Transformations

What you'll learn

How to read and use coordinates on a grid.
How to draw translations, reflections, rotations, and enlargements.
How to describe a transformation fully for exam marks.
How to handle a simple combined transformation.

Coordinate grid basics

A coordinate like $(- 2, 5)$ tells you where a point is on a grid: across first, then up or down. The first number is the $x$ -coordinate, and the second is the $y$ -coordinate.

A vertex is a corner of a shape. The origin, often labelled $O$ , is the point $(0, 0)$ .

Definition

Transformation words

A transformation moves or changes a shape.
The starting shape is called the object.
The new shape after the transformation is called the image.
Matching corners on the object and image are called corresponding vertices.

Example

Reading a movement from coordinates

A point moves from $(- 2, 4)$ to $(3, 1)$ .

The point moves 5 squares right and 3 squares down from (-2,4) to (3,1).

Compare the $x$ -coordinates: from -2 to 3 is 5 spaces right.
Compare the $y$ -coordinates: from 4 to 1 is 3 spaces down.
So the movement is 5 right and 3 down.

The four main transformations

Definition

Four transformation types

A translation slides a shape without turning or flipping it.
A reflection flips a shape over a mirror line.
A rotation turns a shape around a fixed point.
An enlargement changes the size of a shape from a fixed centre.

Translations

A vector is a column of two numbers that describes a translation. The top number tells you the horizontal movement. The bottom number tells you the vertical movement.

Key Idea

Translation vectors

Positive top means right, negative top means left. Positive bottom means up, negative bottom means down.

Example

Describing a translation fully

A triangle has a vertex at $(- 2, - 1)$ . Its image has the matching vertex at $(4, 2)$ . Describe the translation.

A corresponding vertex moves 6 squares right and 3 squares up, giving the translation vector.

Compare the $x$ -coordinates: from -2 to 4 is 6 right.
Compare the $y$ -coordinates: from -1 to 2 is 3 up.
Write the movement as a vector:
$(63)\begin{pmatrix} 6 \\ 3 \end{pmatrix}$
The full answer is: translation by the vector shown above.

Common Mistake

Mixing up vector order

The top number is left/right. The bottom number is up/down. Do not swap them.

Reflections

A mirror line is the line that a shape is reflected in. It might be a line such as $x = 1$ , $y = - 2$ , or $y = x$ .

For the line $y = x$ , the coordinates swap places:

\to (y,x)

Example

Reflecting in the line y = x

Reflect a quadrilateral with vertices $(2, - 4)$ , $(5, - 4)$ , $(4, - 2)$ , and $(1, - 2)$ in the line $y = x$ .

The quadrilateral is reflected in the mirror line y=x, so each point swaps its coordinates.

Use the rule for reflection in $y = x$ : swap each pair of coordinates.
Apply the rule to each vertex:
$(2,−4)→(−4,2)(5,−4)→(−4,5)(4,−2)→(−2,4)(1,−2)→(−2,1)\begin{aligned} (2,-4)&\to(-4,2)\\ (5,-4)&\to(-4,5)\\ (4,-2)&\to(-2,4)\\ (1,-2)&\to(-2,1) \end{aligned}$
Plot the four new points and join them in the same order.

Tip

Spotting the mirror line

Corresponding vertices are the same distance from the mirror line, on opposite sides of it.

In a reflection, each pair of corresponding vertices lies on opposite sides of the mirror line at equal perpendicular distances.

Common Mistake

Reflecting in the wrong axis

Reflecting in the $x$ -axis changes the sign of the $y$ -coordinate. Reflecting in the $y$ -axis changes the sign of the $x$ -coordinate.

Rotations

A rotation needs three details: the angle, the direction, and the centre of rotation.

For rotations about the origin:

180° rotation: $(x,y)→(−x,−y)(x,y)\to(-x,-y)$
90° anticlockwise rotation: $(x,y)→(−y,x)(x,y)\to(-y,x)$

Example

Rotating 90° anticlockwise about O

Rotate a triangle with vertices $(- 5, 1)$ , $(- 3, 1)$ , and $(- 3, 3)$ by 90° anticlockwise about $O$ .

The triangle is turned 90° anticlockwise about the origin, moving it from quadrant II to quadrant III.

