Parabola Opening Downward

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

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

$$\sqrt{x^2 + (y+a)^2} = a - y \quad (y \leq 0)$$

Squaring:

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

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

The curve lies in $y \leq 0$.