The four transformations
A transformation moves every point of a shape (the pre-image) to a new position (the image). Image points are usually labeled with a prime, so A moves to A′.
Translation
Slide every point the same distance in the same direction. Moving a units horizontally and b units vertically follows the rule (x, y) → (x + a, y + b).
Rotation
Turn the shape around a fixed center. Positive angles are counterclockwise unless stated otherwise. About the origin:
| Rotation (about the origin) | Rule |
|---|---|
| 90° counterclockwise (= 270° clockwise) | (x, y) → (−y, x) |
| 180° | (x, y) → (−x, −y) |
| 270° counterclockwise (= 90° clockwise) | (x, y) → (y, −x) |
To rotate about another point (h, k), shift so the center is at the origin, rotate, then shift back. The calculator shows that rule when you move the center.
Reflection
Flip the shape over a mirror line. Each point and its image are the same distance from the line, on opposite sides, and the segment joining them is perpendicular to the line.
| Reflect across | Rule |
|---|---|
| x-axis | (x, y) → (x, −y) |
| y-axis | (x, y) → (−x, y) |
| y = x | (x, y) → (y, x) |
| y = −x | (x, y) → (−y, −x) |
| x = a | (x, y) → (2a − x, y) |
| y = b | (x, y) → (x, 2b − y) |
Dilation
Stretch or shrink the shape from a center by a scale factor k. About the origin the rule is (x, y) → (kx, ky). If k > 1 the image is an enlargement, if 0 < k < 1 it is a reduction, and a negative k also turns the shape through 180°. Lengths are multiplied by |k| and the area by k².
FAQ
Which transformations are rigid motions?
Translations, rotations and reflections. They keep every length and angle, so the image is congruent to the original. Dilations keep angles but change lengths, so the image is similar.
How can I tell a reflection from a rotation?
Check the order of the vertices. A reflection reverses it (clockwise becomes counterclockwise), while a rotation keeps it. The calculator reports this as the shape's orientation.
Can I use these graphs in a worksheet?
Yes. Set up the transformation and press Download PNG for a high-resolution image. The page link also saves your shape and settings, so you can share it with a class.