Horizontal shear factor k
1k01; x′=x+ky.
🎯 Learning Objectives
- describe horizontal and vertical shears;
- identify shear factor and invariant line;
- apply shear matrices;
- recognise how shapes slant while preserving area.
Introduction
This lesson develops Shears 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.
Shear Matrices
Vertical shear factor k
10k1; y′=kx+y.
Invariant Lines
For a horizontal shear, the x-axis is invariant. For a vertical shear, the y-axis is invariant.
Worked Example
Horizontal shear factor 2 applied to (3,4):
x′=3+2(4)=11, y′=4, so image=(11,4).
Practice Questions
- Apply horizontal shear factor 1 to (2,5).
- Apply vertical shear factor −2 to (3,4).
- State the invariant line for each.
✅ Check Your Work
Show answers and working
- x′=2+1(5)=7, y′=5, so (7,5).
- x′=3; y′=−2(3)+4=−2, so (3,−2).
- Horizontal shear: x-axis. Vertical shear: y-axis.
🧠 Learn It Visually: Shears
A shear slants a figure. One coordinate stays fixed while the other changes by an amount proportional to it.
Invariant axis
A standard shear leaves one axis fixed.
Parallel sliding
Rows or columns of points slide parallel to the invariant axis.
Shape change
Area is preserved for standard unit shears, but angles and lengths generally change.
🎛️ See the Transformation Happen
Select an option and watch the object change. Compare orientation, size and position.
Choose a transformation.
Think Like a Mathematician: What visual clue helps distinguish a shear from a rotation?
🧷 Remember
In an x-shear, y-values stay unchanged; in a y-shear, x-values stay unchanged.
⚠️ Common Mistake
Assuming a shear preserves angles because it preserves area.
🎯 Exam Tip
Check which coordinate remains unchanged to identify the invariant axis.
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.