Parabola Opening Left

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

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

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

Squaring:

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

Summary of elements for $y^2 = -4ax$:

The curve lies entirely in $x \leq 0$ since $y^2 \geq 0$ requires $-4ax \geq 0 \Rightarrow x \leq 0$.