Transformations

Syllabus coverage

Recognise, describe and draw the reflection of a shape in a vertical or horizontal line (Extended: in a straight line)
Recognise, describe and draw the rotation of a shape about the origin, a vertex or the midpoint of an edge, through multiples of 90 degrees (Extended: about any centre)
Recognise, describe and draw the translation of a shape by a column vector
Recognise, describe and draw the enlargement of a shape from a centre by a scale factor, using positive and fractional scale factors (Extended: negative scale factors too)
Questions will not involve combinations of transformations at Core; Extended questions may
Extended: work with the enlargement of an area and with combinations of transformations
A ruler must be used for all straight edges

The same L-shaped object reflected, rotated, translated and enlarged, each on its own grid.

The four transformations, and what each one needs

A transformation moves or changes a shape. The shape you start with is the object, and the shape you finish with is the image. Three of the four transformations in this topic - reflection, rotation and translation - leave the shape exactly the same size and shape, so the image is congruent to the object. An enlargement is the odd one out: it keeps the shape and the angles but multiplies every length by the same factor.

Cambridge marks a transformation question on the description as much as on the drawing. Each of the four has a short fixed list of things you must state, and a description that leaves one of them out is incomplete however good the drawing is.

Transformation	What a full description must include	What happens to the shape
Reflection	the mirror line	congruent, but facing the opposite way
Rotation	the centre, the angle and the direction	congruent and facing the same way
Translation	the column vector	congruent, same way up, simply moved
Enlargement	the scale factor and the centre	similar: every length multiplied by the scale factor

Use a ruler for every straight edge and a sharp pencil. A wobbly line across a grid is a drawing error the examiner can see, and this topic is marked partly on the drawing.

Reflection

A reflection flips a shape over a mirror line. Every point of the image is the same perpendicular distance from the mirror line as the matching point of the object, but on the other side, so the mirror line is the perpendicular bisector of the segment joining a point to its image. The image is congruent to the object and faces the opposite way - reflection is the one transformation that reverses the orientation of a shape.

At Core the mirror line is always vertical or horizontal, written as $x = 3$ or $y = - 1$ . A vertical line leaves the y-coordinate alone and changes x: reflecting $(x, y)$ in $x = a$ gives $(2 a - x, y)$ . A horizontal line does the opposite, so reflecting in $y = b$ gives $(x, 2 b - y)$ . Extended questions also use slanted lines, and the two that appear most often are $y = x$ , which swaps the coordinates to $(y, x)$ , and $y = - x$ , which gives $(- y, - x)$ .

To draw a reflection, count the perpendicular distance from each vertex to the mirror line and mark the image the same distance on the other side, then join the image vertices in the same order. A single mis-counted vertex turns a correct method into a wrong shape.

Reflecting $(2, 5)$ in $x = 3$ : $(2 \times 3 - 2, 5) = (4, 5)$
Reflecting $(2, 5)$ in $y = 1$ : $(2, 2 \times 1 - 5) = (2, - 3)$
Reflecting $(2, 5)$ in $y = x$ : $(5, 2)$
Reflecting $(2, 5)$ in $y = - x$ : $(- 5, - 2)$

The word and the line are both needed. 'Reflection' on its own leaves infinitely many possible images, and 'it has been flipped over' names neither.

Rotation

A rotation turns a shape about a fixed point called the centre of rotation. Three things describe it: the centre, the angle, and the direction, clockwise or anticlockwise. The angles you meet are multiples of $90^{\circ}$ - 90, 180 or 270 - and the shape keeps its size and its way round, so a rotation never reverses orientation.

A 180 degree turn is the same clockwise as anticlockwise, so a 180 degree rotation needs no direction. A 90 degree clockwise turn is the same as a 270 degree anticlockwise turn, so either wording is accepted. About the origin the coordinate rules are worth knowing: 90 degrees anticlockwise takes $(x, y)$ to $(- y, x)$ , 90 degrees clockwise takes $(x, y)$ to $(y, - x)$ , and 180 degrees takes $(x, y)$ to $(- x, - y)$ .

