Reflection in y=x
(x,y)→(y,x)
🎯 Learning Objectives
- reflect points in y=x and y=−x;
- reflect in general straight lines;
- use coordinate mappings and geometry together;
- identify invariant points.
Introduction
This lesson develops Reflections in Diagonal and General Lines 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.
Diagonal Reflection Rules
Reflection in y=−x
(x,y)→(−y,−x)
Worked Examples
(2,5) reflected in y=x becomes (5,2).
(2,5) reflected in y=−x becomes (−5,−2).
General Mirror Lines
For a line such as y=2x+1, draw the line accurately and ensure each object-image segment is perpendicular to it and bisected by it.
Do not merely swap coordinates unless the mirror line is exactly y=x.
Practice Questions
- Reflect (−3,6) in y=x.
- Reflect (4,−1) in y=−x.
- Which points remain fixed under reflection in y=x?
✅ Check Your Work
Show answers and working
- (6,−3).
- (1,−4).
- All points lying on y=x, because they are already on the mirror line.
🧠 Learn It Visually: Diagonal Reflections
Diagonal mirror lines swap the roles of horizontal and vertical movement. Use coordinate rules to avoid guessing.
In y=x
(x,y) → (y,x). Swap the coordinates.
In y=−x
(x,y) → (−y,−x). Swap and change both signs.
Visual check
The segment joining a point and image must meet the diagonal mirror line at 90° and be bisected by it.
🎛️ 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 does reflecting (3,1) in y=x produce (1,3)?
🧷 Remember
For y=x, swap coordinates. For y=−x, swap coordinates and change signs.
⚠️ Common Mistake
Using the y=x rule for y=−x and forgetting the sign changes.
🎯 Exam Tip
Test one vertex with the coordinate rule before plotting the whole image.
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.