## Transformations – Notations and Formulas

A **transformation** is an operation that maps an initial figure (pre-image) onto a final image called an image.

### Line Reflections

A reflection is a **flip**. It is an** opposite isometry** – the image does not change size but the lettering is reversed.

### Point Reflections

A **point reflection** exists when a figure is built around a single point called the center of the figure. It is a **direct isometry**.

**Reflection in the Origin:** While any point in the coordinate plane may be used as a point of reflection, the most commonly used point is the origin.

**P(x,y)→P’(-x,-y) **or** r _{origin}(x,y) = (-x,-y)**

### Rotations (assuming center of rotation to be the origin)

A **rotation** turns a figure through an angle about a fixed point called the** center**. A positive angle of rotation turns the figure **counterclockwise**, and a **negative angle** of rotation turns the figure in a clockwise direction. It is a **direct isometry**.

### Dilations

A **dilation** is a transformation that produces an image that is the **same shape** as the original, but is a **different size**. NOT an isometry. Forms similar figures.

**Dilation of scale factor k:**

The center of the dilation is assumed to be the origin unless otherwise specified.

**D _{k }(x, y) = (kx, ky)**

### Translations

A translation “slides” an object a fixed distance in a given direction. The original object and its translation have the same shape and size, and they face in the same direction. It is a direct isometry.

**Translation of h, k:**

**T _{h,k}(x, y) = (x+h, y+k)**