Final answer:
To find the equation of the parabola, we use the standard form (x-h)^2 = 4p(y-k), where (h,k) is the vertex and p is the distance to the directrix.
Given the vertex (5, -4) and directrix x=2, the final equation is (x-5)^2 = 12(y+4).
Step-by-step explanation:
We are asked to write the equation of a parabola given its vertex and directrix.
The vertex is at (5, -4) and the directrix is the vertical line x = 2.
The general form of the equation for a parabola with a vertical axis of symmetry is (x-h)^2 = 4p(y-k), where (h, k) is the vertex, and p is the distance from the vertex to the focus or to the directrix.
Since the directrix is x = 2 and the vertex's x-coordinate is 5, the distance ;p' is 5 - 2 = 3.
Because the vertex (5, -4) has an x-coordinate greater than that of the directrix, the parabola opens to the right, so p is positive.
Therefore, we now have the equation (x-5)^2 = 4*3(y+4).
Simplifying this, we get the parabola's equation in standard form:
=> (x-5)^2 = 12(y+4)