Transformations

Reflection, rotation, translation and enlargement – describing and performing transformations on shapes.

Transformations move or change shapes. You must be able to perform them on a coordinate grid and describe a transformation fully (type, and the required details).

Reflection

A reflection flips a shape over a mirror line. Common mirror lines: x-axis, y-axis, y = x, y = −x, or x = a / y = b. The image is the same distance from the mirror line as the object, on the opposite side.

Worked example

Describe fully the single transformation that maps triangle A onto triangle B when B is the mirror image of A in the line x = 2.

Solution

**Reflection in the line x = 2**

Rotation

A rotation turns a shape about a fixed point (the centre of rotation). You must state: • Centre of rotation • Angle (e.g. 90°, 180°, 270°) • Direction (clockwise or anticlockwise) 90° clockwise about the origin: (x, y) → (y, −x) 90° anticlockwise about the origin: (x, y) → (−y, x) 180° about the origin: (x, y) → (−x, −y)

Worked example

Rotate the point (3, 1) 90° clockwise about the origin. What are the new coordinates?

Solution

(x, y) → (y, −x) (3, 1) → (1, −3) **New coordinates: (1, −3)**

Translation

A translation slides a shape without turning it. Described by a column vector: [a / b] • a = movement right (positive) or left (negative) • b = movement up (positive) or down (negative)

Worked example

Translate the point (2, 5) by the vector [−3 / 4]. New coordinates?

Solution

(2 − 3, 5 + 4) = **(−1, 9)**

Enlargement

An enlargement changes the size of a shape from a centre of enlargement. You must state: • Centre of enlargement • Scale factor Scale factor > 1 → bigger 0 < scale factor < 1 → smaller Negative scale factor → enlargement on the opposite side of the centre (and inverted). Distance from centre × scale factor = distance of image from centre.

Worked example

Enlarge the point (4, 2) by scale factor 3 from the origin. New coordinates?

Solution

From origin: multiply coordinates by 3 (4 × 3, 2 × 3) = **(12, 6)**

Practice questions

Try these without looking at the answers first.

Q1.Reflect (5, 2) in the y-axis. New coordinates?

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(−5, 2)

Q2.Rotate (2, 4) 180° about the origin.

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(−2, −4)

Q3.Translate (1, 3) by [2 / −5].

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(3, −2)

Q4.Enlarge (3, 1) by scale factor 2 from the origin.

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(6, 2)

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