Parabola Opening Upward

Vertex at origin, axis along positive y-axis. Focus at (0, a), directrix y = −a.

Place the focus at $S(0, a)$ and directrix $y = -a$. For $P(x, y)$ on the parabola, $PS = PM$:

$$\sqrt{x^2 + (y-a)^2} = y + a$$

Squaring:

$$x^2 + (y-a)^2 = (y+a)^2$$ $$x^2 + y^2 - 2ay + a^2 = y^2 + 2ay + a^2$$ $$x^2 = 4ay$$

Summary of elements for $x^2 = 4ay$:

All four standard parabolas ($y^2 = \pm 4ax$, $x^2 = \pm 4ay$) are congruent — they differ only in orientation. Any one is obtained from another by a 90° rotation.