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Focus of a Parabola

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Focus of a Parabola

Study Guide

Key Definition

A parabola is the set of all points in a plane which are an equal distance from a given point (focus) and a given line (directrix).

Important Notes

The focus lies on the axis of symmetry of the parabola.
For a parabola in vertex form $y = a(x-h)^2 + k$, the vertex is $(h, k)$.
For a vertical parabola in vertex form $y = a(x-h)^2 + k$, the focus is $(h, k + \frac{1}{4a})$; if a is negative, the focus lies below the vertex.
The directrix is the line $y = k - \frac{1}{4a}$.
The distance from the vertex to the focus is $\left|\frac{1}{4a}\right|$.

Mathematical Notation

$\frac{1}{2}$ represents a half

$+$ represents addition

$\sqrt{x}$ represents square root of x

$x^2$ is x squared

$\angle$ represents an angle

Remember to use proper notation when solving problems

Why It Works

The focus and directrix define the parabola by ensuring that each point on the parabola is equidistant from both, forming the unique U-shaped curve.

Remember

Keep in mind the formula for the focus $(h, k + \frac{1}{4a})$ when solving problems.

Quick Reference

Focus Formula:$(h, k + \frac{1}{4a})$

Understanding Focus of a Parabola

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Video explanation of this concept

Beginner

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Beginner Explanation

The focus of a parabola is the point that helps define its shape, located at $(h, k + \frac{1}{4a})$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If the equation of a parabola is $y = \frac{1}{8}x^2$, what are the coordinates of the focus?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're designing a satellite dish which forms a parabola that is 2 meters wide at its opening and 1 meter deep. Place the vertex at (0, 0) and the rim points at (±1, 1). Where should you place the receiver at the focus?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given the parabola $y = -x^2 + 3x - 4$, find the coordinates of its focus.

Challenge Quiz

Single Choice Quiz

Advanced

For the parabola $y = -\frac{1}{4}(x-3)^2 + 2$, where is the focus?

Recap

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