Describing and Combining Transformations: Edexcel GCSE Foundation maths revision notes

The four transformations

A transformation moves or changes a shape. The shape you start with is the object, and the shape it becomes is the image. There are four transformations to know.

A translation slides the shape. It does not turn or flip.
A reflection flips the shape over a mirror line.
A rotation turns the shape about a fixed point, the centre of rotation.
An enlargement makes the shape bigger, but keeps the same shape.

In the exam you are often asked to describe fully the single transformation that maps shape A onto shape B, usually for 2 marks. You need the name and every detail to get both marks. The hardest questions give two transformations, one after the other, such as a reflection followed by a translation, and ask you to find the missing column vector, for 3 marks.

Maps A onto B means A is the object and B is the image. Always work from A to B, never from B back to A.

Telling the four apart

Here shape A has been transformed in four different ways. T is a translation of A, M is a reflection, R is a rotation and E is an enlargement.

Ask these questions about the image, in this order.

Is it a different size? Then it is an enlargement.
Is it the same size and facing exactly the same way? Then it is a translation.
Is it the same size but flipped, so it faces the other way? Then it is a reflection.
Is it the same size but turned? Then it is a rotation.

Tracing paper tells a reflection from a rotation, and you can ask for it in the exam. Trace the object. If the tracing fits on the image just by turning the paper, it is a rotation. If you have to turn the paper over, it is a reflection.

What to write when you describe fully

To describe fully, write the name of the transformation and then every detail it needs. Each transformation needs different details.

Translation: the column vector, such as \begin{pmatrix}3\\-2\end{pmatrix}
Reflection: the equation of the mirror line, such as x=2 or y=x
Rotation: the angle, the direction (clockwise or anticlockwise) and the coordinates of the centre.
Enlargement: the scale factor and the coordinates of the centre.

In a column vector, the top number is the move across and the bottom number is the move up or down. Positive means right or up, and negative means left or down. So \begin{pmatrix}3\\-2\end{pmatrix} means 3 right and 2 down.

Describe one transformation only. Writing a rotation and then a translation counts as two, and gets no marks at all, even if one of them is right. Use the proper words: translation, not move or shift; reflection, not flip; rotation, not turn.

Worked example 1: Describing a translation

Triangles A and B are drawn on a grid. Describe fully the single transformation that maps triangle A onto triangle B.

B is the same size as A and faces exactly the same way. It has only slid, so this is a translation.
Pick a corner of A and the matching corner of B. The right angle of A is at (-4,1) and the right angle of B is at (2,-4)
Across: from -4 to 2 is 6 squares to the right, so the top number is 6
Your turn. Now count up or down, from 1 to -4. What is the bottom number of the column vector?

Your answer

From 1 down to -4 is 5 squares down, so the bottom number is -5
Check with another corner: (-2,1) moved 6 right and 5 down lands on (4,-4), a corner of B.
Answer: translation by the vector \begin{pmatrix}6\\-5\end{pmatrix}

Describing a reflection

A reflection needs one detail: the equation of the mirror line. The mirror line is exactly halfway between each corner of the object and the matching corner of the image.

A vertical line has an equation such as x=2. Every point on it has x-coordinate 2
A horizontal line has an equation such as y=-3. Every point on it has y-coordinate -3
The y-axis is the line x=0 and the x-axis is the line y=0
The sloping line through (1,1), (2,2) and (3,3) is the line y=x
The sloping line through (1,-1), (2,-2) and (3,-3) is the line y=-x

For example, a corner at (5,3) has its image at (-1,3). They are at the same height, so the mirror line is vertical. Halfway between 5 and -1 is 2, so the mirror line is x=2

Describing a rotation

A rotation needs three details: the angle, the direction and the centre. The angle is usually 90^\circ, a quarter turn, or 180^\circ, a half turn. For 180^\circ you don't need a direction, because turning either way ends in the same place.

To find the angle and direction, pick a side of the object and look at the matching side of the image. Going clockwise, up turns to right, right turns to down, down turns to left and left turns to up. So a side that pointed up and now points right has made a quarter turn clockwise.

The centre of rotation is the one point that does not move. For a quarter turn in the exam it is the origin, O. For a half turn it can also be another point where grid lines cross. Test a point with tracing paper: hold your pencil on it, turn the tracing, and see whether the object lands on the image. Write the centre as coordinates, x first, such as (-1,0)

Check the direction from the object to the image. A turn of 90^\circ clockwise from A to B is a turn of 90^\circ anticlockwise from B back to A.

Worked example 2: Describing a quarter turn

Triangles A and B are drawn on a grid. Describe fully the single transformation that maps triangle A onto triangle B.

B is the same size as A. A tracing of A fits on B by turning the paper, without turning it over, so this is a rotation.
In A, the side from (1,1) to (1,4) points up. In B, the matching side, from (1,-1) to (4,-1), points right. Up has turned to right, so the turn is 90^\circ clockwise.
Try the origin as the centre. Corner (1,4) is 1 right and 4 up from O. Turned a quarter turn clockwise, up becomes right and right becomes down, so its image is 4 right and 1 down: (4,-1), a corner of B.
Your turn. Turn corner (3,1) a quarter turn clockwise about O in the same way. Write down the coordinates of its image.

Your answer

Write your answer as (x, y)

(3,1) is 3 right and 1 up from O, so its image is 1 right and 3 down: (1,-3), the third corner of B.
Every corner of A lands on a corner of B, so the centre is O.
Answer: rotation 90^\circ clockwise about (0,0)

Worked example 3: Finding the centre of a half turn

Triangles A and B are drawn on a grid. Describe fully the single transformation that maps triangle A onto triangle B.

