Function notation and evaluation
Use f(x), substitute values, interpret outputs and connect functions to graphs.
Key skill: Substitute accurately and state outputs clearly.
Back to Higher Mathematics
Higher Mathematics
Topic practice
Practise Higher Mathematics by section: functions, coordinate geometry, polynomials, calculus, trigonometry, exponentials, logarithms, vectors, recurrence relations and exam-style problem solving.
Composite and inverse functions
Build composite functions, find inverses and state suitable domains and ranges.
Key skill: Work in the correct order and check the inverse.
Domain, range and transformations
State domains and ranges, then sketch translations, reflections and stretches of functions.
Key skill: Link function notation to graph movement.
Quadratic equations and discriminant
Solve quadratic equations and use the discriminant to determine or control the nature of roots.
Key skill: Use b² - 4ac and interpret its sign.
Completing the square and quadratic inequalities
Complete the square for unitary and non-unitary quadratics, then solve quadratic inequalities.
Key skill: Expose graph features and use a sign diagram.
Factor and remainder theorem
Evaluate f(a), identify linear factors and find remainders of polynomial division.
Key skill: Test roots by substitution.
Cubic and quartic equations
Factorise and solve cubic or quartic polynomial equations using known or discoverable factors.
Key skill: Find one factor, divide, then solve the reduced equation.
Intersections of functions
Find the coordinates where a line and curve, or two curves, have equal outputs.
Key skill: Set the functions equal and find every coordinate.
Exponential and logarithmic functions
Connect exponential and logarithmic functions as inverses and interpret their graphs.
Key skill: Switch between index and logarithmic form.
Log laws and equations
Simplify logarithmic expressions and solve exponential or logarithmic equations with valid domains.
Key skill: Use one law at a time and check arguments.
Exponential modelling and linearisation
Model growth and decay, find constants from data and linearise power or exponential relationships using logarithms.
Key skill: Match a straight-line form to transformed variables.
Sketching polynomials
Use roots, intercepts, stationary points and end behaviour to sketch polynomial graphs.
Key skill: Mark roots and end behaviour first.