Stretch in one direction while keeping an invariant line
Identify the stretch direction and invariant line · Scale one coordinate relative to that line · Distinguish a stretch from uniform enlargement
Section E5.6 · Extended · Plane transformations
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
- Dilations
- Axis-aligned coordinates
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
- Identify the stretch direction and invariant line
- Scale one coordinate relative to that line
- Distinguish a stretch from uniform enlargement
Direction and invariant line
A horizontal stretch with factor s about the y-axis sends (x,y) to (sx,y): movement is parallel to the x-axis, and points on x=0 stay fixed. A vertical stretch about the x-axis sends (x,y) to (x,sy). State direction, factor and invariant line.
For a horizontal stretch with invariant line x=a, use x′=a+s(x−a), with y unchanged. For a vertical stretch about y=b, use y′=b+s(y−b), with x unchanged.
One direction is different from all directions
A horizontal factor 2 makes a square twice as wide without changing its height; the image is not similar to the original square. Unlike a uniform dilation, a stretch can change angle measures and side-length ratios. This lesson's examples use positive factors; a zero factor collapses one dimension and cannot have a unique inverse across the plane.
Sources for this lesson
Pause and explain
In a horizontal stretch about the y-axis, which coordinate stays unchanged?
Worked example
Stretch P(−3,1) horizontally by factor 2 about the y-axis.
Show the worked solution
- Double x but keep y.
- P′=(−6,1).
Answer P′(−6,1).
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.
Stretch (3,1) horizontally by factor 2 with invariant line x=0.
Give me a hint
Scale x only.
Compare my reasoning
- x′=6.
- y′=1.
(6,1).
Look for these in your work
- Use the stated direction
- Keep the other coordinate
Stretch (−2,6) vertically by factor 1/2 about the x-axis.
Give me a hint
Scale distance from y=0.
Compare my reasoning
- x stays −2.
- y becomes 3.
(−2,3).
Look for these in your work
- Scale one dimension
- Use the invariant line
Stretch P(2,4) horizontally by factor 3 with invariant line x=1. Would applying x′=3x be correct?
Give me a hint
Measure x from the given invariant line.
Compare my reasoning
- x′=1+3(2−1)=4; y′=4.
- 3x would use x=0 as the invariant line, a different transformation.
P′(4,4); x′=3x would use the wrong invariant line.
Look for these in your work
- Shift to the actual invariant line
- State the full transformation
Common mistakes
- The named direction is not the invariant axis.
- Do not multiply both coordinates in a one-direction stretch.
What can you explain now?
In a horizontal stretch about the y-axis, which coordinate stays unchanged?
Compare with the explanation
The y coordinate.
Only horizontal distances from the invariant y-axis are scaled.
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Make it stick
In a horizontal stretch about the y-axis, which coordinate stays unchanged?
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.