IB Math AA HL Topics (2026)

This is the official IB Math AA HL syllabus. I have expanded some points to be clearer and more specific, from years of teaching AA HL. Every subtopic shows the exact formulas, notation, and depth you need to know, so you know exactly what the IB expects at each point.

Topic 1

Number & Algebra

1.1 Operations with Numbers

Operations with numbers in the form a × 10^k
Where 1 ≤ a < 10 and k is an integer

1.2 Arithmetic Sequences

Formulae for the nth term: u_n = u₁ + (n − 1)d
Sum of first n terms: S_n = n2(2u₁ + (n − 1)d)
Use of sigma (Σ) notation, e.g. Σ_k=1ⁿ (3k + 2)
Applications: analysis and prediction in real models

1.3 Geometric Sequences

Formulae for the nth term: u_n = u₁ rⁿ⁻¹
Sum of first n terms: S_n = u₁(rⁿ − 1)r − 1
Use of sigma (Σ) notation for sums
Applications of the topics mentioned above

1.4 Financial Applications

Compound interest formula: A = P(1 + r)ⁿ
Annual depreciation calculations

1.6 Proofs

Simple deductive proof: numerical and algebraic
Layout of LHS to RHS proof
Symbols: equality (=) and identity (≡)

1.7 Exponents & Logarithms

Laws of exponents with rational exponents
Laws of logarithms:
log_a(xy) = log_ax + log_ay
log_a(xy) = log_ax − log_ay
log_a(x^m) = m · log_ax
Change of base: log_ax = log_bxlog_ba
Solving exponential equations using logarithms

1.8 Infinite Geometric Sequences

Sum of infinite convergent sequences
Formula: S_∞ = a1 − r where |r| < 1

1.9 The Binomial Theorem

Expansion of (a + b)ⁿ, where n ∈ ℕ
General term formula: ⁿC_r a^n−r b^r
Use of Pascal’s triangle and ⁿC_r values

1.10 Counting & Binomial Extension

Counting principles for arrangement and selection
Permutations: ⁿP_r = n!(n − r)!
Combinations: ⁿC_r = n!r!(n − r)!
Extension of binomial theorem to fractional/negative indices: (a + b)ⁿ, where n ∈ ℚ

1.11 Partial Fractions

Decomposition of rational expressions for integration and simplification
Notation: P(x)Q(x) = Ax − a + Bx − b

1.12 Complex Numbers

The imaginary unit i, where i² = −1
Cartesian form: z = a + b·i
Terms: real part, imaginary part, conjugate, modulus, and argument
Representation in the complex plane (Argand diagram)

1.13 Polar Form & Operations

Polar form: z = r(cos θ + i·sin θ) = r·cis θ
Exponential (Euler) form: z = r·e^iθ
Operations: sum, product, and quotient in Cartesian or polar form
Geometric interpretation of operations

1.15 Proof Methods

Proof by mathematical induction
Proof by contradiction
Use of counterexamples to disprove statements

1.16 Systems of Equations

Solving systems with max 3 equations and 3 unknowns
Cases: unique solution, infinite solutions, or no solution

Topic 2

Functions

2.1 Equations of a Straight Line

Equation: y = mx + c
Lines with gradients m₁ and m₂:
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = −1

2.2 Concept of a Function

Domain, range, and graph
Function notation: f(x), v(t)
Concept of a function as a mathematical model
Inverse function f⁻¹(x) as a reflection in y = x

2.3 The Graph of a Function

Equation y = f(x)
Creating a sketch from given information or a context
Using technology to graph functions, including sums and differences

2.4 Key Features of Graphs

Determine key features of graphs
Finding intersections of two curves or lines using technology

2.5 Composite Functions

(f ∘ g)(x) = f(g(x))
Identity function and inverse function: f⁻¹(x), (f ∘ f⁻¹)(x) = x

