Translation

Here we will learn about translation, including how to translate 2D shapes and how to describe translations.

There are also translation worksheets and problem solving questions based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is translation?

Translation is a type of transformation that moves a shape in a horizontal direction (left and right) and in a vertical direction (up and down).

We use a column vector to help record the movement.

E.g.

Shape A has been translated to shape B by the column vector

\begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix}

The column vector gives instructions how to move each point of the original shape.

\begin{pmatrix} \; 3 \; \\ \; 2 \; \end{pmatrix} \; \text{is} \; \begin{pmatrix} 3 \ \text{right}\\ 2 \ \text{up}\\ \end{pmatrix}

The inverse transformation would translate shape B back to shape A using the column vector:

\begin{pmatrix} \; -3 \; \\ \; -2 \; \end{pmatrix} \; \text{is} \; \begin{pmatrix} 3 \ \text{left}\\ 2 \ \text{down}\\ \end{pmatrix}

The object is the name of the original shape. The image is the name of the shape after it had been translated.  When an object is translated the object and the image have the same orientation. There is no rotation or reflection.

The object and the image are congruent because they are the same shape and the same size.

Using tracing paper can be useful when translating shapes.

Step-by-step guide: Column vector

How to use translation

In order to translate a 2D shape:

1. Choose a starting point on the shape.
2. Move it across.
3. Move it up or down.
4. Complete the rest of the shape.

Translation maths examples

Example 1: translate a 2D shape

Translate shape A by the column vector \begin{pmatrix} \; 4 \; \\ \; 1 \; \end{pmatrix} and label the image B

1. Choose a starting point on the shape.

Choose one of the vertices of the shape as the point you are going to move. Let’s use the bottom right hand point of the shape.

2Move it across.

The top number in the column vector is 4 so move the point 4 squares to the right.

3Move it up or down.

The bottom number in the column vector is 1 so move the point 1 square up.

4Complete the rest of the shape.

You can move each of the other points in the same way and draw in the rest of the 2D shape.

Example 2: translate a 2D shape

Translate shape A by the column vector and label the image B

\begin{pmatrix} \; -1 \;\\ \; 3 \; \end{pmatrix}

Choose one of the vertices of the shape as the point you are going to move. Let’s use the bottom left hand corner of the shape.

The top number in the column vector is -1 so move the point 1 square to the left.

The bottom number in the column vector is 3 so move the point 3 squares up.

You can move each of the other points in the same way and draw in the rest of the 2D shape.

Example 3: translate a 2D shape

Translate shape A by the column vector and label the image B

\begin{pmatrix} \; 0 \;\\ \; -4 \; \end{pmatrix}

Choose one of the vertices of the shape as the point you are going to move. Let’s use the bottom left hand corner of the shape.

The top number in the column vector is 0 so there is no horizontal movement of the point.

The bottom number in the column vector is -4 so move the point 4 squares down.

You can move each of the other points in the same way and draw in the rest of the 2D shape.

How to describe translations

In order to describe a translation of a shape:

1. Pair up the points.
2. Work out the horizontal movement.
3. Work out the vertical movement.
4. State the column vector.

Describing translations examples

Example 4: describe a translation

Describe the translation of shape P to shape Q

Pair up a point from the object and the corresponding point on the image. Here one pair of points is chosen. It should not matter which pair of points are chosen.

From the object P the point is (1,1)

For the image Q the point is (3,2)

Using the grid or the x-coordinates, work out the horizontal movement. There is a shift of 2 to the right.

Using the grid or the y-coordinates, work out the vertical movement. There is a shift of 1 upwards.

Now we need to write the horizontal shift and the vertical shift in a column vector.

\begin{pmatrix} 2 \ \text{right}\\ 1 \ \text{up}\\ \end{pmatrix} \text{is} \begin{pmatrix} \; 2 \;\\ \; 1 \; \end{pmatrix}

Shape P has been translated to shape Q by the column vector

\begin{pmatrix} \; 2 \;\\ \; 1 \; \end{pmatrix}

Example 5: describe a translation

Describe the translation of shape A to shape B

Pair up a point from the object and the corresponding point on the image. Here one pair of points is chosen. It should not matter which pair of points are chosen.

From the object A the point is (4,6)

For the image B the point is (1,2)

Using the grid or the x-coordinates, work out the horizontal movement. There is a shift of 3  to the left.

Using the grid or the y-coordinates, work out the vertical movement. There is a shift of 4 downwards.

Now we need to write the horizontal shift and the vertical shift in a column vector.

\begin{pmatrix} 3 \ \text{left}\\ 4 \ \text{down}\\ \end{pmatrix} \text{is} \begin{pmatrix} \; -3 \;\\ \; -4 \; \end{pmatrix}

Shape A has been translated to shape B by the column vector

\begin{pmatrix} \; -3 \;\\ \; -4 \; \end{pmatrix}

Example 6: describe a translation

Describe the translation of shape A to shape B

Pair up a point from the object and the corresponding point on the image. Here one pair of points is chosen. It should not matter which pair of points are chosen.

From the object A the point is (1,1)

For the image B the point is (1,6)

Using the grid or the x-coordinates, work out the horizontal movement. There is no horizontal shift.

Using the grid or the y-coordinates, work out the vertical movement. There is a shift of 5 upwards.

Now we need to write the horizontal shift and the vertical shift in a column vector.

\begin{pmatrix} 0 \ \text{left or right}\\ 5 \ \text{up}\\ \end{pmatrix} \text{is} \begin{pmatrix} \; 0 \;\\ \; 5 \; \end{pmatrix}

Shape A has been translated to shape B by the column vector

\begin{pmatrix} \; 0 \;\\ \; 5 \; \end{pmatrix}

Common misconceptions

• Check the scale of the coordinate grid

Be careful to consider the scaling on the axes.

