Lesson 15.2

The Parabola: Focus and Directrix

Defining the parabola through the locus property and its standard equations.

Introduction

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

1

Prerequisite Connection

You know the four conic sections and can identify them by their equations.

2

Today's Increment

We derive the standard form, locate the focus and directrix, and graph parabolas in all orientations.

3

Why This Matters

Parabolas focus light and sound—used in satellite dishes, car headlights, and reflecting telescopes.

Focus and Directrix

Locus Definition

Distance to focus = Distance to directrix

Vertical Axis (Opens Up/Down)

Focus:
Directrix:

Horizontal Axis (Opens Left/Right)

Focus:
Directrix:

The Parameter p

  • • : Opens up (vertical) or right (horizontal)
  • • : Opens down (vertical) or left (horizontal)
  • • is the distance from vertex to focus (and vertex to directrix)

Interactive: Adjust the Focus Parameter p

x² = 4yFocus: (0, 1)Directrix: y = -1

Worked Examples

Example 1: Finding Focus and Directrix

Find the focus and directrix of .

Step 1: Identify the form

This matches (horizontal axis, opens right).

Step 2: Find p

Solution

Focus: , Directrix:

Example 2: Writing the Equation from Focus

Write the equation of a parabola with vertex at the origin and focus at .

Step 1: Determine orientation

Focus below vertex means vertical axis, opening down. So .

Step 2: Apply formula

Solution

Example 3: Translated Parabola

Find the vertex, focus, and directrix of .

Step 1: Identify vertex

Vertex is at .

Step 2: Find p

Step 3: Calculate focus and directrix

Focus:
Directrix:

Solution

Vertex: , Focus: , Directrix:

Common Pitfalls

Confusing 4p with p

The equation uses , not . Divide by 4 to find the actual focal distance.

Wrong direction for negative p

When , the parabola opens toward the negative axis—down or left.

Mixing up x² vs y²

= vertical axis (up/down). = horizontal axis (left/right).

Real-World Application

Satellite Dishes and Telescopes

Parabolic reflectors have a remarkable property: all incoming parallel rays (like signals from a distant satellite) reflect to the focus.

This is why satellite dishes, radio telescopes, and car headlights all use parabolic shapes—to concentrate energy at a single point.

Practice Quiz

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