Moving shapes
A transformation moves or changes a shape. The original is the pre-image; the result is the image, often labelled with primes (A becomes A′).
This lesson covers three transformations that keep size and shape the same:
| Transformation | Everyday word | Changes |
|---|---|---|
| Translation | slide | position |
| Reflection | flip | position and orientation (mirror image) |
| Rotation | turn | position and direction it faces |
Translations
Every point moves the same distance in the same direction.
On a grid, describe it with a rule like "3 right, 2 down", written (x, y) → (x + 3, y − 2).
Example: A(1, 4) → A′(4, 2).
Reflections
Every point flips across a line of reflection to the same distance on the other side.
| Mirror line | Rule |
|---|---|
| x-axis | (x, y) → (x, −y) |
| y-axis | (x, y) → (−x, y) |
| y = x | (x, y) → (y, x) |
Example: Reflect B(3, 5) in the x-axis → B′(3, −5).
A reflection reverses the orientation: if the vertices went A, B, C clockwise, the image goes counterclockwise.
Rotations
The shape turns around a centre of rotation by a given angle and direction (clockwise or counterclockwise).
Rotations about the origin:
| Rotation | Rule |
|---|---|
| 90° counterclockwise | (x, y) → (−y, x) |
| 180° | (x, y) → (−x, −y) |
| 90° clockwise (= 270° counterclockwise) | (x, y) → (y, −x) |
Example: Rotate C(2, 3) 90° counterclockwise about the origin → C′(−3, 2).
Tip: tracing paper makes rotations easy. Trace the shape, pin the centre with your pencil and turn the paper.
Rigid transformations
All three are rigid: lengths and angles don’t change, so the image is congruent to the original.
Combining transformations
You can do one transformation after another. A reflection in the x-axis followed by a reflection in the y-axis is the same as a 180° rotation about the origin.
Worked examples
Example 1: Translate (−2, 1) by 5 right and 3 up.
(3, 4).
Example 2: Reflect (4, −1) in the y-axis.
(−4, −1).
Example 3: Rotate (5, 2) by 180° about the origin.
(−5, −2).
Common mistakes
- Mixing up x and y signs in reflections.
- Rotating the wrong direction.
- Forgetting the centre of rotation when it isn’t the origin.
Try it yourself
- Translate (0, 6) by 4 left and 2 down.
- Reflect (−3, 7) in the x-axis.
- Rotate (1, 4) 90° clockwise about the origin.
- Which transformation produces a mirror image?
Answers: 1) (−4, 4) 2) (−3, −7) 3) (4, −1) 4) reflection
Quick recap
- Translation = slide; reflection = flip; rotation = turn.
- They keep size and shape: the image is congruent.
- Use coordinate rules to transform points on a grid.