A translation moves every point of a shape by the same distance in the same direction. No rotation or reflection occurs — the shape keeps its orientation.
Translations are described using a column vector $\begin{pmatrix} a \\ b \end{pmatrix}$, where:
- $a$ is the horizontal movement (positive = right, negative = left)
- $b$ is the vertical movement (positive = up, negative = down)
Example: Translating point $(3, 1)$ by $\begin{pmatrix} -2 \\ 4 \end{pmatrix}$: $$x: 3 + (-2) = 1, \quad y: 1 + 4 = 5 \quad \Rightarrow \quad (1, 5)$$
Apply the same vector to every vertex of the shape.
The object and image are congruent — same size and shape. Translations are one of the four standard transformations (along with rotations, reflections, and enlargements).
Common error: adding the vector components the wrong way round, or applying a positive $b$ downwards rather than upwards.