Question

Select the correct answer.Given the focus and directrix shown on the graph, what is the vertex form of the equation of the parabola?

Answer 1

The directrix of the parabola is a vertical line at x = -4 and the focus is at the right of the directrix at (1, 3). This means the parabola is horizontal and it opens to the right.

The equation of a vertical parabola is:

$x=\frac{1}{4p}(y-k)^2+h$

Where (h, k) are the vertex coordinates, and the vertex is at the midpoint between the focus and the directrix.

The y-coordinate of the vertex is k = 3 and the x-coordinate is:

$h=\frac{-4+1}{2}=-\frac{3}{2}$

p is the distance from the vertex to the focus:

$p=-\frac{3}{2}+4=\frac{5}{2}$

Thus:

$4p=4\cdot\frac{5}{2}=10$

Now we can substitute all the values into the equation:

$x=\frac{1}{10}(y-3)^2-\frac{3}{2}$

This is the required vertex form of the parabola.