Understand and use the laws of indices for all rational exponents.

Laws of indices

Use and manipulate surds, including rationalising the denominator.

Surds

Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown.

Quadratic functions and graphs Discriminant of a quadratic function Completing the square Solving quadratic equations

Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation.

Solving simultaneous equations

Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions.

Express solutions through correct use of ‘and’ and ‘or’, or through set notation.

Represent linear and quadratic inequalities such as \(y > x + 1\) and \(y > ax^2 + bx + c\) graphically.

Solving linear inequalities Solving quadratic inequalities Represent inequalities graphically

Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem.

Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only).

Polynomials Algebraic division Factor theorem

Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function, \(y = \frac{a}{x} \) and \(y = \frac{a}{x^2} \) (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 Modulus

Understand and use composite functions; inverse functions and their graphs.

Composite functions Inverse functions Functions, mappings, domain and range

Understand the effect of simple transformations on the graph of \(y=f(x)\) including sketching associated graphs: \(y=af(x)\), \(y=f(x)+a\), \(y=f(x+a)\), \(y=f(ax)\), and combinations of these transformations.

Transformations of graphs

Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear).

Partial fractions

Use of functions in modelling, including consideration of limitations and refinements of the models.

Use of functions in modelling