GCSE Maths: Functions and Graphs
Functions and Graphs is where algebra becomes visual. The same rules — y = mx + c, completing the square, transforming graphs — appear again and again, so a small set of techniques unlocks a large slice of the higher-tier paper.
Straight-line graphs come down to y = mx + c. You need to find the gradient between two points (m = (y₂−y₁)/(x₂−x₁)), find the equation of a line through two points, and write the equation of a perpendicular line (gradient is −1/m). Parallel lines have the same gradient.
Quadratic graphs — y = ax² + bx + c — are examined for their roots (where the curve crosses the x-axis), turning point (from completing the square: y = a(x + p)² + q has turning point (−p, q)) and y-intercept (the constant). Higher tier asks for these from the equation, not just the graph.
Cubic, reciprocal and exponential graphs each have a recognisable shape. You should be able to sketch them from the equation, identify asymptotes (for reciprocal and exponential graphs) and read the key features.
Graph transformations follow four rules: y = f(x) + a (vertical shift), y = f(x + a) (horizontal shift in opposite direction), y = af(x) (vertical stretch), y = f(ax) (horizontal stretch by 1/a). Recognise these in unfamiliar contexts; questions often use trig graphs to test transformations.
Function notation f(x), composite functions fg(x) and inverse functions f⁻¹(x) appear most papers. For composite functions, work inside-out (apply g first, then f). For inverse functions, swap x and y in y = f(x), then make y the subject.
What you'll learn
- The full Functions and Graphs sub-topic map and how it sits within GCSE Maths
- What examiners reward in Functions and Graphs questions, command word by command word
- The most common Functions and Graphs mistakes — and how to avoid them in your answers
- How to revise Functions and Graphs with notes, flashcards, exam-style questions and the AI Study Coach
- How OES generates Maths resources without using official copyrighted past papers
Who this is for
- Students revising Functions and Graphs as part of GCSE Maths
- Anyone retaking GCSE Maths who needs a focused review of Functions and Graphs
- Parents and tutors supporting Maths students on Functions and Graphs
Learning objectives
By the end of this topic you should be able to:
- Find the equation of a straight line through two points, including parallel and perpendicular lines
- Sketch quadratic, cubic, reciprocal and exponential graphs
- Find the turning point of a quadratic by completing the square
- Apply graph transformations f(x) + a, f(x + a), af(x) and f(ax)
- Use function notation including composite and inverse functions
Common mistakes
- Confusing the direction of a horizontal shift (y = f(x + 3) shifts left, not right)
- Forgetting the negative reciprocal for perpendicular gradients
- Mixing up roots, turning point and y-intercept of a quadratic
- Working out fg(x) as gf(x) instead
- Forgetting to make y the subject when finding the inverse function
Revision tips
- Memorise the four transformation rules with a single graph example each
- Drill 'find the equation through two points' until it takes under a minute
- Practise quadratic sketches with all three key features labelled
- Build a flashcard for f, fg, gf, f⁻¹ with a worked example on the back
- Use OES to drill composite-function questions — they are common and easy to lose marks on
How OES helps
AI Revision Notes
Concise, spec-aligned notes for every Maths topic, written to match what examiners reward.
Smart Flashcards
Adaptive Maths flashcards that prioritise the cards you keep forgetting using spaced repetition.
Exam-Style Papers
AI-generated Maths practice papers in the style of real exams — not official copyrighted past papers — with mark-scheme aware feedback.
Personal AI Study Coach
A 24/7 coach that plans every Maths session, explains anything you ask and adapts to your weakest topics.
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Frequently asked questions
Is Functions and Graphs on every GCSE Maths specification?
Yes. Functions and Graphs is a core part of the GCSE Maths specification across AQA, Edexcel and OCR. Some boards weight sub-topics slightly differently, but the content on this page maps to all three.
How long should I spend revising Functions and Graphs?
Most students need 4–6 focused sessions to feel confident on Functions and Graphs: one for the notes, one or two for flashcards, and two for exam-style questions. The AI Study Coach can build a schedule for you that fits the time you have left before the exam.
Does OES use official Maths past papers for Functions and Graphs questions?
No. OES does not redistribute official copyrighted past papers. We generate fresh exam-style questions in the style of real AQA, Edexcel and OCR papers, marked against the published mark scheme conventions.
What are the most common mistakes on Functions and Graphs questions?
Confusing the direction of a horizontal shift (y = f(x + 3) shifts left, not right) Forgetting the negative reciprocal for perpendicular gradients The full list — and how to avoid each — is in the "Common mistakes" section above.
Is this topic harder for higher-tier or foundation students?
Most of Functions and Graphs is on both tiers. Higher-tier papers add a few stretch sub-topics and more demanding command words ("evaluate", "explain in detail"). The OES content is tier-aware — questions can be generated at foundation, higher or mixed difficulty.
How do I know if I've actually understood Functions and Graphs, not just memorised it?
The fastest test is to attempt a fresh exam-style question and self-mark it against the mark scheme. If you can hit every mark point in your own words, you understand it. If you keep dropping the same marks, the AI Study Coach can target those gaps directly.
Where should I go next after Functions and Graphs?
Related topics like algebra, trigonometry, geometry build directly on Functions and Graphs, so they are a strong next step. You can also drill Maths exam-style papers to combine Functions and Graphs with everything else you have revised.
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