If a question gives you a shape and its image and asks for the centre, join two pairs of corresponding vertices and draw the perpendicular bisector of each joining segment: the bisectors cross at the centre. For a 180 degree rotation there is a shortcut - the centre is the midpoint of the segment joining any point to its image.

Rotation about the origin	Rule	Image of $(2, 5)$
$90^{\circ}$ anticlockwise	$(x, y) \to (- y, x)$	$(- 5, 2)$
$90^{\circ}$ clockwise	$(x, y) \to (y, - x)$	$(5, - 2)$
$180^{\circ}$	$(x, y) \to (- x, - y)$	$(- 2, - 5)$
$270^{\circ}$ anticlockwise	$(x, y) \to (y, - x)$	$(5, - 2)$

A rotation keeps the shape the same way round. If the image is a mirror image of the object, the transformation is a reflection, whatever the angle.

Translation

A translation slides a shape without turning it or changing its size. Every point moves the same distance in the same direction, and that movement is written as a column vector: the top number is the movement in the x-direction, right positive and left negative, and the bottom number is the movement in the y-direction, up positive and down negative.

To find the vector of a translation from a point to its image, subtract the coordinates of the object point from those of the image point. If $A (1, 4)$ maps to $B (5, 2)$ , the vector is $(\begin{matrix} 5 - 1 \\ 2 - 4 \end{matrix}) = (\begin{matrix} 4 \\ - 2 \end{matrix})$ : 4 right and 2 down. The vector of the translation is the same whichever vertex of the shape you use, which is a useful check on a drawing.

Moving 3 right and 5 down is $(\begin{matrix} 3 \\ - 5 \end{matrix})$
Moving 4 left and 2 up is $(\begin{matrix} - 4 \\ 2 \end{matrix})$
Two shapes related by a translation are congruent and the same way up, which is how you tell a translation from a rotation or a reflection
The zero vector $(\begin{matrix} 0 \\ 0 \end{matrix})$ would leave the shape where it is, so it is never the answer to a translation question

The order of the two numbers is the answer. Writing

(\begin{matrix} - 2 \\ 4 \end{matrix})

for '4 right and 2 down' describes a completely different movement.

Enlargement

An enlargement changes the size of a shape from a fixed point called the centre of enlargement. Every length is multiplied by the scale factor, so the image is similar to the object: the angles are unchanged and the sides stay in the same ratio. Draw it by ruling rays from the centre through each vertex of the object and marking each image vertex on its ray at the scale factor times the distance.

A scale factor greater than 1 makes the image bigger, and a fractional scale factor between 0 and 1 makes it smaller, with the image lying between the centre and the object. Extended questions may use a negative scale factor: the image then lies on the opposite side of the centre and is turned through $180^{\circ}$ , with its size given by the numerical value of the scale factor. A scale factor of $- 2$ gives an image twice as big, upside down on the other side of the centre.

With the centre at the origin the arithmetic is easy, because $(x, y)$ goes to $(k x, k y)$ . For any other centre, work from the centre: find the vector from the centre to a vertex, multiply that vector by the scale factor, and add it back on to the centre.

Scale factor 3, centre $(0, 0)$ : $(1, 2)$ becomes $(3, 6)$
Scale factor $\frac{1}{2}$ , centre $(0, 0)$ : $(4, 6)$ becomes $(2, 3)$
Scale factor 2, centre $(1, 1)$ : the vertex $(3, 1)$ is 2 to the right of the centre, so its image is 4 to the right, at $(5, 1)$
Scale factor $- 1$ , centre $(0, 0)$ : $(3, 1)$ becomes $(- 3, - 1)$ , the same distance on the other side

Never write 'the shape has been enlarged by 2' and stop. Without a centre there are infinitely many possible images, so the description is incomplete and the mark is lost.

Describing a transformation fully: the wording that scores

Most transformation questions end with 'describe fully the single transformation that maps shape A onto shape B', and it is worth two or three marks. The mark scheme wants the type of transformation and then the measurements that pin it down, so write one sentence in a fixed order: the type, then the measurement, then the line or the centre.

