Apply transformations in order and reverse an inverse sequence
Use the image of the first map as input to the second · Demonstrate non-commuting transformations with a counterexample · Undo an invertible sequence in reverse order
Section Paper 4 · Historical Extended · calculator
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Plane transformations and their inverses
- Substitution
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Use the image of the first map as input to the second
- Demonstrate non-commuting transformations with a counterexample
- Undo an invertible sequence in reverse order
Inside step, then outside step
If the instruction says R followed by T, first apply R, then T to the result. For reflection in the x-axis then translation [2; 1], (x,y) becomes (x,−y), then (x+2,−y+1). Reversing the order gives (x+2,−y−1), a different rule.
One counterexample disproves a claim that two orders always agree. Some special pairs commute, such as two translations, so do not claim that every reversal changes the result.
Undo the last step first
The inverse of T after R applies T's inverse first, then R's inverse. For any sequence of invertible maps, reverse the list and undo each step. A collapsed scale cannot be repaired by later steps into a unique whole-plane inverse.
Apply each operation to every vertex. A composition involving a one-direction stretch need not preserve similarity or lengths, so do not assume it is a single rigid motion.
Sources for this lesson
Pause and explain
How do you undo R followed by T when both are invertible?
Worked example
For P(1,2), compare reflection in the x-axis then translation [2; 1] with the reversed order.
Show the worked solution
- Reflect then translate: (1,−2) becomes (3,−1).
- Translate then reflect: (3,3) becomes (3,−3).
Answer (3,−1) versus (3,−3); the orders differ.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Reflect P(2,3) in the x-axis, then translate by [1; 2]. Find the result.
Give me a hint
Feed the reflected point into the translation.
Compare my reasoning
- Reflection gives (2,−3).
- Translation gives (3,−1).
(3,−1).
Look for these in your work
- Use the stated order
- Work from the intermediate image
Rotate P(1,2) 90° counterclockwise about O, then stretch horizontally by factor 2 about the y-axis. Compare with the reverse order.
Give me a hint
Use the intermediate point for each sequence.
Compare my reasoning
- Rotate then stretch: (−2,1) becomes (−4,1).
- Stretch then rotate: (2,2) becomes (−2,2).
(−4,1) versus (−2,2).
Look for these in your work
- Track both stages
- Do not assume the maps commute
A reflection in the x-axis followed by translation [2; 1] gives image (5,−3). Recover the original and name the reverse sequence.
Give me a hint
Undo the translation before reflecting.
Compare my reasoning
- Translate back to (3,−4).
- Reflect back to (3,4).
Original (3,4); inverse translation [−2; −1] followed by the same reflection.
Look for these in your work
- Reverse the sequence
- Check the forward round trip
Common mistakes
- Do not apply both maps separately to the original point.
- Undo a sequence from its last step backwards.
What can you explain now?
How do you undo R followed by T when both are invertible?
Compare with the explanation
Undo T first, then undo R.
The last operation is the first one that must be reversed.
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Make it stick
How do you undo R followed by T when both are invertible?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.