Translations

What Is a Translation?

A translation is a type of transformation that moves a shape without changing its size, shape, or orientation. It's like picking up a shape and sliding it to a new location.

• The original shape is called the object.
• The new shape is called the image.

Translations are described using a vector.

Understanding the Translation Vector

A vector tells you how far to move the shape horizontally and vertically. It’s written as a column:

$\begin{pmatrix} x \\ y \end{pmatrix}$

• The top number tells you how far left or right to move:
  • Positive = right
  • Negative = left
• The bottom number tells you how far up or down to move:
  • Positive = up
  • Negative = down

Example: $\begin{pmatrix} -3 \\ 4 \end{pmatrix}$ means move 3 left and 4 up.

How To Translate a Shape

Step-by-Step:

Read the vector carefully.

Move each vertex (corner) of the shape according to the vector.

Draw the new shape using the new positions of the points.

Label the image with prime notation (e.g., $A'$, $B'$, $C'$).

Describing a Translation

To describe a translation, you need to:

1. Say it is a translation.
2. Write the vector used.

To find the vector:

• Pick a point on the object and find its corresponding point on the image.
• Count the movement:
  • Left/Right for the top number.
  • Up/Down for the bottom number.

Reversing a Translation

To undo a translation, reverse both directions:

If the vector is: $\begin{pmatrix} 5 \\ -7 \end{pmatrix}$

Then the reverse is: $\begin{pmatrix} -5 \\ 7 \end{pmatrix}$

Example

Translate shape A by the vector $\begin{pmatrix} -2 \\ 3 \end{pmatrix}$.

Step 1: The vector means “2 to the left” and “3 up.”

Step 2: Move each point on shape A by this amount. If one vertex is at $(4, 1)$, it moves to:

$$(4 - 2, 1 + 3) = (2, 4)$$

Step 3: Repeat for all vertices and redraw the shape in the new position. 

Final Answer: Shape A has been translated by $\begin{pmatrix} -2 \\ 3 \end{pmatrix}$.

Example

Question: Describe the transformation that maps shape A onto shape A'.

• Point X on shape A is at (2, 6)
• Point Y on shape A' is at (-4, 2)

Solution:

1. Compare the coordinates:
   • Horizontal: $3 \to -4$ = move 6 left
   • Vertical: $6 \to 2$ = move 4 down
2. Write the vector: $$\begin{pmatrix} -6 \\ -4 \end{pmatrix}$$

Final Answer: Translation by $\begin{pmatrix} -6 \\ -4 \end{pmatrix}$