Algebra I / Elem. Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems
Algebra II / E&I Algebra
Exponents & Polynomials Intermediate Algebra starts here!

Factoring Rational Expressions Rational Equations and Applications
Algebra III / Inter. Algebra
Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
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
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
Calculus IV
15.Functions of Several Variables
16.Multiple Integration
17.Vector Analysis

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

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 (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".