Quick revision notes · Mathematics 4024 · Chapter 29 of 35
Transformations
Reflections, rotations, translations and enlargements: drawing, describing and combining them.
5 sectionsWorked examples
Common mistakes
At a glance
Reflections, rotations, translations and enlargements: drawing, describing and combining them.
1Reflection
- Every point moves to the same distance on the other side of the mirror line.
- Describe it with the equation of the mirror line: x = 2, y = −1, y = x, y = −x.
- In the x-axis: (x, y) → (x, −y). In the y-axis: (x, y) → (−x, y). In y = x: (x, y) → (y, x).
Worked example
Reflect the point (2, 7) in the line y = x.
- Swap the coordinates.
(7, 2)
2Rotation
- Describe with centre, angle and direction (180° needs no direction).
- Use tracing paper: hold the centre with your pencil and turn.
- About the origin: 90° anticlockwise (x, y) → (−y, x); 90° clockwise (x, y) → (y, −x); 180° (x, y) → (−x, −y).
Worked example
Rotate (4, 1) through 90° anticlockwise about the origin.
- (x, y) → (−y, x).
(−1, 4)
3Translation
- Every point moves the same distance in the same direction.
- Describe with a column vector: top number = right (+) / left (−); bottom number = up (+) / down (−).
Worked example
Translate (−2, 3) by the vector with top 5 and bottom −4.
- x: −2 + 5 = 3.
- y: 3 − 4 = −1.
(3, −1)
4Enlargement
- Describe with centre and scale factor.
- Distance from the centre × scale factor gives the new position.
- Scale factor between 0 and 1 makes it smaller; negative puts the image on the other side of the centre, upside down.
- Find the centre: join matching corners of the shape and image and extend the lines to where they meet.
How to do it
- Find the vector from the centre to a corner.
- Multiply it by the scale factor.
- Add the result to the centre.
- Repeat for each corner.
Worked example
Triangle A has vertices (1, 1), (2, 1), (1, 3). Triangle B has (3, 3), (5, 3), (3, 7). Describe the transformation.
- Lengths double: scale factor 2.
- Joining matching corners meets at (−1, −1).
Enlargement, scale factor 2, centre (−1, −1)
5Describing and combining
- Name only one single transformation when asked — two answers score nothing.
- Use the exact words: reflection, rotation, translation, enlargement.
- Combinations: do the first, then the second, then describe the single transformation that does both.
- Reflection in the x-axis then the y-axis = rotation of 180° about the origin.
| Transformation | You must give |
|---|---|
| Reflection | mirror line equation |
| Rotation | centre, angle, direction |
| Translation | column vector |
| Enlargement | centre, scale factor |
Worked example
Reflect in x = 1, then in x = 4. What single transformation is this?
- A point at x = 0 → x = 2 → x = 6: it moved 6 right; every point moves the same.
Translation by the vector with top 6 and bottom 0
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| 'Turn' or 'move' as the name | Use rotation, translation, reflection, enlargement |
| Reflecting in y = x by changing the signs | Swap the coordinates |
| Enlarging from the origin when the centre is elsewhere | Measure every distance from the given centre |
| Giving two transformations when asked for one | Find the single transformation |
★In the exam
- Check your image with tracing paper.
- Label images as instructed (A, B, C) — unlabelled shapes can lose marks.
- For enlargements, check the image size: every length × scale factor.