IGCSE Mathematics: Algebra and Graphs (Cambridge 0580)

This guide covers Topic 2 Algebra and graphs, for Cambridge IGCSE Mathematics 0580, 2025–2027 series. The Core subtopics (C2.1, C2.2, C2.4, C2.5, C2.6, C2.7, C2.9, C2.10, C2.11) are examined at all entry levels; the Extended-only subtopics (E2.2 additions, E2.3, E2.4 additions, E2.5 additions, E2.6 additions, E2.7 additions, E2.8, E2.9 additions, E2.10 additions, E2.11 additions, E2.12, E2.13) are required only for the Extended tier. Extended candidates are eligible for grades A* to E and Core candidates for grades C to G, so grades A* to B need the Extended tier (the syllabus advises Extended for anyone aiming at grades A* to C).

Where this fits in 0580

Algebra and graphs is a long topic in 0580, and it is cumulative — later subtopics lean on earlier ones rather than standing alone. Solving equations depends on confident algebraic manipulation; sketching and interpreting graphs of functions depends on being able to substitute into and rearrange those same functions; sequences depend on spotting and describing algebraic patterns. A gap in the manipulation subtopics (C2.2) tends to resurface as lost marks throughout the rest of the topic, and again later in coordinate geometry and calculus.

Syllabus coverage

CAMBRIDGE IGCSE MATHEMATICS 0580 — TOPIC 2 ALGEBRA AND GRAPHS

Core

• C2.1 Introduction to algebra — knowing that letters can represent generalised numbers; substituting numbers into expressions and formulae
• C2.2 Algebraic manipulation — simplifying expressions by collecting like terms; expanding products of algebraic expressions (including two brackets in one variable); factorising by extracting common factors
• C2.4 Indices II — understanding and using indices (positive, zero and negative); understanding and using the rules of indices
• C2.5 Equations — constructing simple expressions, equations and formulae; solving linear equations in one unknown; solving simultaneous linear equations in two unknowns; changing the subject of simple formulae (where the subject appears only once, with no power or root of the subject)
• C2.6 Inequalities — representing and interpreting inequalities, including on a number line
• C2.7 Sequences — continuing a given number sequence or pattern; recognising patterns in sequences, including the term-to-term rule and relationships between different sequences; finding and using the nth term of linear, simple quadratic and simple cubic sequences
• C2.9 Graphs in practical situations — using and interpreting graphs in practical situations, including travel graphs and conversion graphs; drawing graphs from given data
• C2.10 Graphs of functions — constructing tables of values and drawing, recognising and interpreting graphs for functions of the form ax + b, ±x² + ax + b and a/x (a, b integer constants); solving associated equations graphically, including finding and interpreting roots
• C2.11 Sketching curves — recognising, sketching and interpreting graphs of linear and quadratic functions (symmetry and roots are required; turning points are not)

Note that C2.3, C2.8, C2.12 and C2.13 exist only as Extended-tier subtopics (E2.3, E2.8, E2.12, E2.13 below) — there is no Core content at those numbers.

Extended only (in addition to the Core content above)

• E2.2 Algebraic manipulation (Extended) — as C2.2, plus factorising expressions of the form ax + bx + kay + kby, a²x² − b²y², a² + 2ab + b², ax² + bx + c and ax³ + bx² + cx; completing the square for ax² + bx + c
• E2.3 Algebraic fractions — manipulating algebraic fractions; factorising and simplifying rational expressions
• E2.4 Indices II (Extended) — as C2.4, plus fractional indices
• E2.5 Equations (Extended) — as C2.5, plus solving fractional equations with numerical and linear algebraic denominators; solving simultaneous equations where one equation is linear and one is non-linear; solving quadratic equations by factorisation, completing the square and the quadratic formula; changing the subject of a formula where the subject appears twice or has a power or root
• E2.6 Inequalities (Extended) — constructing, solving and interpreting linear inequalities; representing linear inequalities in two variables graphically and listing the inequalities that define a given region
• E2.7 Sequences (Extended) — finding and using the nth term of sequences including linear, quadratic, cubic and exponential sequences and simple combinations of these
• E2.8 Proportion — expressing direct and inverse proportion in algebraic terms and using this form of expression to find unknown quantities
• E2.9 Graphs in practical situations (Extended) — as C2.9, plus applying rate of change to simple kinematics (distance–time and speed–time graphs, acceleration and deceleration) and calculating distance travelled as the area under a speed–time graph
• E2.10 Graphs of functions (Extended) — constructing tables of values and drawing, recognising and interpreting graphs for functions of the form axⁿ (sums of up to three such terms) and abˣ + c; solving associated equations graphically; drawing and interpreting exponential growth and decay graphs
• E2.11 Sketching curves (Extended) — recognising, sketching and interpreting graphs of linear, quadratic, cubic, reciprocal and exponential functions, with turning points, roots, symmetry and asymptotes required
• E2.12 Differentiation — estimating gradients of curves by drawing tangents; using the derivative of functions of the form axⁿ, and simple sums of up to three of these, to calculate gradients; determining turning points (maxima and minima) by calculus
• E2.13 Functions — using function notation and understanding domain and range; finding the inverse function f⁻¹(x); forming the composite function gf(x) = g(f(x))

