The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?

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Question: The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?

Options:

Correct Answer: x^2 = -12y

Solution:

The distance from the vertex to the directrix is 3, so the equation is x^2 = -12y.

The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?

The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?

Step 1: Identify the vertex of the parabola, which is given as (0, 0).
Step 2: Identify the directrix, which is given as y = -3.
Step 3: Calculate the distance from the vertex to the directrix. The vertex is at y = 0 and the directrix is at y = -3, so the distance is 0 - (-3) = 3.
Step 4: Since the directrix is below the vertex, the parabola opens downwards.
Step 5: Use the formula for the equation of a parabola that opens downwards: x^2 = -4py, where p is the distance from the vertex to the directrix.
Step 6: Substitute p = 3 into the formula: x^2 = -4(3)y.
Step 7: Simplify the equation: x^2 = -12y.

Parabola Properties – Understanding the relationship between the vertex, directrix, and the focus of a parabola.
Standard Form of Parabola – Knowledge of the standard equation of a parabola and how to derive it from given parameters.