Combine and reverse transformations
Apply transformations in their stated order, track corresponding vertices and undo each step in reverse order.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Is a reflection its own inverse?
What you will learn
- Apply two transformations in the specified order
- Describe a combined transformation sequence clearly
- Find an original point by applying inverse transformations in reverse order
Order changes the image
Transformations are generally not commutative: translating then reflecting may give a different result from reflecting then translating. Label each intermediate image so points are not confused.
A ruler should be used for straight edges when drawing a shape. Check the final coordinates or measurements against each step in sequence.
Undo the last action first
The inverse of a translation uses the opposite vector; reflection is its own inverse; reverse a rotation by the same angle in the opposite direction. An enlargement is reversed by the reciprocal scale factor about the same centre.
To recover an original point, reverse the order of the transformations and apply each inverse. For an enlargement by −2, the inverse factor is −1/2, not +1/2.
Pause and explain
What is the inverse scale factor for enlargement by 3?
Worked example
Point P(1, 2) is translated by (3, −1), then reflected in the y-axis. Find its final image.
Show the worked solution
- After translation: (1+3, 2−1) = (4, 1).
- Reflection in the y-axis changes the sign of x.
- Final image is (−4, 1).
Answer (−4, 1).
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.
Point A(2, 1) is reflected in the x-axis, then rotated 90° anticlockwise about the origin. Find the image.
Give me a hint
Apply the reflection first; use its image as the input to the rotation.
Compare my reasoning
- Reflect: (2, 1) → (2, −1).
- Rotate 90° anticlockwise: (x, y) → (−y, x).
- (2, −1) → (1, 2).
A′ = (1, 2).
Look for these in your work
- First image recorded
- Order respected
- Second mapping applied
A point is enlarged by factor 2 about the origin, then translated by (−1, 3), to give Q(7, 9). Find the original point.
Give me a hint
Undo the translation, then undo the enlargement.
Compare my reasoning
- Undo translation: (7, 9) − (−1, 3) = (8, 6).
- Undo enlargement by dividing coordinates by 2.
- Original point = (4, 3).
(4, 3).
Look for these in your work
- Last step reversed first
- Opposite translation applied
- Reciprocal scale applied
A photographer enlarges a landmark image by scale factor −2 about C(1, 1), then reflects it in the x-axis. A landmark image ends at Q(−5, −7). Find its original position.
Give me a hint
Undo the reflection, then reverse the enlargement about the same centre.
Compare my reasoning
- Undo the x-axis reflection: (−5, −7) → (−5, 7).
- Vector from C to this point is (−6, 6).
- Undo factor −2 by multiplying the vector by −1/2: (3, −3). Original point = (4, −2).
Original point (4, −2).
Look for these in your work
- Last transformation undone first
- Centre-based vector found
- Negative reciprocal factor applied
Common mistakes
- Applying transformations in a different order
- Reflecting the original when the second step uses the first image
- Undoing steps in the forward order
- Using a scale factor rather than its reciprocal for the inverse enlargement
What can you explain now?
Which transformation must be undone first when reversing a sequence?
Compare with the explanation
The last transformation applied
Reversal proceeds in reverse order.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why do you reverse both the order and the scale factor when undoing a transformation sequence?
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.