Cambridge IGCSE Maths (0580) · Revision notes · Topic 2 of 9

Cambridge IGCSE Maths 0580 Algebra and Graphs Notes

Revision notes for topic 2, Algebra and graphs, of the Cambridge IGCSE Mathematics (0580) syllabus for exams in 2025, 2026 and 2027. Each of the 13 sub-topics lists what the syllabus asks, marked Core or Extended, with a short explanation and a worked example. Cover the answer and try each example first.

13 sub-topics
9 Core and Extended
4 Extended only
3 with notes written for 0580

Core Core students (Papers 1 and 3) learn the Core statements. Extended Extended students (Papers 2 and 4) learn everything: the Core statements, the Extended additions and the Extended-only sub-topics. Papers 1 and 2 are non-calculator.

2.1 Introduction to algebra

Core C2.1 Extended E2.1

Letters represent generalised numbers
Substitute numbers into expressions and formulas

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

Simplify by collecting like terms; expand brackets before combining; factor common terms out.

Example: Simplify 4x + 5y − 2(x − 3y).

Answer: = 4x + 5y − 2x + 6y = 2x + 11y.

0580 practice questions

6 questions written for 0580 in the Algebra and graphs practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Find the value of $3a + 2b$ when $a = 4$ and $b = -5$.

Show the answer and mark scheme

M1 3 × 4 + 2 × (−5)
A1 2

Answer: $2$

Practice question (Core, non-calculator, 2 marks):

Pens cost 45 cents each and pencils cost 20 cents each. Write an expression for the total cost, in cents, of $x$ pens and $y$ pencils.

Show the answer and mark scheme

B2 45x + 20y (B1 for 45x or 20y)

Answer: $45x + 20y$

Edexcel 4MA1 notes: Algebraic Expressions · STEM Formulae
Practise (4MA1 questions, same mathematics): Algebraic Expressions questions · Rearranging Formulae questions

2.2 Algebraic manipulation

Core C2.2 Extended E2.2

Simplify by collecting like terms
Expand products of algebraic expressions, including two brackets in one variable, e.g. (2x + 1)(x − 4)
Factorise by taking out common factors

Extended also: Expand products of more than two brackets; Factorise expressions such as ax + bx + kay + kby, a²x² − b²y², a² + 2ab + b², ax² + bx + c and ax³ + bx² + cx; Complete the square.

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

‘Double brackets’ (e.g. (x+a)(x+b)) expand to a quadratic using FOIL.

Example: Expand and simplify (x + 4)(x + 7).

Answer: = x² + 7x + 4x + 28 = x² + 11x + 28.

0580 practice questions

6 questions written for 0580 in the Algebra and graphs practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Simplify $5a + 3b - 2a + 4b$.

Show the answer and mark scheme

B2 3a + 7b (B1 for 3a or 7b)

Answer: $3a + 7b$

Practice question (Core, non-calculator, 2 marks):

Expand

(a) $4(2x - 3)$ (b) $x(x + 5)$

Show the answer and mark scheme

B1 (a) 8x − 12
B1 (b) x² + 5x

Answer: (a) $8x - 12$ (b) $x^2 + 5x$

Edexcel 4MA1 notes: Expanding Brackets · Factorising Quadratics · Completing the Square
Practise (4MA1 questions, same mathematics): Algebraic Expressions questions · Expanding and Factorising questions · Quadratic Equations questions · Completing the Square questions

2.3 Algebraic fractions

Extended only E2.3

Extended: Add, subtract, multiply and divide algebraic fractions; Factorise and simplify rational expressions.

Same mathematics as Edexcel 4MA1.

Edexcel 4MA1 notes: Algebra (all notes)
Practise (4MA1 questions, same mathematics): Algebraic Fractions questions

2.4 Indices II

Core C2.4 Extended E2.4

Positive, zero and negative indices in algebra and the rules of indices

Extended also: Fractional indices in algebra.

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

Laws of indices tell you how to multiply, divide and raise powers to other powers.

Example: Simplify ((x⁴)³ × x⁻⁵) ÷ x².

Answer: = x¹² × x⁻⁵ ÷ x² = x⁷ ÷ x² = x⁵.

