In Function Notation, a function takes an input and gives exactly one output. A transformation is a function whose inputs and outputs are points: it sends each point $P$ of the plane to an image point $T(P)$, usually written $P'$. The slides, turns and flips from Rotations and the enlargements from Dilations are all transformations.
| Rule $T(x, y)$ | What it does | Keeps every distance? |
|---|---|---|
| $(x + 3, y - 2)$ | translation: right 3, down 2 | yes |
| $(-x, y)$ | reflection over the $y$-axis | yes |
| $(-y, x)$ | rotation $90^\circ$ counterclockwise about the origin | yes |
| $(2x, 2y)$ | dilation, scale factor 2, center the origin | no: all lengths double |
| $(x, 3y)$ | vertical stretch | no: the shape changes too |
Por qué funciona
A rigid motion is a transformation that keeps every distance the same: for any two points, $P'Q' = PQ$. Keeping distances keeps everything else too: side lengths, angle measures, area, straightness. To show a rule is not rigid, one pair of points whose distance changes is enough.
Idea clave
A transformation is a function on points: $T(P) = P'$. Rigid motions (translations, reflections, rotations) keep every distance and angle; dilations and stretches do not.
Ejemplo resuelto: Applying a rule
$T(x, y) = (y, -x)$. Find the images of $A(1, 2)$, $B(4, 2)$ and $C(1, 5)$.
- Put each point into the rule: the new $x$ is the old $y$; the new $y$ is the opposite of the old $x$.
- $A' = (2, -1)$, $B' = (2, -4)$, $C' = (5, -1)$.
- This rule turns the plane $90^\circ$ clockwise about the origin.
Respuesta: $A'(2, -1)$, $B'(2, -4)$, $C'(5, -1)$
Ejemplo resuelto: Is it rigid?
Is $T(x, y) = (2x, y)$ a rigid motion?
- Test two points: $A(0, 0)$ and $B(1, 1)$, with $AB = \sqrt{2}$.
- Images: $A' = (0, 0)$, $B' = (2, 1)$, so $A'B' = \sqrt{2^2 + 1^2} = \sqrt{5}$.
- $\sqrt{5} \ne \sqrt{2}$: the distance changed.
Respuesta: No. It stretches the plane sideways, so it is not rigid.
Ejemplo resuelto: Working backward
$T(x, y) = (x - 4, y + 1)$ and $T(P) = (2, 5)$. Find $P$.
- $T$ subtracts 4 from $x$ and adds 1 to $y$; undo both.
- $P = (2 + 4, 5 - 1) = (6, 4)$. Check: $T(6, 4) = (2, 5)$.
Respuesta: $P = (6, 4)$
Error común
Thinking any transformation that keeps the shape is rigid. A dilation keeps the shape but changes the size, so it is not rigid. A stretch like $(x, 3y)$ changes both size and shape. Check by computing one length before and after: a rigid motion never changes it.