This page maps the syllabus of IB Mathematics: Analysis and Approaches HL to the Brilliant lessons that teach it. Search by syllabus item or skill to find the closest match. An item is listed only where Brilliant content fully or in part addresses it directly.
Higher Level is a superset. It shares its core syllabus with Standard Level, then adds further content — complex numbers, vectors, further calculus and series among it — that SL students never meet. The first section below is the shared core; the sections after it are the Higher Level additions, one per topic, so a student choosing between the two courses can see which is which.
Core content — studied by both SL and HL students
| IB syllabus item | What the syllabus item asks | Brilliant targeted skill |
|---|---|---|
| SL.1.1 | Operations with numbers in the form a × 10^k where 1 ≤ a < 10 and k is an integer (scientific notation). (p. 29) | Exponents and Radicals — Estimating with scientific notation, Expressing numbers in scientific notation, Multiplying and dividing in scientific notation |
| SL.1.3 | Geometric sequences and series: use of the formulae for the nth term and the sum of the first n terms; use of sigma notation for the sums of geometric sequences; applications. (p. 29) | Exponential Functions — Classify a relationship as linear or exponential, Evaluate an exponential expression, Identify exponential decay from a dot pattern, Identify exponential growth from a dot pattern, Write a function for a growth or decay scenario Limits — Finding a formula from a graph |
| SL.1.4 | Financial applications of geometric sequences and series: compound interest and annual depreciation; calculating the real value of an investment given an interest rate and an inflation rate. (p. 30) | Exponential Functions — Graph and compare compound interest functions, Write a compound growth or decay expression |
| SL.1.5 | Laws of exponents with integer exponents; introduction to logarithms with base 10 and e; numerical evaluation of logarithms using technology; awareness that a^x = b is equivalent to log_a b = x, that b > 0, and that log_e x = ln x. (p. 30) | Exponents and Radicals — Applying the power rule for exponents, Applying the product rule for exponents, Applying the quotient rule for exponents, Combining exponent rules to simplify, Dividing with prime factorizations, Evaluating zero-exponent expressions, Multiplying with prime factorizations, Rewriting with negative exponents, Simplifying multi-base quotients, Simplifying negative exponents in denominators, Simplifying products with exponent zero, Simplifying with negative exponents Logarithms — Convert between exponential and logarithmic form, Estimate a logarithm, Evaluate a logarithm exactly, Find the exponent in an exponential equation Exponential Functions — Evaluate an exponential at negative x-values, Express decay with a negative exponent |
| SL.1.7 | Laws of exponents with rational exponents; laws of logarithms (product, quotient and power rules); change of base of a logarithm; solving exponential equations, including using logarithms. (p. 31) | Logarithms — Change the base of a logarithm, Check solutions of logarithmic equations, Convert between log bases graphically, Rewrite logarithms with the power rule, Rewrite logarithms with the product and quotient rules, Solve exponential equations with logarithms, Solve logarithmic equations, Transform a log graph using identities Exponents and Radicals — Connecting fractional exponents to radicals, Evaluating general fractional exponents, Evaluating unit fraction exponents, Multiplying square root expressions, Simplifying and combining unlike radicals Exponential Functions — Evaluate an exponential at non-integer x-values Equations and Curves — Evaluating roots and inverse compositions, Evaluating square roots of squares, Simplifying fractional exponents |
| SL.2.1 | Different forms of the equation of a straight line — y = mx + c (gradient-intercept), ax + by + d = 0 (general), y − y₁ = m(x − x₁) (point-gradient); gradient and intercepts; parallel lines m₁ = m₂ and perpendicular lines m₁ × m₂ = −1. (p. 37) | Linear Relationships — Calculating rates of change, Computing slope, Finding constant change, Finding perpendicular slopes, Finding starting values, Graphing from point-slope form, Graphing lines from intercepts, Graphing lines from slope, Graphing real-world relationships, Interpreting intercepts, Slopes of horizontal and vertical lines, Writing parallel line equations, Writing perpendicular line equations, Writing point-slope equations, Writing slope-intercept equations, Writing standard form equations |