0580 practice questions

6 questions written for 0580 in the Algebra and graphs practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Simplify

(a) $x^4 \times x^3$ (b) $y^9 \div y^2$

Show the answer and mark scheme

B1 (a) x⁷
B1 (b) y⁷

Answer: (a) $x^7$ (b) $y^7$

Practice question (Core, non-calculator, 2 marks):

Simplify

(a) $(m^3)^4$ (b) $a^0$

Show the answer and mark scheme

B1 (a) m¹²
B1 (b) 1

Answer: (a) $m^{12}$ (b) $1$

Edexcel 4MA1 notes: Laws of Indices
Practise (4MA1 questions, same mathematics): Indices and Standard Form questions

2.5 Equations

Core C2.5 Extended E2.5

Construct expressions, equations and formulas
Solve linear equations in one unknown
Solve simultaneous linear equations in two unknowns
Change the subject of simple formulas

Extended also: Fractional equations with numerical and linear algebraic denominators; Simultaneous linear and non-linear equations; Quadratic equations by factorising, completing the square and the quadratic formula; Change the subject where the subject appears twice or as a power or root.

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

Simultaneous equations are two equations in two unknowns. Solve by elimination, substitution, or graphically.

Example: Solve x + y = 7 and 2x − y = 5.

Answer: Add: 3x = 12 → x = 4, y = 3.

0580 practice questions

6 questions written for 0580 in the Algebra and graphs practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Solve $5x - 7 = 18$.

Show the answer and mark scheme

M1 5x = 25
A1 5

Answer: $x = 5$

Practice question (Core, non-calculator, 2 marks):

Solve $3(x + 4) = 33$.

Show the answer and mark scheme

M1 3x + 12 = 33 or x + 4 = 11
A1 7

Answer: $x = 7$

Edexcel 4MA1 notes: Linear Simultaneous Equations · Quadratic Equations · Non-linear Simultaneous Equations · Linear Equations & Inequalities · STEM Formulae
Practise (4MA1 questions, same mathematics): Rearranging Formulae questions · Algebraic Fractions questions · Linear Equations questions · Simultaneous Equations questions · Quadratic Equations questions

2.6 Inequalities

Core C2.6 Extended E2.6

Represent and interpret inequalities, including on a number line

Extended also: Construct, solve and interpret linear inequalities; Represent linear inequalities in two variables on a graph and list the inequalities that define a region.

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

An inequality replaces ‘=’ with ‘<, >, ≤, ≥’. Solve like an equation — but flip the inequality if you multiply/divide by a negative.

Example: Solve 4x − 5 ≥ 11.

Answer: 4x ≥ 16 → x ≥ 4.

0580 practice questions

6 questions written for 0580 in the Algebra and graphs practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Write down the inequality shown on each number line.

(a) An open circle at $-2$ with the line going to the right.

(b) A filled (solid) circle at $3$ with the line going to the left.

Show the answer and mark scheme

B1 (a) x > −2
B1 (b) x ≤ 3

Answer: (a) $x \gt -2$ (b) $x \le 3$

Practice question (Core, non-calculator, 2 marks):

List the integers $x$ such that $-1 \lt x \le 4$.

Show the answer and mark scheme

B2 0, 1, 2, 3, 4 and no others (B1 for four correct values with at most one extra)

Answer: $0, 1, 2, 3, 4$

Edexcel 4MA1 notes: Linear Equations & Inequalities · Regions on Graphs
Practise (4MA1 questions, same mathematics): Inequalities questions · Inequalities on Graphs questions

2.7 Sequences

Core C2.7 Extended E2.7

Continue a sequence; term-to-term rules
Find and use the nth term of linear, simple quadratic and simple cubic sequences (e.g. 2, 5, 10, 17, …)

Extended also: nth term of linear, quadratic, cubic and exponential sequences and simple combinations of these; subscript notation Tₙ may be used.

Partly in Edexcel 4MA1: 4MA1 teaches linear nth terms (and arithmetic series); quadratic, cubic and exponential nth terms are not practised.

Notes for 0580

Look at the differences between terms. Linear: the first differences are all the same, d, so the nth term is dn + (first term − d). Quadratic: the second differences are the same; the n² coefficient is half the second difference. Take that n² part away from each term and find the linear nth term of what is left.

Cubic: compare with the cubes 1, 8, 27, 64, …; many Cambridge questions are a simple change of n³, such as n³ + 1. Exponential: each term is the one before times the same number r, so the nth term is a × r^(n − 1), where a is the first term.

Example (Core, non-calculator): Find the nth term of 3, 8, 15, 24, 35, …

First differences: 5, 7, 9, 11. Second differences: all 2, so the sequence is quadratic.
Half of 2 is 1, so start with 1n² = n²: 1, 4, 9, 16, 25.
Sequence − n²: 2, 4, 6, 8, 10, which is 2n.

