Transformations

A transformation moves or changes a shape. The original shape is the object; the result is the image.

The four types: Translation (slide), Reflection (flip), Rotation (turn), Enlargement (scale). The first three are isometric — they preserve shape and size.

📖 Learn

Translation

A translation slides every point the same distance in the same direction. Described by a column vector.

Vector (a, b) moves right a, up b. Negative = left/down.
Image point: (x + a, y + b)

Properties preserved: shape, size, orientation. No rotation or reflection.

💡 To describe a translation, give the column vector. Do NOT say "moved right 3, up 2" — say vector (3, 2).

Reflection

A reflection flips a shape in a mirror line. Each point is the same perpendicular distance from the mirror on the other side.

y-axis (x=0): (x, y) → (−x, y)
x-axis (y=0): (x, y) → (x, −y)
y = x: (x, y) → (y, x)
y = −x: (x, y) → (−y, −x)

Properties preserved: shape, size. Orientation is reversed (mirror image).

💡 To describe a reflection, state the mirror line equation (e.g. "reflection in y = 2").

Rotation

A rotation turns a shape around a fixed point called the centre of rotation.

90° clockwise about origin: (x, y) → (y, −x)
90° anticlockwise about origin: (x, y) → (−y, x)
180° about origin: (x, y) → (−x, −y)

To describe: state angle, direction (clockwise/anticlockwise), and centre of rotation.

💡 A 270° clockwise = 90° anticlockwise. Always use the smaller angle (≤180°).

Enlargement

An enlargement scales a shape by a scale factor from a centre of enlargement.

Image point = centre + k × (object point − centre)
k > 1: larger · 0 < k < 1: smaller · k negative: on opposite side

To find scale factor: k = image length ÷ object length

Area scale factor = k² (e.g. k=3 → area ×9)

💡 To describe: state "enlargement, scale factor k, centre (a, b)".

Combined Transformations

Transformations can be combined. The order matters! Apply the first transformation, then the second to the image.

Transformation	Isometric?	Describe using…
Translation	Yes	Column vector
Reflection	Yes	Mirror line equation
Rotation	Yes	Angle, direction, centre
Enlargement	No (unless k=±1)	Scale factor, centre

✏️ Worked Examples

Example 1: Translation

Triangle with vertices A(1,2), B(3,2), C(2,4) is translated by vector (4, −3). Find the image.

Add vector to each vertex: A' = (1+4, 2−3) = (5, −1)

B' = (3+4, 2−3) = (7, −1)

C' = (2+4, 4−3) = (6, 1)

Image: A'(5,−1), B'(7,−1), C'(6,1)

Example 2: Reflection in y = x

Reflect point P(3, 7) in the line y = x.

Rule for y = x: (x, y) → (y, x)

P(3, 7) → P'(7, 3)

Example 3: Rotation 90° clockwise about the origin

Rotate A(2, 5) by 90° clockwise about the origin.

Rule: (x, y) → (y, −x)

A(2, 5) → A'(5, −2)

Example 4: Enlargement

Enlarge triangle with vertices P(1,1), Q(3,1), R(3,4) by scale factor 2, centre (0,0).

🔄 Transformations