Algebra and Functions: Structure, Manipulation, and Graphs
Core concepts
A candidate must be fluent with linear and quadratic forms, simultaneous equations, inequalities, function notation, domain and range, and the algebra of exponents and factorisation. You should recognise equivalent forms of a quadratic (standard, factored, completed-square) and know what each reveals: the factored form gives roots, the completed-square form gives the vertex.
Traps to watch
- Sign slips when expanding negatives, especially
-(x - 3)and squaring(x - 5)^2. - Dividing an inequality by a negative without reversing the inequality symbol.
- Extraneous solutions after squaring both sides of a radical equation; always substitute back.
- Domain restrictions hidden in denominators and even roots; a simplified expression may look defined where the original was not.
- Reading a graph's intercepts as solutions when the question asks for the vertex or the axis of symmetry.
Worked example
Solve x^2 - 6x + 5 = 0 and state the vertex of y = x^2 - 6x + 5.
Factor: (x - 1)(x - 5) = 0, so roots are 1 and 5. The axis of symmetry sits midway: x = 3. Substituting gives y = 9 - 18 + 5 = -4, so the vertex is (3, -4).
Rule of thumb
For any quadratic, the axis of symmetry is x = -b/(2a); find it first and half the graph's behaviour follows. When a question says "solve", check whether it wants a value, a set, or an interval - the algebra is the same but the expected answer format differs.