Algebra 2 is the course in which American high school math stops being arithmetic with letters and becomes the study of functions, and it is the course students most often describe as the one that got away from them. It is also the course most students are in when they sit the SAT, which draws about a third of its math section from the first half of it. Parents search for the Algebra 2 syllabus because a 10th or 11th grader is about to start it, or is in it and sinking, or is choosing between Algebra 2, Pre-Calculus and Statistics for next year. This guide is the Algebra 2 curriculum unit by unit as US schools teach it, what each unit assumes, the pacing, what the SAT and the Regents test from it, the four places the course is lost, a readiness check, and the overlap with CBSE Class 11 for families from India.
The ten units, in order
As with Algebra 1, the textbooks (Big Ideas, enVision, Illustrative Mathematics, Eureka, Khan Academy's course) group the material a little differently, and the state standards contain the same content. The order below is the commonest and the one the Regents course follows closely.
| Unit | What it covers | What it assumes | Weeks |
|---|---|---|---|
| 1. Functions and transformations | Function notation, domain and range, intervals of increase and decrease; parent functions (linear, absolute value, quadratic, square root, cube root, cubic); translations, reflections, stretches and compressions as changes to the equation; piecewise functions; inverses and compositions; solving linear systems in three variables, sometimes with matrices | Algebra 1 units 4 and 9 | 3 to 4 |
| 2. Quadratics revisited and complex numbers | Completing the square and vertex form in full; the quadratic formula and the discriminant; the imaginary unit i, operations with complex numbers, complex solutions of quadratics; quadratic inequalities; systems of a line and a parabola | Algebra 1 unit 9 | 3 |
| 3. Polynomial functions | Operations with polynomials; end behaviour and the shape of graphs by degree; the remainder and factor theorems; synthetic and long division; finding zeros and their multiplicity; the fundamental theorem of algebra; graphing polynomials from factored form; solving polynomial equations | Units 1 and 2; factoring from Algebra 1 | 4 |
| 4. Radical functions and rational exponents | Rational exponents and their laws; simplifying radicals of any index; operations with radicals; graphing square-root and cube-root functions with transformations; solving radical equations and checking for extraneous solutions; inverses of radical and power functions | Algebra 1 unit 10 | 2 to 3 |
| 5. Exponential and logarithmic functions | Exponential growth and decay, compound interest, the number e; logarithms as the inverse of exponentials; the laws of logarithms; change of base; solving exponential and logarithmic equations; graphing both families with transformations; modelling with half-life and doubling time | Algebra 1 unit 6; unit 1 inverses | 4 |
| 6. Rational functions | Simplifying, multiplying, dividing, adding and subtracting rational expressions; graphing rational functions with vertical and horizontal asymptotes and holes; solving rational equations and checking for extraneous solutions; inverse and joint variation | Unit 3 factoring and division | 3 |
| 7. Sequences and series | Arithmetic and geometric sequences, explicit and recursive; sigma notation; sums of finite arithmetic and geometric series; the sum of an infinite geometric series | Algebra 1 unit 10 | 2 |
| 8. Trigonometric functions | Radians and degrees; the unit circle and the values at the special angles; sine, cosine and tangent as functions; their graphs with amplitude, period, phase shift and midline; the Pythagorean identity and the basic reciprocal and quotient identities; solving simple trigonometric equations; modelling periodic situations | Right-triangle trigonometry from Geometry | 3 to 4 |
| 9. Probability and statistics | Conditional probability and independence; two-way tables; permutations and combinations; the binomial setting; the normal distribution and z-scores; sampling methods, margin of error and the design of surveys and experiments; using simulation to judge a claim | Algebra 1 unit 11 | 3 |
| 10. Conic sections and matrices (some schools) | Circles, parabolas, ellipses and hyperbolas from their equations; matrix operations and inverses; some schools include these, some leave them to Pre-Calculus | Unit 2 | 0 to 2 |
New York's Regents Algebra II course includes everything above except conics and matrices, with particular weight on functions, exponentials and logarithms, and statistics. Texas's Algebra II TEKS include conics and matrices and less statistics.
How the year is paced
The first semester is usually units 1 to 5: functions, quadratics, polynomials, radicals, exponentials and logarithms. The second is units 6 to 9 or 10. The pace is the shock for most students: Algebra 1 spent four weeks on linear functions; Algebra 2 spends three weeks on each of six different function families and expects the student to transform, invert, graph and solve each of them. A student who falls behind in unit 3 (polynomials) is behind in units 4 and 6 (radicals and rationals) automatically, because they are built on it.
The honest advice on pacing: Algebra 2 is the one course where a student should be a week ahead of the class rather than a week behind, because every unit's first lesson assumes the previous unit is finished.
What the SAT tests from Algebra 2
Most students sit the SAT in the spring of 11th grade, in the middle of this course, which is why it matters here. The SAT math section has four domains, and one of them, Advanced Math, about 35 percent of the section, is built from the first half of Algebra 2: quadratics in every form, polynomial factoring and division, exponential functions and their equations, radicals and rational exponents, rational expressions, systems with one non-linear equation, and function notation and transformations. The second half of Algebra 2 (logarithms, trigonometric functions, sequences and series, the normal distribution) is not on the SAT at all.