2.6 The Quadratic Function

Standard form: f(x) = ax² + bx + c, y-intercept (0, c), axis of symmetry x = −b2a
Factor form: f(x) = a(x − p)(x − q), x-intercepts (p, 0) and (q, 0)
Vertex form: f(x) = a(x − h)² + k, vertex (h, k)

2.7 Quadratic Equations

Solution of quadratic equations and inequalities
Quadratic formula: x = −b ± √Δ2a
Discriminant: Δ = b² − 4ac
Nature of roots: two distinct real roots, two equal real roots, or no real roots

2.8 Reciprocal and Rational Functions

Reciprocal: f(x) = 1x, x ≠ 0, self-inverse nature
Rational: f(x) = ax + bcx + d, graphs, vertical and horizontal asymptotes

2.9 Exponential and Logarithmic

Exponential: f(x) = a^x, f(x) = e^x
Logarithmic: f(x) = log_ax, f(x) = ln x

2.10 Solving Equations

Solving equations graphically and analytically, use of technology
Applications to real-life situations

2.11 Transformations of Graphs

Translations: y = f(x) + b, y = f(x − a)
Reflections: y = −f(x), y = f(−x)
Vertical stretch: y = p·f(x), horizontal stretch: y = f(qx)
Composite transformations

2.12 Polynomial Functions

Polynomial functions, their graphs and equations
Zeros, roots, and factors
The factor and remainder theorems
Sum and product of the roots of polynomial equations

2.13 Rational Functions

Forms: f(x) = ax + bcx² + dx + e and f(x) = ax² + bx + cdx + e
Analysis of graphs, asymptotes (vertical, horizontal, oblique), and key features

2.14 Odd and Even Functions

Definition and identification:
Odd: f(−x) = −f(x)
Even: f(−x) = f(x)
Finding the inverse function, f⁻¹(x), including domain restriction
Self-inverse functions: f⁻¹(x) = f(x)

2.15 Solutions of Inequalities

Solving g(x) ≥ f(x) both graphically and analytically

2.16 Special Function Graphs

Graphs of y = |f(x)| and y = f(|x|)
Transformations: y = 1f(x), y = f(ax + b), y = [f(x)]²

Topic 3

Geometry & Trigonometry

3.1 3D Geometry

Volume and surface area of three-dimensional solids, including right pyramids, cones, spheres, hemispheres, and combinations of these solids
Size of an angle between two intersecting lines or between a line and a plane

3.2 Trigonometry in Right-Angled Triangles

Use of sine, cosine, and tangent ratios:
sin θ = opphyp, cos θ = adjhyp, tan θ = oppadj
Sine rule: asin A = bsin B = csin C
Cosine rule: c² = a² + b² − 2ab · cos C
Area of a triangle: 12 · a · b · sin C

3.3 Applications of Trigonometry

Right-angled and non-right-angled problems, including Pythagoras’ theorem: a² + b² = c²
Angles of elevation and depression
Construction of labelled diagrams from written statements

3.4 The Circle

Radian measure of angles
Length of an arc: s = r · θ
Area of a sector: A = 12 · r² · θ

3.5 Unit Circle Definitions

cos θ and sin θ in terms of the unit circle
tan θ = sin θcos θ
Extension of the sine rule to the ambiguous case

3.6 Trigonometric Identities

Pythagorean identity: sin² θ + cos² θ = 1
Double-angle formulas:
sin 2θ = 2 sin θ cos θ
cos 2θ = cos² θ − sin² θ
cos 2θ = 2cos² θ − 1
cos 2θ = 1 − 2sin² θ
Relationships between trigonometric ratios

3.7 Circular Functions

sin x, cos x, tan x, amplitude, periodic nature, and graphs
Composite functions: f(x) = a · sin (bx + c) + d
Transformations and real-life contexts

3.8 Solving Trigonometric Equations

Graphically and analytically in finite intervals
Equations leading to quadratic equations in sin x, cos x, or tan x

