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Translations

FoundationHigherAQAEdexcelOCR

This free Foundation and Higher GCSE Maths worksheet on translations helps you revise translating shapes using vectors. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. A translation slides a shape using a column vector, with no turning.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A translation slides a shape without turning or resizing it. Every point moves the same distance in the same direction, so the image is identical to the original.

The movement is described by a column vector. The top number gives the horizontal movement, positive for right and negative for left, and the bottom number gives the vertical movement, positive for up and negative for down.

Because nothing changes except position, the object and image are congruent. A full description of a translation needs only the vector, which is why these questions are usually quick marks if the notation is right.

Revision notes

Column vectors

A translation is written as \(\begin{pmatrix} x \\ y \end{pmatrix}\), with \(x\) horizontal and \(y\) vertical.

So \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\) means three right and two down.

Performing a translation

Add the vector to each vertex, then join the new points.

A point at \((2, 5)\) translated by \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\) moves to \((6, 2)\). Moving one vertex at a time is more reliable than sliding the whole shape by eye.

Describing a translation

Pick a vertex on the object and its matching vertex on the image, then work out the movement between them.

A full description states that it is a translation and gives the vector. Naming the transformation is part of the mark.

Key points

  • A translation slides a shape without turning it.
  • It is described by a column vector.
  • The top number is horizontal movement.
  • The bottom number is vertical movement.
  • Object and image are congruent.
  • Say translation and give the vector for a full description.

Worked examples

Example 1

Translate the point \((3, 4)\) by \(\begin{pmatrix} 2 \\ -5 \end{pmatrix}\).

Working

\[3 + 2 = 5\]add the horizontal component
\[4 + (-5) = -1\]add the vertical component
\[(5, -1)\]state the image point

Example 2

A shape moves from \((1, 2)\) to \((7, 5)\). Describe the translation.

Working

\[7 - 1 = 6\]find the horizontal movement
\[5 - 2 = 3\]find the vertical movement
\[\text{Translation by } \begin{pmatrix} 6 \\ 3 \end{pmatrix}\]name it and give the vector

Example 3

Describe the translation taking \((4, 6)\) to \((1, 6)\).

Working

\[1 - 4 = -3\]the shape moves left
\[6 - 6 = 0\]there is no vertical movement
\[\text{Translation by } \begin{pmatrix} -3 \\ 0 \end{pmatrix}\]state the transformation

Common mistakes

  • Writing the vector as a coordinate.

    A column vector is written vertically in brackets, not as (x, y).

  • Reversing the components.

    The top number is horizontal and the bottom is vertical.

  • Getting the direction of the sign wrong.

    Negative means left or down.

  • Not naming the transformation.

    A full description says translation and gives the vector.

Exam tips

  • Translate one vertex at a time rather than sliding by eye.
  • Write vectors vertically in brackets.
  • Check the sign matches the direction of movement.
  • Always name the transformation as well as describing it.

Key terms

Translation
A slide with no turning or resizing.
Column vector
A vertical pair of numbers describing movement.
Congruent
Identical in shape and size.
Image
The shape after the transformation.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.