Worked examples

Solving a quadratic by factorisation (Extended). Solve x² + 5x + 6 = 0.

x^2 + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = -2  or  x = -3

Solving simultaneous linear equations. Solve 2x + y = 7 and x − y = 2.

Adding the two equations eliminates y:
(2x + y) + (x - y) = 7 + 2
3x = 9,  so x = 3

Substituting back: 2(3) + y = 7, so y = 1

Finding the nth term of a linear sequence. Find the nth term of 5, 8, 11, 14, …

Differences: 3, 3, 3 -- constant, so the sequence is linear: an + b
a = 3 (the common difference)
When n = 1, the term is 5, so 3(1) + b = 5, giving b = 2
nth term = 3n + 2

Solving a double inequality (Extended, E2.6). Solve −5 < 2x + 3 ⩽ 11, list the integer solutions and show the solution on a number line.

-5 < 2x + 3 <= 11
Subtract 3 from every part:   -8 < 2x <= 8
Divide every part by 2:       -4 < x  <= 4

Integer solutions: -3, -2, -1, 0, 1, 2, 3, 4
Number line: open circle at -4 (strict <), closed circle at 4 (<=),
line joining them.

Whatever you do to one part, do to all three. Dividing by a negative number would reverse the inequality signs; here we divide by +2, so they stay the same.

Finding turning points by differentiation (Extended, E2.12). Find the turning points of y = x³ − 12x and decide which is a maximum.

dy/dx = 3x^2 - 12
At a turning point dy/dx = 0:  3x^2 = 12,  x^2 = 4,  x = 2 or x = -2

x = 2:   y = 8 - 24   = -16   ->  (2, -16)
x = -2:  y = -8 + 24  =  16   ->  (-2, 16)

d2y/dx2 = 6x
At x = 2:   6(2)  = 12 > 0,  so (2, -16) is a minimum
At x = -2:  6(-2) = -12 < 0, so (-2, 16) is a maximum

The syllabus accepts any method to tell a maximum from a minimum: the second derivative (as here), an accurate sketch, or the sign of the gradient either side of the point.

For more practice, use the Algebra and graphs practice questions, the Extended algebra practice and the Algebra and graphs revision notes.

How to approach it

Treat C2.2 (algebraic manipulation) as the foundation to fix before anything else in this topic — every equation, inequality and sequence question depends on being able to expand, factorise and simplify without hesitation. When solving equations, always check which method is expected: linear equations by rearrangement (Core); solving a quadratic equation by any method is Extended-only content (E2.5), and within that, factorisation is usually tried first (fast, and often what the question wants), reaching for the formula or completing the square only when factorising clearly won’t work cleanly. With sequences, get into the habit of writing out the differences between terms before hunting for the nth term — a constant difference means linear (an + b), a constant second difference means quadratic, and a constant third difference means cubic; Core (C2.7) only asks you to find the nth term for these three types, while Extended (E2.7) adds exponential sequences and combinations of these. For graphs, keep “reading a graph” and “sketching a graph” as separate skills: reading (finding a gradient, an intercept, or a solution from a drawn graph) uses the axes as given, while sketching (Core C2.11, linear and quadratic only) needs the general shape and key features — origin, roots, symmetry — not a precise plot; Extended sketching (E2.11) adds cubic, reciprocal and exponential shapes and requires turning points and asymptotes as well. Function notation (Extended, E2.13) trips students up mostly through notation, not difficulty — practise translating f(x) = … into “substitute x” before attempting fg(x) or f⁻¹(x) composite questions.

Official syllabus

Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027, Topic 2 verified against the PDF on 29 August 2026 — cambridgeinternational.org.