Parabola

Beyond being the graph of a quadratic function, the parabola has an elegant geometric definition that reveals why it appears in optics, engineering, and physics.

Geometric Definition

📐Definition — Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed point called the focus $F$ and a fixed line called the directrix $\ell$ .

For any point $P$ on the parabola: $\text{dist}(P, F) = \text{dist}(P, \ell)$ .

The vertex is the point on the parabola closest to the directrix — it lies exactly halfway between the focus and the directrix.

Standard Form (Vertical Axis)

⚡Theorem — Standard Parabola Equations

A parabola with vertex at the origin and vertical axis of symmetry has the equation:

$x^2 = 4py$

Focus: $(0, p)$
Directrix: $y = -p$
$p > 0$ : opens upward; $p < 0$ : opens downward

Equivalently, solving for $y$ : $y = \dfrac{1}{4p}x^2$ . Comparing with $y = ax^2$ , we get $a = \dfrac{1}{4p}$ , so:

$p = \frac{1}{4a}$

This connects the algebraic coefficient $a$ directly to the geometric parameter $p$ .

Worked Examples

✎Example — From Equation to Focus

For $y = 2x^2$ , find the focus and directrix.

Solution. Here $a = 2$ , so:

$p = \frac{1}{4 \cdot 2} = \frac{1}{8}$

Focus: $\left(0, \dfrac{1}{8}\right)$
Directrix: $y = -\dfrac{1}{8}$

✎Example — From Standard Form

Write $x^2 = 12y$ in focus-directrix form.

Solution. $4p = 12$ , so $p = 3$ .

Focus: $(0, 3)$
Directrix: $y = -3$

The parabola opens upward since $p > 0$ .

Horizontal Parabola

A parabola can also open left or right. The standard form with a horizontal axis is:

$y^2 = 4px$

Focus at $(p, 0)$ , directrix $x = -p$ .
$p > 0$ : opens right; $p < 0$ : opens left.

The Reflective Property

A remarkable property of the parabola: any ray traveling parallel to the axis of symmetry reflects off the parabola and passes through the focus. Conversely, light emanating from the focus reflects off the parabola as a parallel beam.

This principle is used in:

Satellite dishes and radio telescopes — incoming parallel signals concentrate at the focus.
Car headlights — a bulb at the focus produces a parallel beam.
Solar collectors — sunlight is focused to a single point to generate heat.

✏Exercise

Find the focus and directrix of $y = \dfrac{x^2}{4}$ . Hint: express in the form $x^2 = 4py$ .