Transformations

Maths · Transformations and vectors. 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 starting shape is the object and the result is the image. There are four kinds: reflection (a flip in a mirror line), rotation (a turn about a centre), translation (a slide given by a vector) and enlargement (a change of size from a centre).

Reflection, rotation and translation keep the shape and size the same, so the image is congruent to the object. Enlargement changes the size, so the image is similar to the object.

To describe a transformation fully, name it and give every detail it needs: the mirror line; the centre, angle and direction; the vector; or the centre and scale factor.

Rules to remember

Reflection: in the x-axis (x, y) → (x, −y); in the y-axis (x, y) → (−x, y); in y = x (x, y) → (y, x); in y = −x (x, y) → (−y, −x)
Reflection in the line x = a: (x, y) → (2a − x, y). In the line y = b: (x, y) → (x, 2b − y)
Rotation about the origin: 90° anticlockwise (x, y) → (−y, x); 90° clockwise (x, y) → (y, −x); 180° (x, y) → (−x, −y)
Translation by the vector (a, b): (x, y) → (x + a, y + b)
Enlargement, scale factor k: image = centre + k × (vector from centre to point). Lengths × k, area × k²
Scale factor between 0 and 1 makes the image smaller; a negative scale factor puts the image on the opposite side of the centre
The inverse of a rotation is the same angle about the same centre in the opposite direction

Common mistake

For a rotation or enlargement about a centre that is not the origin, students apply the origin rule straight to the coordinates. First find the vector from the centre to the point, transform that vector, then add it back to the centre.

Worked example

The point (5, 3) is rotated 90° anticlockwise about the point (2, 1). Find its image.

Vector from the centre to the point: (5 − 2, 3 − 1) = (3, 2).
Rotate the vector 90° anticlockwise, (x, y) → (−y, x): (3, 2) → (−2, 3).
Add to the centre: (2 + (−2), 1 + 3) = (0, 4).

Answer: (0, 4)

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