Lesson 15.2

The Parabola: Focus and Directrix

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

Introduction

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

Past Knowledge

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

Today's Goal

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

Future Success

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

Practice Quiz

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