Invert non-zero scales and recognise a collapse
Use a reciprocal scale at the same center or invariant line · Retain the sign of a stated non-zero scale · Explain why a zero-scale map has no unique inverse on the plane
Section Paper 2 · Historical Extended · non-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
- Dilation and stretch rules
- Reciprocals
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 a reciprocal scale at the same center or invariant line
- Retain the sign of a stated non-zero scale
- Explain why a zero-scale map has no unique inverse on the plane
Reciprocal with the same reference
For a non-zero scale k about C, the inverse is P=C+(1/k)(P′−C). For a stretch, invert its factor and retain its direction and invariant line. A factor 2 is undone by 1/2; an explicitly signed factor −2 is undone by −1/2, not +1/2.
Zero loses information
A scale k=0 sends every point to its center; a horizontal stretch s=0 sends every point with a given y to the invariant vertical line. Many original points therefore have the same image. On the whole plane there is no unique inverse that recovers the original point.
Check both non-zero conditions and the geometric reference before dividing. An inverse rule must recover the same source from the forward image, not merely produce a plausible point.
Sources for this lesson
Pause and explain
Can a zero horizontal stretch have a unique inverse on the whole plane?
Worked example
A factor-2 dilation about (1,−1) gives image (5,3). Recover the original.
Show the worked solution
- Image displacement from the center is (4,4).
- Halve it and restore the center to obtain (3,1).
Answer Original point (3,1).
Try a topic test
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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.
A positive dilation of factor 3 about O gives image (6,−3). Find the original.
Give me a hint
Apply factor 1/3.
Compare my reasoning
- 6/3=2.
- −3/3=−1.
(2,−1).
Look for these in your work
- Use the reciprocal
- Keep the original center
A horizontal factor-2 stretch with invariant line x=1 gives image (7,4). Recover the source point.
Give me a hint
Halve the image's distance from x=1.
Compare my reasoning
- Source x=1+(7−1)/2=4.
- y stays 4.
(4,4).
Look for these in your work
- Use the actual invariant line
- Scale only the specified direction
The defined map P′=C−2(P−C), C(1,1), gives image (−3,5). Recover P. Compare with a factor-zero map at C.
Give me a hint
Use reciprocal −1/2 for the signed map.
Compare my reasoning
- Image offset (−4,4) becomes (2,−2); add C to get (3,−1).
- A zero map would give C for every source, so it has no unique inverse on the plane.
P(3,−1); the zero map cannot recover a unique source.
Look for these in your work
- Retain the reciprocal's sign
- Explain the information lost at zero
Common mistakes
- A reciprocal factor does not change the center or invariant line.
- Do not divide by zero or drop a negative sign.
What can you explain now?
Can a zero horizontal stretch have a unique inverse on the whole plane?
Compare with the explanation
No; it merges multiple points onto the invariant line.
Their original horizontal coordinates cannot be recovered uniquely.
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Make it stick
Can a zero horizontal stretch have a unique inverse on the whole plane?
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.