A-level Mathematics₇₃₅₇

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

Use and manipulate surds, including rationalising the denominator.

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.

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

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 $\mathit{y}\mathsf{>}\mathit{x}\mathsf{+}\mathsf{1}$ and $\mathit{y}\mathsf{>}{\mathit{a}\mathit{x}}^{\mathsf{2}}\mathsf{+}\mathit{b}\mathit{x}\mathsf{+}\mathit{c}$ 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).

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

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

Understand the effect of simple transformations on the graph of $\mathit{y}\mathsf{=}\mathrm{f}\left(\mathit{x}\right)$ including sketching associated graphs:

$\mathit{y}\mathsf{=}\mathit{a}\mathrm{f}\left(\mathit{x}\right)$ , $\mathit{y}\mathsf{=}\mathrm{f}\left(\mathit{x}\right)\mathsf{+}\mathit{a}$ , $\mathit{y}\mathsf{=}\mathrm{f}\mathsf{(}\mathit{x}\mathsf{+}\mathit{a}\mathsf{)}$ , $\mathit{y}\mathsf{=}\mathrm{f}\mathsf{\left(}\mathit{a}\mathit{x}\mathsf{\right)}$ , and combinations of these transformations.

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).

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