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: General and Degenerate Conics

Overview

In calculus you will be expected to know how to identify and graph conics from both the general form of a conic, ax^2+bxy+cy^2+dx+ey+f=0, and the standard form. The key to understanding the relationship between these two forms of a conic's equation and the graph is the process of converting from one to the other including via completing the square.

Sometimes, depending on the a b c d e f constants, the equation will result in a point, line, or two intersecting lines. These cases are called the degenerate conics and occur when the plane slicing the double cone does so at particularly interesting places or at specific angles. These degenerates provide a great connection between the algebra of conics and the visual experience. Cool stuff!

Objectives

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

• 7.5.1 Understand the general equation of a conic section
• 7.5.2 Algebraically convert from the general form and standard form of a conic and visa versa using CTS as needed
• 7.5.3 Identify a conic from the general form equation
• 7.5.4 Understand degenerate conics including their equations, graphs, formation (in terms of slicing the double cone), and how to identify them from the general equation of a conic

Terminology

Terms you should be able to define: general form of the equation of a conic section, degenerate conic

Supplemental Resources

Download/Print one of the following formula Sheets, whichever works best for you. You may annotate your formula sheet and refer to it during a test.

Summary of Conic Sections provides a thorough list of all the conic sections and their equations.

rev. 2020-11-29