Ask an IB maths teacher which topic predicts success in the whole course and many will say functions. Calculus is about functions changing; trigonometry, exponentials and logarithms are families of functions; and HL Paper 3 questions are usually built around one. Students who are fluent with functions find the rest of AA much easier.
What AA SL covers
- Domain and range, and function notation.
- Composite functions, f(g(x)), and the order they are applied in.
- Inverse functions: finding f⁻¹(x), and the graph as a reflection in y = x.
- Key graph features: intercepts, vertices, asymptotes, maxima and minima – sketching by hand and with a GDC.
- Quadratics in all forms, the discriminant, and solving quadratic equations and inequalities.
- Rational functions of the form (ax + b)/(cx + d), with their asymptotes.
- Exponential and logarithmic functions and their graphs.
- Transformations: translations, stretches and reflections – and combinations of them.
What AA HL adds
- Polynomial functions: the factor and remainder theorems, and the sum and product of roots.
- Rational functions with a quadratic numerator or denominator, including oblique asymptotes.
- Odd and even functions, and self-inverse functions.
- Graphs of |f(x)|, f(|x|), 1/f(x) and [f(x)]² from the graph of f(x).
- Solving inequalities graphically and algebraically.
Transformations without confusion
Most transformation errors come from one idea: changes inside the bracket act on x and work "the opposite way".
| Change | Effect on the graph of y = f(x) |
|---|---|
| f(x) + b | Up by b |
| f(x − a) | Right by a (not left) |
| p f(x) | Vertical stretch, factor p |
| f(qx) | Horizontal stretch, factor 1/q |
| −f(x) | Reflect in the x-axis |
| f(−x) | Reflect in the y-axis |
When several transformations are combined, order matters. A reliable habit: apply horizontal changes in the order that undoes the bracket, then vertical ones. Track one point – such as a maximum – through each step to check.
The errors that cost marks
- Composite functions in the wrong order. f(g(x)) means apply g first.
- Forgetting to restrict the domain before finding an inverse (a function must be one-to-one).
- Asymptotes missing from sketches, or drawn but not labelled with their equations.
- Copying a GDC graph without key features – intercepts and turning points need coordinates.
- Range errors when the graph has a minimum or an asymptote.
How to get fluent
- Sketch by hand before using the GDC. Then compare. Paper 1 needs hand sketches.
- Learn the "parent" graphs (x², x³, 1/x, eˣ, ln x, sin x) so well you can draw them in seconds.
- Track points through transformations rather than memorising rules.
- Mix functions into other topics – most exam questions do.
How Maths Bodhi helps
Functions are where we start with most new DP students, because gaps here show up everywhere later:
- A diagnostic of graph sense, algebra and notation in the first session.
- Sketch-first practice: hand sketches, then GDC checks.
- Mixed IB-style questions where functions meet calculus, trigonometry and logs, marked against the markscheme.
- For HL, focused work on polynomials, rational functions and inequalities.
Ajay Vatsyayan (M.Tech, 14+ years teaching maths) teaches IB DP maths at Maths Bodhi and has taught students from schools across Gurgaon, including Pathways World School, Scottish High International School, Lancers International School and DPS International. Maths Bodhi is independent and not affiliated with any school. Read Ajay's profile, or see our IB functions tutor, IB functions exam preparation, AA SL and AA HL pages.
Home sessions across Gurgaon – including Golf Course Road and Sector 54 – or online. Book a free demo class.
Related: AA calculus, exponentials and logarithms and the IB DP maths guide for Gurgaon families.
Topic lists checked against the current IB AA guide by the Maths Bodhi team (October 2026). Teaching approach reviewed by Ajay Vatsyayan.
Frequently asked questions
- What is the most common mistake with graph transformations in IB maths?
- Moving the graph the wrong way for changes inside the bracket: f(x − a) moves the graph right by a, and f(qx) stretches it horizontally by a factor of 1/q.
- What extra functions content is in AA HL?
- HL adds the factor and remainder theorems, sum and product of roots, rational functions with quadratics, odd, even and self-inverse functions, graphs such as |f(x)| and 1/f(x), and inequalities.
- Why must a function be one-to-one to have an inverse?
- Because each output must come from exactly one input; if not, the domain must be restricted before finding the inverse.