Learn Precalculus – Free Course with Videos and Guided Notes
Build a strong foundation in precalculus with structured video lessons, guided notes, chapter tests, and study resources. Covering functions, trigonometry, exponential and logarithmic models, matrices, conic sections, and sequences, this free course develops understanding and confidence through clear, step-by-step instruction.
Precalculus Foundations
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
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
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
Looking for completed guided notes? The notes for this course are available through the Algebra and Trigonometry note packs in the store.
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 Precalculus Topics
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 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.
Looking for completed guided notes? The notes for this course are available through the Algebra and Trigonometry note packs in the store.
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.
Trigonometry
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 - Basic Trigonometric Concepts
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In this video, you'll learn how angles are measured using degrees and radians, the two most important units in trigonometry. We'll explore the meaning of each unit, examine how they relate to circles, and develop an understanding of why radians play such an important role in mathematics.
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In this video, you'll learn the properties of similar triangles and how they can be used to solve geometric and real-world application problems. We'll determine unknown side lengths and angle measures using proportions and explore practical examples involving similar triangles.
Chapter 2 - Right Triangle Trigonometry
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In this video, you'll learn how the six trigonometric functions relate the angles of a right triangle to ratios of its sides. We'll define sine, cosine, tangent, cosecant, secant, and cotangent, develop their reciprocal relationships, and use right triangles to evaluate each trigonometric function.
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In this video, you'll learn the special side relationships found in 45-45-90 and 30-60-90 triangles. We'll develop formulas for finding unknown side lengths, explore the exact values of related trigonometric functions, and see how these special triangles simplify many trigonometry problems.
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Chapter 3 - Trigonometric Functions
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In this video, you'll learn the fundamental ideas behind the unit circle and how it connects geometry and trigonometry. We'll review the equation of a circle, develop the equation of the unit circle, identify quadrant points, and use the unit circle equation to find coordinates of points on the circle.
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In this video, you'll learn the special angles and coordinate points that appear most frequently in trigonometry. We'll review the quadrants of the unit circle, develop the coordinates associated with 30°, 45°, and 60°, and identify the key angles and points that are essential to memorize.
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In this video, you'll learn a systematic approach for memorizing the unit circle. We'll organize the degree and radian measures of common angles, use patterns to remember their locations, and develop strategies for recalling coordinate points and quadrant signs.
▶️ Video 📝 Notes 📝 Fillable Unit Circle
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In this video, you'll learn what a reference angle is and why it plays an important role in trigonometry. We'll examine how reference angles are defined in each quadrant, develop formulas for finding them, and work through several examples involving angles measured in both degrees and radians.
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In this video, you'll learn how trigonometric functions can be extended from right triangles to real numbers using the unit circle. We'll develop the definitions of the six trigonometric functions in terms of unit circle coordinates and evaluate trigonometric functions for points on the unit circle.
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In this video, you'll learn how the signs of trigonometric functions depend on the quadrant in which an angle is located. We'll examine the signs of sine, cosine, tangent, and their reciprocal functions, introduce helpful memory aids, and use sign information to determine the location of points on the unit circle.
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In this video, you'll learn a step-by-step process for finding the exact value of trigonometric functions for angles outside the first quadrant. We'll determine quadrants, find reference angles, identify the correct signs, and evaluate trigonometric functions using special-angle values from the unit circle.
Chapter 4 - Graphs of Trigonometric Functions
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In this video, you'll learn how the sine and cosine graphs are connected to the unit circle and how they form the foundation of sinusoidal functions. We'll develop the graphs of sine and cosine, introduce amplitude, period, phase shift, and midline, and use these ideas to graph sinusoidal functions.
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In this video, you'll learn how sinusoidal functions can be used to model real-world periodic behavior. We'll analyze the motion of a Ferris wheel rider, determine the amplitude, period, phase shift, and midline of the model, and develop both sine and cosine equations that describe the rider's height over time.
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In this video, you'll learn how to graph the tangent function and identify its key features. We'll develop the graph using the unit circle, determine where tangent is undefined, locate vertical asymptotes, examine its period, and graph general tangent functions involving horizontal and vertical transformations.
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In this video, you'll learn how to graph the reciprocal trigonometric functions. We'll express cosecant, secant, and cotangent in terms of sine, cosine, and tangent, determine where each function is undefined, identify vertical asymptotes and periods, and use these properties to sketch their graphs.
Chapter 5 - Inverse Trigonometric Functions
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In this video, you'll learn what it means for a function to have an inverse and how to determine whether an inverse exists. We'll introduce one-to-one functions, use the horizontal line test to identify functions with inverses, and see how restricting the domain can create an inverse for functions that are not originally one-to-one.
