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2. Algebra and graphs

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  1. E2.1 Introduction to algebra

    1. E2.1.1Know that letters can be used to represent generalised numbers.

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

    2. E2.1.2Substitute numbers into expressions and formulas.

      • Substitute numbers into expressions and formulas.

  2. E2.2 Algebraic manipulation

    1. E2.2.1Simplify expressions by collecting like terms.

      • Simplify expressions by collecting like terms.

    2. E2.2.2Expand products of algebraic expressions.

      • Expand products of algebraic expressions.

    3. E2.2.3Factorise by extracting common factors.

      • Factorise by extracting common factors.

    4. E2.2.4Factorise expressions of the form: • ax + bx + kay + kby • a2x2 − b2y2

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

    5. E2.2.5Complete the square for expressions in the form ax2 + bx + c

      • 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).

  3. E2.3 Algebraic fractions

    1. E2.3.1Manipulate algebraic fractions

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

    2. E2.3.2Simplify rational expressions

      • Factorise and simplify rational expressions.

  4. E2.4 Indices II

    1. E2.4.1Understand and use indices (positive, zero, negative and fractional).

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

    2. E2.4.2Understand and use the rules of indices

      • 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.

  5. E2.5 Equations

    1. E2.5.1Construct expressions, equations and formulas

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

    2. E2.5.2Solve linear equations

      • Solve linear equations in one unknown.

    3. E2.5.3Solve fractional equations

      • Solve fractional equations with numerical and linear algebraic denominators.

    4. E2.5.4Solve simultaneous linear equations

      • Solve simultaneous linear equations in two unknowns.

    5. E2.5.5Solve linear and non-linear simultaneous equations

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

    6. E2.5.6Solve quadratic equations

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

    7. E2.5.7Change the subject of formulas

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

  6. E2.6 Inequalities

    1. E2.6.1Represent and interpret inequalities, including on a number line.

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

    2. E2.6.2Construct, solve and interpret linear inequalities.

      • Construct, solve and interpret linear inequalities.

    3. E2.6.3Represent and interpret linear inequalities in two variables

      • Represent and interpret linear inequalities in two variables graphically.

    4. E2.6.4List inequalities that define a given region

      • 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

    5. E2.6.5Linear programming problems are not included.

      • Linear programming problems are not included.

  7. E2.7 Sequences

    1. E2.7.1Continue sequences

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

    2. E2.7.2Recognise sequence patterns

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

    3. E2.7.3Find and use the nth term

      • Find and use the nth term of sequences.

  8. E2.8 Proportion

    1. E2.8.1

      • 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.

  9. E2.9 Graphs in practical situations

    1. E2.9.1Use and interpret graphs in practical situations including travel

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

    2. E2.9.2Draw graphs from given data.

      • Draw graphs from given data.

    3. E2.9.3Apply the idea of rate of change to simple kinematics involving

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

    4. E2.9.4Calculate distance travelled as area under a speed–time graph

      • 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.

  10. E2.10 Graphs of functions

    1. E2.10.1Draw and interpret function graphs

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

    2. E2.10.2Solve equations graphically

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

    3. E2.10.3Exponential growth and decay graphs

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

  11. E2.11 Sketching curves

    1. E2.11.1

      • 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.

  12. E2.12 Differentiation

    1. E2.12.1Estimate gradients using tangents

      • Estimate gradients of curves by drawing tangents.

    2. E2.12.2Differentiate simple power functions

      • 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.

    3. E2.12.3Gradients and stationary points

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

    4. E2.12.4Distinguish maxima and minima

      • 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.

  13. E2.13 Functions

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

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

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

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

    3. 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.