Math Made Easy’s precalculus lessons cover all 18 chapters, from domain, range and inverse functions to expected value. Every precalculus lesson is free and aligned to the Common Core functions and trigonometry standards. Mathbot, our blue robot coach, guides twelfth graders preparing for calculus through worked examples first.
What the precalculus lessons teach
- Functions: Domain, Range, Composition and Inverses
- Angles, Radians and the Unit Circle
- Laws of Sines and Cosines and Triangle Area
- Polar Coordinates and Parametric Equations
- Series, Binomial Theorem and an Introduction to Limits
Work in order: read a lesson such as Functions: Domain, Range, Composition and Inverses, solve the matching worksheet, check it against the answer key, take the chapter test, and finish with the 10-question quiz. Vectors and matrices are tied to the geometry of the plane, which makes the notation feel purposeful.
Keep a one-page sheet of unit circle values and identities, and rewrite it from memory each week. Domains covered by the precalculus lessons: Functions, Trigonometric Functions, Vector & Matrix Quantities, The Complex Number System, and Expressing Geometric Properties with Equations.
Chapter 1 · Precalculus · F-IF.A.1, F-BF.A.1c, F-BF.B.4b, F-BF.B.4c, F-BF.B.4d
Domains, Composition and Inverse Functions Explained
Write domains in interval notation, compose and decompose functions, and find inverses, checking with f(g(x)) = x and reading inverse values from graphs and tables. For f(x) = 3x - 5 you will confirm the inverse (x + 5)/3. Read the lesson on functions, solve the worksheet and check it with the answer key, then take the quiz and the test.
- Interval notation
- Composition
- Inverse functions
Chapter 2 · Precalculus · A-APR.B.2, A-APR.B.3, F-IF.C.7c, F-IF.C.7d, N-CN.C.9
Polynomial and Rational Functions: Zeros and Asymptotes
Use the Rational Root Theorem and the Fundamental Theorem of Algebra to find real and complex zeros, including conjugate pairs, and handle slant asymptotes and inequalities. Candidate roots of 2x^3 - 3x^2 - 8x - 3 are factors of 3 over factors of 2. Work through the polynomial and rational functions lesson, test yourself on the worksheet with its answer key, and finish with the quiz and the scored test.
- Rational Root Theorem
- Complex zeros
- Slant asymptotes
Chapter 3 · Precalculus · F-LE.A.4, F-BF.B.5, F-IF.C.8b
Exponents, Logarithms and Growth Models Explained
Treat exponents and logarithms as inverses, apply the laws of logarithms, solve equations, and compare logistic and continuous growth models and logarithmic scales. A quake of magnitude 7 is ten times stronger than one of magnitude 6. Mathbot suggests the lesson first, then the worksheet with answer key, and finally the quiz and test to prove your exponential and log models skills.
- Log laws
- Logistic growth
- Log scales
Chapter 4 · Precalculus · F-TF.A.1, F-TF.A.2, F-TF.A.3, F-TF.A.4
Radians, Arc Length and the Unit Circle
Convert between degrees and radians, compute arc length and angular speed, and know the special angles pi/6, pi/4, and pi/3 with reference angles and symmetry. A wheel turning 2 pi/3 radians per second has a definite speed at its rim. Read, solve the worksheet, grade it with the answer key; the quiz is a quick check on the unit circle, and the test comes with a scoring guide.
- Radian conversion
- Special angles
- Reference angles
Chapter 5 · Precalculus · F-TF.B.5, F-IF.C.7e, F-BF.B.3
Graphing Sine, Cosine and Tangent Functions
Graph sine, cosine, and tangent with amplitude, period, phase shift, and vertical shift, and write a sinusoidal model from data. If daylight swings between 9 and 15 hours across the year, the midline is 12 and the amplitude 3. Learn trig graphs in the lesson, build speed on the worksheet, correct mistakes with the answer key, and confirm mastery with the quiz and test.
