Transformations

Maths · Geometry and measures. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A transformation moves or changes a shape. The start shape is the object and the result is the image. A reflection flips the shape in a mirror line. A rotation turns it about a centre. A translation slides it without turning. These three keep the size and shape the same, so the image is congruent to the object.

An enlargement changes the size by a scale factor from a centre. A fractional scale factor between 0 and 1 makes the shape smaller. A negative scale factor puts the image on the opposite side of the centre, turned upside down.

A point that does not move is called invariant.

Rules to remember

Reflection: x-axis (x, y) → (x, −y); y-axis (x, y) → (−x, y); line y = x (x, y) → (y, x); line y = −x (x, y) → (−y, −x).
Rotation about the origin: 90° clockwise (x, y) → (y, −x); 90° anticlockwise (x, y) → (−y, x); 180° (x, y) → (−x, −y).
Translation by the vector with top entry a and bottom entry b: (x, y) → (x + a, y + b).
Enlargement, scale factor k: image = centre + k × (point − centre).
To describe fully: reflection needs the mirror line; rotation needs angle, direction and centre; translation needs the vector; enlargement needs scale factor and centre.
Points on the mirror line are invariant in a reflection; the centre is invariant in a rotation or an enlargement.

Common mistake

In an enlargement, students multiply the coordinates by the scale factor even when the centre is not the origin. Multiply the distances from the centre, then add them to the centre.

Worked example

The point (5, 4) is enlarged by scale factor −2 with centre (1, 2). Find the coordinates of its image.

From the centre to the point: 5 − 1 = 4 across and 4 − 2 = 2 up.
Multiply by −2: 4 × (−2) = −8 across and 2 × (−2) = −4 up.
Add to the centre: (1 − 8, 2 − 4) = (−7, −2).

Answer: (−7, −2)

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