Horizontal
Positive means right; negative means left.
🎯 Learning Objectives
- translate points and figures using column vectors;
- find the translation vector from an object and image;
- describe translations completely;
- reverse a translation.
Introduction
This lesson develops Translations 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.
Translation Vector
The vector (a,b) moves every point a units horizontally and b units vertically.
Vertical
Positive means up; negative means down.
Worked Example
A(1,2) maps to A′(5,−1).
Horizontal movement: 5−1=4.
Vertical movement: −1−2=−3.
Translation vector=(4,−3).
Coordinate Rule
(x,y) → (x+a,y+b).
Practice Questions
- Translate P(2,−1) by (4,3).
- Find the vector taking A(−2,5) to A′(3,1).
- What vector reverses translation (6,−2)?
✅ Check Your Work
Show answers and working
- P′=(2+4,−1+3)=(6,2).
- (3−(−2),1−5)=(5,−4).
- (−6,2).
🧠 Learn It Visually: Translations
A translation is a slide. Every point moves the same distance in the same direction, so the image keeps the same size, shape and orientation.
Column vector
Write the movement vertically: top number = horizontal, bottom number = vertical.
Signs matter
Positive horizontal means right; negative means left. Positive vertical means up; negative means down.
Finding the vector
Subtract object coordinates from image coordinates: image − object.
🎛️ Translation Animator
Choose a horizontal and vertical movement. The same movement is applied to every vertex.
A positive horizontal value moves right; a positive vertical value moves up.
Think Like a Mathematician: Why is it enough to calculate the movement of just one correctly matched vertex?
🧷 Remember
A translation has no centre and no mirror line. It is fully described by its vector.
⚠️ Common Mistake
Writing the vector in the wrong order. Horizontal movement must be written above vertical movement.
🎯 Exam Tip
State “translation by the vector …”, including signs. Do not write only “translation”.
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.