Maths: How Algebra Shapes a Rational Function's Behaviour

Rational function analysis is a cornerstone of the Number and Algebra topic in Maths AA HL, where the goal is to understand how algebraic structure shapes a curve's behaviour. A rational function is a ratio of polynomials, and its most revealing features often hide inside factorisation. Factoring numerator and denominator exposes shared factors that cancel, simplifying the expression while restricting the domain wherever those factors vanish. The simplified form then governs the function's graph, revealing vertical asymptotes, horizontal asymptotes, and intercepts. Solving equations like g(x) = 3 becomes a matter of clearing the denominator, while comparing a function to its limiting value shows why a graph can approach a line without ever touching it. Recognising these connections between algebra and geometry turns a complicated-looking expression into a clear picture of its behaviour.