Decide the type first, from the shape of the image. If the image is the same way round as the object it is a rotation, a translation or an enlargement; if it is a mirror image it must be a reflection. Then find the measurement: for a rotation, measure the angle turned between a line from the centre to a vertex and the matching line to its image; for an enlargement, divide the distance from the centre to an image vertex by the distance from the centre to the object vertex.

Same way round, image the same size, shape turned: a rotation
Same way round, image the same size, shape not turned: a translation
Same way round, image a different size: an enlargement
Mirror image, however it is placed: a reflection

Transformation	A complete description	What the marks are for
Reflection	Reflection in the line $x = 3$	the word reflection, and the line
Rotation	Rotation of $90^{\circ}$ clockwise about $(0, 0)$	the word rotation, the angle and direction, and the centre
Translation	Translation by the column vector $(\begin{matrix} - 3 \\ 4 \end{matrix})$	the word translation, and the column vector
Enlargement	Enlargement, scale factor 2, centre $(1, - 2)$	the word enlargement, the scale factor, and the centre

The command word is 'Describe'. A bare 'rotation' with no angle, no direction and no centre scores at most one mark out of three.

Extended: the area of an image, and combinations of transformations

An enlargement of scale factor $k$ multiplies every length by $k$ . Areas behave differently, because an area is two-dimensional: it is multiplied by $k^{2}$ . A rectangle 4 cm by 3 cm has an area of $12 {cm}^{2}$ ; enlarged by scale factor 2.5 it becomes 10 cm by 7.5 cm, an area of $75 {cm}^{2}$ , and $75 = 12 \times {2.5}^{2}$ . The rule holds for a fractional or a negative scale factor too, since squaring removes the sign.

Extended questions may also ask what single transformation has the same effect as two transformations performed one after the other. Two reflections in parallel lines always give a translation: the distance moved is twice the distance between the lines, at right angles to them. Two reflections in lines that meet give a rotation about the point where the lines cross, through twice the angle between them. Two rotations about the same centre give a rotation whose angle is the sum of the two angles.

Reflect in $x = 1$ , then in $x = 4$ : the lines are 3 apart, so the pair is a translation of $(\begin{matrix} 6 \\ 0 \end{matrix})$
Reflect in $x = 2$ , then in $y = 2$ : the lines cross at $(2, 2)$ at $90^{\circ}$ , so the pair is a rotation of $180^{\circ}$ about $(2, 2)$
A shape of area $20 {cm}^{2}$ enlarged by scale factor 1.5 has an image of area $20 \times 2.25 = 45 {cm}^{2}$
A shape of area $36 {cm}^{2}$ enlarged by scale factor $\frac{1}{3}$ has an image of area $36 \times \frac{1}{9} = 4 {cm}^{2}$

Read the question carefully. If it asks for the single transformation, describing the two separate steps scores nothing; if it asks you to draw the image after each step, the intermediate shape must be drawn.

Key terms you must be able to define

Object: The shape you start with, before the transformation is applied.

Image: The shape you finish with, after the transformation has been applied to the object.

Transformation: A rule that moves or changes a shape. This topic covers reflection, rotation, translation and enlargement.

Reflection: A flip of a shape over a mirror line. The image is congruent to the object and faces the opposite way.

Mirror line: The line a shape is reflected in. It is the perpendicular bisector of the segment joining any point to its image.

Rotation: A turn of a shape about a fixed point, described fully by the centre, the angle and the direction.

Centre of rotation: The fixed point a shape turns about. It is the only point that does not move.

Angle of rotation: How far the shape turns, in multiples of 90 degrees: 90, 180 or 270 degrees.

Clockwise and anticlockwise: The two directions of a rotation. A 90 degree clockwise turn is the same as a 270 degree anticlockwise turn.

Translation: A slide: every point of the shape moves the same distance in the same direction.

Column vector: A movement written as two numbers in a column: the top number is across, the bottom number is up. A negative number means left or down.

Enlargement: A change of size from a centre by a scale factor. The image is similar to the object, not congruent to it.

