graph transformations
4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.
Lesson Overview
Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Learning Objectives
- Explain the key ideas of graph transformations
- Apply graph transformations to exam-style questions
- Develop independent study skills
- Build cross-topic connections
Prerequisites
- Read the revision notes for graph transformations first (GCSE Revise summary)
- Key vocab to pre-teach: Transformations
- Basic skills: reading the summary notes and answering the practice questions there
Materials & Equipment
- Exercise book, pencil, ruler
- Scientific calculator
- Pair of compasses and protractor (geometry topics)
- Graph paper
- Printed revision notes (link below)
Lesson 1: Introduction: graph transformations
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Write down everything you already know about graph transformations. Then check against the key terms: Transformations. Use a mini-whiteboard or paper.
Main Content (35 minutes)
Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Transformations.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Before lesson: Read the script below. Pre-teach key vocab: Transformations.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: graph transformations. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
Mins 0-5 - Hook: "Today: graph transformations. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
First Look
Start with the revision notes summary, then attempt: y = f(x) is transformed to y = f(x) + 5. Describe the transformation.
Plenary (5 minutes)
Check Out
Your student states one thing they learned and one question they still have about graph transformations.
Lesson 2: Core Concepts: graph transformations
Duration: 50 minutes
Starter Activity (5 minutes)
Review Previous Lesson
Quick recap: write 3 key points from Lesson 1 on graph transformations. Check them against the notes below.
Main Content (35 minutes)
Transformations: Moving or reflecting a graph without changing its basic shape.
| Term | Meaning | Example |
|---|---|---|
| Translation up | y = f(x) + a | Move up by a units |
| Translation down | y = f(x) - a | Move down by a units |
| Translation right | y = f(x - a) | Move right by a units |
| Translation left | y = f(x + a) | Move left by a units |
| Reflection in x-axis | y = -f(x) | Flip vertically |
| Reflection in y-axis | y = f(-x) | Flip horizontally |
Practice (10 minutes)
Q: y = f(x) is transformed to y = f(x) + 5. Describe the transformation.
Answer: Translation 5 units UP
Plenary (5 minutes)
Explain Back
Your student teaches the key points back to you without looking. Fill any gaps immediately.
Lesson 3: Application: graph transformations
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Recall the key terms: Transformations. Define each in one sentence.
Main Content (35 minutes)
Parent/Teacher Guide: Let your student attempt each question alone first, then compare with the model answer. Award method marks for correct working even if the final answer is wrong.
Q1: y = f(x) is transformed to y = f(x) + 5. Describe the transformation.
Answer: Translation 5 units UP
Q2: y = f(x) is transformed to y = f(x - 2). Describe the transformation.
Answer: Translation 2 units RIGHT
Q3: y = f(x) is transformed to y = -f(x). Describe the transformation.
Answer: Reflection in the x-axis
Q4: y = f(x) is transformed to y = f(-x). Describe the transformation.
Answer: Reflection in the y-axis
Q5: y = x² has vertex (0, 0). Where is the vertex of y = (x - 3)² + 4?
Answer: (3, 4)
Q6: y = f(x) passes through (2, 4). Where does y = -f(x) pass through?
Answer: (2, -4)
Plenary (5 minutes)
Error Review
Review any questions answered incorrectly. Identify whether the error was knowledge, method, or reading the question.
Lesson 4: Exam Practice: graph transformations
Duration: 50 minutes
Starter Activity (5 minutes)
Command Words
Review what these command words require: state (one point), describe (say what happens), explain (say why), compare (both sides), evaluate (judgement).
Main Content (35 minutes)
Extended Answer
Extended question: Extended Answer 6 marks: The graph of y = x² has vertex (0, 0) and passes through (2, 4) and (-1, 1). (a) Describe fully the transformation that maps y = x² to y = (x - 3)² + 4. (b) State the new vertex and the images of (2, 4) and (-1, 1). (c) The graph is then reflected in the x-axis. Write the new equation and find the vertex. <div class="
(a) Translation right by 3, up by 4. Vector [ 3 ⁄ 4 ] (b) New vertex: (3, 4). (2, 4) → (5, 8). (-1, 1) → (2, 5) (c) New equation: y = -(x - 3)² - 4. Vertex: (3, -4) Mark scheme: (a) 2 marks for full description with direction. (b) 2 marks for all three points. (c) 2 marks for equation and vertex.
Exam Tips: Inside the bracket: horizontal transformation (opposite direction) | Outside the bracket: vertical transformation (same direction) | Minus sign outside: reflection in x-axis | Minus sign inside: reflection in y-axis | x - a means right; x + a means left | Always state the transformation type AND direction
Common Errors: Watch Out! 1. Wrong: y = f(x - 3) shifts 3 units LEFT Correct: y = f(x - 3) shifts 3 units RIGHT (opposite direction inside brackets) 2. Wrong: y = -f(x) reflects in the y-axis Correct: y = -f(x) reflects in the x-axis (y-values change sign) 3. Wrong: y = f(x) + 2 shifts 2 units right Correct: y = f(x) + 2 shifts 2 units UP (outside the function = vertical)
AO3 - Reasoning & Interpretation: Reasoning and Interpretation The height h of a tide over time t is h = 5 + 3sin(t). A new model shifts the peak tide 2 hours later and raises the mean by 1 metre. (a) Write the new equation. (b) How does the amplitude change? (c) Explain why the new model might be more realistic. Answers: (a) h = 6 + 3sin(t - 2) — shift right 2 (inside) and up 1 (outside). (b) Amplitude is still 3 — unchanged by translations. (c) Shifting the peak accounts for later high tide; higher mean accounts for rising sea levels or seasonal effects.
Stretch & Challenge (Grade 8-9):
- Synoptic links: explain how graph transformations connects to another Mathematics topic you have studied
- Real-world: research one real-world use or example of graph transformations
- Critical: "What are the limitations of the models used in graph transformations?"
Plenary (5 minutes)
Assessment Criteria
- Got it: Confident explanation + correct worked examples
- Getting there: Main points OK, needs support with detail
- Not yet: Confused on key concepts - re-run Lesson 2
Homework & Consolidation
- Consolidation: Re-answer any Lesson 3 practice questions answered incorrectly (20 mins)
- Retrieval: Write flashcards for the key terms: Transformations (10 mins)
- Exam practice: One past-paper question on graph transformations from the board websites (15 mins)
- Extension: Explain graph transformations to someone else in your own words (10 mins)