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Hyperbolas

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Hyperbolas

Study Guide

Key Definition

A hyperbola is the set of all points where the difference of the distances from two fixed points, called foci, is constant. The simplest rectangular hyperbola has the graph of the function y = 1/x, while the general hyperbola in standard form is given by (x–h)^2/a^2 – (y–k)^2/b^2 = 1 or (y–k)^2/a^2 – (x–h)^2/b^2 = 1.

Important Notes

The center of a hyperbola is the point of symmetry for its two branches.
A hyperbola approaches its asymptotes as x or y becomes large in magnitude.
The standard form of a hyperbola centered at (h, k) with horizontal transverse axis is (x–h)^2/a^2 – (y–k)^2/b^2 = 1, and with vertical transverse axis is (y–k)^2/a^2 – (x–h)^2/b^2 = 1.
Asymptotes are lines that a graph approaches but never touches.
Hyperbolas can be obtained by intersecting a plane with both halves of a double cone.

Mathematical Notation

$(x - h)^2/a^2 - (y - k)^2/b^2 = 1$ standard hyperbola equation with horizontal transverse axis

$(y - k)^2/a^2 - (x - h)^2/b^2 = 1$ standard hyperbola equation with vertical transverse axis

$a$ = semi-transverse axis length

$b$ = semi-conjugate axis length

$c$ = focal distance from center

Use proper notation for a, b, and c when analyzing a hyperbola

Why It Works

Hyperbolas have asymptotic behavior and reflective properties, which make them useful in navigation systems like LORAN and in certain optical mirror designs.

Remember

A hyperbola's graph consists of two disjoint curves (branches) that are mirror images across its center.

Quick Reference

Standard Form:$(x - h)^2/a^2 - (y - k)^2/b^2 = 1$

Center:$(h, k)$

Understanding Hyperbolas

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A hyperbola is two mirror-image curves, like the graph of y = 1/x, opening in opposite quadrants.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the standard form of a hyperbola centered at $(h, k)$ with horizontal transverse axis?

Real-World Problem

Question Exercise

Intermediate

Navigation Scenario

Hyperbolic navigation systems locate a point by the difference in distances to two fixed stations. Describe the shape traced by all points for which the distance difference is constant.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the hyperbola $y = \frac{2}{x}$. What happens to $y$ as $x$ approaches zero?

Challenge Quiz

Single Choice Quiz

Advanced

Which of these is the horizontal asymptote of the hyperbola $y = \frac{3}{x-2} + 5$?

Recap

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Review key concepts and takeaways