Master

Finding the Equation of a Parabola Given Focus and Directrix

Master finding the equation of a parabola given focus and directrix with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Finding the Equation of a Parabola Given Focus and Directrix

Study Guide

Key Definition

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

Important Notes

The standard form of a parabola opening upwards or downwards is $(x - h)^2 = 4p(y - k)$
The vertex is halfway between the focus and the directrix.
The distance from the vertex to the focus is the same as from the vertex to the directrix.
$p$ represents the distance from the vertex to the focus or directrix.
For a focus at (h, k) and directrix y = d, set $(x - h)^2 + (y - k)^2 = (y - d)^2$ and simplify to $(x - h)^2 = 4p(y - k)$.

Mathematical Notation

$h, k$ are the coordinates of the parabola's vertex

$p$ is the distance from the vertex to the focus (positive if opening upward/right, negative if downward/left)

$d$ is the constant in the directrix equation y = d or x = d

Use proper notation for parabola parameters

Why It Works

The equation of a parabola is derived by setting the distance from any point on the parabola to the focus equal to the distance from the point to the directrix.

Remember

The focus and directrix give a unique parabola, and their positions determine the shape and orientation of the parabola.

Quick Reference

Vertex Form:$(x - h)^2 = 4p(y - k)$

Distance Formula:$\sqrt{(x - h)^2 + (y - k)^2}$

Understanding Finding the Equation of a Parabola Given Focus and Directrix

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A parabola is a U-shaped curve that can open upwards or downwards. You can find its equation using the focus and directrix.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the equation of a parabola with focus at $(0, 2)$ and directrix $y = -2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A teenage skateboarder wants to design a ramp with a parabolic path. The ramp's highest point is the focus at $(3, 5)$ and the ground is the directrix at $y = 1$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a parabola with focus at $(0, 3)$ and directrix $y = 1$, find the equation using the formula.

Challenge Quiz

Single Choice Quiz

Advanced

Find the equation of the parabola with focus $(-2, 2)$ and directrix $y = 4$.

Recap

Watch & Learn

Review key concepts and takeaways