Learn Precalculus (Online Precalculus Tutorials)

by Dave

In this article, I take the student on a journey where we’ll learn precalculus. I’ll describe each step for you and provide tutorials and videos where you get the necessary understanding. You will understand the process and goals for each chapter, and you’ll gain valuable insights from a seasoned math educator. Precalculus is a gateway to many other subjects. Let’s get started.

What You’ll Learn in Fundamental Concepts of Algebra

Master the fundamental concepts of algebra. Learn all about real numbers, radical expressions, polynomials, and solving equations.

  • Real Numbers
  • Ordering Real Numbers
  • Absolute Value and Distance
  • Algebraic Expressions
  • Basic Rules of Algebra
  • Integer Exponents
  • Scientific Notation
  • Radicals and Their Properties
  • Simplifying Radicals
  • Rational Exponents
  • Polynomials
  • Operations with Polynomials
  • Special Products
  • Factoring
  • Factoring Special Polynomial Forms
  • Trinomials with Binomial Factors
  • Factoring by Grouping
  • Domain of an Algebraic Expression
  • Simplifying Rational Expressions
  • Operations with Rational Expressions
  • Complex Fractions
  • The Cartesian Plane
  • The Midpoint Formula
  • The Equation of a Circle

What You’ll Learn in Functions and Their Graphs

Learn functions and their graphs. Master function notation, graphing functions, the transformations of functions, and inverse functions.

  • The Graph of an Equation
  • Using a Graphing Utility
  • Applications
  • The Slope of a Line
  • The Point-Slope Form
  • Sketching Graphs of Lines
  • Parallel and Perpendicular Lines
  • Introduction to Functions
  • Function Notation
  • The Domain of a Function
  • Applications
  • The Graph of a Function
  • Increasing and Decreasing Functions
  • Relative Minimum and Maximum Values
  • Graphing Step Functions
  • Graphing Piecewise-Defined Functions
  • Even and Odd Functions
  • Summary of Graphs of Parent Functions
  • Vertical and Horizontal Shifts
  • Reflecting Graphs
  • Arithmetic Combinations of Functions
  • Compositions of Functions
  • Inverse Functions
  • The Graph of an Inverse Function
  • The Existence of an Inverse Function
  • Finding Inverse Functions Algebraically

This course begins with an explanation of the Cartesian plane. After that, I discuss plotting points, the distance formula, the midpoint formula, and the Pythagorean Theorem.

Next, we study graphing equations. For example, we are sketching the graph of an equation by plotting points. Then I show that finding the intercepts of a graph and graphical (and algebraic) tests for symmetry are important.

Now we specialize in linear equations in two variables. First, we examine the slope-intercept form of the equation of a line and finding the slope of a line. Then, the slope of a line passing through two points, and of course, the point-slope form of an equation of a line. While working through many examples, we also consider parallel and perpendicular lines.

After these beginnings, we now introduce our main topic: functions. We first discuss domain and range and then drive home the concept of a function by examining the characteristics of a function from one set to another. Importantly, we discuss four ways to represent a function, verbally, numerically, graphically, and algebraically. We discuss each way at length while also explaining function notation.

With a thorough understanding of what a function is, we now being to analyze their graphs. In particular, we next study the zeros of a function, where the graph of a function is increasing (decreasing), and the relative extrema.

Next, we switch gears and move to study a library of parent functions. We will use these parent functions to build more complex functions useful in a large number of applications. For example, we consider parent functions such as linear, squaring, cubic, square root, and reciprocal functions. After that, we investigate certain transformations of functions such as shifting graphs, reflecting graphs, and non-rigid transformations.

Now that we understand some parent functions and how to transform them, we consider combining them into more complex functions. So we consider the sum, difference, product, and quotients of functions. In the end, the result is to have a useful, more extensive array of scenarios, where we can understand a function from its basic building blocks.

When we began our study of functions early in the course, we discussed relations. So it is easy to define an inverse relation, but now we focus on inverse functions. We define inverse functions, learn how to verify inverse functions, and how to graph inverse functions. After that, we characterize one-to-one functions and work algebraically to find the inverse of a function.

