PSK Lesson Notes

LESSON NOTES MENU

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 I / Elementary Algebra
Topic: Functions and Graphs I
Subtopic: Functions I - Introduction

Overview

Today we introduce the concepts of functions and relations. Relations will be given in the form of sets of ordered pairs, mappings, graphs, or equations. Try to seem these as just different ways to visualize/communicate the same information.

This is a vital section since the remainder of this course and intermediate algebra deal with different types of functions and their graphs. There is lots of important terminology here. Be sure you well versed in recognizing functions given a set of ordered pairs (e.g., why is {(0,1),(0,2),(1,3),(2,3)} not a function?), a mapping, a graph (using the VLT), or an equation (which you can just graph and determine visually).

A key thing to remember is that if a relation has "same x different y" then it is NOT a function.

Notation Caution: f(x) ... that is "f of x" ... does not mean f times x! It does mean the equation is a function called f and the input variable is x. You can think of "f(x)" as meaning "y". In other words, "Evaluate y=x³ or x=2" means the same thing as "Given f(x)=x³ find f(2)".

Objectives

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

3.3.1 Determine if a relation is a function or not given: set of ordered pairs, mapping, graph, or equation
3.3.2 Use the vertical line test to determine if a curve is a graph of a function or not
3.3.3 Understand function notation and use/write it properly
3.3.4 Evaluate functions algebraically for given values of the input variable
3.3.5 Evaluate functions by observation of its graph

Terminology

Define: set, mapping, relation, function, vertical line test (VLT)

Text Notes
Text: Intro & Inter Algebra for CS 3ed by Blitzer, sect. 8.1

This section contains lots of important new terminology, notation, and processes. Plan to spend significant time studying it!
ch 8.1-8.3 Caution: The text covers foil before functions, but the Clark elementary algebra curriculum covers functions first. So we jumped ahead to chapter 8 to pick up functions but the text includes foil in some of the exercises. THERE WILL BE NO FOIL PROBLEMS ON THIS WEEK'S QUIZ! We'll leave those until after we cover foil later in the course. So there will be NO problems such as simplify a function that has a binomial times a binomial like (x+2)(x-1) because to do that multiplication involves foil. You can SKIP any exercises that involve multiplying a binomial times a binomial.

Prof. Keely's Lesson Notes for Mathematics Online