Question

Identify the equations of parabolas that have the directrix x = -4.

Answer 1

Suppose that the point A(x,y) belongs to this parabola. The defining point of the parabola is that the distance between a point F and the directrix. Suppose that the chosen point F is at (a,b). Then, the distance from A to the directrix is |b+4| while the distance from A to the focus is given by the pythagorean theorem:
$d= \sqrt{(x-a)^2+(y-b)^2}$.
We have that these two have to be equal. Squaring both sides we get:
$b^2+8b+16=(x-a)^2+(y-b)^2$.
This is the equation that describes all equations of parabolas with that directrix; you just need to choose the focus and substituting a and b will yield the equation.