Since the centre is $O$ , use the origin rule for 90° anticlockwise:
$(x,y)→(−y,x)(x,y)\to(-y,x)$
Apply the rule to each vertex:
$(−5,1)→(−1,−5)(−3,1)→(−1,−3)(−3,3)→(−3,−3)\begin{aligned} (-5,1)&\to(-1,-5)\\ (-3,1)&\to(-1,-3)\\ (-3,3)&\to(-3,-3) \end{aligned}$
Plot the three new points and join them to make the rotated triangle.

Common Mistake

90 degree rotations

A 90° clockwise turn and a 90° anticlockwise turn land in different places, so read the direction carefully.

Enlargements

An enlargement changes the size of a shape. The scale factor tells you how many times bigger the distances become. The centre of enlargement is the fixed point the shape grows from.

When the centre is $O$ , multiply both coordinates by the scale factor.

Example

Enlarging from the origin

Enlarge a triangle with vertices $(1, 0)$ , $(2, 0)$ , and $(2, 2)$ by scale factor 3, centre $O$ .

Each vertex lies on the same ray from O as its image, with distances multiplied by 3.

The centre is $O$ , so multiply each coordinate by 3.
Work out the new vertices:
$(1,0)→(3,0)(2,0)→(6,0)(2,2)→(6,6)\begin{aligned} (1,0)&\to(3,0)\\ (2,0)&\to(6,0)\\ (2,2)&\to(6,6) \end{aligned}$
Plot the new vertices and join them.

Common Mistake

Enlarging from the wrong place

Multiplying coordinates only works when the centre is $O$ . If the centre is somewhere else, you must enlarge from that centre instead.

Combined transformations

Sometimes a question gives two transformations in order, such as “reflect in the $x$ -axis followed by a translation”.

Example

Finding p and q after two transformations

A vertex starts at $(2, 3)$ . It is reflected in the $x$ -axis, then translated to $(- 1, - 4)$ by the vector $(pq)\begin{pmatrix} p \\ q \end{pmatrix}$ .

The point first reflects in the x-axis, then translates 3 left and 1 down to the final position.

First reflect in the $x$ -axis. The $y$ -coordinate changes sign:
$(2,3)→(2,−3)(2,3)\to(2,-3)$
Now compare $(2, - 3)$ with the final point $(- 1, - 4)$ .
The movement is 3 left and 1 down, so $p = - 3$ and $q = - 1$ .

Common Mistake

Doing the moves in the wrong order

“Reflect then translate” is not always the same as “translate then reflect”. Follow the order written in the question.

Describing a transformation fully

Key Idea

Full descriptions

Translation: give the vector.
Reflection: give the mirror line.
Rotation: give the angle, direction, and centre.
Enlargement: give the scale factor and centre.

Example

Choosing the correct full description

A shape has a vertex at $(- 3, 1)$ . Its image has the matching vertex at $(3, 1)$ . Other matching vertices also have opposite $x$ -coordinates and the same $y$ -coordinates.

Opposite x-coordinates and the same y-coordinates indicate reflection in the y-axis, or x=0.

The shape has been flipped, not just slid.
The matching points are the same distance from the $y$ -axis.
The mirror line is the $y$ -axis, which has equation $x = 0$ .
The full answer is: reflection in the line $x = 0$ .

Exam technique

In the exam

Check whether the shape has been slid, flipped, turned, or resized.
Use one pair of corresponding vertices, then check another pair to avoid mistakes.
For “describe fully”, always include the extra detail: vector, mirror line, centre, angle, direction, or scale factor.

Self review

Check yourself

What does the top number in a translation vector tell you?
If a point is reflected in the $x$ -axis, which coordinate changes sign?
What extra details are needed to describe a rotation fully?

Transformations

What you'll learn

How to read and use coordinates on a grid.
How to draw translations, reflections, rotations, and enlargements.
How to describe a transformation fully for exam marks.
How to handle a simple combined transformation.

Coordinate grid basics

A coordinate like $(- 2, 5)$ tells you where a point is on a grid: across first, then up or down. The first number is the $x$ -coordinate, and the second is the $y$ -coordinate.

