Translation

In a Nutshell

A translation slides every point of a shape the same distance in the same direction, without turning or flipping it. It is described by a column vector.

A translation moves a shape without rotating or reflecting it. You describe a translation using a column vector:

\begin{pmatrix} a \\ b \end{pmatrix}

where $a$ is the horizontal movement (positive = right, negative = left) and $b$ is the vertical movement (positive = up, negative = down).

For example, the vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ moves every point 3 units to the right and 2 units down.

Every point moves by the same amount: $(x, y) \to (x + a,\, y + b)$ .

Explore: translation

Right / Left Up / Down

Enter horizontal and vertical values and press Transform to slide the shape. Positive values move right and up; negative values move left and down.

Watch it work

Question: Translate the point $A(2, 5)$ by the vector $\begin{pmatrix} -4 \\ 1 \end{pmatrix}$ .

Have a go

Q1. Translate $(3, 4)$ by $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$ .

Q2. Translate $(-1, 2)$ by $\begin{pmatrix} -3 \\ -4 \end{pmatrix}$ .

Q3. A triangle has vertices $A(1, 1)$ , $B(4, 1)$ and $C(4, 3)$ . Translate it by $\begin{pmatrix} -5 \\ 2 \end{pmatrix}$ .

Q4. Point $P$ is at $(6, -2)$ and its image $P'$ is at $(3, 1)$ . What column vector describes this translation?

Q5. Translate $(0, -3)$ by $\begin{pmatrix} 4 \\ 5 \end{pmatrix}$ .