Transformations move and modify geometric shapes. There are several types of transformations that all transform figures in different ways. These transformations can be rigid (isometric) or non-rigid.

**Transformation: **An operation that moves, flips, or changes a figure to create a new figure.

**Rigid: **A transformation that preserves size and shape.

**Isometry: **Another word for rigid transformation, a transformation that does not change the shape or size of a figure.

**Non-Rigid: **A transformation that does not preserve size

**Preimage: **The original figure before a transformation.

**Image: **The figure after a transformation.

**Vector: **A quantity that has direction and size

Specifically, **rigid transformations** (or congruence transformations) transform a figure without changing its size or shape.

The three types of rigid transformations are:

- Reflection: A transformation that turns a figure into its mirror image by flipping it over a line.
- Rotation: A transformation that turns a figure around a fixed point to create an image.
- Translation: A transformation that moves every point in a figure the same distance in the same direction.

To distinguish between the **preimage** and the** image, **primes are used to label the image. The arrow \(\rightarrow\) is used to describe a transformation

- Example: If the preimage is A, then the image would be \(A’\). \(A \rightarrow A’\)

Another way to describe a reflection is a “flip.”

- Image is the same distance away from the reflection line as the preimage.

x-coordinates stay the same while y-coordinates change

- If (x, y) is reflected over the x-axis, then the image is (x, -y).
- If (x, y) is reflected over y = b, then the image is (x, 2b - y).

x-coordinates change while y-coordinates stay constant

- If (x, y) is reflected over the y-axis, then the image is (-x, y).
- If (x, y) is reflected over y = b, then the image is (x, 2b - y).

Any line can be a line of reflection, but the lines y = x and y = -x are special cases.

- If (x, y) is reflected over the y = x, then the image is (y, x).
- If (x, y) is reflected over y = -x, then the image is (-x, -y).

Same as a “reflection in the origin,” a figure is spun around the origin.

Rotation of 90°

- If (x, y) is rotated 90° around the origin, then the image will be (-y, x).

Rotation of 180°

- If (x, y) is rotated 180° around the origin, then the image will be (-x, -y).

Rotation of 270°

- If (x, y) is rotated 270° around the origin, then the image will be (y, -x).

Vectors can be used to represent a translation.

- A vector has a length and a direction.
- A vector going from point A to B is labeled \(\overrightarrow{AB}\). A is the initial point and B is the terminal point. The terminal point has the arrow pointing towards it.
- The component form of the vector combines the horizontal and vertical distances traveled.
- In the diagram above, the component form of \(\overrightarrow{AB}\) is <3, 7>.

A translation can be described by the notation: \((x, y) \rightarrow (x + a, y + b)\), where the point (x, y) is translated horizontally a units and vertically b units.

The translation rule for this figure is: (x, y) \(\rightarrow\) (x + 6, y + 4).

- The vector for this translation is <6, -4>.

A dilation is a **non-rigid transformation** that preserves shape but not size.

- Has a center and scale factor
- A center is the point of reference for the dilation
- The scale factor relates how much a figure stretches or shrinks
- Enlargement or expansion: scale factor > 1
- Reduction or contraction: 0 < scale factor < 1
- Always produce a similar shape to the original

In a coordinate plane, a non-rigid transformation transforms \((x, y) \rightarrow (kx, ky)\), where k is the scale factor. If k < 0, then the preimage is rotated around 180° about the center of dilation.