3.9 Reciprocal Trigonometric Ratios

Definitions:
sec θ = 1cos θ
cosec θ = 1sin θ
cot θ = 1tan θ
Pythagorean identities:
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
Inverse functions: arcsin x, arccos x, arctan x; domains, ranges, and graphs

3.10 Compound Angle Identities

Compound angle formulas:
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = tan A ± tan B1 ∓ tan A tan B
Double-angle identity for tan:
tan 2θ = 2tan θ1 − tan² θ

3.11 Symmetry & Trig Graphs

sin(π − θ) = sin θ
cos(π − θ) = −cos θ
tan(π − θ) = −tan θ
Relationships between trigonometric functions and symmetry properties of their graphs

3.12 Vectors: Basic Concepts

Concept of a vector; position vectors; displacement vectors
Base vectors i, j, k
Components: v = v₁i + v₂j + v₃k
Algebraic and geometric approaches to: sum/difference, zero vector, −v, scalar multiplication, parallel vectors, magnitude |v|, unit vectors v|v|
Position vectors OA = a, OB = b; displacement AB = b − a
Proofs of geometrical properties using vectors

3.13 Scalar Product

Definition: a · b = |a| |b| cos θ
Angle between two vectors
Perpendicular (dot product = 0) and parallel vectors

3.14 Vector Equation of a Line

Vector equation in 2D and 3D: r = a + λb
Angle between two lines
Simple applications to kinematics

3.15 Lines in Space

Coincident, parallel, intersecting, and skew lines
Distinguishing between these cases and points of intersection

3.16 Vector Product

Definition: v × w
Properties of the vector product
Geometric interpretation of |v × w| (Area of parallelogram)

3.17 Vector Equations of a Plane

r = a + λb + μc (parametric form)
r · n = a · n (scalar/normal form)
Cartesian equation: ax + by + cz = d

3.18 Intersections and Angles

Intersections of: a line with a plane, two planes, three planes
Angle between: a line and a plane, two planes

Topic 4

Statistics & Probability

4.1 Concepts of Data

Population, sample, random sample, discrete and continuous data
Reliability of data sources and bias in sampling
Interpreting outliers, sampling techniques

4.2 Presentation of Data

Frequency distributions, histograms
Cumulative frequency and cumulative frequency graphs
Using graphs to find median, quartiles, percentiles, range, and interquartile range (IQR)
Box-and-whisker diagrams

4.3 Measures of Central Tendency

Mean, median, and mode
Estimation of mean from grouped data
Modal class
Measures of dispersion: IQR, standard deviation, variance
Effect of constant changes on original data
Quartiles of discrete data

4.4 Linear Correlation of Bivariate Data

Pearson’s correlation coefficient r
Scatter diagrams, lines of best fit by eye through mean point
Regression line of y on x: y = a · x + b, interpretation of a and b

4.5 Probability Concepts

Trial, outcome, equally likely outcomes, relative frequency, sample space U, and event
Probability of event A: P(A) = n(A)n(U)
Complementary events: A and A′
Expected number of occurrences

4.6 Using Diagrams for Probability

Venn diagrams, tree diagrams, sample space diagrams, and tables of outcomes
Combined events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Mutually exclusive events: P(A ∩ B) = 0
Conditional probability: P(A|B) = P(A ∩ B)P(B)
Independent events: P(A ∩ B) = P(A) · P(B)

4.7 Discrete Random Variables

Probability distributions
Expected value (mean) E(X) and applications

4.8 Binomial Distribution

General notation: X ~ B(n, p)
Mean and variance of the binomial distribution

4.9 Normal Distribution

General notation: X ~ N(μ, σ²)
Properties and diagrammatic representation
Normal probability calculations and inverse normal calculations

4.10 Regression Line of x on y

Use for prediction purposes

4.11 Conditional Probability (Formal)

P(A|B) = P(A ∩ B)P(B) for conditional probabilities
Independent events: P(A|B) = P(A) = P(A|B′)

