Math Made Easy’s precalculus course has 18 chapters, from domain, range and inverse functions to expected value. Each precalculus chapter has a lesson, a worksheet and test with answer keys, a scoring guide and a 10-question quiz. It follows the Common Core functions and trigonometry standards and is free to use.
What is in the course
- Functions: Domain, Range, Composition and Inverses
- Angles, Radians and the Unit Circle
- Trigonometric Identities
- Complex Numbers in Polar Form
- Solving Systems with Matrices
- Counting, Permutations, Combinations and Probability
Functions and trigonometry take the first half, vectors, complex numbers and matrices the middle, and conics, series and probability the rest. Trigonometry occupies five chapters, which is why it is the single largest strand in the course.
Take each precalculus chapter in the same order: lesson, worksheet, answer key, test, quiz. Mathbot, the blue robot coach, marks the key idea in every lesson, including the one on angles and the unit circle. Chapters such as vectors and conic sections are typical stopping points for review.
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, Worksheets, Tests and Quizzes: frequently asked questions
How many chapters are in precalculus?
The precalculus course has 18 chapters, so there are 18 lessons, 18 worksheets, 18 tests and 18 quizzes. The list starts with domain, range and inverse functions and ends with expected value.
What order should I study precalculus in?
Follow the chapter list from domain, range and inverse functions to expected value, passing through angles and the unit circle and vectors on the way. Within each precalculus chapter use the lesson, the worksheet, its answer key, the test and then the quiz.
Which standards does the precalculus course follow?
The course follows 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 chapter is tagged with codes such as F-TF.B.6, N-VM.A.1 and N-VM.C.6.
Who is the precalculus course written for?
It is written for twelfth graders preparing for calculus, and it also works as a free reference for parents and teachers. Chapters such as angles and the unit circle and conic sections can be used on their own for review.
How do I know I am ready to move on in precalculus?
Score well on the precalculus chapter quiz and test, then compare every miss with the answer keys. Chapters on angles and the unit circle and conic sections are good checkpoints before moving on.