use the equation of a parabola and the distance formula to derive the standard equation of a parabola whoch has the origin for its vertex, the x-axis as the

Let (x, y) be some point on the parabola.......since the distance from the point to the focus  = the distance of the point from the directrix, we have that

√[ (x - p)^2 + (y - 0)^2 ]   =  √ [ (x + p)^2 + (y - y)^2 ]    square both sides and simplify.....

x^2 - 2px + p^2 + y^2  =  x^2 + 2px + p^2

y^2  = 4px