry is a vertical line passing through the vertex. In this case, the vertex is (-5, 2), so the axis of symmetry is given by x = -5.

Step 2: Find the distance 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 (since the focus is equidistant from the vertex and the directrix). In this case, the vertex is at y = 2, and the directrix is y = -1. So, the distance is 2 units.

Step 3: Determine the coordinates of the focus.
Since the focus is equidistant from the vertex and the directrix, and the directrix is a horizontal line, the focus will have the same x-coordinate as the vertex and a y-coordinate that is 2 units greater. So, the coordinates of the focus are (-5, 4).

Step 4: Determine the equation of the parabola.
The general equation of a parabola with its vertex at (h, k) and its focus at (h, k + p) is given by:

(y - k)^2 = 4p(x - h)

In this case, the vertex is (-5, 2), and the focus is (-5, 4). Thus, the equation of the parabola is:

(y - 2)^2 = 4(2)(x + 5)

Simplifying this equation gives:

(y - 2)^2 = 8(x + 5)

Therefore, the equation of the parabola with a vertex at (-5, 2) and a directrix y = -1 is (y - 2)^2 = 8(x + 5).