These are important terms and notations for this section.

This section aims to introduce the idea of mathematical reasoning and give an example of how it is used.

This section analyzes the previous example in detail to develop a three phase deductive process to develop a mathematical model.

This section aims to explore and explain different types of information.

This section aims to show how mathematical reasoning is different than ‘typical reasoning’, as well as showing how what we are doing is mathematical.

This section contains important points about the analogy of mathematics as a language.

This section aims to name and explain the common sets of numbers.

This is a detailed numeric model example and walkthrough.

This section aims to show the virtues, and techniques, in generalizing numeric models into ‘generalized’ models.

This section explains types and interactions between variables.

This is an example of a detailed generalized model walkthrough

These are important terms and notations for this section.

In this section we discuss a very subtle but profoundly important difference between a relationship between information, and an equation with information.

In this section we discuss what makes a relation into a function.

In this section we demonstrate that a relation requires context to be considered a function.

In this section we cover Domain, Codomain and Range.

In this section we cover how to actual write sets and specifically domains, codomains, and ranges.

This section covers function notation, why and how it is written.

This section covers $f(x)$ notation.

We cover the idea of function composition and it’s effects on domains and ranges.

These are important terms and notations for this section.

This section introduces the origin an application of graphing.

This section describes how we will use graphing in this course; as a tool to visually depict a relation between variables.

This section describes how accuracy and precision are different things, and how that relates to graphs.

This section covers what graphs should be used for, despite being imprecise.

This section describes the vertical line test and why it works.

This section provides the specific parent functions you should know.

These are important terms and notations for this section.

We discuss what Geometric and Analytic views of mathematics are and the different roles they play in learning and practicing mathematics.

We discuss the geometric perspective and what its role is in learning and practicing mathematics.

We discuss the analytic view of mathematics such as when and where it is most useful or appropriate.

This section describes the geometry and useful symmetry of even functions, as well as how to test for them analytically.

This section describes the geometry and useful symmetry of odd functions, as well as how to test for them analytically.

An introduction to the ideas of rigid translations.

This section describes the geometric perspective of Rigid Translations.

This section describes the analytic perspective of what makes a Rigid Translation.

This section describes the geometric interpretation of what makes a transformation

This section describes the analytic interpretation of what makes a transformation and how to use the function notation to perform (or read) a transformation quickly and easily.

This covers doing transformations and translations at the same time. In particular we discuss how to determine what order to do the translations/transformations in.

This section describes types of points of interest (PoI) in general and covers zeros of functions as one such type.

This section describes one of the most useful tools in mathematics - especially in calculus - the sign chart!

This section describes extrema of a function as points of interest (PoI) on a graph.

This section describes one of the most important concepts in calculus - continuity!

This section describes discontinuities of a function as points of interest (PoI) on a graph.

This section describes how to perform the familiar operations from algebra (eg add, subtract, multiply, and divide) on functions instead of numbers or variables.

This section describes the very special and often overlooked virtues of the ‘equals sign’. It also includes when and why you should “set something equal to zero” which is often overused or used incorrectly.

This section describes the very special and often overlooked virtue of the numbers Zero and One.

This section introduces the geometric viewpoint of invertability.

This section discusses the Horizontal Line Test

This section introduces the analytic viewpoint of invertability, as well as one-to-one functions.

These are important terms and notations for this section.

We know an awful lot about polynomials, but it relies on the very specific structure of a polynomial, and thus it is paramount that one can correctly recognize what is, and isn’t, a polynomial to use these tools.

This section covers one of the most important results in the last couple centuries in algebra; the so-called “Fundamental Theorem of Algebra.”

This section is a quick foray into math history, and the history of polynomials!

This section contains a demonstration of how odd versus even powers can effect extrema.

This section contains information on how exponents effect local extrema

This section shows and explains graphical examples of function curvature.

Some information on factoring before we delve into the specifics.

We discuss step 0 of factoring, extracting any common factors to simplify the process going forward.

First dive into factoring polynomials. This section covers factoring quadratics with leading coefficient of $1$ by factoring the coefficients.

Factor higher polynomials by grouping terms

How to factor when the leading coefficient isn’t one.

Factor polynomials quickly when they are in special forms

This section introduces the technique of completing the square.

In this section we explore how to factor a polynomial out of another polynomial using polynomial long division

Factor one polynomial by another polynomial using polynomial synthetic division

Find factors via rational root theorem

Intro to complex numbers and conjugates

Simplifying complex numbers

Exploring the usefulness and (mostly) non-usefulness of the quadratic formula

A Comprehensive Factoring Practice Quiz.

These are important terms and notations for this section.

This section introduces radicals and some common uses for them.

This section discusses how to simplify numeric and odd root-value radicals.

This section discusses some of the complications that arise when simplifying radicals.

This section views the square root function as an inverse function of a monomial. This is used to explain the dreaded $\pm$ symbol and when to use (and not use) absolute values.

This section shows techniques to solve an equality that has a radical that can’t be simplified into a non radical form. This has potential drawbacks which is also covered in this section.

These are important terms and notations for this section.

This section introduces inequalities and what they are (and aren’t!)

This section discusses how to compute and/or verify inequalities

This section discusses how to algebraically manipulate inequalities.

This section discusses how to solve inequalities.

This section discusses how to graph linear and non-linear inequalities

These are important terms and notations for this section.

This section reviews the basics of exponential functions and how to compute numeric exponentials.

This section gives the properties of exponential expressions. Most of these should be familiar, although we go into slightly more details as to how and why these properties hold in some cases.

This section gives the properties of exponential functions. There is a subtlety between the function and the expression form which will be explored, as well as common errors made with exponential functions.

This section discusses the two main modeling uses of exponentials; exponential growth, and exponential decay.

These are important terms and notations for this section.

This section is a quick introduction to logarithms and notation (and ways to avoid the notation).

We discuss logarithms as inverse functions, and how to understand logs from this perspective.

This is one of the most vital sections for logarithms. We cover primary and secondary properties of logs, which are pivotal in future math classes as these properties are often exploited in otherwise difficult mechanical situations.

This is one of the most vital sections for logarithms. We cover primary and secondary properties of logs, which are pivotal in future math classes as these properties are often exploited in otherwise difficult mechanical situations.

This is one of the most vital sections for logarithms. We cover primary and secondary properties of logs, which are pivotal in future math classes as these properties are often exploited in otherwise difficult mechanical situations.

This is a demonstration of several examples of using log rules to handle logs mechanically.

This section discusses the geometric view of piecewise functions.

This section discusses the analytic view of piecewise functions.

We discuss the domains of piecewise functions - which must be explicitly given.

This section discusses how to compute values using a piecewise function

This discusses Absolute Value as a geometric idea, in terms of lengths and distances.

This discusses the absolute value analytically, ie how to manipulate absolute values algebraically.

We discuss how to compute absolute values, as well as some techniques to speed up the process.

We discuss how to graph absolute values, as well as some key features of the graph.

This section is on how to solve absolute value equalities.

These are important terms and notations for this section.

We discuss what makes a rational function, and why they are useful.

We discuss one of the most important aspects of rational functions; the domain restrictions.

We discuss the circumstances that generate vertical asymptotes in rational functions.

We discuss the circumstances that generate holes in the domain of rational functions rather than vertical asymptotes.

We discuss the circumstances that generate horizontal asymptotes and what they mean.