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Algebra I / Elem. Algebra

The Basics

1.1 Intro to Algebra
1.2 Real Number System and Calculators

Linear Equations and Inequalities

Functions and Graphs I

3.1^s Introduction to Graphing
3.2 Solving Equations Graphically
3.3 Functions I - Introduction
3.4 Functions II - Domain, Range, & Operations

Lines and thier Graphs

4.1 Intercept Points
4.2^s Slopes
4.3 Equations of Lines
4.4 Graphing Linear Inequalities

Algebra II / E&I Algebra

Linear Systems

5.1 2x2 Systems of Equations I - Graphically
5.2 2x2 Systems of Equations II - Substitution Method
5.3 2x2 Systems of Equations III - Elimination Method
5.4^s 3x3 Systems of Equations
5.5 Applications
5.6 Systems of Linear Inequalities

Exponents & Polynomials

Intermediate Algebra starts here!

Factoring

Rational Expressions

Rational Equations and Applications

Algebra III / Inter. Algebra

Radical Expressions

10.1 Roots & Radicals
10.2 Fractional Exponents
10.3 Operations I - Simplify & Multiply
10.4 Operations II - Add, Subtract, & Divide
10.5 Operations III - Distribute, Foil, & Factor
10.6^s Complex Numbers

Nonlinear Equations and Applications

11.1 Radical Equations
11.2 Quadratic Equations I - Root Method
11.3^s Quadratic Equations II - Complete the Square
11.4^s Quadratic Equations III - Quadratic Formula
11.5^s Applications - Geometry & Pythagorean Theorem

Functions and Graphs II

12.1 Parabolas I - Algebraic Approach
12.2 Pareabolas II - Graphical Approach
12.3^s Functions III - Composite & Inverse

Exponential and Logarithmic Functions

13.1^s Exponential Functions & Graphs
13.2 Logarithmic Functions & Graphs
13.3 Exponential & Logarithmic Equations

Liberal Arts Maths

Under Construction

College Algebra

Equations and Inequalitites

Functions and Graphs

Polynomial and Rational Functions

3.1 Polynomial Functions I - Introduction, Zeros
3.2 Polynomial Functions II - Division, Remainder Thm, Factor Thm
3.3 Polynomial Functions III - Complex Zeros, Fundamental Thm of Algebra, Descartes Rule
3.4 Rational Functions & Graphs
3.5 Partial Fractions

Exponential and Logarithmic Functions

Systems and Matrices

Conic Sections

Sequences and Series

7.1^s Introduction to Sequences and Series
7.2 Arithmetic Sequences and Series
7.3 Geometric Sequences and Series

Trigonometry

The Six Trigonometric Functions

Right Triangle Trigonometry

Circular Functions

Graphs of Trigonometric Functions

4.1^s Basic Graphs
4.2^s Amplitude & Period
4.3^s Vertical Translations & Phase Shift
4.4 General Equation & Graph to Equation
4.5^s Combination Graphs & Harmonic Motion
4.6^s Inverse Trigonometric Functions

Trigonometric Identities

5.1 Proving Trig Identities
5.2 Sum/Difference Identities
5.3 Double-Angle Identities
5.4^s Half-Angle Identities
5.5 IDs Involving Inverse Trig Functions

Trigonometric Equations

Oblique Triangles and the Laws

Vectors

Complex, Parametric, and Polar Forms

9.1 Trigonometric Form of Complex Numbers
9.2 Products and Quotients in Trig Form
9.3 Powers and Roots in Trig Form
9.4 Parametric Equations and Graphs
9.5^s Polar Equations and Graphs

Calculus I

Limits and Continuity

Derivatives

2.1 Definition of Derivative
2.2 Derivatives of Basic Functions
2.3 Motion and the Derivative
2.4 Product Rule, Quotient Rules, & Deriv. of Trig Functions
2.5 Chain Rule
2.6 Implicit Differentiation
2.7 Derivatives of Exponential and Logarithmic Functions
2.8 Logarithmic Differentiation
2.9^s Derivatives of Inverse Functions
2.10^s Derivatives of Inverse Trig Functions
2.11 Related Rate Applications

Analysis of Curves

3.1 Extreme Values
3.2 Mean Value Theorem for Derivatives
3.3 First Derivative Test & Increasing-Decreasing Functions
3.4 Second Derivative Test & Concavity
3.5^s Curve Sketching
3.6 Fun with Reform Calculus
3.7 Optimization Applications

Antiderivatives

Calculus II

Transcendental Functions

5.0^s Review of Transcendental Functions and thier Derivatives
5.1 Integrals Resulting in Logarithmic Functions
5.2 Integrals Resulting in Inverse Trig Functions
5.3 Hyperbolic Trigonometric Functions
5.4 Numeric and Electronic Integration

Geometry

Physics

Integration Techniques

Calculus of Infinity

Parametric, Polar, and Conic Curves

Calculus III

Under Construction

S = contains supplemental resources

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.

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 V_x=-b/(2a) (memorize it!)

Text Notes
Text: Intro & Inter Algebra for CS 3ed by Blitzer, sect. 11.3

This text section covers too much material so is split into two lecture notes, the first focusing on the algebraic approach to parabolas and the second focusing on the graphic approach.
examples 1-2 Given the standard form of the parabola, be sure that you can find the vertex and all intercept points algebraically. You should also be able to verify this information graphically too. Practice both approaches!
examples 3-4 Given the general form of the parabola be sure that you can find the vertex and all intercept points algebraically. You should also be able to verify this information graphically too. Practice both approaches!

Prof. Keely's Lesson Notes for Mathematics Online