Pre-calculus is the course with the least agreed syllabus in the American sequence, which is why parents and students search for it: it is a different course in every district, it exists in two versions since the College Board added AP Precalculus in 2023, and nobody tells a student which parts of it the calculus course next year will actually use. This guide is the pre-calculus curriculum unit by unit as US schools teach it, the AP version compared, what calculus will use and what it will not, the three places students lose the course, a readiness check from Algebra 2, and the overlap with CBSE Classes 11 and 12 for families from India.
The ten units of a traditional pre-calculus course
A traditional pre-calculus course (Larson, Sullivan, Blitzer, Stewart's "Precalculus", or the open Khan Academy course) is two-thirds Algebra 2 done deeper and one-third trigonometry done properly, with a few topics (conics, matrices, limits) that calculus or later courses will need. The order below is the commonest.
| Unit | What it covers | New, or deeper? | Weeks |
|---|---|---|---|
| 1. Functions and their analysis | Domain and range from a graph or formula; even and odd functions; increasing, decreasing and constant intervals; local and absolute extrema; average rate of change; transformations in full; compositions and their domains; inverses and their graphs; piecewise functions | Deeper than Algebra 2 unit 1 | 3 |
| 2. Polynomial and rational functions | End behaviour, zeros and multiplicity, turning points; the remainder and factor theorems; complex zeros and the fundamental theorem of algebra; rational functions with vertical, horizontal and slant asymptotes and holes; polynomial and rational inequalities | Deeper; slant asymptotes and inequalities are new | 4 |
| 3. Exponential and logarithmic functions | The function families and their transformations; the number e and continuous growth; the laws of logarithms; exponential and logarithmic equations of every type; modelling: compound interest, half-life, Newton's cooling, logistic growth | Deeper; logistic models are new | 3 |
| 4. Trigonometric functions | Angles in radians and degrees, arc length and area of a sector; the unit circle built from the special triangles; the six trigonometric functions of any angle; their graphs with amplitude, period, phase shift and vertical shift; the inverse trigonometric functions and their restricted domains | New in most of it, even for students who met the unit circle in Algebra 2 | 4 |
| 5. Trigonometric identities and equations | The Pythagorean, reciprocal, quotient, cofunction and even-odd identities; sum and difference formulas; double-angle and half-angle formulas; product-to-sum; proving identities; solving trigonometric equations on an interval and in general | New | 3 to 4 |
| 6. Applications of trigonometry | The law of sines (with the ambiguous case) and the law of cosines; area of a triangle; vectors in two dimensions, components, the dot product and the angle between vectors; polar coordinates and polar graphs; complex numbers in polar (trigonometric) form and De Moivre's theorem; parametric equations | New | 3 to 4 |
| 7. Conic sections | Parabolas, ellipses and hyperbolas from their equations and geometric definitions; completing the square to find the standard form; in some courses, rotation of axes and the polar forms | New for most; some schools do it in Algebra 2 | 2 |
| 8. Sequences, series and the binomial theorem | Arithmetic and geometric sequences and series; sigma notation; infinite geometric series; mathematical induction in some courses; the binomial theorem and Pascal's triangle | Deeper; induction and the binomial theorem are new | 2 |
| 9. Matrices and systems | Systems of three or more equations; Gaussian elimination; matrix operations, determinants and inverses; solving systems with inverses or Cramer's rule; partial fractions in some courses | New, or from Algebra 2 unit 10 | 2 |
| 10. An introduction to limits | Limits from graphs and tables; limit laws; one-sided limits; limits at infinity; continuity; the average rate of change becoming the derivative at the end of the year in some courses | New; the first page of calculus | 1 to 2 |
Two-semester college pre-calculus courses, and the "Precalculus with Trigonometry" or "Trigonometry/Math Analysis" courses in some states, are the same content with the trigonometry units sometimes a separate course.
