If the directrix of a parabola is given by the equation y = -p, what is the equation of the parabola?

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Q: If the directrix of a parabola is given by the equation y = -p, what is the equation of the parabola?

Solution: The equation of a parabola with directrix y = -p is y^2 = -4px.

Steps: 6

Step 1: Understand what a directrix is. A directrix is a line used to define a parabola. For this problem, the directrix is given as y = -p.
Step 2: Identify the orientation of the parabola. Since the directrix is a horizontal line (y = -p), the parabola opens either upwards or downwards.
Step 3: Determine the vertex of the parabola. The vertex is located at the point (0, 0) when the directrix is y = -p.
Step 4: Use the formula for the equation of a parabola. The standard form of a parabola that opens downwards is y^2 = -4px.
Step 5: Identify the value of p. In this case, p is the distance from the vertex to the directrix, which is p units.
Step 6: Substitute p into the equation. Since the directrix is y = -p, the equation of the parabola becomes y^2 = -4px.