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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 I / Elementary Algebra
Topic: Lines and their Graphs
Subtopic: Slopes

Overview

Here we'll focus on the concept of slope which measures the steepness of a line. Slope is an important aspect of linear functions and has applications to real-life situations in the form of "rate" of growth. Some cautions to note:

Watch your sign when you are finding slope! Anytime you move from point to point and you go down or left the sign will be negative on that side of the slope triangle.
As you move from point to point, be sure to do just that in one smooth motion. Don't start at the right angle and move to one point, then start back at the right angle and move to the other point. If you do the signs may get messed up.
When using the formula to find slope, it doesn't matter which point is (x₁,y₁) and which point is (x₂,y₂). But you do have to be careful that you don't mix the order up and use, say, y₂-y₁ on the top and x₁-x₂ on the bottom. If you do your signs will get messed up.
Some textbooks say that vertical lines have "no slope". Don't confuse that with a "zero" slope! Horizontal lines have a slope of zero; vertical lines have "no slope" or better yet "undefined slope".

Objectives

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

4.2.1 Given the graph of a line, find its slope
4.2.2 Given two points on a line, use the slope to find a third point on the line
4.2.3 Given two points on a line, find the slope of the line
4.2.4 Recognize lines that have positive slope, negative slope, zero slope, and undefined slope
4.2.5 Know that two parallel lines have the same slope
4.2.6 Know that two perpendicular lines have opposite reciprocal slopes
4.2.7 For horizontal and vertical lines know their equations, slopes, and axis-intercept points
4.2.8 Graph horizontal and vertical lines algebraically
4.2.9 Understand applications of slopes and rates

Terminology

Define: steepness, slope, slope triangle, rate of change (a.k.a. rate, rate of growth/decay), horizontal line, vertical line, zero slope, undefined slope

Text Notes
Text: Intro & Inter Algebra for CS 3ed by Blitzer, sect. 3.3

pg 222 It is strongly recommended that you memorize the formulas given in the "definition of slope" box.
pg 222 Take note of the caution/common error in the "study tip" box.

Supplementary Resources

Read Ask Dr Math's Why is the letter m used for slope? and PurpleMath's The Meaning of Slope and Y-Intercept. Both highly recommended!

Prof. Keely's Lesson Notes for Mathematics Online