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
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Course: Algebra III / Intermediate Algebra
Topic: Functions and Graphs II
Subtopic: Parabolas I - Algebraic Approach

Overview

This topic is really fun and useful! We are going to explore the graphs of quadratic functions. These functions all have the same basic U-shape graph called a parabola. A parabola's equation takes on two forms: the standard form of a parabola y=a(x-h)2+k and the general form of a parabola y=ax2+bx+c. This lesson covers analyzing these equations algebraically and producing the graphs manually. Be sure that you can find all the information (vertex, intercepts, etc.) no matter which form of the equation you are given.

Objectives

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

• 12.1.1 Know the graph of the basic parabola y=x2 including specific points that lie on it
• 12.1.2 Recognize and be able to perform reflections (flips across the x-axis), stretches, vertical shifts (up or down), and horizontal shifts (left or right)
• 12.1.3 Know what values in the parabola's equation controls the above features (shifts, etc.) of the graph
• 12.1.4 Given either form of a parabola, know that the a-value gives you information about which way the parabola opens and how wide/narrow that opening is.
• 12.1.5 Given the standard form of a parabola, be able to identify the a, h, and k-values and know that the k-term is the shift up/down, the h-term is the shift left/right, and the vertex is the point (h,k).
• 12.1.6 Given the general form of a parabola, be able to identify the a, b, and c-values and know the vertex is the point (-b/(2a),f(-b/(2a))).
• 12.1.7 Given either form of a parabola, know how to algebraically find the y-intercept (by letting x=0) and the x-intercepts (by letting y=0 and solving for x using factoring, the root method, CTS, or the quadratic formula as needed)
• 12.1.8 Pull all the known information about a parabola together and manually produce its graph

Terminology

Define: parabola, standard vs. general form of a parabola, vertex, reflection about the x-axis, vertical translation (shift), horizontal shift, general form's vertex formula Vx=-b/(2a) (memorize it!)