What is the column vector for the translation of shape P to shape Q?

If we count the squares it may appear that shape P has been translated by the column vector

\begin{pmatrix} \; 6 \;\\ \; 2 \; \end{pmatrix}

But if by considering the scale on the axes, the correct column vector is

\begin{pmatrix} \; 3 \;\\ \; 1 \; \end{pmatrix}

• Interpreting the column vector

Remember, the top number is for horizontal movement.  A positive number moves the shape to the right and a negative number moves the shape to the left.

The bottom number is for vertical movement.  A positive moves the shape upwards and a negative number moves the shape downwards.

• Object and image

The original shape is the object and the translated shape is the image. Make sure you know which shape is the original shape and start there when describing transformations such as translations.

Practice translation maths questions

1. Translate the shaded shape by the column vector

\begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix}

Choose a point on the vertex of the shape and move it 3 to the right and 2 up.

Do the same with all the points, or carefully draw in the rest of the image.

\begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix} \text{is} \begin{pmatrix} 3 \ \text{right}\\ 2 \ \text{up}\\ \end{pmatrix}

2. Translate the shaded shape by the column vector

\begin{pmatrix} \; -4 \;\\ \; 1 \; \end{pmatrix}

Choose a point on the vertex of the shape and move it 4 to the left and 1 up.

Do the same with all the points, or carefully draw in the rest of the image.

\begin{pmatrix} \; -4 \;\\ \; 1 \; \end{pmatrix} \text{is} \begin{pmatrix} 4 \ \text{left}\\ 1 \ \text{up}\\ \end{pmatrix}

3. Translate the shaded shape by the column vector

\begin{pmatrix} \; 3 \;\\ \; 0 \; \end{pmatrix}

Choose a point on the vertex of the shape and move it 3 to the right and 0 up or down.

Do the same with all the points, or carefully draw in the rest of the image.

\begin{pmatrix} \; 3 \;\\ \; 0 \; \end{pmatrix} \text{is} \begin{pmatrix} 3 \ \text{right}\\ 0 \ \text{up or down}\\ \end{pmatrix}

4. Describe the transformation of shape A to shape B.

Translation by

\begin{pmatrix} \; -3 \;\\ \; -2 \; \end{pmatrix}

Translation by

\begin{pmatrix} \; 2 \;\\ \; 3 \; \end{pmatrix}

Translation by

\begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix}

Translation by

\begin{pmatrix} \; -2 \;\\ \; -3 \; \end{pmatrix}

Choose a point on the object shape A and its corresponding point on the image shape B.  Count the shift in the horizontal direction.  Count the shift in the vertical direction.

\begin{pmatrix} 2 \ \text{right}\\ 3 \ \text{up}\\ \end{pmatrix} \text{is} \begin{pmatrix} \; 2 \;\\ \; 3 \; \end{pmatrix}

5. Describe the transformation of shape A to shape B.

Translation by

\begin{pmatrix} \; 4 \;\\ \; 3 \; \end{pmatrix}

Translation by

\begin{pmatrix} \; -3 \;\\ \; 4 \; \end{pmatrix}

Translation by

\begin{pmatrix} \; 4 \;\\ \; -3 \; \end{pmatrix}

Translation by

\begin{pmatrix} \; -3 \;\\ \; -4 \; \end{pmatrix}

Choose a point on the object shape A and its corresponding point on the image shape B.  Count the shift in the horizontal direction.  Count the shift in the vertical direction.

\begin{pmatrix} 4 \ \text{right}\\ 3 \ \text{up}\\ \end{pmatrix} \text{is} \begin{pmatrix} \; 4 \;\\ \; -3 \; \end{pmatrix}

6. Describe the transformation of shape P to shape Q.

Translation by

\begin{pmatrix} \; 1 \;\\ \; -4 \; \end{pmatrix}

Translation by

\begin{pmatrix} \; 4 \;\\ \; -1 \; \end{pmatrix}

Translation by

\begin{pmatrix} \; 1 \;\\ \; 4 \; \end{pmatrix}

Translation by

\begin{pmatrix} \; 4 \;\\ \; 1 \; \end{pmatrix}

Choose a point on the object shape P and its corresponding point on the image shape Q. Count the shift in the horizontal direction. Count the shift in the vertical direction.

\begin{pmatrix} 1 \ \text{right}\\ 4 \ \text{down}\\ \end{pmatrix} \text{is} \begin{pmatrix} \; 1 \;\\ \; -4 \; \end{pmatrix}

Translation maths GCSE questions

1. A shape is translated by the vector

\begin{pmatrix} \; 0 \;\\ \; -3 \; \end{pmatrix}

In which direction does the shape move?

Up          Down          Left          Right

(1 mark)

Down

(1)

2. Translate trapezium P by the vector

\begin{pmatrix} \; 1 \;\\ \; -3 \; \end{pmatrix}

(2 marks)

For a translation (new shape is same orientation as the original shape)

(1)

For the correct translation

(1)

3. Describe the single transformation that maps Shape A onto shape B.

(2 marks)

Translation

(For stating translation)

(1)

\begin{pmatrix} \; 2 \;\\ \; 3 \; \end{pmatrix}

(For the correct vector)

(1)

Learning checklist

You have now learned how to:

• Translate a 2D shape by a vector
• Describe translations as 2D vectors

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