# Parabolas part 1

A Parabola is the set of all points (locus) equal distance from a fixed line (Directrix) and a fixed point (Focus).  The Vertex is equal distance from the directrix and the focus. The equation of a parabola is derived from the distance formula. The standard equation of a parabola with vertex at the origin and directrix horizontal is

$X^2 = 4pY$

solving for Y

$y=\frac{x^2}{4p}$

$p>0$  the curve opens up

$p<0$  the curve opens down

A parabola has the useful feature that a line parallel to the symmetry axis striking the parabola at any point is reflected onto the focus.

recasting our most common parabola

$y=x^2$

$y=\frac{x^2}{4p}$

becomes

$y=\frac{x^2}{4*0.25}$

the focus is at (0, 0.25)

the directrix is at Y=-0.25