An alternative way to define a parabola is as a locus of points.
Focus and directrix
The locus defining a parabola depends on a focus and a directrix.
The focus is a point. For a standard parabola, the focus is located on the x axis a distance
$a$ from the origin, that is at the
point (a, 0).
$a$ is the constant in the parabola equation
$y^2 = 4 a x$
The directrix is a line. For a standard parabola it is a line perpendicular to the x axis passing through (-a, 0), that is
$x = -a$
The vertex of the parabola is the turning point. It is always halfway between the focus and directrix, which is always at the origin for a standard parabola.
A parabola is the locus of all points which are an equal distance from the focus and directrix. The diagram below illustrates this:
FP = PD
Press the play button to run it. As the purple dot traces the curve, notice that the two purple lines are alway equal length.