12.6 Parabolas: from definition to equation

We have talked about the geometric definition of a parabola: “all the points in a plane that are the same distance from a given point (the focus ) that they are from a given line (the directrix ).” And we have talked about the general equations for a parabola:

Vertical parabola: $y=a(x–h{)}^2+k$

Horizontal parabola: $x=a(y–k{)}^2+h$

What we haven’t done is connect these two things—the definition of a parabola, and the equation for a parabola. We’re going to do it the exact same way we did it for a circle—start with the geometric definition, and turn it into an equation.

In the drawing above, I show a parabola whose focus is the origin (0,0) and directrix is the line $y=-4$ . On the parabola is a point ( $x$ , $y$ ) which represents any point on the parabola.