Understand and use graphs of functions; sketch curves defined by simple equations including polynomials; the modulus of a linear function; y = a/x and y = a/x² (including their vertical and horizontal asymptotes); interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations; understand and use proportional relationships and their graphs

Graphs of functions are a cornerstone of A-Level Mathematics, providing a visual representation of algebraic relationships. This topic covers sketching curves defined by polynomial equations, the modulus of a linear function, and reciprocal graphs like y = a/x and y = a/x². Understanding these graphs allows you to interpret solutions to equations graphically, such as finding intersection points to solve simultaneous equations. Mastery of this topic is essential for calculus, where graphical understanding underpins concepts like limits, derivatives, and areas under curves.

Proportional relationships are also explored, where y is directly proportional to x (y = kx) or inversely proportional to x (y = k/x). Their graphs are straight lines through the origin or hyperbolas, respectively. Recognising these relationships helps in modelling real-world phenomena, from physics (e.g., Hooke's law) to economics (e.g., supply and demand). The modulus function, |ax + b|, introduces a V-shaped graph, which is crucial for solving equations involving absolute values and understanding piecewise functions.

In the Edexcel A-Level syllabus, this topic appears in Pure Mathematics Paper 1 and Paper 2. It builds on GCSE knowledge of linear and quadratic graphs and prepares you for more advanced topics like transformations of functions, differentiation, and integration. By the end of this unit, you should be able to sketch curves accurately, identify key features like asymptotes and intercepts, and use graphs to solve equations both algebraically and graphically.