| SL.2.2 | Concept of a function, domain, range and graph; function notation such as f(x), v(t), C(n); the concept of a function as a mathematical model; the informal concept that an inverse function reverses or undoes the effect of a function, the inverse as a reflection in the line y = x, and the notation f⁻¹(x). (p. 37) | Equations and Curves — Analyzing implicit equations, Connecting transformations to domain and range, Determining domain and range, Finding domain and range after transformations, Graphing the square root function, Identifying power function graphs Introduction to Functions — Applying the vertical line test, Finding function domains, Finding function ranges, Following conditional rules, Following function rules, Identifying conditional rules, Identifying function rules, Matching input-output pairs, Recognizing non-functions, Writing numeric function rules |
| SL.2.3 | The graph of a function and its equation y = f(x); creating a sketch from information given or from a context, including transferring a graph from screen to paper; using technology to graph functions including their sums and differences. (p. 38) | Introduction to Functions — Combining functions with operations, Plotting conditional functions, Plotting numeric functions Equations and Curves — Matching equations to graphs |
| SL.2.4 | Determine key features of graphs: maximum and minimum values, intercepts, symmetry, vertex, zeros of functions or roots of equations, and vertical and horizontal asymptotes, using graphing technology; finding the points of intersection of two curves or lines using technology. (p. 38) | Equations and Curves — Controlling the number of intersections, Finding intersection points, Solving systems and unions of curves Introduction to Functions — Reading continuous function graphs, Reading discontinuous function graphs Quadratics — Reading solutions from graphs |
| SL.2.5 | Composite functions (f ∘ g)(x) = f(g(x)); the identity function; finding the inverse function f⁻¹(x) with (f ∘ f⁻¹)(x) = (f⁻¹ ∘ f)(x) = x; the existence of an inverse for one-to-one functions. (p. 39) | Introduction to Functions — Determining invertibility, Evaluating composite functions, Finding inverse functions, Writing composite function expressions Equations and Curves — Evaluating squaring and square roots |
| SL.2.6 | The quadratic function f(x) = ax² + bx + c: its graph, y-intercept (0, c) and axis of symmetry; the factored form f(x) = a(x − p)(x − q) with x-intercepts (p, 0) and (q, 0); the vertex form f(x) = a(x − h)² + k with vertex (h, k). (p. 39) | Quadratics — Completing factors with negatives, Completing partial factorizations, Connecting intercepts to factors, Converting to vertex form, Determining signs in products, Expanding binomial products, Factoring signed quadratics fully, Factoring simple quadratics, Factoring with leading coefficients, Factoring with negative constants, Finding maximum and minimum values, Finding the axis of symmetry, Finding the best possible value, Finding the vertex, Finding vertices using symmetry, Graphing by factoring, Graphing parabolas from vertex form, Recognizing perfect square patterns, Writing equations from graphs, Writing vertex form equations |
| SL.2.7 | Solution of quadratic equations and inequalities; the quadratic formula; the discriminant Δ = b² − 4ac and the nature of the roots (two distinct real roots, two equal real roots, no real roots). (p. 39) | Quadratics — Checking solutions by substitution, Classifying solution types, Completing squares with fractions, Determining number of solutions, Finding when objects arrive, Isolating squared variables, Predicting whether a target is reached, Solving by completing the square, Solving equations by factoring, Solving shifted square equations, Solving with square roots, Using the discriminant, Using the quadratic formula |
| SL.2.8 | The reciprocal function f(x) = 1/x, x ≠ 0, its graph and self-inverse nature; rational functions of the form f(x) = (ax + b)/(cx + d) and their graphs; equations of vertical and horizontal asymptotes. (p. 40) | Equations and Curves — Completing hyperbola equations, Finding equation parameters, Identifying symmetric points, Locating asymptotes, Reading values from graphs |
| SL.2.9 | Exponential functions and their graphs f(x) = a^x (a > 0) and f(x) = e^x; logarithmic functions and their graphs f(x) = log_a x (x > 0) and f(x) = ln x; the relationships a^x = e^(x ln a) and log_a a^x = x, and exponential and logarithmic functions as inverses of each other. (p. 40) | Exponential Functions — Build a decay function with a fractional base, Build an exponential function from non-integer points, Build an exponential function from plotted points, Build an exponential function over negative domain, Deduce an exponential function from two points, Find the base of an exponential function, Set the coefficient of an exponential function Logarithms — Find the domain of a logarithmic function, Graph a logarithmic function, Undo a logarithm or an exponential with its inverse |
