Combinations of transformations
OCR J560 tests the four basic transformations (rotation, reflection, translation, enlargement) and asks students to combine them — and to identify properties that stay the same (invariant) under each.
The four transformations
| Type | What it does | Specified by |
|---|---|---|
| Translation | Slides the shape | Vector (a, b) |
| Reflection | Flips across a line | Mirror line equation |
| Rotation | Turns about a point | Centre + angle + direction |
| Enlargement | Scales from a centre | Centre + scale factor |
What stays the same (invariant)
| Property | Translation | Reflection | Rotation | Enlargement |
|---|---|---|---|---|
| Shape | ✓ | ✓ | ✓ | ✓ |
| Size | ✓ | ✓ | ✓ | ✗ |
| Angles | ✓ | ✓ | ✓ | ✓ |
| Orientation | ✓ | ✗ | ✓* | ✓** |
| Parallelism | ✓ | ✓ | ✓ | ✓ |
*Rotation preserves orientation if seen as same handedness. **Enlargement preserves orientation unless scale factor is negative.
Combinations
When combining transformations, the order matters. "Rotate 90° about (0,0), then translate by (3,1)" usually gives a different result from doing the translate first.
A composition of two transformations of the same type can sometimes be replaced by a single equivalent transformation:
- Two reflections in parallel mirrors = a translation.
- Two reflections in intersecting mirrors = a rotation about the point of intersection.
- Two rotations about the same centre = a single rotation (angles add).
- Two enlargements about the same centre = a single enlargement (scale factors multiply).
Identifying a single transformation
If you see "before" and "after" on a grid:
- Check sizes — different = enlargement (need centre & SF).
- Same size, mirror image — reflection (find the mirror line).
- Same size, turned around — rotation (find centre, angle, direction).
- Same size, slid over — translation (find the vector).
OCR mark scheme conventions
- "Describe fully the single transformation..." needs ALL the parameters:
- Translation: type + vector.
- Reflection: type + mirror line equation.
- Rotation: type + centre + angle + direction (clockwise/anticlockwise).
- Enlargement: type + centre + scale factor.
- 1 mark for the type, 1–2 marks for the parameters, total typically 2–3 marks per transformation.
⚠Common mistakes
- Saying "reflection" without giving the mirror line.
- Forgetting the direction of rotation (clockwise vs anticlockwise).
- Confusing the order in compositions.
- Treating "enlargement" as always making bigger — fractional or negative SFs are still enlargements.
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