Learn Algebra – Free Course with Videos and Guided Notes
Build a strong foundation in algebra with structured video lessons, guided notes, and chapter tests. This free course covers functions, equations, inequalities, polynomials, rational expressions, exponentials, logarithms, and more—all taught with clear, step-by-step instruction. Whether you're learning algebra for the first time, reviewing key concepts, or preparing for college-level mathematics, these resources are designed to help you succeed.
Algebra Foundations
Functions, equations, inequalities, and graphing.
Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams
Chapter 1 – Prerequisite Algebra and Geometry Concepts
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In this video, you'll learn how numbers are organized into different classification systems. We'll distinguish between real and complex numbers, rational and irrational numbers, and natural numbers, whole numbers, and integers. You'll also practice classifying numbers using examples.
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In this video, you'll learn how to correctly evaluate mathematical expressions using the order of operations. We'll review the meaning of PEMDAS, discuss the proper order for performing operations, and work through examples involving parentheses, exponents, multiplication, division, addition, and subtraction.
Chapter 2 – Functions
Function Basics
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In this video, you'll learn what a function is, how to identify the input and output of a function, and how to determine whether a graph represents a function using the Vertical Line Test. You'll also learn key function terminology, including independent and dependent variables, domain, and range.
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Analyzing Graphs of Functions
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In this video, you'll learn how to calculate the average rate of change of a function over an interval. We'll connect average rate of change to the slope of a secant line, develop the average rate of change formula, and work through examples involving functions represented by equations and graphs.
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In this video, you'll learn how to determine whether a graph is concave up or concave down and how to identify inflection points. We'll explore how concavity relates to the behavior of a function's rate of change and examine graphs to determine intervals of concavity and points where the concavity changes.
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In this video, you'll learn how to analyze the graph of a function by identifying key characteristics. We'll determine intervals where a function is increasing or decreasing, locate local maxima and minima, identify intervals of increasing and decreasing rate of change, and find inflection points.
Composite Functions
Parent Functions and Transformations
Inverse Functions
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In this video, you'll learn how to determine whether a function is one-to-one and therefore has an inverse. We'll develop the Horizontal Line Test, examine graphical and algebraic interpretations of one-to-one functions, and see how restricting the domain can create an inverse function.
Chapter 3 – Linear Functions
Linear Functions and Equations
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In this video, you'll learn how to solve linear equations in one variable. We'll develop a step-by-step process for simplifying equations, collecting like terms, and isolating the variable while working through examples involving fractions, decimals, and variables on both sides of the equation.
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In this video, you'll learn how slopes can be used to identify and write equations of parallel and perpendicular lines. We'll develop the relationships between their slopes, write equations through given points, and determine whether two lines are parallel, perpendicular, or neither.
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Absolute Value Equations
Inequalities
Copyright & Usage
© Understand The Math LLC. All Rights Reserved. These materials are freely provided for non-commercial educational use. Redistribution is permitted with proper attribution to UnderstandTheMath.com.
Quadratics and Polynomis
Factoring, graphing, and solving equations
Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams
Chapter 4 – Quadratic Functions and Equations
Complex Numbers
Quadratic Functions
Factoring Quadratics
Solving Quadratic Equations and Inequalities
Chapter 5 – Polynomial Functions
Polynomial Basics
Polynomial Operations
Factoring Polynomials
Finding Zeros of Polynomial Functions
Graphing Polynomial Functions
Copyright & Usage
© Understand The Math LLC. All Rights Reserved. These materials are freely provided for non-commercial educational use. Redistribution is permitted with proper attribution to UnderstandTheMath.com.
Advanced Functions
Rational, rational, exponential, and logarithmic functions.
Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams
Chapter 6 - Radical and Rational Functions
Radicals and Rational Exponents
Rational Expressions
Rational Equations and Inequalities
Rational Functions
Chapter 7 - Exponential and Logarithmic Functions
Exponential Functions
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In this video, you'll learn what exponential functions are and how they differ from linear functions. We'll explore the general form of an exponential function, compare exponential and linear growth, distinguish between exponential growth and decay, and discuss real-world applications of exponential models.
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In this video, you'll learn the seven fundamental laws of exponents. We'll simplify expressions using the product, quotient, power, zero exponent, negative exponent, power of a product, and power of a quotient rules, then apply multiple exponent rules together in more complex examples.