| SL.2.10 | Solving equations, both graphically and analytically; use of technology to solve a variety of equations, including those where there is no appropriate analytic approach; applications of graphing skills and solving equations that relate to real-life situations. (p. 41) | Linear Relationships — Find the input that reaches a target value |
| SL.2.11 | Transformations of graphs: translations y = f(x) + b and y = f(x − a); reflections in both axes y = −f(x) and y = f(−x); vertical stretch with scale factor p, y = p f(x); horizontal stretch with scale factor 1/q, y = f(qx); composite transformations. (p. 41) | Equations and Curves — Applying reflections and equivalent stretches, Applying shifts and stretches, Combining multiple transformations, Identifying transformed graphs, Shifting the square root graph, Stretching and reflecting root graphs, Transforming hyperbola graphs, Transforming the square root graph, Writing transformations from constraints Polynomials — Applying single transformations, Combining transformations, Identifying transformed features Exponential Functions — Build a shifted exponential from its graph, Write a shifted exponential for a scenario approaching a limit Introduction to Functions — Combining reflections and stretches, Reflecting functions across axes, Sliding Function Graphs, Stretching Function Graphs |
| SL.3.2 | Use of the sine, cosine and tangent ratios to find the sides and angles of right-angled triangles; the sine rule a/sin A = b/sin B = c/sin C; the cosine rule c² = a² + b² − 2ab cos C and cos C = (a² + b² − c²)/(2ab); the area of a triangle as ½ ab sin C. (p. 45) | Trigonometric Functions — Finding special right triangle sides |
| SL.3.3 | Applications of right and non-right-angled trigonometry, including Pythagoras's theorem; angles of elevation and depression; construction of labelled diagrams from written statements. (p. 45) | Coordinate Geometry — Applying the Pythagorean theorem to areas, Calculating a circle's radius, Calculating distance from coordinates, Finding side lengths with the Pythagorean theorem |
| SL.3.4 | The circle: radian measure of angles; length of an arc; area of a sector. (p. 46) | Trigonometric Functions — Computing arc length, Constructing angles in radians, Constructing unit circle arcs, Converting degrees and radians, Identifying angles from geometry Polar Coordinate Plane — Computing arc lengths from angles, Computing arc lengths from fractions, Describing features in radians, Finding angles from arc lengths, Plotting points using radians |
| SL.3.5 | Definition of cos θ and sin θ in terms of the unit circle, including the relationship between angles in different quadrants; definition of tan θ as sin θ / cos θ; exact values of the trigonometric ratios of 0, π/6, π/4, π/3, π/2 and their multiples; extension of the sine rule to the ambiguous case. (p. 46) | Polar Coordinate Plane — Computing angles with inverse tangent, Computing x and y from polar, Converting to polar coordinates, Finding quadrants from angles Trigonometric Functions — Determining tangent sign by quadrant, Evaluating sine and cosine, Evaluating tangent at an angle, Evaluating trig functions in radians, Evaluating trig values at key angles, Finding angle pairs for trig values, Finding angles beyond one revolution, Reading cosine from circular motion, Reading sine from circular motion |
| SL.3.6 | The Pythagorean identity cos²θ + sin²θ = 1; double angle identities for sine and cosine; the relationship between trigonometric ratios. (p. 47) | Trigonometric Functions — Using the Pythagorean identity |
| SL.3.7 | The circular functions sin x, cos x and tan x: amplitude, their periodic nature and their graphs; composite functions of the form f(x) = a sin(b(x + c)) + d; transformations; real-life contexts. (p. 47) | Trigonometric Functions — Computing period from rotation speed, Describing how oscillations change, Finding transformed function periods, Identifying amplitude and midline, Identifying periodic function features, Interpreting changing oscillation models, Interpreting trig models, Matching functions with transformations, Modeling changing oscillations, Modeling situations with trig functions, Predicting periodic values, Transforming periodic functions |
