# Parabola - Definition, Formula, Graph, Examples

## TL;DR

A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). It's one of four classical conic sections — created by slicing a cone with a plane parallel to its slant side. The standard equation is y² = 4ax (horizontal opening) or (x − h)² = 4p(y − k) (vertex form).

## What Is a Parabola?

A **parabola** is the locus (set of points) in a plane equidistant from a fixed point — the **focus** — and a fixed line — the **directrix**. This definition is purely geometric — no algebra required.

Equivalently, a parabola is a **conic section**: the curve formed when a flat plane slices through a cone parallel to the cone's slant side. The other three conic sections are the _circle_, the _ellipse_, and the _hyperbola_.

Algebraically, the standard parabola opens horizontally to the right:

y² = 4ax

with focus at (a,0) and directrix x = -a.

## What Is the Parabola Equation?

Parabolas come in three standard algebraic forms.

### Standard Form (Opens Up)

y = ax² + bx + c,  a ≠ 0

The parabola opens _upward_ when a > 0, _downward_ when a < 0.

### Vertex Form

y = a(x − h)² + k

The vertex sits at (h,k). The axis of symmetry is x = h.

### Conic Form (Vertex at Origin)

y² = 4ax (opens right),  x² = 4ay (opens up)

The number a is the distance from the vertex to the focus. The directrix is x = -a or y = -a respectively.

## What Are the Parts of a Parabola?

| Part                    | Definition                                               |
|-------------------------|---------------------------------------------------------|
| **Vertex**              | The "tip" of the parabola — point where it changes direction |
| **Focus**               | A fixed point inside the curve                          |
| **Directrix**           | A fixed line outside the curve                          |
| **Axis of symmetry**    | Line through vertex and focus, perpendicular to directrix |
| **Latus rectum**        | Chord through the focus perpendicular to the axis — length 4a |

## What Are the Properties of a Parabola?

1. **Symmetry.** A parabola is symmetric about its axis of symmetry.
2. **One axis of symmetry only.** A parabola has exactly one axis.
3. **Single vertex.** The unique point where the curve meets its axis of symmetry.
4. **Eccentricity = 1.** Ratio of distance-to-focus over distance-to-directrix is equal by definition.
5. **Latus rectum length 4a.** Useful for sketching and checking the standard form.
6. **Reflective property.** Rays entering parallel to the axis reflect through the focus.

## How Do You Derive the Parabola Equation?

The algebraic equation y² = 4ax comes directly from the focus-directrix definition. Here's the derivation:

**Setup.** Place the vertex at the origin, focus at (a,0), and directrix as the vertical line x = -a.

**Distance condition.** Distance from P to focus equals distance from P to directrix:

√((x − a)² + y²) = |x + a|

**Square both sides:**

(x − a)² + y² = (x + a)²

**Expand:**

x² − 2ax + a² + y² = x² + 2ax + a²

**Cancel the x² and a²:**

y² = 4ax

## How Do You Graph a Parabola?

For a parabola y = ax² + bx + c:

1. Find the vertex using x = −b/2a, then plug into the equation to find y.
2. Determine direction. If a > 0, opens up; if a < 0, opens down.
3. Find the y-intercept by setting x = 0.
4. Find the x-intercepts using the quadratic formula.
5. Plot the vertex, intercepts, and a few other points then draw the curve.

## Why Are Parabolas Important?

Parabolas are important in nature and engineering because they describe the trajectory of objects under uniform gravity and have a fundamental reflective property. Applications include:
- **Satellite dishes and radio telescopes**
- **Car headlights and flashlights**
- **Solar concentrators**
- **Suspension bridge cables**
- **Projectile motion**

The term _parabola_ comes from the Greek _parabolē_, named by Apollonius of Perga, who studied conics without algebra.

## Common Mistakes With Parabolas

### **Mistake 1:** Confusing the y-intercept with the vertex's y-coordinate.
### **Mistake 2:** Forgetting to take the sign of a into account when sketching.
### **Mistake 3:** Confusing focus and vertex in conic form.

## The Mathematicians Who Shaped Parabola Theory

- **Apollonius of Perga** - Wrote _Conics_, defining parabolas.
- **Galileo Galilei** - Proved that projectiles follow parabolic paths.
- **René Descartes** - Standardized the algebraic form of parabolas.