Parabola by focus and directix

Problem:


use given information to write the equation of each parabola

Solution:

The distances (therefore, the square of them) between a point to the vertex is the same as the distance of the point from the directix.

That works for every point, so let equate the distance squares

$ (x + 1)^2 + (y - 1)^2 = (y - 4)^2 $

Simplify, we get $ y = \frac{-1}{6}(x^2 + 2x - 14) $, that's the parabola we are looking for, it opens downwards as follow: