Functions and graphs
Connect a function’s rule, its allowed inputs and the shape of its graph.
Choose an available lesson. Read the explanation at your own pace and move the models yourself.
- The modulus functionUnderstand the modulus (absolute value) function, sketch |ax + b| and solve basic modulus equations. Move the graph yourself and practise with worked solutions.
- Modulus equations and inequalitiesSolve A-level modulus equations and inequalities using branches, graphs and sign checks. Find valid roots, reject extraneous answers and practise with full solutions.
- Functions, domain and rangeUnderstand functions, domain and range, one-to-one and many-to-one mappings. Explore restricted quadratics and practise ranges of roots, reciprocals, exponentials and logarithms.
- Piecewise functionsEvaluate piecewise functions, sketch open and closed endpoints, find ranges and solve equations with branch restrictions. Explore a jump using a manual graph.
- Composite functionsLearn composite functions fg(x), gf(x) and f²(x), find composite domains, and solve equations. Follow each function machine with a manual model and worked practice.
- Inverse functionsFind inverse functions, choose the correct quadratic branch and exchange domain and range. Explore reflection in y = x, verify compositions and practise with solutions.
- Graphs of |f(x)| and f(|x|)Sketch y = |f(x)| and y = f(|x|), transform points, and check domains and asymptotes. Compare both rules interactively and practise original graph questions.
- Combined graph transformationsCombine graph translations, stretches and reflections. Track points, domains, ranges, asymptotes and stationary points, and compare transformation order interactively.
- Modulus graphs and problem solvingSolve modulus graph problems with parameters, restricted inverses, inequalities and areas. Distinguish zero, one, two and infinitely many intersections using an interactive model.
Build on the basics
Revisit the earlier graph skills whenever a transformation or sketch feels unfamiliar.