IGCSE Maths: Algebra and Graphs

Overview

Algebra and Graphs is the largest topic in the IGCSE Mathematics syllabus, covering everything from basic equations and algebraic manipulation to function notation, sequences, graph sketching, inequalities, and introductory differentiation. Extended students must handle quadratic equations, composite and inverse functions, and calculus concepts.

Core Syllabus Topics

• Simplifying expressions
• Solving linear equations
• Substitution
• Expanding brackets
• Factorising (common factors)
• Changing the subject of a formula
• Recognising number patterns
• Finding terms of a sequence
• Linear (arithmetic) sequences
• nth term of a linear sequence
• Reading and writing inequalities
• Solving simple linear inequalities
• Representing inequalities on a number line

Extended Syllabus Topics

• Quadratic equations (factorising, formula, completing the square)
• Simultaneous equations (linear and non-linear)
• Algebraic fractions
• Surds and indices
• Inequalities (solving and graphing)
• Function notation f(x)
• Domain and range
• Composite functions f(g(x))
• Inverse functions f⁻¹(x)
• Piecewise functions
• Function graphs and transformations
• Quadratic sequences (an² + bn + c)
• Geometric sequences
• Exponential sequences
• Sum of arithmetic and geometric sequences
• Solving compound inequalities
• Quadratic inequalities
• Graphing linear inequalities in two variables
• Identifying feasible regions
• Integer solutions within regions
• Differentiating powers of x
• Finding gradients at specific points
• Equations of tangents
• Turning points (maxima and minima)
• Increasing and decreasing functions
• Simple optimisation problems

Common Mistakes to Avoid

• Sign errors when expanding negative brackets
• Forgetting to factorise completely
• Errors in algebraic fraction simplification
• Not checking solutions by substituting back
• Getting composite function order wrong (fg(x) ≠ gf(x))
• Errors when finding the inverse of complex functions
• Confusing f(2) with 2f(x)
• Not restricting domain for inverse functions
• Using first differences for quadratic sequences instead of second differences
• Errors in geometric sequence common ratio
• Confusing nth term with sum to n terms
• Forgetting to flip the inequality sign when multiplying/dividing by a negative
• Not using a dashed line for strict inequalities on graphs
• Shading the wrong region
• Forgetting to multiply by the power before reducing it
• Not setting dy/dx = 0 for turning points
• Confusing maximum and minimum — check second derivative or nature
• Errors when differentiating negative powers