Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: College Algebra
Topic: Conic Sections
Subtopic: Hyperbolas

Overview

One of the conic section curves is the hyperbola. A hyperbola has two "branches" and is created by slicing a "double cone" (one atop the other touching at their vertices) by a plane to create the two branches. The curve of a hyperbola looks somewhat like a parabola but it is not the same curve at all! A hyperbola is bounded by intersecting asymptote lines, but a parabola is unrestrained. This changes the way the hyperbola curve grows in subtle but important ways. The reflective properties of the hyperbola are distinct from those of a parabola (important difference to note!). In comparison to an ellipse, the definition of the hyperbola differs from the definition of the ellipse in terms of distance from the foci to points on the curve.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 7.4.1 Understand the standard equation of a hyperbola including those that are horizontal, vertical, or whose center is shifted to a point not at the origin
• 7.4.2 Algebraically convert the general form of a hyperbola to the standard form by completing the square as needed
• 7.4.3 Given the standard equation of a hyperbola, produce its graph both manually and electronically
• 7.4.4 Understand how to use the "defining rectangle" to locate teh asymptotes and branches of the curve when sketching a hyperbola
• 7.4.5 Given the graph of a hyperbola or information about the graph, find its equation in standard form
• 7.4.6 Find the eccentricity "e" of a hyperbola and understand its effect on the graph of the hyperbola
• 7.4.7 Understand the definition of a hyperbola as the set of points at a required distance from the two foci
• 7.4.8 Appreciate that hyperbolas have a variety of applications in science, engineering, and architecture

Terminology

Terms you should be able to define: ellipse, standard form, focal points (foci), vertices, transverse axis, conjugate axis, defining rectangle, asymptote lines, branches, eccentricity.

Formulae: You should know how to use these formulae!

Supplemental Resources (recommended)

Read Importance of Hyperbolas in Life for a brief overview of applications from Sciencing.

Explore the interactive graphs at Interactive Hyperbola Graphs which show how the hyperbola's focal points form the hyperbola's two branches.

Supplemental Resources (optional)

If you need supplemental tutorial videos with examples relevant to this section go to James Sousa's MathIsPower4U and search for topics: "Graphing and Writing Equations of Hyperbolas".

The Hyperbola has lots of examples if you need more practice working with equations and graphs of hyperbolas.

rev. 2020-11-29