2. Algebra and graphs

• Know that letters can be used to represent generalised numbers.

• Substitute numbers into expressions and formulas.

• Simplify expressions by collecting like terms.

• Expand products of algebraic expressions.

• Factorise by extracting common factors.

• Factorise expressions of the form: • ax + bx + kay + kby • a2x2 − b2y2 • a2 + 2ab + b2 • ax2 + bx + c • ax3 + bx2 + cx.

• Complete the square for expressions in the form ax2 + bx + c. Simplify means give the answer in its simplest form, e.g. 2a2 + 3ab – 1 + 5a2 – 9ab + 4 = 7a2 – 6ab + 3. e.g. expand 3x(2x – 4y), (3x + y)(x – 4y). Includes products of more than two brackets, e.g. expand (x – 2)(x + 3)(2x + 1). Factorise means factorise fully, e.g. 9x2 + 15xy = 3x(3x + 5y).

• Manipulate algebraic fractions by adding, subtracting, multiplying and dividing.

• Factorise and simplify rational expressions.

• Understand and use indices (positive, zero, negative and fractional).

• Understand and use positive, zero, negative and fractional indices. Solve simple exponential equations and apply the laws of indices to simplify expressions. Knowledge of logarithms is not required.

• Construct expressions, equations and formulas, including simultaneous equations.

• Solve linear equations in one unknown.

• Solve fractional equations with numerical and linear algebraic denominators.

• Solve simultaneous linear equations in two unknowns.

• Solve simultaneous equations involving one linear and one non-linear equation, with powers no higher than two.

• Solve quadratic equations by factorisation, completing the square and the quadratic formula; write completed-square forms and give solutions in surd form when required.

• Change the subject of formulas, including cases where the subject appears twice or is raised to a power or under a root.

• Represent and interpret inequalities, including on a number line.

• Construct, solve and interpret linear inequalities.

• Represent and interpret linear inequalities in two variables graphically.

• List inequalities that define a given region. When representing and interpreting inequalities on a number line: • open circles should be used to represent strict inequalities (<, >) • closed circles should be used to represent inclusive inequalities (⩽, ⩾). e.g. – 3 ⩽ x < 1 –/uni202F3– /uni202F2– /uni202F101 x Examples include: • 3x < 2x + 4 • –3 ⩽ 3x – 2 < 7. The following conventions should be used: • broken lines should be used to represent strict inequalities (<, >) • solid lines should be used to represent inclusive inequalities (⩽, ⩾) • shading should be used to represent unwanted regions (unless otherwise directed in the question). e.g. 0 12 0 12 xx yy x < 1 y /uni2A7E

• Linear programming problems are not included.

• Continue a given number sequence or pattern; subscript notation may be used.

• Recognise term-to-term rules and relationships in linear, quadratic, cubic and exponential sequences and simple combinations of them.

• Find and use the nth term of sequences.

• Express direct and inverse proportion in algebraic terms and use this form of expression to find unknown quantities. Includes linear, square, square root, cube and cube root proportion. Knowledge of proportional symbol (∝) is required.

• Use and interpret graphs in practical situations including travel graphs and conversion graphs.

• Draw graphs from given data.

• Apply the idea of rate of change to simple kinematics involving distance–time and speed–time graphs, acceleration and deceleration.

• Calculate distance travelled as area under a speed–time graph. Includes estimation and interpretation of the gradient of a tangent at a point. Areas will involve linear sections of the graph only.

• Construct tables of values and draw, recognise and interpret linear, quadratic, cubic, reciprocal, exponential and other specified power-function graphs.

• Solve associated equations graphically, including finding and interpreting roots and intersections of lines and curves.

• Draw and interpret graphs representing exponential growth and decay problems.

• Recognise, sketch and interpret graphs of the following functions: (a) linear (b) quadratic (c) cubic (d) reciprocal (e) exponential. Functions will be equivalent to: • ax + by = c • y = ax2 + bx + c • y = ax3 + b • y = ax3 + bx2 + cx • y x a b= + • y = arx + b where a, b and c are rational numbers and r is a rational, positive number. Knowledge of turning points, roots and symmetry is required. Knowledge of vertical and horizontal asymptotes is required. Finding turning points of quadratics by completing the square is required.

• Estimate gradients of curves by drawing tangents.

• Use derivatives of functions of the form axⁿ, where a is rational and n is a non-negative integer, and simple sums of no more than three such terms; dy/dx notation is required.

• Apply differentiation to gradients and stationary points or turning points.

• Distinguish maxima and minima using an accurate sketch, the second derivative or the gradient on either side of a turning point; points of inflection are not required.

E2.13.1Understand functions, domain and range and use function notation.

• Understand functions, domain and range and use function notation.

E2.13.2Understand and find inverse functions f –1(x).

• Understand and find inverse functions f –1(x).

E2.13.3Form composite functions as defined by gf(x) = g(f(x))

• Form composite functions as defined by gf(x) = g(f(x)). Examples include: • f (x) = 3x – 5 • g(x) = 3(x + 4) • h(x) = 2x2 + 3. e.g. f(x) = 3 x + 2 and g(x) = (3x + 5)2. Find fg(x). Give your answer as a fraction in its simplest form. Candidates are not expected to find the domains and ranges of composite functions. This topic may include mapping diagrams.