AP Precalculus: the four units and the exam
The College Board's AP Precalculus course, first examined in May 2024, is organised into four units and built around modelling: functions are studied as descriptions of change, and most questions give a situation or a data set and ask which function fits it and why.
| AP unit | What it covers | Share of the exam |
|---|---|---|
| 1. Polynomial and rational functions | Change in tandem, rates of change, polynomial and rational functions, their zeros, end behaviour and asymptotes, modelling with them | 30 to 40 percent |
| 2. Exponential and logarithmic functions | Arithmetic and geometric sequences as functions, exponential and logarithmic functions, their inverses, modelling growth and decay, semi-log plots | 27 to 40 percent |
| 3. Trigonometric and polar functions | Periodic phenomena, the unit circle, sinusoidal functions and their transformations, the tangent and reciprocal functions, inverse trigonometric functions, trigonometric equations and the basic identities, polar coordinates and polar graphs, rates of change in polar functions | 30 to 35 percent |
| 4. Functions involving parameters, vectors and matrices | Parametric functions, implicitly defined functions, conic sections in parametric form, vectors, matrices and linear transformations | Taught, not examined |
The exam is three hours: forty multiple-choice questions (28 without a calculator, 12 with a graphing calculator) and four free-response questions (two with the calculator, two without), each free-response question in a fixed form: function concepts, modelling a non-periodic context, modelling a periodic context, and symbolic manipulations. Compared with a traditional course, AP Precalculus is heavier on interpreting rates of change from tables and graphs, on residuals and the choice of model, and on polar functions; and lighter on conic sections, sequences and series as a topic, trigonometric identities beyond the basic ones, the law of sines and cosines, and matrices. A student who takes AP Precalculus and then AP Calculus is well prepared; a student who takes it and then a traditional college calculus course may need to pick up the identities.
What calculus will actually use
This is the question nobody answers, and it decides where a student short of time should spend it.
Used constantly in calculus: function notation, domain, transformations, compositions and inverses (unit 1); the algebra of polynomial and rational functions, factoring, rational expressions, asymptotes (unit 2); exponential and logarithmic functions and every law of logarithms (unit 3); the unit circle, the graphs of sine, cosine and tangent, the inverse trigonometric functions (unit 4); the Pythagorean, double-angle and half-angle identities (unit 5); limits and continuity (unit 10).
Used occasionally: the law of sines and cosines (related-rates problems), vectors (Calculus 3 and physics), parametric and polar equations (Calculus BC and 2), sequences and series (Calculus BC and 2, where they return in full).
Barely used in calculus: conic sections beyond the circle, matrices and determinants, Cramer's rule, mathematical induction, the binomial theorem beyond expanding (a + b)² and (a + b)³, the sum-to-product identities.
A student with two months before AP Calculus and a weak pre-calculus year should spend them on units 1 to 5 and 10 and not open units 7 to 9. Our article on whether pre-calculus is harder than calculus has the ten-question check that tells a student which of those units needs the time.
The three places students lose the course
- The unit circle learned as a table. Sixteen angles, three functions each, memorised as forty-eight facts, forgotten by November. Learned the other way, from the two special triangles and the symmetry of the circle, it is built in a minute from a sketch and never lost. Every trigonometry unit and all of calculus depends on it.
- Identities as a list. Students see twenty identities and try to memorise them. Eight are enough (the three Pythagorean forms, the two quotient forms, sine and cosine of a sum, and the double angle), and the rest are derived from those in a line or two. Proving identities is then a game of choosing which of eight tools to use, which is teachable; memorising twenty is not.
- The algebra underneath. Rational expressions, factoring and exponent laws from Algebra 1 and 2, used at speed inside every pre-calculus problem. A student who makes an algebra error in every third line never finds out whether they understand the trigonometry. It is the first thing checked in the free class, because it is the commonest cause of a pre-calculus grade below the student's ability.
A readiness check from Algebra 2
Twelve questions. No calculator. Answers follow.
- If f(x) = x² + 1 and g(x) = 2x − 3, find f(g(x)).
- Find the inverse of f(x) = (x − 4)/3.
- Describe how the graph of y = −2(x + 1)² + 5 relates to y = x².
