In geometry you met the parabola as the set of points equally far from a focus and a directrix. Here we turn that definition into standard forms that open in any of four directions, so you can read the focus and directrix straight from an equation.
Idea clave
With vertex $(h, k)$: $$(x - h)^2 = 4p(y - k) \text{ opens up } (p > 0) \text{ or down } (p < 0);$$ $$(y - k)^2 = 4p(x - h) \text{ opens right } (p > 0) \text{ or left } (p < 0).$$ The focus is $|p|$ units from the vertex, inside the curve; the directrix is $|p|$ units from the vertex on the other side.
Por qué funciona
Take focus $(0, p)$ and directrix $y = -p$. A point $(x, y)$ is equally far from both when $\sqrt{x^2 + (y - p)^2} = |y + p|$. Square both sides: $$x^2 + y^2 - 2py + p^2 = y^2 + 2py + p^2,$$ which simplifies to $x^2 = 4py$. The vertex, halfway between focus and directrix, is the origin, and shifting it to $(h, k)$ gives the general form.
Worked examples
Ejemplo resuelto: Vertex at the origin
Find the focus and directrix of $x^2 = 12y$.
- Match $x^2 = 4py$: $4p = 12$, so $p = 3$. It opens up.
- Focus $3$ units above the vertex: $(0, 3)$. Directrix $3$ units below: $y = -3$.
Respuesta: Focus $(0, 3)$; directrix $y = -3$
Ejemplo resuelto: Shifted, opening left
Find the vertex, focus and directrix of $(y - 2)^2 = -8(x + 1)$.
- Vertex $(-1, 2)$. $4p = -8$, so $p = -2$: it opens left.
- Focus: $2$ units left of the vertex, $(-3, 2)$. Directrix: $2$ units right, $x = 1$.
Respuesta: Vertex $(-1, 2)$; focus $(-3, 2)$; directrix $x = 1$
Ejemplo resuelto: From focus and directrix
Write the parabola with focus $(2, 5)$ and directrix $y = 1$.
- The vertex is halfway between: $(2, 3)$.
- The focus is $2$ units above the vertex, so $p = 2$ and the parabola opens up.
- $(x - 2)^2 = 4 \cdot 2\,(y - 3)$.
Respuesta: $(x - 2)^2 = 8(y - 3)$
Error común
Taking $p$ as the distance from the focus to the directrix. That distance is $2|p|$; $p$ is only the distance from the vertex to the focus. In the last example the focus and directrix are $4$ apart, but $p = 2$. Check that your vertex sits exactly halfway between them.