# Types of Transformations

## Big Picture

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.

## Key Terms

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

## Rigid Transformations

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 $$A’$$.  $$A \rightarrow A’$$

## Reflections

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

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

### Reflection Over Horizontal Lines

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).

### Reflection Over Vertical Lines

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).

### Reflections over y=x and y=-x

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).

# Types of Transformations Cont.

## Rotations

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).

## Translations

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>.

### Translations in the Coordinate Plane

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>.

## Dilations

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.