Parabola Orientations and Applications

HSG.GPE.A.2 — Lesson 2 of 2

In this lesson:

• Write equations for all four parabola orientations
• Connect vertex form to focus and directrix
• Apply parabolas to real-world problems

Learning Objectives

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

1. Write the equation of a parabola for all four orientations
2. Identify focus, directrix, and direction from any standard-form equation
3. Connect vertex form to geometric form using
4. Find the focus and directrix of any quadratic function

Four Directions, One Definition

The equidistance condition works in any orientation.

The direction the parabola opens depends on where the focus is relative to the directrix:

• Focus above a horizontal directrix → opens up
• Focus below a horizontal directrix → opens down
• Focus right of a vertical directrix → opens right
• Focus left of a vertical directrix → opens left

All Four Orientations at a Glance

→ vertical (up/down) · → horizontal (left/right) · sign → direction

Downward Case: Reflection Argument

Horizontal Case:

Rightward: Focus · Directrix · Vertex

Leftward: Focus · Directrix →

All Four Orientations: Reference Table

→ vertical · → horizontal · positive sign → focus in positive direction

Worked Example 1: ID from Equation

→ horizontal parabola. Positive coefficient → opens right. , so .

Given:

Worked Example 2: ID from Equation

→ vertical parabola. Negative coefficient → opens down. , so .

Given:

Worked Example 3: Write from Focus and Directrix

Focus on -axis → vertical. Focus above directrix → opens up. .

Result: or

Check-In: Mixed Orientations

A. Given :

• What direction does the parabola open?
• Find the focus and directrix.

B. A parabola with vertex at the origin has focus .

• Write its equation.

Pause and try both before the next slide.

Check-In: Answers

A. → vertical, opens down, :
Focus · Directrix ✓

B. Focus → opens left, :

From Standard to General

Standard form (vertex at origin) → General form (vertex at any point ):

The shift: replace with and with .

This moves the vertex from to without changing the shape.

• Focus offsets from vertex by — in the opening direction
• Directrix offsets from vertex by — in the opposite direction

General Case: Vertex at

The Relationship

Key formula: · Never use directly as

Worked Example 1: Vertex Form → Focus/Directrix

Opens upward. Focus: · Directrix:

Worked Example 2: From Standard Form

Vertex , → opens down,

Focus: · Directrix:

Application: Satellite Dish Receiver Placement

2 m wide, 0.5 m deep parabolic dish — where does the receiver go?

• Model: vertex at origin, rim at ; use
• Substitute : m
• Place receiver 0.5 m above the vertex — at the focus
• Parallel rays reflect to the focus; all signals converge there

Guided Practice

Given:

Find: vertex, , direction, focus, and directrix.

Use , then offset focus from the vertex by .

Guided Practice: Answer

Vertex: · Direction: up ()

Focus: · Directrix:

Practice Set

1. — find focus, directrix, and direction.

2. Vertex at origin, directrix — write the equation.

3. — find , focus, and directrix.

4. — complete the square, then find focus and directrix.

Pause and try before advancing.

Key Results and Watch-Outs

→ vertical · → horizontal · positive sign → focus in positive direction

	Mistake	Fix
M4	" always opens up"	Check the sign: opens down
M5		— different quantities
M3	Focus at	Focus at — offset from vertex by

What's Next: HSG.GPE.A.3

Ellipses and Hyperbolas

• Ellipse: sum of distances to two foci = constant →
• Hyperbola: difference of distances to two foci = constant
• Same strategy: geometric condition → distance formula → simplify

Circle ✓ · Parabola ✓ · Ellipse / Hyperbola →