How the functions topic fits together, from straight lines and the idea of a function to quadratic, rational, exponential and logarithmic models, transformations and, at HL, polynomials and modulus graphs.
Each one has its own page once something is published for it.
A function is a rule that turns each input into exactly one output. Functions are the language in which relationships between quantities are written: the cost of a journey and its length, the height of a ball and time, the value of an investment and the years since it was made. This topic builds the vocabulary and the toolkit for working with them: formulas, graphs, domains and ranges, inverses, the main families of functions, and the ways in which their graphs can be moved and reshaped.
The topic underpins the rest of the course. Sequences are functions defined on whole numbers, the circular functions of trigonometry are functions of an angle, probability distributions are functions, and calculus is the study of how functions change. Graphing technology plays a large part throughout, both for exploring functions and for solving equations that algebra cannot solve.
The topic starts with straight lines and the general idea of a function, then studies families of functions one at a time, and ends with methods that apply to all of them.
Subtopic | What it covers |
|---|---|
2.1 Equations of straight lines | Gradient, the three forms of a line, parallel and perpendicular lines, linear models |
2.2 Functions, domain, range and inverse | What a function is, notation, domain and range, models, the idea of an inverse |
2.3 The graph of a function | Plotting and sketching, choosing a viewing window, graphs of and |
2.4 Key features of graphs | Intercepts, zeros, maximum and minimum points, symmetry, asymptotes, intersections |
2.5 Composite functions | , the identity function, finding inverses algebraically |
2.6 Quadratic functions | The three forms of a quadratic and the features of a parabola |
2.7 Solving quadratic equations and inequalities | Factorising, the quadratic formula, the discriminant |
2.8 Reciprocal and rational functions | and , their asymptotes and inverses |
2.9 Exponential and logarithmic functions | Growth and decay models, and as inverses |
2.10 Solving equations | Graphical and analytical methods, equations that need technology |
2.11 Transformations of graphs | Translations, reflections and stretches, and combinations of them |
2.12 Polynomial functions (HL) | Factor and remainder theorems, sums and products of roots |
2.13 Rational functions (HL) | Quadratic denominators or numerators, oblique asymptotes |
2.14 Odd, even and self-inverse functions (HL) | Symmetry of functions, restricting domains to find inverses |
2.15 Solving inequalities (HL) | Polynomial and rational inequalities, graphically and analytically |
2.16 Modulus graphs and transformations (HL) | , , and modulus equations |
The first four subtopics provide tools used everywhere else: the equation of a line returns for tangents and normals in calculus, and key features such as zeros, turning points and asymptotes are what every later sketch must show. Subtopics 2.5 to 2.9 apply these tools to the main families of functions. Subtopics 2.10 and 2.11 then work across all families, solving equations and transforming graphs. The HL subtopics extend the same ideas to polynomials of higher degree, more general rational functions, inequalities and modulus graphs.
Relationships and representation. One function can be shown as a formula, a table, a mapping or a graph. Moving between them, and choosing the one that makes a problem easiest, is the central skill of the topic.
Modelling and change. Linear functions model a constant rate of change, quadratics model projectiles and areas, and exponentials model growth and decay. Every model has a domain limited by its context, and its validity must be checked against reality.
Equivalence. The same line or parabola can be written in several equivalent forms, each showing different features: shows the gradient and intercept, and shows the vertex.
Patterns and generalization. Transformations act the same way on every function, so understanding them once for a parent graph such as or explains whole families of graphs.
SL students study subtopics 2.1 to 2.11. HL students study all of these and the five HL subtopics, 2.12 to 2.16, which go deeper into polynomial and rational functions, symmetry, inequalities and the modulus function. The SL subtopics are the foundation for HL: for example, the inverse functions of 2.2 and 2.5 lead to the domain restrictions and self-inverse functions of 2.14.
The entries of the mathematics: analysis and approaches formula booklet that 2 uses. Each opens in the explained booklet.
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