A vertex is a corner of a shape. The origin, often labelled $O$ , is the point $(0, 0)$ .

Definition

Transformation words

A transformation moves or changes a shape.
The starting shape is called the object.
The new shape after the transformation is called the image.
Matching corners on the object and image are called corresponding vertices.

Example

Reading a movement from coordinates

A point moves from $(- 2, 4)$ to $(3, 1)$ .

The point moves 5 squares right and 3 squares down from (-2,4) to (3,1).

Compare the $x$ -coordinates: from -2 to 3 is 5 spaces right.
Compare the $y$ -coordinates: from 4 to 1 is 3 spaces down.
So the movement is 5 right and 3 down.

The four main transformations

Definition

Four transformation types

A translation slides a shape without turning or flipping it.
A reflection flips a shape over a mirror line.
A rotation turns a shape around a fixed point.
An enlargement changes the size of a shape from a fixed centre.

Translations

A vector is a column of two numbers that describes a translation. The top number tells you the horizontal movement. The bottom number tells you the vertical movement.

Key Idea

Translation vectors

Positive top means right, negative top means left. Positive bottom means up, negative bottom means down.

Example

Describing a translation fully

A triangle has a vertex at $(- 2, - 1)$ . Its image has the matching vertex at $(4, 2)$ . Describe the translation.

A corresponding vertex moves 6 squares right and 3 squares up, giving the translation vector.

Compare the $x$ -coordinates: from -2 to 4 is 6 right.
Compare the $y$ -coordinates: from -1 to 2 is 3 up.
Write the movement as a vector:
$(63)\begin{pmatrix} 6 \\ 3 \end{pmatrix}$
The full answer is: translation by the vector shown above.

Common Mistake

Mixing up vector order

The top number is left/right. The bottom number is up/down. Do not swap them.

Reflections

A mirror line is the line that a shape is reflected in. It might be a line such as $x = 1$ , $y = - 2$ , or $y = x$ .

For the line $y = x$ , the coordinates swap places:

\to (y,x)

Example

Reflecting in the line y = x

Reflect a quadrilateral with vertices $(2, - 4)$ , $(5, - 4)$ , $(4, - 2)$ , and $(1, - 2)$ in the line $y = x$ .

The quadrilateral is reflected in the mirror line y=x, so each point swaps its coordinates.

Use the rule for reflection in $y = x$ : swap each pair of coordinates.
Apply the rule to each vertex:
$(2,−4)→(−4,2)(5,−4)→(−4,5)(4,−2)→(−2,4)(1,−2)→(−2,1)\begin{aligned} (2,-4)&\to(-4,2)\\ (5,-4)&\to(-4,5)\\ (4,-2)&\to(-2,4)\\ (1,-2)&\to(-2,1) \end{aligned}$
Plot the four new points and join them in the same order.

Tip

Spotting the mirror line

Corresponding vertices are the same distance from the mirror line, on opposite sides of it.

In a reflection, each pair of corresponding vertices lies on opposite sides of the mirror line at equal perpendicular distances.

Common Mistake

Reflecting in the wrong axis

Reflecting in the $x$ -axis changes the sign of the $y$ -coordinate. Reflecting in the $y$ -axis changes the sign of the $x$ -coordinate.

Rotations

A rotation needs three details: the angle, the direction, and the centre of rotation.

For rotations about the origin:

180° rotation: $(x,y)→(−x,−y)(x,y)\to(-x,-y)$
90° anticlockwise rotation: $(x,y)→(−y,x)(x,y)\to(-y,x)$

Example

Rotating 90° anticlockwise about O

Rotate a triangle with vertices $(- 5, 1)$ , $(- 3, 1)$ , and $(- 3, 3)$ by 90° anticlockwise about $O$ .

The triangle is turned 90° anticlockwise about the origin, moving it from quadrant II to quadrant III.

Since the centre is $O$ , use the origin rule for 90° anticlockwise:
$(x,y)→(−y,x)(x,y)\to(-y,x)$
Apply the rule to each vertex:
$(−5,1)→(−1,−5)(−3,1)→(−1,−3)(−3,3)→(−3,−3)\begin{aligned} (-5,1)&\to(-1,-5)\\ (-3,1)&\to(-1,-3)\\ (-3,3)&\to(-3,-3) \end{aligned}$
Plot the three new points and join them to make the rotated triangle.

