Algebra and functions

Algebra and functions is Pure Topic 2 of Edexcel A level Mathematics (9MA0), and it's the topic almost everything else leans on. Indices, surds, quadratics, simultaneous equations, inequalities and polynomials are built in Year 12; rational expressions, modulus, composite and inverse functions, combined transformations and partial fractions extend it in Year 2. Because so much other content assumes this fluency, it's typically credited with around 32 of the 300 marks directly — an estimate, since a shaky quadratic or mishandled inequality also costs marks in questions nominally about calculus, trigonometry or exponentials.

AS content is assessed across Papers 1 and 2; the Year 2 additions sit mostly in Paper 1. Calculators are allowed throughout, but these questions test whether you can manipulate expressions and justify a method on paper, not just produce a number.

The spec at a glance

The specification statements this topic covers. AS = Year-1 content, also assessed in the standalone AS course (8MA0); A2 = full A level only. Typical share of a 300-mark series: ≈32 marks — our estimate from the 2018–2025 papers, not an official weighting.

Ref	Spec statement	Level
2.1	Laws of indices	AS
2.2	Surds	AS
2.3	Quadratic functions	AS
2.4	Simultaneous equations	AS
2.5	Linear and quadratic inequalities	AS
2.6	Polynomials and factor theorem	AS
2.6	Rational expressions	A2
2.7	Graphs of functions	AS
2.7	Modulus of a linear function	A2
2.8	Composite and inverse functions	A2
2.9	Graph transformations	AS
2.9	Combined graph transformations	A2
2.10	Partial fractions	A2
2.11	Functions in modelling	A2

What examiners keep reporting

'Show that' chains left incomplete

Examiners repeatedly report answers that jump from an intermediate line straight to the printed result — dropping a final '= 0', or starting from the answer and working backwards. Treat the given result as a destination, not a tool, and write down every line that leads to it.

Method marks need algebra on the page, not a calculator shortcut

For a three-term quadratic, examiners look for a genuine method — factorising, the formula, or completing the square — and a calculator-only answer with no working can lose the method mark even when the number is right. Write the equation down before solving it.

Answers left in the wrong form

Students often do the algebra correctly and lose the mark on presentation: a decimal where 'exact' demands a surd, or a model's constants stated without ever writing the complete equation asked for. Re-read the question's last sentence once you have an answer.

Sketches missing the features that carry the marks

A 'sketch' is marked on shape and key features — intercepts, asymptotes, turning points — not on scale, and examiners flag curves that cross an asymptote they should only approach. Work out the behaviour as x approaches zero and infinity before drawing anything.

Common mistakes

Invented index laws

Writing x^a·x^b as x^(ab) instead of x^(a+b), or mishandling negative and fractional powers such as x^(−2) as −x². Test a rule on small numbers, like 2³·2², before trusting it on algebra.

Splitting a surd over a sum

√(a+b) is not √a + √b, and rationalising only works if the whole fraction — top and bottom — is multiplied by the conjugate, not just one part.

Wrong discriminant inequality

Using b²−4ac ≥ 0 for two distinct roots (it needs strictly greater than zero) is an easy sign to get backwards, especially once a, b and c must be identified after rearranging. Sketch what the condition means for the graph first.

Forgetting to reverse an inequality

Multiplying or dividing by a negative number, or an unknown that might be negative, without flipping the inequality sign turns a correct method into a wrong answer. Check by substituting a value back into the original.

fg(x) applied in the wrong order

fg(x) means g acts on x first and f acts on the result, not the reverse — treating fg as f(x)g(x), or evaluating f before g, gives a different function altogether.

Stretch factor and order in combined transformations

f(ax) stretches a graph horizontally by 1/a, not a, and combinations like y = 2f(3x) only work if applied in the right order. Track one point through each step rather than guessing the combined effect.

Sample questions

Worked in full, with every line’s method and accuracy marks shown the way an Edexcel scheme awards them. How we check our questions.

Sample question 1 · Edexcel 9MA0 style(4 marks)

Find the value of

16^{-\frac{3}{4}}

. (2 marks)

Worked solution and mark scheme

$16^{-\frac{3}{4}} = \dfrac{1}{16^{\frac{3}{4}}} = \dfrac{1}{\left(16^{\frac{1}{4}}\right)^{3}}$
M1A negative index means take the reciprocal; the index $\tfrac{3}{4}$ means the fourth root, raised to the power $3$ .
$= \dfrac{1}{2^{3}} = \dfrac{1}{8}$
A1

Given that

\dfrac{\sqrt{x^{5}}\times x^{2}}{x^{-1}} = x^{k}

, where

x > 0

, find the value of the constant

k

. (2 marks)

Worked solution and mark scheme

$\dfrac{\sqrt{x^{5}}\times x^{2}}{x^{-1}} = \dfrac{x^{\frac{5}{2}}\times x^{2}}{x^{-1}} = \dfrac{x^{\frac{9}{2}}}{x^{-1}}$
M1 $\sqrt{x^5}=x^{5/2}$ , and multiplying powers of the same base adds the indices: $\tfrac{5}{2}+2=\tfrac{9}{2}$ .
$= x^{\frac{9}{2}-(-1)} = x^{\frac{11}{2}}$
A1Dividing subtracts the indices, so $\tfrac{9}{2}-(-1)=\tfrac{11}{2}$ . Hence $k=\tfrac{11}{2}$ .

Sample question 2 · Edexcel 9MA0 style(3 marks)

Solve the equation

(\sqrt{x} + 2)^2 = x + 16,

showing each step of your algebraic working. (3 marks)

Worked solution and mark scheme

$(\sqrt{x} + 2)^2 = (\sqrt{x} + 2)(\sqrt{x} + 2) = x + 4\sqrt{x} + 4$
M1Every product counts — the 4√x cross term is the one most students drop.
$x + 4\sqrt{x} + 4 = x + 16 \;\Rightarrow\; 4\sqrt{x} = 12 \;\Rightarrow\; \sqrt{x} = 3$
A1The x on each side cancels, leaving a short equation in √x.
$\sqrt{x} = 3 \;\Rightarrow\; x = 9$
A2Square both sides to undo the root, then check: (√9 + 2)² = 25 and 9 + 16 = 25.

How to revise it

The laws of indices and the quadratic formula aren't given in the formula booklet, so keep them secure from memory, along with surd conventions and the direction of a discriminant condition. Most of the rest — curve shapes, transformation effects, partial-fractions forms — is faster to re-derive from a quick sketch or small example than to memorise as a list.

Calculator discipline matters even though every paper allows one: write the equation down before solving it, since the method mark depends on visible algebra, not just the final root. For 'sketch' and 'show that' questions, check your last line against exactly what was asked — form and justification are marked separately from the working.

Because algebra underpins calculus, trigonometry and exponential work throughout the course, mixed practice pays off more than drilling this topic alone — marks lost elsewhere in a paper often trace back to a slip in factorising or rearranging rather than to whatever the question is nominally about.

All 19 topics: Edexcel A level Maths topic guides. Reference: formula booklet vs memorise and grade boundaries.

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Original questions written for the Pearson Edexcel A Level Mathematics (9MA0) specification. Not affiliated with or endorsed by Pearson.