- Sinusoidal models
- Transformations
- Tangent asymptotes
Chapter 6 · Precalculus · F-TF.B.6, F-TF.B.7
Inverse Trigonometric Functions and Trig Equations
Restrict domains to define arcsin, arccos, and arctan, then solve trigonometric equations on an interval and in general. Solving 2 sin x = 1 gives pi/6 and 5 pi/6 on 0 to 2 pi, and you will interpret answers in context. Learn inverse trigonometry in the lesson, build speed on the worksheet, correct mistakes with the answer key, and confirm mastery with the quiz and test.
- Inverse trig functions
- Equation solving
- General solutions
Chapter 7 · Precalculus · F-TF.C.8, F-TF.C.9
Proving and Using Trigonometric Identities
Use reciprocal, quotient, and Pythagorean identities to verify new identities, then apply sum, difference, and double-angle formulas. From sin(a + b) you can find sin 75 degrees exactly, as (sqrt 6 + sqrt 2)/4. After the lesson, the worksheet and answer key let you fix errors in trig identities before you attempt the quiz and the test.
- Verifying identities
- Sum formulas
- Double-angle formulas
Chapter 8 · Precalculus · G-SRT.D.9, G-SRT.D.10, G-SRT.D.11
Law of Sines, Law of Cosines and Area
Find area with (1/2)ab sin C, solve triangles with the Law of Sines, including the ambiguous SSA case, and with the Law of Cosines, and use Heron's formula. Navigation problems with bearings show why each tool exists. Take the quiz first to see what you know about laws of sines and cosines, then study the lesson, finish the worksheet, and use the test scoring guide.
- Ambiguous case
- Heron's formula
- Bearings
Chapter 9 · Precalculus · N-VM.A.1, N-VM.A.2, N-VM.A.3, N-VM.B.4, N-VM.B.5
Working with Vectors in the Plane
Describe vectors by magnitude, direction, and components, add and subtract them with the parallelogram rule, scale them, and form unit vectors. A plane flying at 200 mph into a 30 mph crosswind needs vector addition, and the dot product gives angles. Do the lesson, then the worksheet and answer key, then the scored test on vectors; the short quiz gives a fast check on the way.
- Component form
- Vector addition
- Dot product
Chapter 10 · Precalculus · N-CN.B.4, N-CN.B.5, N-CN.B.6, N-CN.A.3
Polar Form and Powers of Complex Numbers
Plot complex numbers in the plane, find modulus and argument, and convert between rectangular and polar form. Multiply and divide in polar form and apply De Moivre's theorem, such as raising 1 + i to the 8th power to get 16. Lesson first, worksheet next: the answer key confirms each polar complex numbers result, and the quiz and test with scoring guide close the chapter.
- Modulus and argument
- Polar form
- De Moivre's theorem
Chapter 11 · Precalculus · N-CN.B.4, F-IF.C.7
Polar Coordinates and Parametric Curves
Plot points in polar coordinates, convert between polar and rectangular equations, and graph circles, roses, and limacons. Parametric equations then describe motion, like a thrown ball with x = 20t and y = -16t^2 + 40t. Read the lesson on polar and parametric equations, solve the worksheet and check it with the answer key, then take the quiz and the test.
- Polar graphs
- Coordinate conversion
- Parametric motion
Chapter 12 · Precalculus · N-VM.C.6, N-VM.C.7, N-VM.C.8, N-VM.C.9, N-VM.C.10
Matrix Operations, Determinants and Inverses
Store data in matrices, add them, scale them, and multiply 2x2 and 3x3 matrices, using identity and zero matrices. You will compute a 2x2 determinant, as ad - bc, and find an inverse. Work through the matrices lesson, test yourself on the worksheet with its answer key, and finish with the quiz and the scored test.
- Matrix multiplication
- Determinants
- Inverse matrices
Chapter 13 · Precalculus · A-REI.C.8, A-REI.C.9, N-VM.C.11, N-VM.C.12
Solving Systems and Transforming the Plane
Write a system as AX = B and solve it with an inverse matrix, then see matrices as transformations of the plane, with the absolute value of the determinant giving area scaling. A three-variable mixture story makes the method concrete. Mathbot suggests the lesson first, then the worksheet with answer key, and finally the quiz and test to prove your matrix systems skills.