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In this video, you'll learn about the natural exponential function and the natural logarithm. We'll explore the number e, examine the relationship between y = e^x and y = ln(x), discuss their properties, and see why these functions are important in mathematics, science, and engineering.
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Logarithmic Functions
Solving Exponential and Logarithmic Functions
Applications of Exponential Functions
Copyright & Usage
© Understand The Math LLC. All Rights Reserved. These materials are freely provided for non-commercial educational use. Redistribution is permitted with proper attribution to UnderstandTheMath.com.
Advanced Topics
Systems, matrices, conics, and sequences.
Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.
Chapter 8 - Systems of Equations
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In this video, you'll learn two common methods for solving systems of equations in two variables: substitution and elimination. We'll follow a step-by-step process for each method, solve systems of linear equations, and verify solutions by checking them in the original equations.
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In this video, you'll learn how to determine the type of solution a system of equations has. We'll examine systems with one solution, no solution, and infinitely many solutions, relate these cases to the graphs of the lines, and classify systems as consistent or inconsistent and independent or dependent.
Chapter 9 - Matrices and Determinants
Introduction
Matrix Operations
Determinants
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In this video, you'll learn what a determinant is and how to compute the determinant of a 2 × 2 matrix. We'll develop the determinant formula, interpret determinant notation, determine when a determinant exists, and work through examples that illustrate how determinants provide important information about a matrix.
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Matrix Inverses
Solving Systems of Equations
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In this video, you'll learn how Gaussian elimination is used to solve systems of linear equations. We'll write systems as augmented matrices, perform elementary row operations, develop row echelon form, use back-substitution to find solutions, and solve systems with both two and three unknowns.
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In this video, you'll learn how Gauss-Jordan elimination is used to solve systems of linear equations. We'll write systems as augmented matrices, perform elementary row operations, develop reduced row echelon form, and solve systems with both two and three unknowns using a systematic matrix-based approach.
Chapter 9 Test
Chapter 10 - Conic Sections
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In this video, you'll learn the essential geometry and equations of parabolas. We'll define a parabola using its focus and directrix, identify vertices, foci, axes of symmetry, and directrices, develop the standard forms of parabola equations, and work through examples involving graphing parabolas and finding equations from given geometric information.
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In this video, you'll learn the essential geometry and equations of ellipses. We'll define ellipses using their foci, identify major and minor axes, vertices, foci, and eccentricity, develop the standard forms of ellipse equations, and work through examples involving graphing ellipses and finding equations from given geometric properties.
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In this video, you'll learn the essential geometry and equations of hyperbolas. We'll define hyperbolas using their foci, identify vertices, foci, asymptotes, and transverse axes, develop the standard forms of hyperbola equations, and work through examples involving graphing hyperbolas and finding equations from given vertices, foci, asymptotes, and points on the graph.
Chapter 11 - Sequences and Series
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In this video, you'll learn the fundamentals of sequences, including how to identify terms, find the first several terms of a sequence, develop formulas for the nth term, and calculate partial sums. We'll work with explicit and recursive formulas, recognize common sequence patterns such as arithmetic, geometric, and Fibonacci sequences, and solve examples involving sequence notation and partial sums.
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In this video, you'll learn how to use summation (sigma) notation to represent and evaluate series. We'll define the components of sigma notation, evaluate finite sums, write series using summation notation, identify patterns in sequences, and work through examples involving powers, fractions, alternating series, and polynomial expressions.
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In this video, you'll learn how arithmetic sequences are formed and how to work with them using formulas and examples. We'll develop the arithmetic sequence formula, find terms and nth-term formulas, determine whether a sequence is arithmetic, identify common differences, and calculate partial sums of finite arithmetic sequences using the arithmetic series formula.
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In this video, you'll learn how geometric sequences and geometric series work. We'll develop the geometric sequence formula, find terms and nth-term formulas, identify common ratios, determine whether a sequence is geometric, calculate partial sums of finite geometric sequences, and find the sums of infinite geometric series when they converge.
Copyright & Usage
© Understand The Math LLC. All Rights Reserved. These materials are freely provided for non-commercial educational use. Redistribution is permitted with proper attribution to UnderstandTheMath.com.
About Dr. Cheryl Hile
Understand The Math was created by Dr. Cheryl Hile, who earned a Ph.D. in Engineering Science and Applied Mathematics from Northwestern University. She has more than 25 years of experience teaching mathematics at the university level, including 20 years at Penn State. The lessons and guided notes on this site are based on materials she developed for her university courses.