Answer: n² + 2n

Example (Core, non-calculator): Find the nth term of 2, 9, 28, 65, …

Compare with the cubes n³: 1, 8, 27, 64.
Each term is 1 more than the cube.

Answer: n³ + 1

Example (Extended, non-calculator): Find the nth term of 5, 15, 45, 135, … and the 8th term.

Each term is 3 times the one before, so r = 3 and the first term is 5.
nth term = 5 × 3^(n − 1)
8th term = 5 × 3⁷ = 5 × 2187 = 10 935

Answer: 5 × 3^(n − 1); the 8th term is 10 935

0580 practice questions

16 questions written for 0580 in the Algebra and graphs practice: 6 Core and 10 Extended, 12 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Here are the first five terms of a sequence: $1, \ 4, \ 9, \ 16, \ 25, \ \ldots$

(a) Write down the next two terms.

(b) Write down an expression for the $n$th term.

Show the answer and mark scheme

B1 (a) 36, 49
B1 (b) n²

Answer: (a) $36, 49$ (b) $n^2$

Practice question (Extended, non-calculator, 4 marks):

Here are the first five terms of a sequence: $5, \ 8, \ 9, \ 8, \ 5, \ \ldots$

(a) Find an expression for the $n$th term.

(b) Which term of the sequence is the largest?

Show the answer and mark scheme

M1 (a) second difference −2 seen
M1 (a) −n² seen, or subtracts −n² to get 6, 12, 18, 24, 30
A1 (a) −n² + 6n oe
B1 (b) the 3rd term (value 9)

Answer: (a) $6n - n^2$ (b) the 3rd term

Edexcel 4MA1 notes: Sequences
Practise the shared part (4MA1 questions): Sequences questions

2.8 Proportion

Extended only E2.8

Extended: Express direct and inverse proportion algebraically and use it to find unknown quantities.

Same mathematics as Edexcel 4MA1.

Edexcel 4MA1 notes: Ratio & Proportion
Practise (4MA1 questions, same mathematics): Direct and Inverse Proportion questions

2.9 Graphs in practical situations

Core C2.9 Extended E2.9

Use and interpret travel graphs and conversion graphs
Draw graphs from given data

Extended also: Rate of change on distance-time and speed-time graphs: acceleration and deceleration; Distance travelled as the area under a speed-time graph.

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

‘Real-life graphs’ include distance–time and speed–time — the gradient tells you speed or acceleration.

Example: On a distance-time graph a runner covers 300 m in 60 s. What's their speed?

Answer: Speed = gradient = 300/60 = 5 m/s.

0580 practice questions

6 questions written for 0580 in the Algebra and graphs practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

A straight-line conversion graph passes through $(0, 0)$ and shows that $\$50$ is worth 45 euros.

(a) Change $\$20$ into euros.

(b) Change 90 euros into dollars.

Show the answer and mark scheme

B1 (a) 18
B1 (b) 100

Answer: (a) 18 euros (b) $\$100$

Practice question (Core, non-calculator, 2 marks):

A travel graph shows Sam's walk. He leaves home at 10 00 and walks 3 km in 45 minutes. He stops for 30 minutes, then walks home in 1 hour.

(a) Find his speed on the way out, in km/h.

(b) At what time does he arrive home?

Show the answer and mark scheme

B1 (a) 4
B1 (b) 12 15

Answer: (a) $4$ km/h (b) 12 15

Edexcel 4MA1 notes: Real-Life Graphs
Practise (4MA1 questions, same mathematics): Quadratic and Cubic Graphs questions · Real-Life Graphs questions

2.10 Graphs of functions

Core C2.10 Extended E2.10

Tables of values and graphs of ax + b, ±x² + ax + b and a/x (x ≠ 0), with a and b integers
Solve the associated equations graphically, including roots

Extended also: Graphs of axⁿ for n = −2, −1, −½, 0, ½, 1, 2, 3 and sums of up to three such terms, and of exponential functions ab^x + c (a, c rational, b a positive integer); Solve equations graphically, including where a curve meets a line; Graphs of exponential growth and decay problems.

Partly in Edexcel 4MA1: Exponential graphs (ab^x + c) and powers such as x⁻² and x^(1/2) are outside 4MA1.

Notes for 0580

For an exponential function y = ab^x + c, make a table of values (b⁰ = 1, and b^(−x) = 1/b^x), plot the points and join them with a smooth curve. The curve gets closer and closer to the line y = c but never reaches it.

