Precalculus / Unit 9: Conic Sections and Parametric Equations
9.2 Parabolas
Notes
A parabola is the set of points equally distant from a focus and a directrix.
- $x^2 = 4py$: vertex $(0, 0)$, focus $(0, p)$, directrix $y = -p$. Opens up if $p > 0$, down if $p < 0$.
- $y^2 = 4px$: focus $(p, 0)$, directrix $x = -p$. Opens right if $p > 0$, left if $p < 0$.
- With vertex $(h, k)$: $(x - h)^2 = 4p(y - k)$ or $(y - k)^2 = 4p(x - h)$.
- Example: $x^2 = 8y$ has $4p = 8$, so $p = 2$, focus $(0, 2)$ and directrix $y = -2$.
Practice
What is the focus of $x^2 = 8y$?
$4p = 8$ so $p = 2$, and the focus is $(0, p)$.
What is the directrix of $y^2 = 12x$?
$4p = 12$ so $p = 3$, and the directrix is $x = -p$.
Describe the parabola $(x - 1)^2 = -8(y + 2)$.
The vertex is $(h, k) = (1, -2)$, and $4p = -8$ is negative, so it opens down.
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