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In this video, you'll learn the fundamental ideas behind inverse trigonometric functions. We'll introduce arcsine, arccosine, and arctangent, discuss why trigonometric functions require restricted domains to have inverses, examine their graphs, domains, and ranges, and develop the key properties that connect inverse trigonometric functions to their corresponding trigonometric functions.
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In this video, you'll learn how to evaluate inverse trigonometric functions using special angles and the unit circle. We'll determine exact values of inverse sine, inverse cosine, and inverse tangent functions, identify when inverse trig expressions are undefined, and use the domains and ranges of inverse trig functions to select the correct angle.
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In this video, you'll learn how trigonometric functions and inverse trigonometric functions interact through composition. We'll use inverse function properties to simplify expressions, evaluate compositions such as sin(sin⁻¹(x)) and sin⁻¹(sin(x)), determine when inverse properties apply, and use reference angles and right triangles to simplify more complicated inverse trigonometric expressions.
Chapter 6 - Trigonometric Identities
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In this video, you'll learn the most important trigonometric identities used throughout trigonometry, calculus, physics, and engineering. We'll introduce reciprocal identities, ratio identities, Pythagorean identities, periodicity identities, even-odd identities, cofunction identities, sum and difference formulas, product-to-sum formulas, sum-to-product formulas, and double-angle formulas.
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In this video, you'll learn the Pythagorean identities and how they are used to relate trigonometric functions. We'll derive the identity sin²θ + cos²θ = 1, develop equivalent forms involving tangent, secant, cotangent, and cosecant, and use these identities to determine unknown trigonometric function values.
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In this video, you'll learn how periodicity allows trigonometric functions to repeat their values. We'll determine the periods of the six trigonometric functions, develop periodicity identities, and use them to evaluate trigonometric functions involving large positive and negative angles.
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In this video, you'll learn how symmetry properties lead to important trigonometric identities. We'll define even and odd functions, identify which trigonometric functions are even or odd, develop the corresponding identities, and use them to rewrite trigonometric expressions involving negative angles.
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In this video, you'll learn how complementary and supplementary angles affect trigonometric function values. We'll define complementary and supplementary angles, develop the cofunction identities, derive supplementary angle identities, and use them to evaluate and simplify trigonometric expressions.
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In this video, you'll learn how to evaluate trigonometric functions involving sums and differences of angles. We'll develop the sum and difference formulas for sine, cosine, and tangent and use them to find exact values of trigonometric expressions that cannot be evaluated directly from the unit circle.
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In this video, you'll learn three important families of identities that are widely used in calculus, physics, and engineering. We'll develop the double-angle formulas, derive the power reduction formulas, introduce the half-angle formulas, and use each set of identities to evaluate trigonometric expressions.
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In this video, you'll learn strategies for rewriting complicated trigonometric expressions in simpler forms. We'll combine trigonometric identities with algebraic techniques such as factoring, common denominators, conjugates, and substitutions to simplify a variety of trigonometric expressions.
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In this video, you'll learn a systematic process for verifying trigonometric identities. We'll discuss how to choose the best side to work on, identify useful algebraic and trigonometric techniques, and apply a step-by-step strategy to verify a variety of increasingly challenging identities.
Chapter 7 - Trigonometric Equations
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In this video, you'll learn what trigonometric equations are and how they differ from trigonometric identities. We'll classify the major types of trigonometric equations, introduce the basic forms involving sine, cosine, and tangent, and develop a framework for solving all trigonometric equations covered in this chapter.
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In this video, you'll learn how to solve trigonometric equations that contain only one trigonometric function. We'll review the solutions of the basic equations sin θ = c, cos θ = c, and tan θ = c, use the unit circle to determine solution angles, and find both all solutions and solutions on a specified interval.
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In this video, you'll learn how to solve trigonometric equations written in quadratic form. We'll use substitution techniques to rewrite equations as quadratics, factor or solve the resulting polynomial, identify valid trigonometric solutions, and reject any extraneous values outside the range of the trigonometric function.
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In this video, you'll learn how trigonometric identities can transform difficult equations into forms that can be solved using factoring and other algebraic techniques. We'll apply Pythagorean identities, double-angle identities, and other common identities to simplify equations and find all solutions.
Chapter 8 - Additional Topics
Oblique Triangles
Polar Coordinates
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In this video, you'll learn how to convert equations between polar and rectangular form. We'll develop the relationships between polar and rectangular coordinates, identify the common forms of polar and rectangular equations, convert rectangular equations to polar equations, and convert polar equations to rectangular equations using coordinate relationships and trigonometric identities.
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Complex Numbers
Looking for completed guided notes? The notes for this course are available through the Algebra and Trigonometry note packs in the store.
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.