Combinations and the order of transformations
Apply transformations in the stated order and compare the overall mapping.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Translate, reflect, rotate and enlarge individual points.
- Keep original and image labels distinct.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which point does the second transformation use?
What you will learn
- Carry an intermediate image into the second transformation.
- Show why changing order can change the final image.
The second transformation acts on the first image
Start with P(1,2), translate by (3,0) to P′(4,2), then reflect in the y-axis to P″(−4,2). If reflection comes first, (1,2) goes to (−1,2) and translation then gives (2,2). The two orders have different final images.
Use an intermediate sketch or coordinate table for every vertex. Keep the stated centre or mirror for the second operation; it does not automatically move with the shape. Combinations are Extended requirements, while Core focuses on individual transformations.
Describe an equivalent single transformation when possible
Two translations combine by adding their displacement vectors. Translations (2,−1) then (−5,4) give a single translation (−3,3). Two reflections in the same line restore the original shape. Two half-turns about the same centre also restore it.
Do not assume a combination is always a translation or that transformations commute. Test multiple corresponding vertices and state any equivalent single transformation completely. One coincident point alone does not determine the whole shape's transformation.
Pause and explain
Translate (1,2) by (3,0), then reflect in the y-axis. What is the result?
Worked example
Translate (1,2) by (3,0), then reflect in the y-axis.
Show the worked solution
- The translation gives (4,2).
- Reflecting in x = 0 changes its x sign.
- The final point is (−4,2); order is essential.
Answer (−4,2).
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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.
Combine translations (2,−1) and (−5,4).
Give me a hint
Add component by component.
Compare my reasoning
- Horizontal total is 2 − 5 = −3.
- Vertical total is −1 + 4 = 3.
- The equivalent single translation is (−3,3).
(−3,3).
Look for these in your work
- Added corresponding components.
- Named the equivalent transformation.
Reflect (2,−3) in the x-axis, then rotate 180° about the origin.
Give me a hint
Record the intermediate image.
Compare my reasoning
- Reflection gives (2,3).
- Half-turn gives (−2,−3).
- For all points this combination is equivalent to reflection in the y-axis.
(−2,−3); equivalent to reflection in the y-axis.
Look for these in your work
- Used the intermediate point.
- Described the equivalent mapping.
Compare translating (1,0) by (2,0) then enlarging by 2 about the origin with the reversed order.
Give me a hint
The enlargement scales the translation only in the first order.
Compare my reasoning
- Translation then enlargement gives (3,0) then (6,0).
- Enlargement then translation gives (2,0) then (4,0).
- The different results show that these operations do not commute.
(6,0) versus (4,0); order changes the result.
Look for these in your work
- Computed both orders.
- Explained the difference.
Common mistakes
- Skipping the intermediate image.
- Moving the second centre without instruction.
- Assuming every combination commutes.
What can you explain now?
Can the order of two translations be reversed without changing the result?
Compare with the explanation
Yes.
Their displacement components add, and numerical addition is commutative.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, find one combination that commutes and one that does not, using a labelled point.
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.