Parabolas: Focus, Directrix, and Standard Forms

By the end of this lesson, you will be able to find the focus, directrix, vertex, and axis of symmetry for any parabola; convert between standard form and focus-directrix form; and apply the reflective property of parabolas.

The Geometric Definition

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

This definition is different from the Algebra 1 "vertex form" — it reveals why parabolas are used in satellite dishes, headlights, and solar collectors.

Standard Forms

Vertical parabola (opens up/down):

$$(x - h)^2 = 4p(y - k)$$

• Vertex: $(h, k)$
• Focus: $(h, k + p)$
• Directrix: $y = k - p$
• Opens up if $p > 0$; down if $p < 0$

Horizontal parabola (opens right/left):

$$(y - k)^2 = 4p(x - h)$$

• Vertex: $(h, k)$
• Focus: $(h + p, k)$
• Directrix: $x = h - p$
• Opens right if $p > 0$; left if $p < 0$

The Value of $p$

$|p|$ is the distance from the vertex to the focus (and from the vertex to the directrix).

$$|p| = \frac{1}{4|a|} \quad \text{if parabola is } y = a(x-h)^2 + k$$

Example 1 — Finding Focus and Directrix

$(x - 2)^2 = 8(y + 1)$

$4p = 8 \implies p = 2$

Vertex: $(2, -1)$; Focus: $(2, -1+2) = (2, 1)$; Directrix: $y = -1-2 = -3$

Opens up (p > 0).

Example 2 — Writing from Focus/Directrix

Focus $(3, 5)$, Directrix $y = 1$. Find the equation.

Vertex = midpoint of focus and foot of perpendicular to directrix:

$k = \dfrac{5 + 1}{2} = 3$; $h = 3$ (same $x$ as focus for vertical parabola)

Vertex: $(3, 3)$; $p = 5 - 3 = 2$

$(x - 3)^2 = 4(2)(y - 3) = 8(y - 3)$

Example 3 — Converting from Vertex Form

$y = 3x^2 - 6x + 5$. Find the focus.

Complete the square: $y = 3(x-1)^2 + 2$, or $(x-1)^2 = \frac{1}{3}(y-2)$

$4p = \frac{1}{3} \implies p = \frac{1}{12}$

Vertex: $(1, 2)$; Focus: $\left(1, 2 + \frac{1}{12}\right) = \left(1, \frac{25}{12}\right)$