- Find all zeros of p(x) = x³ − 4x² + x + 6, given that 2 is one of them.
- Give the vertical and horizontal asymptotes of y = (3x + 1)/(x − 2).
- Solve 5 · 2ˣ = 80.
- Write log₃ 81 as a number, and write log a + 2 log b as a single logarithm.
- Convert 150° to radians, and 5π/4 to degrees.
- Give the exact values of sin 120° and cos(−π/3).
- Simplify (1 − cos²θ)/sin θ.
- Write the first four terms of the sequence aₙ = 3 · 2ⁿ⁻¹ and the sum of its first ten terms.
- Solve the system x + y + z = 6, x − y = 1, y + z = 4.
Answers: 1. 4x² − 12x + 10. 2. f⁻¹(x) = 3x + 4. 3. Shifted left 1, stretched vertically by 2, reflected in the x-axis, shifted up 5. 4. 2, 3, −1. 5. x = 2 and y = 3. 6. x = 4. 7. 4; log(ab²). 8. 5π/6; 225°. 9. √3/2; 1/2. 10. sin θ. 11. 3, 6, 12, 24; 3,069. 12. x = 2, y = 1, z = 3.
Ten or more right: ready. Seven to nine: ready, with the missed questions as September's warm-ups. Fewer than seven: four weeks on Algebra 2 functions, logarithms and the unit circle before the course, because pre-calculus will revisit none of them slowly.
Pre-calculus against CBSE Classes 11 and 12, for families from India
| Pre-calculus unit | Where CBSE covers it | The student from India is… |
|---|---|---|
| Functions and their analysis | Class 11 (relations and functions), Class 12 (types of functions, composition, inverse) | Ahead on definitions; behind on analysis from graphs, rate of change and modelling |
| Polynomial and rational functions | Class 9 and 10 (polynomials), Class 11 (quadratics, complex roots) | Ahead on algebra; behind on graphing, asymptotes and inequalities |
| Exponential and logarithmic functions | Class 11 (limited), ICSE Class 9 (logarithms) | Behind on the functions and modelling; the laws are often known |
| Trigonometric functions | Class 11 (trigonometric functions, radians, graphs) | Ahead |
| Identities and equations | Class 11 (sum, difference, multiple angles, general solutions) | Ahead, usually by a wide margin |
| Applications: sines, cosines, vectors, polar, complex | Class 11 (complex numbers in polar form), Class 12 (vectors) | Ahead on complex numbers and vectors; behind on polar graphs and parametric equations |
| Conic sections | Class 11 | Ahead |
| Sequences, series, binomial theorem, induction | Class 11 | Ahead |
| Matrices and systems | Class 12 | Ahead or level, depending on timing |
| Limits | Class 11 (limits and derivatives) | Ahead |
A student from Class 11 is ahead on most of pre-calculus and behind on exactly the things AP Precalculus examines most: reading a function from a table or a graph, choosing a model, interpreting a rate of change in a sentence, and polar functions. For a traditional pre-calculus course the student is usually comfortable; for AP Precalculus the first term should go to the modelling questions and the free-response form.
Where to get help
Our pre-calculus classes teach the units above in the school's order, with the unit circle built from the triangles and the eight identities on one page, one tutor with one student on a shared whiteboard, every line read, the AP Precalculus free-response questions scored to the rubric where the student is in that course, at 7 am before school, after 4 pm and on weekend mornings US time, from US$10 to $13 an hour. The first 30-minute class is free and runs the readiness check. The Pre-Calculus and AP Precalculus pages say what the class works on each term; our math tutors from India to USA page says how the arrangement works for US families. The Algebra 2 syllabus is the guide before this one, and what grade you take calculus says where this course sits on the two tracks.
Questions parents ask
1What is in the pre-calculus syllabus?
2What is the difference between Pre-Calculus and AP Precalculus?
3What grade is pre-calculus?
4What does calculus actually use from pre-calculus?
5Which CBSE class covers pre-calculus?
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