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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: Exponential and Logarithmic Functions
Subtopic: Exponential Functions & Graphs

Overview

We have studied linear and quadratic functions both of which are simple polynomial functions. But not all functions are polynomial. In this lesson we investigate exponential functions. These functions have tons of real-world applications! For instance, you've heard of diseases like AIDS spreading "exponentially"? Spread of diseases, population growth, compounding interest, and carbon dating are all examples of applications that can be modeled using exponential functions. Keep your grapher handy as you explore exponential functions such as y=2^x (note the variable is in the power not the base like the parabola y=x²). Watch what effect the following things have on the graph: a negative in front, different bases, adding or subtracting a number from the basic function, adding or subtracting a number to the x in the power. Watch for the effects on the shape, axis intercept points, asymptote line(s), domain, and range.

Although your text may not do so until later, I think it is useful to introduce the scientific constant "e" at this time. "e" is an irrational number like π except that e is equal to approximately 2.718. Try graphing y=e^x and variations (reflections, shifts, translations) of this function. How does it compare with the graph of y=2^x and y=3^x? Why?

Objectives

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

13.1.1 Evaluate exponential functions algebraically and electronically including those containing "e"
13.1.2 Graph exponential functions manually and electronically
13.1.3 For the basic exponential function f(x)=a^x, know the basic shape of its graph, intercept points, asymptote lines, domain, and range
13.1.4 Recognize and be able to perform reflections (flips across the x-axis), stretches, vertical shifts (up or down), and horizontal shifts (left or right) in regards to exponential functions
13.1.5 Know what values in an exponential function's equation controls the above features (shifts, etc.) of the graph

Terminology

Define: exponential function, horizontal asymptote line, the number "e"

Text Notes
Text: Intro & Inter Algebra for CS 3ed by Blitzer, sect. 12.1

Everything is important here! New terminology, definitions, graphs, connections between equations and graphs, and more. This section builds the foundation for the rest chapter 12. Learn it well! Be sure you are able to work problems both manually and electronically.

Supplementary Resources

A basic introduction to "e" (this lesson) and "ln" (next lesson) is at Algebra Lab: e and ln. Much more info about e is at Math Forum's Ask Dr. Math: FAQs about e including a good collection of Q&A links near the bottom of the article.

Prof. Keely's Lesson Notes for Mathematics Online