To solve an equation from a graph, rearrange it so that one side is the function you have drawn; the solutions are the x-coordinates where the curve meets the other side (a horizontal line or another graph). Graph answers are estimates, so give them to the accuracy you can read, usually 1 decimal place.

Example (Extended, non-calculator): Complete a table of values for y = 2^x − 3 for x = −2, −1, 0, 1, 2 and 3. Use the graph to solve 2^x − 3 = 0.

x = −2: y = 1/4 − 3 = −2.75; x = −1: y = 1/2 − 3 = −2.5; x = 0: y = 1 − 3 = −2
x = 1: y = −1; x = 2: y = 1; x = 3: y = 5
The curve crosses y = 0 between x = 1 and x = 2, a little past halfway.

Answer: x ≈ 1.6

Example (Extended, non-calculator): A colony of 500 bacteria doubles every hour. Write a formula for the number N after t hours, and find N after 6 hours.

Doubling each hour multiplies by 2 once per hour: N = 500 × 2^t
t = 6: N = 500 × 2⁶ = 500 × 64

Answer: N = 500 × 2^t; after 6 hours N = 32 000

0580 practice questions

16 questions written for 0580 in the Algebra and graphs practice: 4 Core and 12 Extended, 8 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Complete the table of values for $y = x^2 - 2x - 3$.

\[\begin{array}{c|ccccccc} x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline y & & & & & & & \end{array}\]

Show the answer and mark scheme

B2 5, 0, −3, −4, −3, 0, 5 (B1 for 5 or 6 correct)

Answer: $5, 0, -3, -4, -3, 0, 5$

Practice question (Extended, non-calculator, 4 marks):

The curve $y = x^2 - 3x - 2$ and the line $y = x + 3$ are drawn on the same grid. Find the coordinates of the points where they meet.

Show the answer and mark scheme

M1 x² − 3x − 2 = x + 3
M1 x² − 4x − 5 = 0 and (x − 5)(x + 1) = 0
A1 x = −1 and x = 5
A1 (−1, 2) and (5, 8)

Answer: $(-1, 2)$ and $(5, 8)$

Edexcel 4MA1 notes: Quadratic Graphs
Practise the shared part (4MA1 questions): Quadratic and Cubic Graphs questions · Real-Life Graphs questions

2.11 Sketching curves

Core C2.11 Extended E2.11

Recognise, sketch and interpret graphs of linear and quadratic functions, showing roots and symmetry (turning points are not required at Core)

Extended also: Sketch functions equivalent to ax + by = c, y = ax² + bx + c, y = ax³ + b, y = ax³ + bx² + cx, y = a/x + b and y = ar^x + b (a, b, c rational; r rational and positive); Show turning points, roots, symmetry and vertical and horizontal asymptotes; Find the turning point of a quadratic by completing the square.

Partly in Edexcel 4MA1: Exponential graph shapes are outside 4MA1.

Notes for 0580

A sketch shows the shape and the key points, not a plotted scale. Quadratic y = ax² + bx + c: a U shape when a > 0 and an upside-down U when a < 0; it crosses the y-axis at c, and the roots come from solving y = 0. The curve is symmetrical: when it has two roots, the line of symmetry is halfway between them. Core sketches show the roots, the y-intercept and the symmetry; turning points are not needed at Core.

Extended: find the turning point of a quadratic by completing the square. y = (x + p)² + q has its turning point at (−p, q), a minimum when the x² term is positive; this works whether or not the curve crosses the x-axis (if it has no real roots, the halfway rule cannot be used, but completing the square still can).

Extended also sketches cubics y = ax³ + b and y = ax³ + bx² + cx (with a positive x³ term the curve rises from bottom left to top right; mark each root), reciprocals y = a/x + b (two branches, asymptotes x = 0 and y = b) and exponentials y = ar^x + b (for a > 0 and r > 1 it rises steeply to the right and approaches the asymptote y = b on the left).

Example (Core, non-calculator): Sketch y = x² − 6x + 5, marking where it meets the axes and its line of symmetry.

x = 0 gives y = 5, so it crosses the y-axis at (0, 5).
x² − 6x + 5 = (x − 1)(x − 5) = 0 gives x = 1 or x = 5.
The line of symmetry is halfway between the roots: x = 3.
The x² term is positive, so it is a U shape.