Throughout this course, we present many applications by working through examples. Towards the end, though, applications take center stage. First, we investigate mathematical modeling and then discuss least square regression. Finally, we discuss direct variation, inverse variation, and joint variation.

What You’ll Learn in Polynomial and Rational Functions

Master polynomial and rational functions. Learn linear, quadratic, polynomials of higher degree, rational functions, and complex numbers.

  • The Graph of a Quadratic Function
  • The Standard Form of a Quadratic Function
  • Finding Minimum and Maximum Values
  • Graphs of Polynomial Functions
  • The Leading Coefficient Test
  • Zeros of Polynomial Functions
  • The Intermediate Value Theorem
  • Long Division of Polynomials
  • Synthetic Division
  • The Remainder and Factor Theorems
  • The Rational Zero Test
  • Other Tests for Zeros of Polynomials
  • The Fundamental Theorem of Algebra
  • Conjugate Pairs
  • Factoring a Polynomial
  • Introduction to Rational Functions
  • Horizontal and Vertical Asymptotes
  • Applications
  • The Graph of a Rational Function
  • Slant Asymptotes
  • More Applications

What You’ll Learn in Exponentials and Logarithmic Functions

Master exponential and logarithmic functions, including learning about their properties, graphs, solving equations, and applications.

  • Exponential Functions
  • Graphs of Exponential Functions
  • The Natural Base e
  • Applications
  • Logarithmic Functions
  • Graphs of Logarithmic Functions
  • Properties of Logarithms
  • Rewriting Logarithmic Expressions
  • Solving Exponential Equations
  • Solving Logarithmic Equation

What You’ll Learn in Introduction to Trigonometry

With this introduction to trigonometry, you’ll learn trigonometric functions, their graphs, and master inverse trigonometric functions.

  • Angles
  • Degree Measure
  • Radian Measure
  • Conversion of Angle Measure
  • Linear and Angular Speed
  • The Six Trigonometric Functions
  • Trigonometric Identities
  • Evaluating Trig Functions with a Calculator
  • Applications Involving Right Triangles
  • Reference Angles
  • Trigonometric Functions of Real Numbers
  • Basic Sine and Cosine Curves
  • Amplitude and Period with Sine and Cosine
  • Translations of Sine and Cosine Curves
  • Graph of the Tangent Function
  • Graph of the Cotangent Function
  • Graphs of the Reciprocal Functions
  • Damped Trigonometric Graphs
  • Inverse Sine Function
  • Other Inverse Trigonometric Functions
  • Compositions of Functions
  • Applications Involving Right Triangles
  • Trigonometry and Bearings
  • Harmonic Motion

What You’ll Learn in Analytic Trigonometry

Master analytic trigonometry with trigonometric identities, solving trigonometric equations and learning many trigonometric formulas.

  • Using Fundamental Identities
  • Verifying Trigonometric Identities
  • Equations of Quadratic Type
  • Functions Involving Multiple Angles
  • Using Inverse Functions
  • Using Sum and Difference Formulas
  • Multiple-Angle Formulas
  • Power-Reducing Formulas
  • Half-Angle Formulas
  • Product-to-Sum Formulas

What You’ll Learn in Additional Topics in Trigonometry

Master additional topics in trigonometry by learning the law of sines and cosines. Including vectors, dots products, and complex numbers.

  • Law of Sines
  • Solving Triangles
  • The Ambiguous Case (SSA)
  • Area of an Oblique Triangle
  • Law of Cosines
  • Heron’s Area
  • Formula Component Form of a Vector
  • Vector Operations
  • Unit Vectors
  • Direction Angles
  • Applications of Vectors
  • The Dot Product of Two Vectors
  • The Angle Between Two Vectors
  • Finding Vector Components Work
  • The Complex Plane
  • Trigonometric Form of a Complex Number
  • Multiplication and Division of Complex Numbers
  • Powers of Complex Numbers Roots of Complex Numbers

What You’ll Learn in Solving Equations and Inequalities

Master solving equations and inequalities. With these lessons you’ll learn solving equations, complex numbers, and solving inequalities.

