Transformations

Types of Transformation

A transformation moves or changes a shape on a coordinate grid. The original shape is called the object and the result is the image.

There are four main types:

1. Translation — slide (no rotation or reflection)
2. Reflection — flip over a mirror line
3. Rotation — turn about a centre point
4. Enlargement — resize by a scale factor

Translation

A translation slides a shape to a new position without rotating or reflecting it. Every point moves the same distance in the same direction.

Translations are described using a column vector: (x, y)

• Top number: positive = right, negative = left
• Bottom number: positive = up, negative = down

Example: Vector (3, -2) means move 3 right, 2 down.

If point A is at (1, 4), after translation by (3, -2): A moves to (1+3, 4-2) = (4, 2)

Apply the same vector to every vertex of the shape.

Reflection

A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.

Common mirror lines and their rules:

Mirror line	Rule
x-axis (y = 0)	(x, y) → (x, -y)
y-axis (x = 0)	(x, y) → (-x, y)
y = x	(x, y) → (y, x)
y = -x	(x, y) → (-y, -x)

Example: Reflect (3, 5) in the x-axis → (3, -5)

Example: Reflect (5, 2) in the line y = x → (2, 5)

To reflect in other lines (e.g. x = 2), count the perpendicular distance from the point to the line and place the image the same distance on the other side.

Rotation

A rotation turns a shape through an angle about a fixed point called the centre of rotation.

To fully describe a rotation, state:

1. Centre of rotation (coordinates)
2. Angle of rotation (in degrees)
3. Direction (clockwise or anticlockwise)

Rotation rules about the origin (0, 0):

Rotation	Rule
90° clockwise	(x, y) → (y, -x)
90° anticlockwise	(x, y) → (-y, x)
180° (either direction)	(x, y) → (-x, -y)

Example: Rotate (2, -3) by 180° about the origin → (-2, 3)

Example: Rotate (4, 1) by 90° anticlockwise about the origin → (-1, 4)

Finding the Centre of Rotation

Draw lines connecting each original vertex to its image. The centre of rotation is where the perpendicular bisectors of these connecting lines meet.

Enlargement

An enlargement changes the size of a shape by a scale factor (k) from a centre of enlargement.

• Scale factor > 1: shape gets larger
• Scale factor between 0 and 1 (a fraction): shape gets smaller
• Negative scale factor: shape is enlarged and rotated 180° (on the other side of the centre)

To enlarge a shape:

1. Draw lines from the centre of enlargement through each vertex
2. Multiply the distance from the centre to each vertex by the scale factor
3. Mark the new vertices along those lines

Example: Centre (0, 0), scale factor 3, point (2, 4): New point = (2 × 3, 4 × 3) = (6, 12)

Example: Centre (0, 0), scale factor 1/2, point (6, 8): New point = (6 × 1/2, 8 × 1/2) = (3, 4)

Congruence and Similarity

Isometric transformations (translation, reflection, rotation) preserve both shape and size — the object and image are congruent.

Enlargements preserve shape but change size — the object and image are similar (corresponding angles equal, sides in the same ratio).

Transformation	Size changes?	Shape changes?	Congruent or Similar?
Translation	No	No	Congruent
Reflection	No	No	Congruent
Rotation	No	No	Congruent
Enlargement	Yes	No	Similar

Combining Transformations

Transformations can be applied one after another. The order matters — applying A then B may give a different result to B then A.

Example: Reflect in the x-axis, then translate by (2, 3). Point (1, 4): reflect → (1, -4), then translate → (3, -1)

Describing Transformations

When asked to fully describe a transformation between an object and image:

• Translation: state the column vector
• Reflection: state the equation of the mirror line
• Rotation: state the centre, angle, and direction
• Enlargement: state the scale factor and centre of enlargement

Always include all required information — a partial description loses marks.

Common Mistakes

• Describing a reflection with just "reflected in the y-axis" without giving the line equation (y-axis: x = 0)
• Forgetting direction for rotations (clockwise vs anticlockwise)
• Applying enlargement from the wrong centre — always measure distances from the given centre
• Confusing column vector notation: (3, -2) is 3 right and 2 down, not 3 up and 2 right

Tips and Tricks

• For 90° rotations, use the coordinate rules rather than drawing — it is faster and more accurate
• For reflections in diagonal lines (y = x), swap the coordinates
• When enlarging, check that all sides of the image are the scale factor times the corresponding sides of the object
• Tracing paper is allowed in most exams and is very useful for rotations and reflections