Knowledge organisers

Algebra and graphs: Cambridge IGCSE Maths (0580) knowledge organiser

Everything to know about algebra and graphs on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Expression, equation, identity: An expression has no equals sign; an equation is true for some values; an identity (≡) is true for all values.
Subject of a formula: The single letter on its own on one side, as v in v = u + at.
nth term: A rule that gives any term of a sequence from its position n.
Inverse function: f⁻¹(x) undoes f(x): swap x and y and rearrange.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Quadratic formula, \(ax^2+bx+c=0\)On the Extended formula list	\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
Completing the square	\(x^2+bx+c=\left(x+\tfrac b2\right)^2-\tfrac{b^2}4+c\)
Difference of two squares	\(a^2-b^2=(a+b)(a-b)\)
Linear sequence, \(n\)th term	\(u_n=a+(n-1)d\)
Exponential sequence	\(u_n=ar^{n-1}\)
Proportion	\(y\propto x^n\Rightarrow y=kx^n;\) \(y\propto\tfrac1{x^n}\Rightarrow y=\tfrac k{x^n}\)
Functions	\(fg(x)=f\big(g(x)\big);\) \(f^{-1}\text{: swap }x,y\text{ and rearrange}\)
Inequalities: multiplying or dividing by a negative number reverses the sign
Straight line: \(y=mx+c\), gradient \(m\), \(y\)-intercept \(c\); vertical line \(x=k\)
Gradient from two points	\(m=\frac{y_2-y_1}{x_2-x_1}\)
Midpoint; length	\(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2\right),\) \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Parallel lines	\(m_1=m_2\)
Perpendicular lines	\(m_1\times m_2=-1\)
Differentiation	\(y=ax^n\Rightarrow\frac{dy}{dx}=anx^{n-1}\)
Turning points: \(\frac{dy}{dx}=0\); the gradient of the curve at \(x=a\) is the value of \(\frac{dy}{dx}\) at \(a\)

Worked example

Solve the simultaneous equations 3x + 2y = 12 and x − y = 4.

From the second equation, x = y + 4
3(y + 4) + 2y = 12, so 5y = 0 and y = 0
x = 0 + 4

Answer: x = 4, y = 0

Common mistakes

Inverse functions: left in terms of y, or a sign slip
Composite functions: gg(x) is not g(x) × g(x)
Expanding brackets with a negative in front
Speed-time graphs: units and the sign of a deceleration

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Complete a short baseline check on number and algebra from earlier years so the teacher can see what each student already knows and plan the first unit.
Solve linear equations, including those with brackets and with unknowns on both sides, and check each solution by substitution.
Write quadratic expressions such as x² − 8x + 3 and 2x² + 6x − 1 in completed-square form.
Draw graphs of linear, quadratic and reciprocal functions from tables of values, using suitable scales and smooth curves.
Use function notation such as f(x) = 2x + 3, evaluate functions and find their domain and range, including values that must be excluded.
Find acceleration and deceleration from speed-time graphs and the distance travelled as the area under the graph.

The printable sheet

Algebra and graphs knowledge organiser for Cambridge IGCSE Maths (0580): one A4 page of key definitions, formulas, a worked example and common mistakes — Algebra and graphs knowledge organiser (Cambridge IGCSE Maths (0580)), A4. Download the PDF.

Revise it next

Other Cambridge IGCSE Maths (0580) topics: Number · Coordinate geometry · Geometry · Mensuration · Trigonometry · Transformations and vectors · Probability · Statistics · All Cambridge IGCSE Maths (0580) organisers