Answer: U-shaped curve through (0, 5), (1, 0) and (5, 0), symmetrical about the line x = 3

Example (Extended, non-calculator): By completing the square, find the turning point of y = x² − 6x + 5 and say whether it is a maximum or a minimum.

x² − 6x = (x − 3)² − 9
y = (x − 3)² − 9 + 5 = (x − 3)² − 4
(x − 3)² is never negative, so y is smallest when x = 3, where y = −4.

Answer: Minimum point (3, −4)

Example (Extended, non-calculator): Sketch y = 2^x + 1.

x = 0: y = 1 + 1 = 2, so it crosses the y-axis at (0, 2).
As x decreases, 2^x gets closer to 0, so y gets closer to 1: the asymptote is y = 1.
It increases more and more steeply as x increases.

Answer: Increasing exponential curve through (0, 2), approaching the asymptote y = 1 on the left

Example (Extended, non-calculator): Sketch y = x(x − 1)(x + 3).

Roots: x = 0, x = 1 and x = −3.
Expanded, y = x³ + 2x² − 3x, so the curve passes through the origin (0, 0).
The x³ term is positive, so the curve rises from bottom left to top right.

Answer: Cubic curve crossing the x-axis at −3, 0 and 1, passing through the origin

0580 practice questions

12 questions written for 0580 in the Algebra and graphs practice: 4 Core and 8 Extended, all non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 3 marks):

The graph of $y = x^2 - 9$ is sketched. Write down

(a) the coordinates of the points where it crosses the $x$-axis

(b) the coordinates of the point where it crosses the $y$-axis

Show the answer and mark scheme

B1 (a) (−3, 0) and (3, 0)
B1 (b) (0, −9)
B1 (c) x = 0

Answer: (a) $(-3, 0), (3, 0)$ (b) $(0, -9)$ (c) $x = 0$

Practice question (Extended, non-calculator, 4 marks):

(a) Write $-x^2 + 6x - 5$ in the form $a - (x - b)^2$.

(b) Sketch $y = -x^2 + 6x - 5$, showing the turning point and the points where it meets the axes.

Show the answer and mark scheme

B1 (a) 4 − (x − 3)²
B1 (b) maximum point (3, 4) FT their (a)
B1 (b) (1, 0) and (5, 0)
B1 (b) ∩-shape through (0, −5)

Answer: (a) $4 - (x - 3)^2$ (b) ∩-shape, maximum $(3, 4)$, through $(1, 0), (5, 0), (0, -5)$

Edexcel 4MA1 notes: Quadratic Graphs · Turning Points
Practise the shared part (4MA1 questions): Quadratic and Cubic Graphs questions

2.12 Differentiation

Extended only E2.12

Extended: Estimate gradients of curves by drawing tangents; Differentiate axⁿ (a rational, n a positive integer or zero) and simple sums of up to three such terms, using dy/dx notation; Gradients and stationary (turning) points; decide between maxima and minima by any method (no points of inflection).

Same mathematics as Edexcel 4MA1.

Revise it

Differentiating gives the gradient function: bring the power down and reduce it by one (xⁿ → nxⁿ⁻¹).

Example: Differentiate y = 4x³ − 5x + 2.

Answer: dy/dx = 12x² − 5.

Edexcel 4MA1 notes: Differentiation · Turning Points
Practise (4MA1 questions, same mathematics): Real-Life Graphs questions · Differentiation questions

2.13 Functions

Extended only E2.13

Extended: Function notation, domain and range; Inverse functions f⁻¹(x); Composite functions gf(x) = g(f(x)).

Same mathematics as Edexcel 4MA1.

Edexcel 4MA1 notes: Graphs, Sequences and Calculus (all notes)
Practise (4MA1 questions, same mathematics): Functions questions

Practise algebra and graphs

80 practice questions are written for 0580 on 2.1, 2.2, 2.4, 2.5, 2.6, 2.7, 2.9, 2.10, 2.11. For the 10 sub-topics that are the same mathematics in Edexcel 4MA1, the practice also has our 4MA1 Higher questions, which are Extended level. Core students: use the Core filter on the practice page; every sub-topic with Core content has questions written for 0580 Core. Practice is part of the IGCSE plan, with a free preview; these notes are free.

Algebra and graphs practice for 0580: the 80 questions written for 0580, tagged Core or Extended and non-calculator or calculator, with the Edexcel 4MA1 Higher questions for the sub-topics that are the same mathematics
Student hub: the Edexcel 4MA1 units (Unit 2, Unit 3, Unit 4, Unit 8), each question marked
Cambridge IGCSE Maths 0580 past papers: the 2025 specimen papers and past papers, with a timer
IGCSE Maths questions by topic with worked answers