| SL.3.8 | Solving trigonometric equations in a finite interval, both graphically and analytically, including equations leading to quadratic equations in sin x, cos x or tan x. (p. 48) | Trigonometric Functions — Describing all solutions with periodicity, Factoring non-linear trig equations, Solving equations from the tangent graph, Solving equations from trig graphs, Solving equations with inverse cosine, Solving equations with inverse sine, Solving trig equations on an interval |
| SL.4.1 | Concepts of population, sample, random sample, discrete and continuous data; reliability of data sources and bias in sampling; interpretation of outliers; sampling techniques and their effectiveness. (p. 53) | Everyday Statistics — Spotting outliers, Using the outlier rule |
| SL.4.2 | Presentation of data (discrete and continuous): frequency distributions (tables); histograms; cumulative frequency and cumulative frequency graphs, used to find median, quartiles, percentiles, range and interquartile range; production and understanding of box and whisker diagrams. (p. 54) | Exploring Data Visually — Compare boxplots across datasets, Find frequencies and percentiles from a histogram, Group a data table to summarize categories, Read median, quartiles, and extremes from a boxplot, Reading histograms Everyday Statistics — Comparing data sets with box plots, Matching data to box plots, Reading box plots |
| SL.4.3 | Measures of central tendency (mean, median and mode), including estimation of the mean from grouped data and the modal class; measures of dispersion (interquartile range, standard deviation and variance); the effect of constant changes on the original data; quartiles of discrete data. (p. 55) | Everyday Statistics — Combining groups to find the mean, Comparing mean, median, and mode, Finding quartiles, Finding the mean (the average) of a data set, Finding the mean of uneven data, Finding the median, Predicting how new data shifts the mean, Seeing how outliers affect the mean, Splitting data into quarters, Updating the mean with new data, Updating the median with new data |
| SL.4.4 | Linear correlation of bivariate data; Pearson's product-moment correlation coefficient r; scatter diagrams and lines of best fit by eye passing through the mean point; the equation of the regression line of y on x and its use for prediction; interpretation of the parameters a and b in y = ax + b. (p. 55) | Regression — Checking the groups inside data, Choosing the best predictor, Fitting a line to data, Interpreting correlations in context, Measuring how well a model fits, Modeling one subgroup at a time, Predicting from a model equation, Predicting from a regression line, Reading correlation from a scatter plot, Spotting nonlinear relationships Exploring Data Visually — Describe the relationship in a scatter plot, Describing how two things relate, Judging how strongly things relate Linear Relationships — Fit a line to data and predict |
| SL.4.5 | Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event, with P(A) = n(A)/n(U); the complementary events A and A′; expected number of occurrences. (p. 56) | Probability and Chance — Calculating probability, Changing probability by removing options, Comparing the likelihood of outcomes, Counting outcomes across overlapping events, Counting possible outcomes, Finding the probability something won't happen Probability in Data — Estimating chances without data, Estimating probabilities from past data |
| SL.4.6 | Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities; 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); probabilities with and without replacement; independent events P(A ∩ B) = P(A)P(B). (p. 57) | Probability in Data — Adding probabilities of separate outcomes, Combining probabilities across cases, Combining probabilities with a formula, Finding conditional probabilities from data, Finding probabilities of multi-step outcomes, Finding probability of two related events, Judging how often a rule fails Probability and Chance — Finding probability given a condition, Finding probability of two unrelated events, Finding the probability of both events, Finding the probability of either event |
| SL.4.7 | Concept of discrete random variables and their probability distributions; expected value (mean) for discrete data; applications. (p. 57) | Probability and Chance — Calculating expected value, Calculating expected value for multiple events, Comparing expected values |
| SL.4.11 | Formal definition and use of P(A|B) = P(A ∩ B)/P(B) for conditional probabilities, and of P(A|B) = P(A) = P(A|B′) for independent events; testing for independence. (p. 59) | Probability in Data — Identifying when events affect each other |
