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: Ellipses

Overview

One of the conic section curves is the ellipse. Ellipses are circles that have been stretched horizontally or vertically to make an oval like shape. They are formed based on the placement of two interior focal points rather than one center point. Ellipses are rich in scientific history. Fun research might include Kepler's Laws of Planetary Motion.

Objectives

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

• 7.3.1 Understand the standard equation of an ellipse including those that are horizontal, vertical, or whose center is shifted to a point not at the origin
• 7.3.2 Algebraically convert the general form of an ellipse to the standard form by completing the square as needed
• 7.3.3 Given the standard equation of an ellipse, produce its graph both manually and electronically
• 7.3.4 Given the graph of an ellipse or information about the graph, find its equation in standard form
• 7.3.5 Find the eccentricity "e" of an ellipse and understand its effect on the graph of the ellipse
• 7.3.6 Understand the definition of an ellipse as the set of points at a required distance from the two foci
• 7.3.7 Know the area of an ellipse formula and how to use it
• 7.3.8 Appreciate that ellipses have a variety of applications in science, engineering, and architecture

Terminology

Terms you should be able to define: ellipse, standard form (e.g. (x-h)^2/a^2+(y-k)^2/b^2=1), focal points (foci), vertices, major axis, minor axis, eccentricity

Text Notes

Be sure that you understand the process of converting a general form conic to a standard form conic via complete the square. This process will be used throughout the chapter on conics. Note that the CTS process shown in this chapter is slightly different than the CTS process of solving a quadratic equation because a conic's equation may contain both x's and y's that are squared and thus require that you CTS on the x and CTS on the y.

Supplemental Resources (recommended)

Explore the interactive graphs at Interactive Ellipse Graphs which show how the ellipses's focal points form the ellipse's oval shape. The Ellipse has a couple nice applications of ellipses, worth skimming.

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 Ellipses".

rev. 2020-11-29