Knowledge organisers

Graphs, sequences and calculus: Edexcel IGCSE Maths (4MA1) knowledge organiser

Everything to know about graphs, sequences and calculus 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

Linear Graphs: A linear graph is y = mx + c where m is the gradient (rise/run) and c is the y-intercept.
Sequences: A sequence is a list of numbers with a pattern. The nth-term rule generates any term from its position.
Quadratic Graphs: A quadratic graph is a parabola y = ax² + bx + c. Its vertex is the min (a>0) or max (a<0).
Differentiation: Differentiating gives the gradient function: bring the power down and reduce it by one (xⁿ → nxⁿ⁻¹).

Key formulas

Straight line	\(y=mx+c,\) \(m=\frac{y_2-y_1}{x_2-x_1}\)
Parallel; perpendicular	\(m_1=m_2;\) \(m_1\times m_2=-1\)
Midpoint; length	\(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2\right),\) \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Differentiation	\(y=ax^n\Rightarrow\frac{dy}{dx}=anx^{n-1}\)
Turning points: \(\frac{dy}{dx}=0\); gradient of the tangent at \(x=a\) is \(\frac{dy}{dx}\) at \(a\)
Kinematics	\(v=\frac{ds}{dt},\) \(a=\frac{dv}{dt}\)
Distance–time gradient = speed; velocity–time gradient = acceleration, area = distance

Worked example

Find the turning point of y = x² − 6x + 1.

dy/dx = 2x − 6 = 0 → x = 3, y = −8. d²y/dx² = 2 > 0 → minimum (3, −8).

Common mistakes

Graph transformations of f(x)
Differentiation: forgetting the negative root and the context
Giving the nth term as n + d instead of dn + c
Reading gradient as change in x over change in y

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

You should be able to…

Use term-to-term rules to continue sequences and find and use the nth term of linear sequences, such as deciding whether 100 is a term.
Find the roots, y-intercept and line of symmetry of a quadratic graph, and use the graph to solve related equations.
Find the sum of an arithmetic series using the formula Sₙ = ½n[2a + (n − 1)d], and find unknown terms from a given sum.
Find the domain and range of functions, and identify values that must be excluded from the domain, such as x = 2 for 1/(x − 2).
Draw and interpret the graph of y = tan x, describe its asymptotes and period, and use symmetry to find several solutions of trigonometric equations.
Find turning points by setting the derivative equal to zero, and solve optimisation and kinematics problems using differentiation.

The printable sheet

Graphs, sequences and calculus knowledge organiser for Edexcel IGCSE Maths (4MA1): one A4 page of key definitions, formulas, a worked example and common mistakes — Graphs, sequences and calculus knowledge organiser (Edexcel IGCSE Maths (4MA1)), A4. Download the PDF.

Revise it next

Other Edexcel IGCSE Maths (4MA1) topics: Number · Algebra · Geometry and trigonometry · Statistics and probability · All Edexcel IGCSE Maths (4MA1) organisers