Translations and Column Vectors –

Translations are one of the four transformations you need to master for GCSE Maths. A translation slides a shape from one position to another without rotating, reflecting, or resizing it. The movement is described using a column vector. This guide covers how to read and write column vectors, apply translations, and describe them fully in exam questions.

What Is a Translation?

A translation moves every point of a shape the same distance in the same direction. The shape does not change size, orientation, or handedness — it simply slides to a new position.

Key Formulas

A positive top number means move right; negative means move left. A positive bottom number means move up; negative means move down.

Step-by-Step Method

1. Read the column vector. The top number is the horizontal shift; the bottom number is the vertical shift.
2. Apply the vector to each vertex. Add the top number to each x-coordinate and the bottom number to each y-coordinate.
3. Plot the new vertices and draw the translated shape.
4. To describe a translation, find the column vector by subtracting the original coordinates from the image coordinates.

Worked Example 1 — Foundation Level

Question: Translate the triangle with vertices A(2, 3), B(5, 3), and C(5, 6) by the vector (3 over −2).

Working:

Add 3 to each x-coordinate and subtract 2 from each y-coordinate.

A(2, 3) maps to A'(2 + 3, 3 − 2) = A'(5, 1)

B(5, 3) maps to B'(5 + 3, 3 − 2) = B'(8, 1)

C(5, 6) maps to C'(5 + 3, 6 − 2) = C'(8, 4)

Answer: The translated vertices are A'(5, 1), B'(8, 1), C'(8, 4).

Worked Example 2 — Higher Level

Question: Shape P is translated to shape Q. A vertex of P is at (−1, 4) and the corresponding vertex of Q is at (3, 1). Describe the translation fully.

Working:

Horizontal shift: 3 − (−1) = 4

Vertical shift: 1 − 4 = −3

The translation vector is (4 over −3).

Answer: Translation by the vector (4 over −3).

Worked Example 3 — Exam Style

Question: A shape is translated by the vector (−5 over 2). A vertex of the image is at (1, 7). Find the coordinates of the corresponding vertex on the original shape.

Working:

To reverse the translation, subtract the vector.

Original x = 1 − (−5) = 1 + 5 = 6

Original y = 7 − 2 = 5

Answer: The original vertex is at (6, 5).

Common Mistakes

• Mixing up the horizontal and vertical components. The top number in the column vector is always horizontal (left/right) and the bottom number is always vertical (up/down). Swapping them moves the shape in the wrong direction.
• Getting the sign wrong. Positive top means right; negative top means left. Positive bottom means up; negative bottom means down. A sign error flips the direction of the translation.
• Not describing the transformation correctly. You must say "Translation by the vector ..." and give the column vector. Writing "the shape moves 3 right and 2 down" without a vector may lose marks.

Exam Tips

• Always describe a translation using a column vector — this is the expected format in GCSE exams.
• Check your answer by verifying that all vertices have been shifted by the same vector.
• If you need to find the vector from a diagram, pick one vertex and count the squares right/left and up/down to its image.
• Translations preserve congruence — the image is identical to the object.

Practice Questions

Q1 (Foundation): Translate the point (4, 2) by the vector (−3 over 5).

Q2 (Foundation): A shape is translated so that the point (6, 1) maps to (2, 4). What is the translation vector?

Q3 (Higher): A shape is translated by the vector (a over −1). A vertex at (3, 5) maps to (−2, 4). Find the value of a.

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Related Topics

• Reflections
• Rotations
• Enlargement and Centre of Enlargement
• Vectors

Summary

• A translation slides a shape without changing its size, orientation, or shape. It is described by a column vector where the top number gives the horizontal shift and the bottom number gives the vertical shift. To apply a translation, add the vector to every vertex. To find the vector, subtract the original coordinates from the image coordinates. Always use column vector notation in exam answers.