Transformations of the unit square Revision Notes for AQA GCSE Further Maths

Transformations of the unit square

Introduction to the unit square

The unit square is a fundamental shape in coordinate geometry that serves as the basis for understanding matrix transformations. When we apply different transformation matrices to the unit square, we can observe how various geometric transformations affect shapes and coordinates.

The unit square has four vertices with specific coordinates that make calculations straightforward. These vertices are positioned at $O(0,0)$ , $A(1,0)$ , $B(1,1)$ , and $C(0,1)$ . The square has sides of length 1 unit and sits in the first quadrant of the coordinate plane, making it an ideal reference shape for studying transformations.

Note

The unit square's simple coordinates make it perfect for understanding how transformations work. Each vertex represents a key position that helps us track exactly how shapes change under different transformations.

Types of transformations

There are several key types of transformations that you need to understand for your GCSE exam. Each transformation changes the position, size, or orientation of the unit square in a specific way.

Reflections

Note

Reflections create mirror images of shapes across different lines:

Reflexion in the x-axis flips the shape vertically
Reflexion in the y-axis flips the shape horizontally
Reflexion in the line y = x swaps x and y coordinates
Reflexion in the line y = -x also swaps coordinates but with sign changes

Rotations about the origin

Rotations turn the shape around the origin by specific angles. The most common rotations you'll encounter are:

90° rotation (anticlockwise)
180° rotation
270° rotation (which is equivalent to 90° clockwise)

Enlargements with centre at the origin

Note

Enlargements change the size of shapes using scale factors:

Positive scale factors increase or decrease size while maintaining orientation
Negative scale factors change size and also rotate the shape 180°

Finding transformation matrices using position vectors

To work out the transformation matrix for any geometric transformation, you need to follow a systematic method using position vectors. This approach ensures you get the correct 2×2 matrix every time.

Important

The Key Method for Finding Transformation Matrices:

The technique involves examining what happens to two specific position vectors under the transformation:

The vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ (which represents moving 1 unit right)
The vector $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ (which represents moving 1 unit up)

Once you determine where these vectors map to after the transformation, you write these new position vectors side by side to create your transformation matrix.

Worked example: rotation and enlargement

Let's work through finding matrices for two different transformations to see this method in action.

Example

Worked Example: Finding Transformation Matrices

Part (i): Rotation through 270° about the origin

When we rotate 270° anticlockwise (or equivalently 90° clockwise), we need to track where our key vectors go:

Starting with vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ : This point moves from $(1, 0)$ to $(0, -1)$

Starting with vector $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ : This point moves from $(0, 1)$ to $(1, 0)$

Writing these new position vectors side by side gives us the transformation matrix: $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

Part (ii): Enlargement with scale factor 3, centre the origin

For an enlargement with scale factor 3:

Vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ maps to $(3, 0)$ , giving position vector $\begin{bmatrix} 3 \\ 0 \end{bmatrix}$

Vector $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ maps to $(0, 3)$ , giving position vector $\begin{bmatrix} 0 \\ 3 \end{bmatrix}$

Writing these side by side gives the transformation matrix: $\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$

Worked example: applying a transformation matrix

When you have a transformation matrix and need to apply it to the unit square, you can either multiply the matrix by each vertex coordinate or recognise the transformation type and sketch directly.

Example

Worked Example: Applying a Transformation Matrix

Method 1: Matrix multiplication approach

Given the transformation matrix $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ , let's apply it to each vertex:

For $O(0,0)$ : $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ → O stays at $(0,0)$

For $A(1,0)$ : $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \end{bmatrix}$ → A maps to $(-1,0)$

For $B(1,1)$ : $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ -1 \end{bmatrix}$ → B maps to $(-1,-1)$

For $C(0,1)$ : $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ -1 \end{bmatrix}$ → C maps to $(0,-1)$

Method 2: Recognition approach

Alternatively, if you recognise that the matrix $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ represents a 180° rotation about the origin, you can sketch the result directly without calculating each point individually.

Common transformation matrices

Note

Essential Transformation Matrices to Remember:

Reflections:

x-axis: $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
y-axis: $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$
Line y = x: $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
Line y = -x: $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$

Rotations about origin:

90° anticlockwise: $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$
180°: $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$
270° anticlockwise: $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

Enlargements (scale factor k):

$\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$

Exam tips and common mistakes

Important

Critical Points to Avoid Common Mistakes:

Always remember that rotations are anticlockwise unless specifically stated otherwise
A rotation of 270° anticlockwise is the same as 90° clockwise
Be careful with signs when applying transformations - negative coordinates often indicate reflections or rotations that flip the shape's orientation
Always check your work by verifying that at least one vertex transforms correctly
Remember that the origin O(0,0) typically remains fixed under rotations and enlargements centred at the origin

Advanced concepts

While this topic focuses on GCSE-level transformations, it's worth noting that invariant points and lines (points and lines that don't move under transformation) are studied in much greater depth at A-Level mathematics.

Summary

Key Points to Remember:

The unit square has vertices at $O(0,0)$ , $A(1,0)$ , $B(1,1)$ , and $C(0,1)$
To find a transformation matrix, determine where vectors $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ map to, then write these new vectors side by side
Rotations are measured anticlockwise from the positive x-axis unless stated otherwise
Matrix multiplication allows you to find the exact coordinates of transformed points
Common transformations have standard matrices that are useful to memorise for quick recognition

Transformations of the unit square (AQA GCSE Further Maths): Revision Notes