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Translation is taught in the note for Geometrical Transformations. Read it below and look for the section on this topic.

Geometrical Transformations

11 min read

Moving shapes

A transformation changes the position or size of a shape. The original is the object and the result is the image. Reflection, translation and rotation keep the size (the image is congruent). Enlargement changes the size (the image is similar).

The four transformationsFour panels. Reflection flips a shape in a mirror line. Rotation turns it about a centre. Translation slides it without turning. Enlargement makes it bigger or smaller from a centre by a scale factor. Blue is the object and red is the image.mirror lineReflectioncentreRotation 90°TranslationEnlargement ×2centre

Reflection

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

Reflecting in the x-axis changes (x, y) into (x, −y).
Reflection rules
Mirror line(x, y) becomes
x-axis(x, −y)
y-axis(−x, y)
line y = x(y, x)

Translation

A translation slides every point by the same amount. We describe it with a vector : units across and units up.

Try itTranslate a triangle

Choose the vector. The grey-blue triangle is the object; the green one is the image.

Rotation

A rotation turns a shape about a fixed point (the centre) through an angle. Anticlockwise is the positive direction. A turn of 90° about the origin sends to .

Try itRotate about the origin

Choose the angle of turn. Notice the image is always the same size.

Enlargement

An enlargement multiplies every distance from a fixed centre by the scale factor . From the origin, becomes . If the image is smaller.

Each point moves twice as far from the origin along the same straight line.
Worked exampleFind the image

Triangle with vertices A(1, 2), B(3, 2), C(1, 5) is reflected in the y-axis. Write the image vertices.

  1. Reflection in the y-axis: (x, y) → (−x, y).
  2. Apply to each vertex.

Answer: A′(−1, 2), B′(−3, 2), C′(−1, 5).

Check yourself Point (4, −3) is reflected in the x-axis. Where is the image?
Check yourself Which transformation changes the size of a shape?

Key terms

Object / image
The shape before and after a transformation.
Vector
Numbers (a, b) describing a movement across and up.
Centre of rotation
The fixed point a shape turns around.
Scale factor
The number lengths are multiplied by in an enlargement.

Summary

  • Reflection, translation and rotation keep size; enlargement changes it.
  • Reflect in the x-axis: (x, −y). In the y-axis: (−x, y). In y = x: (y, x).
  • A translation slides by a vector; a rotation needs a centre and an angle.
  • Enlargement from the origin: (x, y) → (kx, ky).
Reflection Enlargement