Centre of enlargement: The fixed point from which an enlargement is measured. It is unchanged by the enlargement.

Scale factor: The number every length is multiplied by in an enlargement. Greater than 1 enlarges, between 0 and 1 reduces, and a negative value puts the image on the other side of the centre.

Congruent: Exactly the same shape and the same size. The image under a reflection, rotation or translation is congruent to the object.

Orientation: The way round a shape is. A reflection reverses it; a rotation and a translation keep it.

Invariant point: A point that does not move under the transformation. The centre of a rotation and the centre of an enlargement are invariant.

Area scale factor: The factor an area is multiplied by under an enlargement: the square of the scale factor, $k^{2}$ .

In context

Cambridge sets 0580 in real situations - a recipe, a bank statement, a map, a garden, a graph of a car's journey. These show how the mathematics appears in a question, then you write your own working for the numbers you are given.

Reflection in a vertical line

Reflect the point $(2, 5)$ in the line $x = 3$ . The mirror line is vertical, so the y-coordinate stays 5 and only x changes. The point is 1 unit to the left of the line, so its image is 1 unit to the right of it, at $(4, 5)$ . The rule $2 \times 3 - 2 = 4$ does the same job in one line.

Rotation about the origin

Rotate the point $(3, - 1)$ through $90^{\circ}$ anticlockwise about the origin. Use the rule $(x, y) \to (- y, x)$ , which gives $(- (- 1), 3) = (1, 3)$ . Check it with a point you know: $(1, 0)$ goes to $(0, 1)$ , which is a quarter turn anticlockwise.

Translation by a column vector

The point $A (5, - 1)$ is translated by the column vector $(\begin{matrix} - 3 \\ 4 \end{matrix})$ . Add the top number to the x-coordinate and the bottom number to the y-coordinate: $5 + (- 3) = 2$ and $- 1 + 4 = 3$ , so the image is $(2, 3)$ . The same vector moves every other point of the shape, which is what makes it a translation and not a rotation.

Enlargement and its area

The triangle with vertices $(1, 1)$ , $(3, 1)$ and $(1, 2)$ is enlarged by scale factor 3, centre the origin. Multiply every coordinate by 3 to get $(3, 3)$ , $(9, 3)$ and $(3, 6)$ . The object has legs 2 and 1, so its area is 1 square unit, and the image has legs 6 and 3, so its area is 9: nine times as big, which is $3^{2}$ .

Describing a transformation fully

The point $(5, 3)$ is mapped to $(- 1, 3)$ , and the image faces the other way. The y-coordinate has not changed, so the mirror line is vertical; the point has moved 6 to the left, so the line is 3 to the left of it, at $x = 2$ . Check: $2 \times 2 - 5 = - 1$ . The full description is 'reflection in the line $x = 2$ ' - the word and the line, nothing less.

Flashcards

Click a card to flip it. Mark honestly — the tracker uses this.

Q1What three things describe a rotation fully?

The centre, the angle and the direction, for example 90 degrees clockwise about (2, 1). A 180 degree rotation needs no direction.

Q2What do you need to describe a reflection fully?

The mirror line, for example reflection in the line x = 3. The word alone is not enough.

Q3What do you need to describe a translation fully?

The column vector: the top number is the movement across and the bottom number is the movement up or down.

Q4What two things describe an enlargement fully?

The scale factor and the centre of enlargement.

Q5Which transformations produce an image congruent to the object?

Reflection, rotation and translation. An enlargement gives a similar image, not a congruent one.

Q6Which transformation reverses the orientation of a shape?

Reflection. The image faces the opposite way; a rotation and a translation keep the shape the same way round.

Q7Rotate

(2, 5)

through

90^{\circ}

clockwise about the origin.

(5, - 2)

. A 90 degree clockwise turn takes

(x, y)

(y, - x)

Q8Rotate

(2, 5)

through

90^{\circ}

anticlockwise about the origin.

(- 5, 2)

. A 90 degree anticlockwise turn takes

(x, y)

(- y, x)

Q9Reflect

(3, 7)

in the line

y = x

(7, 3)

. Reflection in

y = x

swaps the two coordinates.