- Matrix equations
- Solving with inverses
- Determinant and area
Chapter 14 · Precalculus · G-GPE.A.1, G-GPE.A.2, G-GPE.A.3
Parabolas, Ellipses and Hyperbolas in Standard Form
Review circles, then build parabolas from focus and directrix and ellipses and hyperbolas from foci, using sums and differences of distances. You will read vertices, foci, and asymptotes from standard forms, such as x^2/25 + y^2/9 = 1. Every page pairs a lesson with a worksheet and answer key, plus a quiz and a test whose scoring guide shows where conic sections points were earned.
- Ellipses
- Hyperbolas
- Foci and vertices
Chapter 15 · Precalculus · A-APR.C.5, A-SSE.B.4, F-BF.A.2
Series, Binomial Expansions and Limits
Use sigma notation, sum arithmetic and geometric series, test infinite geometric series for convergence, expand with the binomial theorem and Pascal's triangle, and meet limits informally. The series 1 + 1/2 + 1/4 + ... settles at 2. Read, solve the worksheet, grade it with the answer key; the quiz is a quick check on series and limits, and the test comes with a scoring guide.
- Infinite series
- Binomial theorem
- Intro to limits
Chapter 16 · Precalculus · S-CP.B.8, S-CP.B.9, S-MD.A.1, S-MD.A.2
Counting Methods and Probability Rules
Apply the fundamental counting principle, then permutations and combinations, and fold counting into probability, the general multiplication rule, and expected value. Choosing 3 officers from 10 students in order gives 720 ways, and as a committee, 120. Learn counting and probability in the lesson, build speed on the worksheet, correct mistakes with the answer key, and confirm mastery with the quiz and test.
- Permutations
- Combinations
- Expected value
Chapter 17 · Precalculus · S-MD.A.3, S-MD.A.4, S-MD.B.5
Expected Value and Smart Decisions
Build probability distributions from theory or data, compute the expected value of a random variable, and weigh decisions such as insurance or a game. A $2 ticket with a 1 in 100 chance at $100 has an expected net of -$1. After the lesson, the worksheet and answer key let you fix errors in expected value before you attempt the quiz and the test.
- Probability distributions
- Expected value
- Weighing decisions
Chapter 18 · Precalculus · N-CN.C.8, G-GMD.A.2
Complex Factoring and Sphere Volume
Extend polynomial identities to complex numbers, factoring sums of squares like x^2 + 4 = (x + 2i)(x - 2i), and follow an informal argument for the volume of a sphere. The same ideas link algebra with solids. Take the quiz first to see what you know about complex factoring, then study the lesson, finish the worksheet, and use the test scoring guide.
- Complex factoring
- Polynomial identities
- Sphere volume
Grade 12 Precalculus Lessons: frequently asked questions
Are the precalculus lessons free?
Yes. All 18 precalculus lessons are free to read, with no account needed. Each precalculus lesson points ahead to its worksheet, test and quiz.
Which standards do the precalculus lessons follow?
They follow the Common Core high school standards for Functions, Trigonometry, Vector & Matrix Quantities, and Complex Numbers, which organizes the course into Functions, Trigonometric Functions, Vector & Matrix Quantities, The Complex Number System, and Expressing Geometric Properties with Equations. Each precalculus lesson is tagged with codes such as F-TF.B.6, N-VM.A.1 and N-VM.C.6.
What should I know before starting precalculus?
Students should have completed Algebra 2, including exponentials, logarithms, polynomial functions and right triangle trigonometry. If a gap shows up, begin with the lesson on domain, range and inverse functions.
How are the precalculus lessons organized?
Functions and trigonometry take the first half, vectors, complex numbers and matrices the middle, and conics, series and probability the rest. The precalculus chapters can be read in order or picked by topic.
What comes after precalculus?
Most students continue into calculus, either AP Calculus AB or a college calculus course. Finishing the 18 chapters on domain, range and inverse functions through expected value leaves students ready for that step.