Derive the Equation of a Parabola

HSG.GPE.A.2 — Lesson 1 of 2

In this lesson:

• Define a parabola from its geometric property
• Derive step by step
• Identify focus, directrix, and vertex from an equation

Learning Objectives

By the end of this lesson, you will be able to:

1. Define a parabola: the set of all points equidistant from a fixed point and a fixed line
2. Derive the equation from the geometric definition
3. Identify the focus, directrix, and vertex from a given equation

What Geometrically Defines a Parabola?

You've graphed and — but what makes a curve a parabola?

It's not the equation — the equation is a consequence of the geometry.

A parabola is defined by a condition involving two geometric objects:

• A fixed point called the focus
• A fixed line called the directrix

The Geometric Definition

A parabola is the set of all points with

The Setup: Focus, Directrix, Vertex

Focus · Directrix · Vertex

Equidistance Property in Action

For any point on the parabola:

Verify 3 points with , focus , directrix :

Why Vertex at the Origin?

Strategic placement creates maximum cancellation in the algebra ahead.

• Vertex at , focus at , directrix
• Focus and directrix are each distance from the vertex — symmetric about the -axis
• Focal length : larger → wider parabola; smaller → narrower parabola

Check-In: Vertex Distance

Given: vertex , focus , directrix

What is the perpendicular distance from the vertex to the directrix?

From Geometry to Algebra

The parabola is defined by a distance condition — now we turn it into an equation.

The derivation plan:

1. Take a generic point
2. Write using the distance formula
3. Write as the perpendicular distance to a line
4. Set — the definition
5. Expand and simplify to find the equation

Distance to the Focus

Focus , generic point :

Distance to the Directrix

M1: This is the perpendicular distance to a line, not point-to-point.
Distance from to the line is .

For points above the directrix, , so .

Steps 1–2: Set Equal and Square

M2: — the middle term is required.

Steps 3–4: Expand and Cancel

and cancel from both sides — leaving

The Result: Standard Form

Verification: , Point

Equation: · Focus: · Directrix:

Vertex vs. Focus: A Critical Distinction

The Complete Picture:

• Focus and directrix are each from the vertex
• For every point on this curve:

Guided Practice:

Given: vertex at origin, opening upward, focal length

Write the equation. State the focus and directrix.

Use and , directrix .

Guided Practice: Answer

Focus: · Directrix:

Verify: vertex is distance from both. ✓

Practice Set

1. The equation — find , state focus and directrix.

2. Focus , vertex at origin — write the equation.

3. Parabola , focal length — verify that is on the curve and satisfies equidistance.

Pause and try before advancing.

Key Results and Watch-Outs

	Mistake	Fix
M1	Distance to directrix via distance formula	Perpendicular to : use
M2
M3	Vertex at	Vertex on curve; focus inside

What's Next: Lesson 2 of 2

Orientations and Applications

• Four orientations: up, down, left, right — the full picture
• Reference table: read focus and directrix from any standard form instantly
• Vertex form: every has a hidden focus and directrix
• Real-world: satellite dishes, headlights, solar collectors