2. Algebra and graphs
- Syllabus
- 0580–2028–2029
- Section
- 2
- Level
- Extended
Exam analysis
No tagged past-paper evidence yet
Published Concept pages under this syllabus area do not have tagged past-paper appearances in the selected level yet.
Recent 5 years
In this section
Topic E2.1
E2.1 Introduction to algebra
Objectives in this topic
E2.1.1 Know that letters can be used to represent generalised numbers.
- Know that letters can be used to represent generalised numbers.
E2.1.2 Substitute numbers into expressions and formulas.
- Substitute numbers into expressions and formulas.
Topic E2.2
E2.2 Algebraic manipulation
Objectives in this topic
E2.2.1 Simplify expressions by collecting like terms.
- Simplify expressions by collecting like terms.
E2.2.2 Expand products of algebraic expressions.
- Expand products of algebraic expressions.
E2.2.3 Factorise by extracting common factors.
- Factorise by extracting common factors.
E2.2.4 Factorise 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.
E2.2.5 Complete 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).
Topic E2.3
E2.3 Algebraic fractions
Objectives in this topic
E2.3.1 Manipulate algebraic fractions
- Manipulate algebraic fractions by adding, subtracting, multiplying and dividing.
E2.3.2 Simplify rational expressions
- Factorise and simplify rational expressions.
Topic E2.4
E2.4 Indices II
Objectives in this topic
E2.4.1 Understand and use indices (positive, zero, negative and fractional).
- Understand and use indices (positive, zero, negative and fractional).
E2.4.2 Understand 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.
Topic E2.5
E2.5 Equations
Objectives in this topic
E2.5.1 Construct expressions, equations and formulas
- Construct expressions, equations and formulas, including simultaneous equations.
E2.5.2 Solve linear equations
- Solve linear equations in one unknown.
E2.5.3 Solve fractional equations
- Solve fractional equations with numerical and linear algebraic denominators.
E2.5.4 Solve simultaneous linear equations
- Solve simultaneous linear equations in two unknowns.
E2.5.5 Solve linear and non-linear simultaneous equations
- Solve simultaneous equations involving one linear and one non-linear equation, with powers no higher than two.
E2.5.6 Solve 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.
E2.5.7 Change 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.
Topic E2.6
E2.6 Inequalities
Objectives in this topic
E2.6.1 Represent and interpret inequalities, including on a number line.
- Represent and interpret inequalities, including on a number line.
E2.6.2 Construct, solve and interpret linear inequalities.
- Construct, solve and interpret linear inequalities.
E2.6.3 Represent and interpret linear inequalities in two variables
- Represent and interpret linear inequalities in two variables graphically.
E2.6.4 List 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
E2.6.5 Linear programming problems are not included.
- Linear programming problems are not included.
Topic E2.7
E2.7 Sequences
Objectives in this topic
E2.7.1 Continue sequences
- Continue a given number sequence or pattern; subscript notation may be used.
E2.7.2 Recognise sequence patterns
- Recognise term-to-term rules and relationships in linear, quadratic, cubic and exponential sequences and simple combinations of them.
E2.7.3 Find and use the nth term
- Find and use the nth term of sequences.
Topic E2.8
E2.8 Proportion
Objectives in this topic
E2.8.1 Express direct and inverse proportion in algebraic terms and use this
- 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.
Topic E2.9
E2.9 Graphs in practical situations
Objectives in this topic
E2.9.1 Use and interpret graphs in practical situations including travel
- Use and interpret graphs in practical situations including travel graphs and conversion graphs.
E2.9.2 Draw graphs from given data.
- Draw graphs from given data.
E2.9.3 Apply 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.
E2.9.4 Calculate 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.
Topic E2.10
E2.10 Graphs of functions
Objectives in this topic
E2.10.1 Draw and interpret function graphs
- Construct tables of values and draw, recognise and interpret linear, quadratic, cubic, reciprocal, exponential and other specified power-function graphs.
E2.10.2 Solve equations graphically
- Solve associated equations graphically, including finding and interpreting roots and intersections of lines and curves.
E2.10.3 Exponential growth and decay graphs
- Draw and interpret graphs representing exponential growth and decay problems.
Topic E2.11
E2.11 Sketching curves
Objectives in this topic
E2.11.1 Recognise, sketch and interpret graphs of the following functions: (a)
- 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.
Topic E2.12
E2.12 Differentiation
Objectives in this topic
E2.12.1 Estimate gradients using tangents
- Estimate gradients of curves by drawing tangents.
E2.12.2 Differentiate 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.
E2.12.3 Gradients and stationary points
- Apply differentiation to gradients and stationary points or turning points.
E2.12.4 Distinguish 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.
Topic E2.13
E2.13 Functions
Objectives in this topic
E2.13.1 Understand functions, domain and range and use function notation.
- Understand functions, domain and range and use function notation.
E2.13.2 Understand and find inverse functions f –1(x).
- Understand and find inverse functions f –1(x).
E2.13.3 Form 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.