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

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:
[tex]d= \sqrt{(x-a)^2+(y-b)^2} [/tex].
We have that these two have to be equal. Squaring both sides we get:
[tex]b^2+8b+16=(x-a)^2+(y-b)^2[/tex].
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.