What are Transformations in Math?

Transformations in Maths are helpful tools that enable us to manipulate and analyse geometric figures by changing their position, size, shape, or orientation without changing fundamental properties such as angles and distances. Transformations enable us to study symmetry, congruence, and similarity, and they have many real-world applications in fields like computer graphics, architecture, engineering, and physics. In this article, we will explore the topic from the perspective of math, from their formulas to rules related to these transformations. 

1.0Definition of Transformations

A mathematical transformation is a function that maps a figure and moves its position, size, or orientation in space. It may be to translate the figure to another place, rotate it, reflect it, scale it, or distort it in some manner. The point is that a transformation is a mapping of points or coordinates from one location to another. It either maintains some characteristics (such as size and shape) or deforms them (such as orientation or position).

2.0Types of Transformations

Transformations can be divided into four main types based on the rotation and results of a geometric figure. These types are: 

1. Translation: A translation is a form of transformation in which each point of the figure travels the same distance in the same direction. That is, the figure is "slid" along a straight line, either horizontally, vertically, or along any diagonal line. A translation doesn't alter the shape, size, or position of the figure, so the figure remains congruent to its original.

2. Reflection: A reflection turns a figure over a line, producing a mirror image of the original figure. The line that the reflection takes place over is referred to as the line of reflection. The new figure is congruent to the original, but the orientation is reversed (as if looking in a mirror).

3. Rotation: A rotation is turning a figure about a fixed point, which is called the centre of rotation. The figure is turned through a given angle in a given direction, either clockwise or counterclockwise. A rotation differs from translation and reflection in that it alters the orientation of the figure.

4. Dilation (Scaling): Dilation is resizing a figure either by increasing or decreasing it. The change happens with respect to a fixed point, which is referred to as the centre of dilation, and a factor called the scale factor.

3.0Rules for Transformations

Transformations in maths follow certain rules and regulations to process and visualise the effects of different operations on geometric figures, which include: 

Translation of a Figure

Translation is the gliding of a shape on the plane without changing its size or orientation. When a figure is translated, it moves all its points an equal distance in the same direction. For example, suppose we translate a point (x, y) by moving it to 6 units to the left and 4 units down. The new position for the translated figure can be expressed as (x–6, y–4). 

Let's say, for example, the point (x,y)(8,9) then the new point will become: (x–6, y–4)(8-6,9-4)(2,5)

Reflection of  a Figure

Reflection happens when a figure is "flipped" over a given line, making a mirror image. The vertices of the figure are reflected so that they are the same distance from the line of reflection but on the opposite side. For Instance, if a point (x,y) is reflected over the y-axis, it becomes (−x,y), whereas if it is reflected over the x-axis, it becomes (x,−y).

For example, a point (4,5) is given and reflected across the y-axis. Then the reflected point becomes (-4, 5). Similarly, for the x-axis, the new point would be (4, -5). 

Rotation of a  Figure

Rotation is a process of revolving a figure about a stationary point, referred to as the centre of rotation, through a given angle. The orientation of rotation can either be clockwise or counterclockwise. These are some rotations of geometric figures: 

• 90° Counterclockwise: The points for this rule are (x,y)(-y,x).
• 180° Rotation: The transformation rule says, (x,y)(y,-x)
• 270° Counterclockwise: The rule followed here is, (x,y)(y,-x)

For example, consider a point (2,3) and rotate it 90° counterclockwise, the coordinates will become (-3,2).

Dilation of a Figure

Dilation is the resizing of a figure by making it bigger and smaller but keeping its shape intact. The size of the figure is changed, not the shape or the angles. Dilation is regulated by a scale factor. A scale factor larger than 1 makes the figure larger, while a scale factor smaller than 1 makes the figure smaller.

• Vertical Dilation: If the equation is y=af(x), the function vertically stretches when a >1 or shrinks when 0 < a < 1.
• Horizontal Dilation: If the equation is, y=af(bx) the function horizontally shrinks when b >1 or stretches when 0 < b < 1.

For example, for a point (x,y), let the dilation factor be 2, the coordinate here will become (2x, 2y). Conversely, a scale factor of ½ would decrease the size of the figure to

$\frac{x}{2},\frac{y}{2}.$

4.0Formula of Transformations

The general formula of transformations for a quadratic equation or any other functions that undergo similar transformations can be expressed as: 

$f(x)=a(bx-h{)}^n+k$

Here, 

• a: The vertical stretch/compression factor. If a>1, the graph is vertically stretched. If 0<a<1, the graph is vertically compressed. If a is negative, the graph is reflected over the x-axis.
• b: The horizontal stretch/compression factor. If b>1, the graph is compressed horizontally. If 0<b<1, the graph is stretched horizontally. If b is negative, the graph is reflected over the y-axis.
• h: The horizontal shift. The graph is shifted h units to the right if h>0, and h units to the left if h<0.
• k: It is the vertical shift. The graph is shifted k units up if k>0, and k units down if k<0.