Common Mistake

90 degree rotations

A 90° clockwise turn and a 90° anticlockwise turn land in different places, so read the direction carefully.

Enlargements

An enlargement changes the size of a shape. The scale factor tells you how many times bigger the distances become. The centre of enlargement is the fixed point the shape grows from.

When the centre is $O$ , multiply both coordinates by the scale factor.

Example

Enlarging from the origin

Enlarge a triangle with vertices $(1, 0)$ , $(2, 0)$ , and $(2, 2)$ by scale factor 3, centre $O$ .

Each vertex lies on the same ray from O as its image, with distances multiplied by 3.

The centre is $O$ , so multiply each coordinate by 3.
Work out the new vertices:
$(1,0)→(3,0)(2,0)→(6,0)(2,2)→(6,6)\begin{aligned} (1,0)&\to(3,0)\\ (2,0)&\to(6,0)\\ (2,2)&\to(6,6) \end{aligned}$
Plot the new vertices and join them.

Common Mistake

Enlarging from the wrong place

Multiplying coordinates only works when the centre is $O$ . If the centre is somewhere else, you must enlarge from that centre instead.

Combined transformations

Sometimes a question gives two transformations in order, such as “reflect in the $x$ -axis followed by a translation”.

Example

Finding p and q after two transformations

A vertex starts at $(2, 3)$ . It is reflected in the $x$ -axis, then translated to $(- 1, - 4)$ by the vector $(pq)\begin{pmatrix} p \\ q \end{pmatrix}$ .

The point first reflects in the x-axis, then translates 3 left and 1 down to the final position.

First reflect in the $x$ -axis. The $y$ -coordinate changes sign:
$(2,3)→(2,−3)(2,3)\to(2,-3)$
Now compare $(2, - 3)$ with the final point $(- 1, - 4)$ .
The movement is 3 left and 1 down, so $p = - 3$ and $q = - 1$ .

Common Mistake

Doing the moves in the wrong order

“Reflect then translate” is not always the same as “translate then reflect”. Follow the order written in the question.

Describing a transformation fully

Key Idea

Full descriptions

Translation: give the vector.
Reflection: give the mirror line.
Rotation: give the angle, direction, and centre.
Enlargement: give the scale factor and centre.

Example

Choosing the correct full description

A shape has a vertex at $(- 3, 1)$ . Its image has the matching vertex at $(3, 1)$ . Other matching vertices also have opposite $x$ -coordinates and the same $y$ -coordinates.

Opposite x-coordinates and the same y-coordinates indicate reflection in the y-axis, or x=0.

The shape has been flipped, not just slid.
The matching points are the same distance from the $y$ -axis.
The mirror line is the $y$ -axis, which has equation $x = 0$ .
The full answer is: reflection in the line $x = 0$ .

Exam technique

In the exam

Check whether the shape has been slid, flipped, turned, or resized.
Use one pair of corresponding vertices, then check another pair to avoid mistakes.
For “describe fully”, always include the extra detail: vector, mirror line, centre, angle, direction, or scale factor.

Self review

Check yourself

What does the top number in a translation vector tell you?
If a point is reflected in the $x$ -axis, which coordinate changes sign?
What extra details are needed to describe a rotation fully?

Transformations

What you'll learn

Coordinate grid basics

The four main transformations

Translations

Reflections

Rotations

Enlargements

Combined transformations

Describing a transformation fully

Recap questions

You've reached the end

Transformations

What you'll learn

Coordinate grid basics

The four main transformations

Translations

Reflections

Rotations

Enlargements

Combined transformations

Describing a transformation fully

Recap questions

You've reached the end

Transformations

What you'll learn

Coordinate grid basics

The four main transformations

Translations

Reflections

Rotations

Enlargements

Combined transformations

Describing a transformation fully

Recap questions

You've reached the end

Transformations Revision Guide

Transformations

What you'll learn

Coordinate grid basics

The four main transformations

Translations

Reflections

Rotations

Enlargements

Combined transformations

Describing a transformation fully

Recap questions

You've reached the end

Transformations Revision Guide