B is the same size as A and upside down. A tracing of A fits on B by turning the paper, without turning it over, so this is a rotation.
In A, the side from (1,3) to (1,5) points up. Its match in B, from (-3,-1) to (-3,-3), points down. Up has turned to down, a half turn, so the angle is 180^\circ
For a half turn, the centre is exactly halfway between each corner and its image. Corner (1,3) goes to (-3,-1), and halfway between the x-coordinates 1 and -3 is -1
Your turn. What number is halfway between the y-coordinates 3 and -1?

Your answer

3 and -1 are 4 apart, and 2 down from 3 is 1. So the centre is (-1,1)
Check with another corner: (3,3) goes to (-5,-1), and halfway between them is also (-1,1)
Answer: rotation 180^\circ about (-1,1)

Describing an enlargement

An enlargement needs two details: the scale factor and the centre of enlargement. The scale factor tells you how many times longer every length is. Find it from a pair of matching sides:

\text{scale factor}=\frac{\text{length on the image}}{\text{matching length on the object}}

To find the centre, draw a straight line through a corner of the image and the matching corner of the object, and carry it on past the smaller shape. Do the same for another pair of corners. The centre is where the lines cross. Write it as coordinates.

Check it: with scale factor 3, each corner of the image is 3 times as far from the centre as its matching corner on the object. The centre is not the middle of a shape, and not just any point between the two shapes.

Worked example 4: Describing an enlargement

Triangles A and B are drawn on a grid. Describe fully the single transformation that maps triangle A onto triangle B.

B is the same shape as A, but bigger, so this is an enlargement.
Your turn. The upright side of A is 2 squares long. Count the matching side of B. What is the scale factor?

Your answer

The upright side of B is 6 squares long, and 6 \div 2 = 3, so the scale factor is 3
Check with the bottom sides: A's is 1 square long and B's is 3 squares long, and 3 \div 1 = 3
Draw lines through matching corners: (4,1) through (2,3), (7,1) through (3,3) and (4,7) through (2,5). They all meet at (1,4)
Check: corner (2,3) is 1 right and 1 down from (1,4). Its match, (4,1), is 3 right and 3 down, which is 3 times as far.
Answer: enlargement, scale factor 3, centre (1,4)

Two transformations, one after the other

A question may give two transformations, one after the other, such as a reflection followed by a translation. Do them one at a time. The second transformation starts from the image of the first, not from the original shape.

On a grid, draw the first image and label it, then transform that image. For a single point, write down its coordinates after each step. These shortcuts help with points:

Reflection in the x-axis: the y-coordinate changes sign, so (2,5) goes to (2,-5)
Reflection in the y-axis: the x-coordinate changes sign, so (2,5) goes to (-2,5)
Rotation of 180^\circ about O: both signs change, so (2,5) goes to (-2,-5)
Translation: add the top number of the vector to x and the bottom number to y

To find a missing translation that comes after another transformation, do the first transformation, then find the vector from that image to the final shape. A vector straight from the original shape to the final shape leaves out the first step, so it is wrong.

Worked example 5: Following a point through two transformations

The point D has coordinates (-3,-2). D is reflected in the y-axis. Its image is then translated by the vector \begin{pmatrix}-1\\5\end{pmatrix}. Where does D end up? Write down its coordinates.

Do the reflection first. In the y-axis the x-coordinate changes sign and the y-coordinate stays the same, so (-3,-2) goes to (3,-2)
Now translate (3,-2), not the original point. The vector means 1 left and 5 up.
Your turn. Write down the coordinates of where D ends up.

Your answer

Write your answer as (x, y)

3 - 1 = 2 and -2 + 5 = 3, so D ends at (2,3)
The order matters: translating first and then reflecting would give (4,3), a different point.
Answer: (2,3)

Worked example 6: Finding a missing column vector

Triangles A and B are drawn on a grid. Triangle A is reflected in the y-axis. The image is then translated by the vector \begin{pmatrix}c\\d\end{pmatrix} to give triangle B. Find the value of c and the value of d.

A has its right angle at (-1,3) and B has its right angle at (-4,-1). B faces the other way from A, so A cannot slide straight onto B. It has to be reflected first.
Reflect A in the y-axis and call the image X. Each x-coordinate changes sign, so (-1,3), (-4,3) and (-1,1) go to (1,3), (4,3) and (1,1)
Your turn. Now X is translated onto B. The right angle of X is at (1,3) and the right angle of B is at (-4,-1). Find the value of c.

Your answer

From 1 across to -4 is 5 left, so c=-5
From 3 down to -1 is 4 down, so d=-4
Check: (4,3) moved 5 left and 4 down is (-1,-1), a corner of B. Going straight from A to B would give \begin{pmatrix}-3\\-4\end{pmatrix}, which is wrong because it leaves out the reflection.
Answer: c=-5, d=-4

Common mistakes

Using everyday words, or giving two transformations. Write one of translation, reflection, rotation or enlargement, not moved, shifted, flipped or turned, and never add a second transformation.
Leaving out a detail. A rotation needs the angle, direction and centre, an enlargement the scale factor and centre, and a reflection its mirror line.
Mixing up x= and y= for a mirror line. A vertical line is x= a number and a horizontal line is y= a number, and a number on its own is not an equation.
Getting the column vector wrong. Count from A to B: top number across, bottom number up or down, negative for left or down, and write it as a vector, not as coordinates.
Putting the centre of enlargement in the wrong place. It is where the lines through matching corners meet, not the middle of a shape.
Skipping the first of two transformations. Do the first one, label its image, then start the second from that image.