  • Equations and Solutions of Equations
  • Using Math Models to Solve Problems
  • Common Formulas
  • Intercepts, Zeros, and Solutions
  • Finding Solutions Graphically
  • Points of Intersection of Two Graphs
  • The Imaginary Unit i
  • Operations with Complex Numbers
  • Complex Conjugates
  • Fractals and the Mandelbrot Set
  • Quadratic Equations
  • Polynomial Equations
  • Equations Involving Radicals
  • Equations Involving Fractions
  • Equations Involving Absolute Values
  • Properties of Inequalities
  • Solving a Linear Inequality
  • Inequalities Involving Absolute Values
  • Polynomial Inequalities
  • Rational Inequalities
  • Applications

What You’ll Learn in Linear Systems and Matrices

Master linear systems and matrices. Learn Gaussian elimination, operations with matrices, inverse matrices, and determinants.

  • Substitution Method
  • Graphing Method
  • Points of Intersection
  • The Method of Elimination
  • Applications
  • Interpretation of  2-Variable Systems
  • Row-Echelon Form
  • Back-Substitution
  • Gaussian Elimination
  • Nonsquare Systems
  • Matrices
  • Elementary Row Operations
  • Equality of Matrices
  • Matrix Multiplication
  • Interpretation of 3-Variable Systems
  • Back-Substitution
  • Matrix Addition
  • Scalar Multiplication
  • The Inverse of a Matrix
  • Finding Inverse Matrices
  • The Inverse of a 2 by 2 Matrix
  • Systems of Linear Equations
  • The Determinant of a 2 by 2 Matrix
  • Minors and Cofactors
  • The Determinant of a Square Matrix
  • Triangular Matrices
  • Area of a Triangle
  • Collinear Points
  • Cramer’s Rule
  • Cryptography

What You’ll Learn in Sequences, Series, and Probability

Master sequences, series, and probability with these lessons. Learn partial sums, mathematical induction, and the basics of probability.

  • Sequences
  • Factorial Notation
  • Summation Notation
  • Series
  • Applications
  • Arithmetic Sequences
  • The Sum of a Finite Arithmetic Sequence
  • Geometric Sequences
  • The Sum of a Finite Geometric Sequence
  • Geometric Series
  • Mathematical Induction
  • Sums of Powers of Integers
  • Finite Differences
  • Binomial Coefficients
  • Binomial Expansions
  • Pascal’s Triangle
  • Simple Counting Problems
  • The Fundamental Counting Principle
  • Permutations
  • Combinations
  • The Probability of an Event
  • Mutually Exclusive Events
  • Independent Events
  • The Complement of an Event

What You’ll Learn in Topics in Analytic Geometry

Master topics in analytic geometry. In these lessons, you’ll learn conic sections, polar coordinates, and polar equations of conic sections.

  • Conics
  • Circles
  • Parabolas
  • Reflective Property of Parabolas
  • Ellipses
  • Eccentricity
  • Hyperbolas
  • Asymptotes of a Hyperbola
  • General Equations of Conics
  • Plane Curves
  • Sketching a Plane Curve
  • Eliminating the Parameter
  • Finding Parametric Equations for a Graph
  • Polar Coordinates
  • Coordinate Conversion
  • Equation Conversion
  • Graphs of Polar Equations
  • Symmetry
  • Zeros and Maximum r-Values
  • Special Polar Graphs
  • Alternative Definition of Conics
  • Polar Equations of Conics

Take in everything in precalculus; you need algebra, trigonometry, and geometry. With these articles and videos, you get the skills you need to succeed.

Precalculus is a mathematics course for high-school or undergraduate students whose primary goal is to prepare students for a calculus course. Topics in precalculus are usually functions, and their graphs followed by polynomial and rational functions. Then exponential and logarithmic functions and a detailed study of trigonometry. In this part, students study right-angle trigonometry, analytic trigonometry, and additional topics in trigonometry. From here, the topics can vary widely. Some topics include systems of equations and inequalities, matrices and determinants, sequences, series, and probability.