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
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Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

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Course: College Algebra
Topic: Conic Sections
Subtopic: Parabolas

Overview

One of the conic section curves is the parabola. In our previous study of parabolas (graphs of quadratic equations) the equations were generally solved for y, e.g., y=x^2-3x-4 or y=(x-3/2)^2-25/4 (*). But in today's lesson we learn a different form of the equation of a parabola, one that is solved for the square term, e.g., x^2=2y-5. This form actually gives more information and is the form used most often in calculus. This new approach allows us to graph parabolic functions (those that open up/down) as well as sideways parabolas (ones that open left/right and are non-functions).

* Try me: Both of these parabolas open up from the vertex point (3/2,25/4). Can you complete the square to transform the first equation to the second? Can you verify this vertex from both forms of the equation?

Objectives

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

• 7.2.1 Understand the standard equation of a parabola including those that are vertical, horizontal, or whose vertex is shifted to a point not at the origin
• 7.2.2 Algebraically convert the general form of a parabola to the standard form by completing the square as needed
• 7.2.3 Find the "p" term in the equation and understand its effect on the graph of the parabola
• 7.2.4 Given the equation of a parabola, produce its graph both manually and electronically
• 7.2.5 Given the graph or information about the graph, find the equation of the parabola in standard form
• 7.2.6 Understand the definition of a parabola as the set of points at a required distance from the focal point and directrix line
• 7.2.7 Understand the reflective properties of an ellipse
• 7.2.8 Appreciate that parabolas have a variety of applications in science, engineering, and architecture

Terminology

Terms you should be able to define: parabola, parabolic function, sideways parabola, general form (e.g. y=ax^2+bx+c), standard form (e.g. (x-h)^2=p(y-k)^2), vertex, axis of symmetry, focal point, directrix line, latus rectum

Supplemental Resources (recommended)

Explore the interactive graphs at Interactive Parabola Graphs which show how the parabola's focal point and directrix line work together to form the parabola's unique shape. The Parabola has a couple nice applications of parabolas, 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 Parabolas".

rev. 2020-11-29