| SL.5.1 | Introduction to the concept of a limit; the derivative interpreted as a gradient function and as a rate of change. (p. 61) | Derivatives — Evaluating a derivative function, Finding instantaneous velocity, Writing derivatives in two notations, Zooming in to find a derivative Limits — Finding one-sided limits of piecewise functions, Reading one-sided limits from a graph |
| SL.5.2 | Increasing and decreasing functions; graphical interpretation of f′(x) > 0, f′(x) = 0 and f′(x) < 0. (p. 62) | Introduction to Functions — Identifying function intervals Derivatives — Reading slope on a wave graph, Sorting intervals by direction of change |
| SL.5.3 | Derivative of f(x) = ax^n is f′(x) = anx^(n−1) for n ∈ ℤ, and the derivative of functions of the form f(x) = ax^n + bx^(n−1) + … where all exponents are integers. (p. 62) | Derivatives — Combining derivative rules, Differentiating powers of x, Differentiating sums and multiples |
| SL.5.4 | Tangents and normals at a given point, and their equations. (p. 63) | Derivatives — Writing a tangent line equation |
| SL.5.5 | Introduction to integration as anti-differentiation of functions of the form f(x) = ax^n + bx^(n−1) + … where n ∈ ℤ and n ≠ −1; anti-differentiation with a boundary condition to determine the constant term; definite integrals using technology; the area of a region enclosed by a curve y = f(x) and the x-axis where f(x) > 0. (p. 63) | Integrals — Finding the area under a graph, Finding when a total is reached |
| SL.5.6 | Derivatives of x^n (n ∈ ℚ), sin x, cos x, e^x and ln x; differentiation of a sum and a multiple of these functions; the chain rule for composite functions; the product and quotient rules. (p. 63) | Derivatives — Differentiating sine and cosine, Finding wave slopes and velocities |
| SL.5.7 | The second derivative; graphical behaviour of functions, including the relationship between the graphs of f, f′ and f″. (p. 64) | Derivatives — Sketching a derivative graph, Sketching a function from its derivative |
| SL.5.8 | Local maximum and minimum points; testing for maximum and minimum; optimization; points of inflexion with zero and non-zero gradients. (p. 64) | Derivatives — Finding maximums and minimums |
| SL.5.9 | Kinematic problems involving displacement s, velocity v, acceleration a and total distance travelled. (p. 65) | Derivatives — Finding velocity from position |
| SL.5.11 | Definite integrals, including the analytical approach ∫ from a to b of g′(x)dx = g(b) − g(a); areas of a region enclosed by a curve y = f(x) and the x-axis where f(x) can be positive or negative, without the use of technology; areas between curves. (p. 65) | Integrals — Evaluating definite integrals from graphs, Finding net change from a graph, Finding peak and return times, Tracking a running total, Using properties of definite integrals, Writing definite integral notation |
Higher level only — Number and algebra
| IB syllabus item | What the syllabus item asks | Brilliant targeted skill |
|---|---|---|
| AHL.1.12 | Complex numbers: the number i with i² = −1; Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument; the complex plane. (p. 34) | Complex Numbers — Evaluating expressions with i, Evaluating real and imaginary parts, Finding absolute value, Finding complex conjugates, Finding complex number distance, Plotting complex numbers, Simplifying powers of i |
| AHL.1.13 | Modulus–argument (polar) form z = r(cos θ + i sin θ) = r cis θ and Euler form z = re^(iθ); sums, products and quotients in Cartesian, polar or Euler form and their geometric interpretation. (p. 34) | Complex Numbers — Adding and subtracting complex numbers, Adding arguments to multiply, Dividing by i, Dividing complex numbers, Dividing in polar form, Finding magnitude and argument, Identifying complex transformations, Multiplying complex numbers, Multiplying conjugate pairs, Multiplying in polar form, Rotating by multiplying by i, Scaling complex numbers, Using conjugates in polar form, Writing in polar form |
| AHL.1.14 | Complex conjugate roots of quadratic and polynomial equations with real coefficients (complex roots occur in conjugate pairs); De Moivre's theorem and its extension to rational exponents; powers and roots of complex numbers. (p. 34) | Complex Numbers — Computing powers of conjugates, Finding all complex roots, Finding principal roots, Raising complex numbers to powers, Solving equations with complex solutions Quadratics — Solving with imaginary numbers |
| AHL.1.15 | Proof by mathematical induction; proof by contradiction; use of a counterexample to show that a statement is not always true. (p. 35) | Limits — Building functions through the same points |
