Lesson 1: Parabolas

Property

A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

Examples

• The path of a basketball shot through the air follows a parabolic arc due to gravity.
• A satellite dish is shaped like a parabola to gather incoming signals and reflect them to a single point, the focus, where the receiver is located.
• The reflector in a car's headlight has a parabolic shape to take the light from the bulb (at the focus) and project it forward in a strong, straight beam.

Explanation

Think of a parabola as a perfect U-shape. Every point on this curve is an equal distance away from a special point (the focus) and a special line (the directrix). This unique balance creates the parabola's signature curve.

Property

To derive a parabola equation from focus $F(a, b)$ and directrix line, set the distance from any point $P(x, y)$ on the parabola to the focus equal to the distance from $P$ to the directrix: $d(P, F) = d(P, \text{directrix})$. Apply the distance formula and algebraically solve by squaring both sides and simplifying.

Examples

Property

A parabola with vertex at $(h, k)$ can be written in vertex form as:

where $p$ is the distance from the vertex to the focus, and the vertex is $(h, k)$.

Examples