x-direction stretch factor a
a001
🎯 Learning Objectives
- describe one-way and two-way stretches;
- identify stretch factors and invariant lines;
- calculate images using stretch matrices;
- distinguish stretch from enlargement.
Introduction
This lesson develops Stretches from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Stretch vs Enlargement
An enlargement multiplies both coordinate directions by the same factor. A stretch may affect only one direction or use different factors.
y-direction stretch factor b
100b
Two-way stretch
a00b
Invariant Line
Under an x-direction stretch, the y-axis is invariant point-by-point. Under a y-direction stretch, the x-axis is invariant point-by-point.
Worked Example
Apply 3001 to (2,−4): image=(6,−4).
Practice Questions
- Apply 2001 to (3,5).
- State the invariant line for this stretch.
- Apply 100−2 to (4,3).
✅ Check Your Work
Show answers and working
- (6,5).
- The y-axis, x=0.
- (4,−6).
🧠 Learn It Visually: Stretches
A stretch changes distances in one direction while leaving a whole line fixed. Unlike an enlargement, it does not usually preserve shape.
Invariant line
Every point on this line stays fixed.
Stretch factor
Perpendicular distances from the invariant line are multiplied by the factor.
Direction
A horizontal stretch changes x-distances; a vertical stretch changes y-distances.
🎛️ See the Transformation Happen
Select an option and watch the object change. Compare orientation, size and position.
Choose a transformation.
Think Like a Mathematician: Why is a stretch not the same as an enlargement?
🧷 Remember
Only distances perpendicular to the invariant line are scaled.
⚠️ Common Mistake
Naming an invariant line that the image points do not actually leave fixed.
🎯 Exam Tip
State both the stretch factor and the invariant line.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.