4.12 Standardization of Normal Variables

Formula: z = x − μσ
z-values and inverse normal calculations when mean and standard deviation are unknown

4.13 Use of Bayes’ Theorem

Formula: P(A|B) = P(B|A) · P(A)P(B)
Extended: P(A|B) = P(B|A) · P(A)P(B|A)P(A) + P(B|A′)P(A′)
For a maximum of three events

4.14 Variance of a Random Variable

Variance of a discrete random variable
Continuous random variables and their probability density functions
Mode and median of continuous random variables
Mean, variance, and standard deviation of both discrete and continuous random variables
Effect of linear transformations of X

Topic 5

Calculus

5.1 Introduction to Calculus

Concept of a limit: lim_{x → a} f(x)
Derivative as gradient function and rate of change

5.2 Increasing and Decreasing Functions

Graphical interpretation: f′(x) > 0, f′(x) = 0, f′(x) < 0

5.3 Derivatives of Polynomials

f(x) = a · xⁿ + b · x^m ⇒ f′(x) = a · n · xⁿ⁻¹ + b · m · x^m−1

5.4 Tangents and Normals

Equations of tangent and normal lines at a given point

5.5 Introduction to Integration

Anti-differentiation of functions: ∫ (a · xⁿ + b · x^m) dx
Anti-differentiation with boundary condition to determine constant
Definite integrals using technology: area under y = f(x), f(x) ≥ 0

5.6 Derivatives of Elementary Functions

xⁿ, sin x, cos x, e^x, ln x
Derivative of sums and multiples
Chain rule for composite functions: e.g. sin(3x − 1) or ln(2x + 5)
Product and quotient rules

5.7 Second Derivative

Graphical behavior: relationships between f(x), f′(x), f″(x)

5.8 Local Maxima and Minima

Testing for maxima and minima, optimization
Points of inflection: zero and non-zero gradient

5.9 Kinematics Problems

Displacement (s), velocity (v), acceleration (a)
Total distance traveled

5.10 Indefinite Integrals

xⁿ (n rational), 1x, e^x
Composites with linear functions: ∫ f(ax + b) dx
Integration by inspection (e.g. ∫ cos(3x) dx), reverse chain rule, or substitution

5.11 Definite Integrals

Analytical approach: ∫_a^b f(x) dx
Areas of regions enclosed by curves y = f(x) and x-axis (f(x) positive or negative)
Areas between curves: ∫_a^b (f(x) − g(x)) dx

5.12 Continuity & Differentiability

Understanding of limits (convergence and divergence)
Definition of derivative from first principles: f′(x) = lim_{h → 0} f(x + h) − f(x)h
Higher derivatives

5.13 Evaluation of Limits

Limits of the form lim_{x → a} f(x)g(x) and lim_{x → ∞} f(x)g(x) using l’Hôpital’s rule or Maclaurin series
Repeated use of l’Hôpital’s rule

5.14 Implicit Differentiation

Related rates of change
Optimization problems

5.15 Special Functions

Derivatives of tan x, sec x, cosec x, cot x, a^x, log_ax, arcsin x, arccos x, arctan x
Indefinite integrals of the derivatives of the above functions
Composites with linear functions and use of partial fractions to rearrange integrand

5.16 Integration Techniques

Integration by substitution
Integration by parts, including repeated integration by parts

5.17 Areas and Volumes

Area of region enclosed by a curve and y-axis
Volumes of revolution about the x-axis or y-axis

5.18 Differential Equations

Numerical solution of dydx = f(x, y) using Euler’s method
Variables separable
Homogeneous: dydx = f(yx) using y = vx
Linear: y′ + P(x)y = Q(x) using integrating factor

5.19 Maclaurin Series

Expansions for e^x, sin x, cos x, ln(1 + x), (1 + x)^p (p rational)
Use of simple substitution, products, integration, and differentiation
Series developed from differential equations

Mathematics: analysis and approaches guides

Mathematics: analysis and approaches formula booklet