Translation (Coordinate Rule)

Statement

A translation moves every point of a shape the same distance in a given direction. The coordinate rule for translation is:

\[(x,y) \mapsto (x+a,\; y+b)\]

Here, \(a\) is the horizontal movement (positive right, negative left), and \(b\) is the vertical movement (positive up, negative down).

Why it’s true

• A translation is a rigid transformation — it does not change size, shape, or orientation, only position.
• Adding \(a\) to the x-coordinate shifts the point sideways.
• Adding \(b\) to the y-coordinate shifts the point vertically.

Recipe (how to use it)

1. Identify the translation vector \((a,b)\).
2. Take the original point \((x,y)\).
3. Compute the new point: \((x+a,\;y+b)\).
4. Apply to all points of a shape to get its new position.

Spotting it

Exam questions often state “translate by vector (a,b)” or “move 3 units right and 2 units up”. These directly mean apply \((x,y)\mapsto(x+a,y+b)\).

Common pairings

• Reflections and rotations in combined transformations.
• Vectors and coordinate geometry questions.

Mini examples

1. Given: Translate (2,3) by (4,1). Answer: (6,4).
2. Given: Translate (-5,7) by (-3,-2). Answer: (-8,5).

Pitfalls

• Forgetting signs: a negative \(a\) means left, a negative \(b\) means down.
• Mixing translation with reflection (translations never flip shapes).

Exam strategy

• Always write the vector clearly.
• Apply translation to each coordinate systematically.
• Check by sketching: movement direction must match the vector.

Summary

Translations move shapes without changing them. The rule \((x,y)\mapsto(x+a,y+b)\) is fundamental in coordinate geometry, describing horizontal and vertical shifts in one step.