About the transformations calculator
This is the full graphing calculator with transformations built in. Type points such as P = (1, 1), (4, 1), (2, 3), then press Transform… under them to add a translation, reflection, rotation, dilation or shear. The image appears as a new line, for example Q = rotate(P, 90), with its vertices labeled A′, B′, C′ and the coordinate rule written out.
Transformations can be chained (R = translate(Q, 3, -2)) and everything updates when you drag the original points. You can also type the functions directly: translate(P, a, b), rotate(P, degrees) or rotate(P, degrees, (h, k)), reflect(P, "y = x"), dilate(P, k) and shear(P, k, "x").
Formulas
| Translation by (a, b) | (x, y) → (x + a, y + b) |
| Reflection over the x-axis | (x, y) → (x, −y) |
| Reflection over the y-axis | (x, y) → (−x, y) |
| Reflection over y = x | (x, y) → (y, x) |
| Reflection over y = −x | (x, y) → (−y, −x) |
| Rotation 90° counterclockwise | (x, y) → (−y, x) |
| Rotation 180° | (x, y) → (−x, −y) |
| Rotation 90° clockwise | (x, y) → (y, −x) |
| Rotation by θ about the origin | (x, y) → (x cos θ − y sin θ, x sin θ + y cos θ) |
| Dilation by k about the origin | (x, y) → (kx, ky) |
Frequently asked questions
What is the rule for a 90° rotation?
Counterclockwise about the origin: (x, y) → (−y, x). Clockwise: (x, y) → (y, −x).
How do I reflect a point over the line y = x?
Swap the coordinates: (x, y) → (y, x). The point (2, 5) becomes (5, 2).
Which transformations keep a shape congruent?
Translations, reflections and rotations (rigid motions) keep size and shape. A dilation keeps the shape but changes the size, so the image is similar. A shear changes the shape.
How do I rotate about a point that is not the origin?
Choose “Another point” as the center. The calculator shifts the center to the origin, rotates, and shifts back.