The practical consequence: a student who finishes Algebra 2 unit 5 by February and practises those units in the SAT's own format has covered the whole Advanced Math domain before the spring test date, whatever the rest of the course does. Our SAT math formula sheet guide covers what the test gives and what must be known.
The four places the course is lost
- Transformations. Unit 1 teaches that f(x − 3) + 2 moves a graph right 3 and up 2, that f(2x) compresses it, that −f(x) flips it. Students who memorise this as rules for parabolas and do not understand it as a property of every function meet it again in units 4, 5, 6 and 8, each time as a new mystery. It is the single idea worth getting right in September.
- Logarithms. A new kind of operation, defined as an inverse, with laws that look like the exponent laws turned inside out. Students who are told "a logarithm is an exponent" and practise rewriting log₂ 8 = 3 as 2³ = 8 fifty times, in both directions, before touching the laws, are fine. Students who go straight to the laws are lost for the rest of the year.
- Factoring, again. Algebra 2 assumes the Algebra 1 factoring patterns are automatic, and uses them in polynomials, rational expressions and trigonometric equations. A student who found factoring hard in Algebra 1 and was promoted anyway meets it in unit 3 at twice the speed. Three weeks of twenty minutes a day, before unit 3 starts, is the fix.
- The unit circle. Unit 8 asks the student to know sine and cosine at sixteen angles in radians, from memory, and to read them from a picture rather than a table. Students who learn the two special triangles and build the circle from them have it in a week; students who memorise the table forget it by the exam.
A readiness check, before the course starts
Twelve questions from Algebra 1 and Geometry that Algebra 2 assumes. No calculator. Answers follow.
- Factor x² − 5x − 14.
- Factor 2x² + 7x + 3.
- Solve x² − 6x + 2 = 0 by completing the square or the formula; give exact answers.
- Write the vertex form of y = x² + 4x + 1.
- Simplify √72.
- Simplify (x³)⁴ · x⁻² .
- If f(x) = 2x² − 3, find f(−2).
- Solve the system y = x² and y = 2x + 3.
- Simplify (x² − 9)/(x² + 5x + 6).
- Find the slope of a line perpendicular to y = (2/3)x − 1.
- In a right triangle with legs 5 and 12, find sin of the angle opposite the side of length 5.
- A population of 500 grows 8 percent a year. Write a function for the population after t years.
Answers: 1. (x − 7)(x + 2). 2. (2x + 1)(x + 3). 3. x = 3 ± √7. 4. y = (x + 2)² − 3. 5. 6√2. 6. x¹⁰. 7. 5. 8. (3, 9) and (−1, 1). 9. (x − 3)/(x + 2). 10. −3/2. 11. 5/13. 12. P(t) = 500(1.08)ᵗ.
Ten or more right: ready. Seven to nine: ready, with the missed topics as September's warm-ups. Fewer than seven: a month on Algebra 1 factoring, quadratics and function notation before the course starts, because every unit will assume them; our article on why Algebra 2 feels harder than Algebra 1 says what that month looks like.
Algebra 2 against CBSE Class 11, for families from India
| Algebra 2 unit | Where CBSE covers it | The student from India is… |
|---|---|---|
| Functions and transformations | Class 11 (relations and functions), without transformations | Behind on transformations, inverses and piecewise functions |
| Quadratics and complex numbers | Class 10 (quadratics), Class 11 (complex numbers and quadratic equations) | Ahead |
| Polynomial functions | Class 9 and 10 (polynomials, remainder and factor theorems) | Ahead on algebra; behind on end behaviour and graphing from factored form |
| Radicals and rational exponents | Class 9 (surds, laws of exponents for real numbers) | Ahead on simplifying; behind on radical functions as graphs |
| Exponential and logarithmic functions | Class 11 in some boards; logarithms are often met through ICSE Class 9 or not at all in CBSE school maths | Varies; the laws are often known, the functions and modelling are new |
| Rational functions | Not as a topic; rational expressions appear in Class 8 to 10 algebra | Behind on asymptotes and graphs |
| Sequences and series | Class 10 (AP), Class 11 (AP, GP, sums, infinite GP) | Ahead |
| Trigonometric functions | Class 10 (ratios, identities), Class 11 (trigonometric functions, radians, graphs, general solutions) | Ahead, usually by a wide margin |
| Probability and statistics | Class 10 (probability), Class 11 (statistics: mean deviation, variance), Class 12 (conditional probability, binomial) | Ahead on probability; behind on the normal distribution, sampling and experimental design |
| Conics and matrices | Class 11 (conic sections), Class 12 (matrices) | Ahead |
A student who arrives from Class 11 is often the strongest in the Algebra 2 room on trigonometry and sequences and the weakest on transformations, logarithmic modelling and statistics. The first term should go to those three, and to the SAT's way of asking an Advanced Math question, which is a word problem with a function in it, a form CBSE rarely uses.
Where to get help
Our Algebra 2 classes teach the ten units in this order, a week ahead of the school where we can, one tutor with one student on a shared whiteboard, every line read, the SAT's Advanced Math questions practised from February and the Regents Algebra II from released papers in the spring, 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 Algebra 2 page says 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 1 syllabus is the guide before this one and the Pre-Calculus syllabus the one after.
Questions parents ask
1What is in the Algebra 2 syllabus?
2What grade is Algebra 2?
3Is Algebra 2 on the SAT?
4Why is Algebra 2 harder than Algebra 1?
5Which CBSE class covers Algebra 2?
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