Contents

# Transformation|Definition & Meaning

## Definition

In geometry, a **transformation** is when we manipulate or change a shape by either **rotating**(Turning), **flipping**, **translating** (sliding), or **rescaling** it. It is possible to denote these in the form of equations involving the objectâ€™s coordinates. Note that transformations are not limited to the field of geometry and extend to several other basic and advanced branches of **mathematics**.

Figure 1 – Rotation and Reflection Transformation

A new perspective on geometry was introduced in the 19th century by **Felix** **Klein**, called **transformational** **geometry**.Â Through the use of transformations, any image in a coordinate plane can be altered. By applying the principles of transformations to video games, we can better understand their graphics.It is important to know how to identify the **transformations**, understand the rules that govern the transformations of functions, and examine the different types of **transformations** that we may encounter.Â

**Transformations in Maths**

**Transformations** are functions that map to themselves, i.e.,** f: X â†’ X**. An image **X** is formed by transforming the pre-image **X**. As part of this transformation, you can perform any or a combination of operations such as **dilation**, **rotation**, **reflection**, and **translation**.

In **translation**, a function is moved along a specific direction, in rotation, a function is rotated around a point, in reflection, the function is reflected in its mirror image, and in **dilation**, the function is scaled. The concept of transformation in Math explains how two-dimensional figures are moved around a coordinate plane as they rotate.

**What are the Different Types of Transformations?**

The four most common types of transformation are **dilation**, **translation**, **rotation**, and **reflection**. If we define a transformation as rotation about a point, reflection over a line, and translation along a vector, we can rotate about any point, reflect over any line, and translate along any vector.

The images in these cases are rigid transformations where the pre-image is congruent to the image. This type of transformation is also called an **isometric** **transformation**. **Dilation** is non-isometric and can take place at any point. In this case, the image is the same as its pre-image.

Transformation | Function | Result |

Rotation | A pre-image is rotated or turned around an axis | Size and shape remain the same |

Reflection | Produces a mirror image by flipping the pre-image | Size, shape, or orientation remain the same |

Translation | Moves or slides the pre-image. | Size and shape remain the same. Only a change in direction |

Dilation | Stretches or shrinks pre-images | This function allows you to expand or contract the shape |

## Â

**Transformation Rules**

Suppose we have a **function** **f(x)**. When you use a coordinate grid, you can measure the movement by using the **x-axis** and **y-axis**.Â Following are some rules that can be applied to the graph of functions for transformations of functions.

The representation of **transformations** can be either **algebraic** or **graphic**. Algebraic functions often involve transformations. It is possible to obtain a graph by applying the **formula** of the transformation in graphical functions, and by doing so we can move the graph around rather than **tabulating** the coordinate values of the plot to obtain the graph by transforming the basic or parent function.

**Algebraic** **equations** are visualized and learned more easily with the help of transformations.

Figure 2 – Translation and Dilation Transformation

**Translation Transformation**

**Two**–**dimensional** shapes slide when they are translated. Translations move shapes up, down, or sideways without changing their appearance. As a result of the **translations**, the shape is created by a series of points which are all **translated** in the same distance and the same direction at each point.

**How to Describe Translations**

**Translations **are described in terms of **column vectors**. For example **(2/-4),** this means that translate the shape **2** squares to the right and 3 squares down**. (-4/3)** this means that translate the shape **4 **squares to the left and **3 **squares to the up.

In vectors** (a/b)** **a** represents horizontal movement, and **b** represents vertical movement. A positive value of **a** indicates a rightward movement, while a negative value of** b** indicates a leftward movement. When** b** is positive, the movement is upward, and when **b** is negative, the movement is downward.

**Reflection Transformation**

The **axis of reflection** or line of reflection flips all the points of an object. Whenever an image is reflected, it retains the same shape and size as the original. **Transformations** of this type are known as **isometric transformations**. They are facing opposite directions, which means they are laterally inverted.

**Coordination Plane Reflection**

Let’s examine how the **coordinate plane** reflects points and shapes. When the points are reflected over different lines of reflection, it will be helpful to note the pattern of the coordinates.

- The image of
**(a, b)**is the point**(a, -b)**if it is reflected on the**x-axis** - The image of
**(a, b)**is the point**(-a, b)**if it is reflected on the**y-axis** - The image of
**(a, b)**is the point**(b, a)**if its reflection is on the line**y = x**. - An image of
**(a, b)**can be found at the point**(-b, a)**if it is reflected on the line**y = -x.**

**The Geometry Reflection**

An image and its original are congruent when a reflection occurs, which is usually referred to as a **flip**. Performing a geometry reflection requires a line of reflection; the figures are oriented in **opposite** **directions**.

**Rotation Transformation**

When an object is rotated about a **fixed** **point**, it is considered a transformation. **Rotation** is either in a clockwise manner or anti-clockwise. The center of rotation is a fixed point at which rotation occurs. It is the angle of rotation that determines how much rotation is made.

**Dilation Transformation**

A **dilation** is a form of transformation that involves resizing an object by altering its **size**. As a result of dilation, the objects can be made **larger** or **smaller** in size. “**Scale** **factor**” is the term used to describe this transformation.

- It is known as enlargement when a dilation results in a larger image.
- Reduction occurs when a dilation produces a smaller image.

**An Example of a Reflection**

It is possible to combine transformations of shapes, as there are five different types. Reflection and translation of a polygon produce an image that appears apart from its preimage and is mirrored. By enlarging and shearing a rectangle, it appears as a larger parallelogram.

**The point (-3, 2) Where is the point now if it is reflected over the x-axis?**

Figure 3 – Reflection of a point

### Solution

As in reflection, (x, y) = (x, -y), so the point (-3, 2) will be (3, -2) after reflection.

*All images were created with GeoGebra.*