Higher level only — Functions
Higher level only — Geometry and trigonometry
| IB syllabus item | What the syllabus item asks | Brilliant targeted skill |
|---|---|---|
| AHL.3.9 | Definition of the reciprocal trigonometric ratios sec θ, cosec θ and cot θ; the Pythagorean identities 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ; the inverse functions f(x) = arcsin x, f(x) = arccos x and f(x) = arctan x, their domains and ranges, and their graphs. (p. 48) | Trigonometric Functions — Evaluating inverse trig functions |
| AHL.3.11 | Relationships between trigonometric functions and the symmetry properties of their graphs. (p. 49) | Trigonometric Functions — Applying complementary angle relationships, Applying symmetry and periodicity rules, Identifying phase shift relationships |
| AHL.3.12 | Concept of a vector; position vectors; displacement vectors; representation of vectors using directed line segments; base vectors i, j, k; components of a vector; algebraic and geometric treatment of the sum and difference of two vectors, the zero vector 0 and the vector −v, multiplication by a scalar kv and parallel vectors, magnitude |v| and unit vectors v/|v|, position vectors OA = a and OB = b, and the displacement vector AB = b − a; proofs of geometrical properties using vectors. (p. 49) | Vectors — Adding vectors by components, Adding vectors from angles, Adding velocity vectors, Combining velocities from bearings, Computing relative velocity, Computing vector magnitude, Decomposing vectors with trigonometry, Drawing linear combinations, Drawing vector differences, Drawing vector sums, Drawing vectors from polar form, Evaluating linear combinations, Finding opposite vectors, Representing shifts as vectors, Scaling to a target magnitude, Scaling vectors, Subtracting vectors by components, Using non-polar angle decomposition Matrices — Decomposing into linear combinations, Identifying scalar multiples, Using standard basis vectors |
| AHL.3.13 | The definition of the scalar product of two vectors; the angle between two vectors; perpendicular vectors and parallel vectors; applications of the properties of the scalar product. (p. 50) | Vectors — Comparing unit vectors, Computing dot products, Computing unit vector dot products, Scaling dot products, Using the dot product formula |
Higher level only — Statistics and probability
| IB syllabus item | What the syllabus item asks | Brilliant targeted skill |
|---|---|---|
| AHL.4.13 | Use of Bayes' theorem for a maximum of three events. (p. 60) | Probability in Data — Updating probabilities with new evidence |
Higher level only — Calculus
| IB syllabus item | What the syllabus item asks | Brilliant targeted skill |
|---|---|---|
| AHL.5.12 | Informal understanding of continuity and differentiability of a function at a point; understanding of limits (convergence and divergence); definition of the derivative from first principles, f′(x) = lim(h→0) [f(x + h) − f(x)]/h; higher derivatives. (p. 66) | Limits — Building functions with given limits, Classifying limits at a point, Comparing a limit to the value, Deciding whether a sequence converges, Describing how a sequence diverges, Finding infinite limits at vertical asymptotes, Finding limits at infinity from formulas, Finding limits by plotting terms, Finding limits of geometric sequences, Finding limits of sequences, Finding terms outside a target interval, Reading limits at infinity on graphs Derivatives — Computing a derivative from the limit, Spotting where a derivative is undefined, Writing a derivative as a limit |
| AHL.5.15 | Derivatives of tan x, sec x, cosec x, cot x, a^x, log_a x, arcsin x, arccos x and arctan x; indefinite integrals of the derivatives of any of these functions; the composites of any of these with a linear function; use of partial fractions to rearrange the integrand. (p. 67) | Derivatives — Differentiating exponential functions |
How this page is built
Every skill in the Brilliant math topics that reach IB Diploma level is mapped to the single syllabus item it aligns with most closely, using the IB's Mathematics: Analysis and Approaches guide for first assessment 2021 as the authority for what each syllabus item requires and where its boundaries fall. Skills that build toward a syllabus item without teaching its stated content are excluded, so a skill appears here only where it addresses the syllabus item directly.
Each skill links to a lesson where it is practised, and hovering a skill shows the description learners see for it. This page is regenerated from the mapping, so it reflects the current state rather than a fixed snapshot.