Q10Reflect

(3, 7)

in the line

x = 2

(1, 7)

. The y-coordinate is unchanged, and the x-coordinate becomes

2 \times 2 - 3 = 1

Q11A shape of area

8 {cm}^{2}

is enlarged by scale factor 3. What is the area of the image?

72 {cm}^{2}

. Areas are multiplied by the square of the scale factor:

8 \times 9 = 72

Q12What does a negative scale factor do to a shape?

It puts the image on the opposite side of the centre, turned through 180 degrees, with the size given by the numerical value of the scale factor.

Q13Two reflections in the parallel lines

x = 1

and

x = 4

are equivalent to which single transformation?

A translation of

(\begin{matrix} 6 \\ 0 \end{matrix})

: twice the distance between the two lines, at right angles to them.

Q14How do you find the centre of a rotation when you are given the object and its image?

Join two pairs of corresponding vertices and draw the perpendicular bisector of each joining segment; the bisectors cross at the centre.

0 of 14 marked as known

Exam technique for this topic

Write a description as one sentence in a fixed order: the type of transformation, then the measurement, then the line or the centre. 'Rotation of 90 degrees clockwise about (0, 0)' scores three marks; 'the shape has been turned' scores none.
Before you write anything, look at the way the image faces. Same way round means it cannot be a reflection; a mirror image means it cannot be a rotation or a translation.
Always give the direction of a rotation. A 90 degree clockwise turn and a 90 degree anticlockwise turn send the same point to two different places, so only one of them is right - except for 180 degrees, where the direction does not matter.
Use a ruler and count squares from the mirror line or from the centre for every single vertex. Most marks lost in this topic are drawing errors rather than reasoning errors, and an image with one vertex in the wrong place is a wrong image.
For an enlargement, decide the scale factor first: bigger than 1, between 0 and 1, or negative. It tells you how far from the centre the image sits and on which side, which is the quickest check that your drawing is sensible.

Why students lose marks here

A student wroteThe transformation is a rotation of

90^{\circ}

about the origin.

Why it loses marksNo direction is given. A 90 degree clockwise turn sends

(2, 5)

(5, - 2)

, and a 90 degree anticlockwise turn sends it to

(- 5, 2)

. One angle, two different images, so the description is incomplete.

The answer that scoresRotation of

90^{\circ}

clockwise about the origin, or equivalently

270^{\circ}

anticlockwise.

A student wroteThe shape has been enlarged by scale factor 2.

Why it loses marksThe scale factor alone does not fix an enlargement. Any centre at all would give a different image of the same size, so a description without a centre cannot be marked correct.

The answer that scoresEnlargement, scale factor 2, centre

(1, - 2)

- the type, the scale factor and the centre.

A student wroteThe shape moved 3 squares right and 4 squares down, so the vector is

(\begin{matrix} 4 \\ - 3 \end{matrix})

Why it loses marksThe two numbers have been swapped. The top number of a column vector is the movement across and the bottom number is the movement up or down.

The answer that scores

(\begin{matrix} 3 \\ - 4 \end{matrix})

: 3 to the right and 4 down.

A student wroteA shape of area

5 {cm}^{2}

is enlarged by scale factor 2, so the image has area

10 {cm}^{2}

Why it loses marksLengths are multiplied by the scale factor but areas are multiplied by its square. Doubling the lengths of a shape makes it four times as big in area, not twice.

The answer that scoresThe area scale factor is

2^{2} = 4

, so the image has area

5 \times 4 = 20 {cm}^{2}

. Checking with the sides works too: 2 cm by 2.5 cm becomes 4 cm by 5 cm, which is

20 {cm}^{2}

Syllabus coverage

The four transformations, and what each one needs

Reflection

Rotation

Translation

Enlargement

Describing a transformation fully: the wording that scores

Extended: the area of an image, and combinations of transformations

Key terms you must be able to define

In context

Flashcards

Exam-style questions

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Exam technique for this topic

Why students lose marks here