Find the equation of the parabola with focus at (0, 2) and directrix y = -2.

Find the equation of the parabola with focus at (0, 2) and directrix y = -2.

Step 1: Identify the focus and directrix. The focus is at (0, 2) and the directrix is the line y = -2.
Step 2: Find the vertex of the parabola. The vertex is the midpoint between the focus and the directrix. The y-coordinate of the vertex is the average of 2 (focus) and -2 (directrix), which is (2 + (-2)) / 2 = 0. The x-coordinate is 0, so the vertex is at (0, 0).
Step 3: Determine the value of p. The distance from the vertex to the focus is p. Since the vertex is at (0, 0) and the focus is at (0, 2), p = 2.
Step 4: Write the equation of the parabola. The standard form of a parabola that opens upwards is y^2 = 4px. Here, p = 2, so 4p = 8.
Step 5: Substitute p into the equation. The equation becomes y^2 = 8x.

Parabola Properties – Understanding the relationship between the focus, directrix, vertex, and the standard form of a parabola.
Vertex Calculation – Determining the vertex of the parabola based on the given focus and directrix.
Standard Form of a Parabola – Using the correct standard form of a parabola to derive the equation based on the value of p.