Knowledge organisers

Algebra and functions: A Level Maths knowledge organiser

Everything to know about algebra and functions 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

Discriminant: b² − 4ac decides how many real roots ax² + bx + c = 0 has.
Completing the square: Writing ax² + bx + c as a(x + p)² + q to find the turning point.
Factor theorem: f(a) = 0 exactly when (x − a) is a factor of f(x).
Inverse function: f⁻¹ reverses f; it exists only when f is one-to-one, and its graph reflects f in y = x.

Key formulas

Quadratic formula	\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
Discriminant \(b^2-4ac\): \(>0\) two real roots, \(=0\) one repeated root, \(<0\) no real roots
Completing the square	\(ax^2+bx+c=a\left(x+\tfrac b{2a}\right)^2+c-\tfrac{b^2}{4a}\)
Indices	\(a^ma^n=a^{m+n},\) \((a^m)^n=a^{mn},\) \(a^{-n}=\tfrac1{a^n},\) \(a^{\frac mn}=\sqrt[n]{a^m}\)
Surds	\(\sqrt{ab}=\sqrt a\sqrt b,\) \(\frac1{\sqrt a+\sqrt b}=\frac{\sqrt a-\sqrt b}{a-b}\)
Factor theorem	\(f(a)=0\iff(x-a)\text{ is a factor}\)
Partial fractions	\(\frac{px+q}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b},\) \(\frac{\dots}{(x-a)^2(x-b)}=\frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{x-b}\)
Modulus; composite; inverse	\(\|x\| \(fg(x)=f(g(x));\) \(ff^{-1}(x)=x\)
Transformations: \(f(x)+a\) up \(a\); \(f(x+a)\) left \(a\); \(af(x)\) vertical ×\(a\); \(f(ax)\) horizontal ×\(\tfrac1a\)

Worked example

Write x² − 6x + 11 in the form (x + p)² + q and state the minimum value.

x² − 6x = (x − 3)² − 9
x² − 6x + 11 = (x − 3)² − 9 + 11

Answer: (x − 3)² + 2, so the minimum value is 2, at x = 3

Common mistakes

Inverse and composite functions: domain left out, fg and gf mixed up
Proof: "squares are always positive" and no concluding statement
(ab)ⁿ = abⁿ
A^(−n) treated as −aⁿ

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

You should be able to…

Apply the laws of indices to simplify expressions such as (2x³)² ÷ 4x, then expand products of brackets and factorise by taking out common factors.
Sketch quadratic graphs showing roots, the y-intercept and the turning point, and use function notation f(x) when finding roots and values.
Represent linear and quadratic inequalities as regions on a graph, using dotted and solid boundary lines, and describe a shaded region with inequalities.
Simplify algebraic fractions by factorising and cancelling, and divide polynomials such as x³ + 2x² − 5x − 6 by (x + 1) using algebraic long division.
Classify mappings as one-to-one, many-to-one or not functions, and state the domain and range of functions, including those defined piecewise.
Split an expression into partial fractions and then use binomial expansions of each part to expand it, stating the range of validity of the result.

The printable sheet

Algebra and functions knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Algebra and functions knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

Other A Level Maths topics: Proof · Coordinate geometry · Sequences and series · Trigonometry · Exponentials and logarithms · Differentiation · Integration · Numerical methods · Vectors · Sampling and data · Probability · Binomial and normal distributions · Hypothesis testing · Kinematics · Forces and Newton's laws · Moments · All A Level Maths organisers