Rules of Transformations

Changes made to the position, size, or shape of different geometric figures, such as squares, triangles, circles, or any other shape, are referred to as geometric transformations. The figure is referred to as the ‘pre-image before the transformation, and “image” after the transformation is complete. 

The geometric transformations can be classified into 4 types, depending on the way we move or change the shapes. 

• Translation 
• Reflection 
• Rotation 
• Dilation 
   

Except for dilation, all other transformations are rigid. They do not alter shape and size before or after the changes. Dilation is a non-rigid transformation that alters the size without changing the shape of a geometric figure. Now, let us take a look at each of the transformations. 

Moving or sliding a geometric figure from one place to another without changing its size or rotating is known as translation. Imagine sliding a paper across the floor; the paper moves without changing its shape or orientation. Naturally, when translation happens, every point of the figure moves exactly like every other point; they all move the same distance in the same direction. 

As the title itself implies, the final image is the reflection of the pre-image. In this geometric transformation, we flip a figure over a line known as the line of reflection. The image appears similar to be the pre-image, but is a reversed mirror image. 

In this geometric transformation, the image is rotated around a fixed point known as the center of rotation. The size or shape of the image will not change; it will remain the same. Rotation can be in two directions: clockwise or anticlockwise. The most common degrees of rotations are 90°, 180°, 270°, and 360°. These angles are common because they indicate quarter, half, three-quarter, and full turns of a rotation.   

In dilation, the shape of the geometric figure remains the same, but the size is changed. Due to this geometric transformation, the image can be bigger or smaller. The fixed point used for resizing the figure is called the center of dilation, and it remains stationary. Every point of the figure is pulled away from or closer to the fixed center, enabling the dilation. To determine how much the image becomes larger or smaller, a scale factor is used.  

Transformations change the orientation, position, or size of a shape by applying rules to a function. In a function y = f(x), the x value is the domain, and the y value is the range of f(x). To transform the domain, we change the input variable, x.

For example, x + a, 5x, x - 3, or x/2. To transform the range, we modify the output, f(x). For example, -f(x), f(x) + 3, f(x) - 2, or f(x) / 4.     

The 6 important rules of transformation are given below:
 

• Vertical transformation: If the function f(x) is shifted upward by a unit, then it will become f(x) + a. Similarly, if it is shifted downward by a unit, the function f(x) becomes f(x) - a.    
   
• Horizontal transformation: In a horizontal transformation, when the input is changed to f(x + a), the graph of f(x) shifts units to the left. When the input is f(x-a), the graph shifts units to the right. 
   
• Reflection over the x-axis: If the output is changed to -f(x), the graph of f(x) is flipped across the x-axis. 
   
• Mirror transformation about the y-axis: To get a mirror transformation over the y-axis, the function f(x) is replaced with f(-x). 
   
• Stretched/compressed vertical transformation: When the function f(x) is changed to cf(x), then the graph is either stretched or compressed vertically. It is stretched if ‘c’ is greater than 1 and compressed if c is less than 1 but greater than 0.
   
• Stretched/compressed horizontal transformation: Change the function to c  f(x), and the graph is stretched or compressed horizontally. The graph is compressed if c > 1, and stretched if 0 < c < 1.

The rules can be graphically represented, and the function’s domain and range can be indicated on the x-axis and the y-axis, respectively. The changes in the x-values and y-values represent the changes in the domain or the range of the function. The change in the function’s graph can be used to depict the transformation rules of the function. 

The graph of the function either shifts upward or downward during the vertical transformation. 
 

• If the function f(x) is moved vertically upward by a unit, the new function will be f(x) + a. In this case, with the new function, the point (x, y) becomes (x, y + a).
    
• If the function f(x) is moved vertically downward by a unit, the new function will be f(x) - a. In this case, with the new function, the point (x, y) becomes (x, y-a). 
   

For example, if the function f(x) = x³ + 2x² is vertically transformed by 3 units, then the new function will be f(x) =  x³ + 2x² + 3.   

The graph of a function moves either to the left or right along the x-axis in the horizontal transformation. 
 

• If x is replaced by x - a, the graph shifts to the right, and the new function becomes f(x - a). In this case, the point (x, y) becomes (x + a, y) of the new function. 
   
• If x is replaced by x + a, the graph shifts to the left, and the new function will be f(x + a). In this case, the point (x, y) becomes (x - a, y) of the new function. 
  
   

For example, if a function f(x) = 1x + 2 is shifted horizontally by 3 units to the left, the new function will be f(x + 3) = 1 (x + 3) + 2. In this case, the point (x, y) becomes (x - 3, y). 
 

The function f(x) is flipped over the x-axis, and it is written as -f(x) since it is the mirror reflection of the function. In this instance, the point (x, y) becomes (x, -y) after the flip transformation.

For example, the function f(x) = 2x + 1 is flipped over the x-axis, and the function becomes:
-f(x) = -(2x + 1).
 

If a function is reflected across the y-axis, then it is written as:

   f(x) = f(-x). 

The new function becomes f(-x), and the point (x, y) will be changed to (-x, y). The function is reflected around the y-axis. 

For example, if a function f(x) = 4x + 1 is reflected over the y-axis, then the new function will be:

f(-x) = 4 (-x) + 1 = -4x + 1. 

The point (x, y) becomes (-x, y) on the new function graph. 
 

Using a constant ‘c’, a function f(x) is stretched or compressed vertically to cf(x). 

• If c > 1, the function cf(x) is stretched vertically.
• If 0 < c < 1, cf(x) is compressed vertically. 

In this case, the point (x, y) becomes (x, cy). Take a look at the given example. 

In the above image, the orange curve represents the original function f(x). The blue curve represents the stretched function cf(x). Look at the point (1, 1) on the orange curve; after stretching, it becomes (1, 3) on the blue curve. 
 

Using a constant ‘c’, the function f(x) is stretched or compressed horizontally to f(cx). 
 

•  If 0 < c < 1, the new function f(cx) is stretched horizontally. 
• If c > 1, the function f(cx) is compressed horizontally. 

In this instance, the point (x, y) becomes (x/c, y) in the new function graph. 
 

In the above image, the orange curve represents the original function f(x), and the blue curve indicates the stretched function f(cx). Look at the point (1, 1) on the orange curve; after a horizontal stretch, it becomes (3, 1) on the blue curve. 

The use of the rules of transformations is not limited to mathematics classes. We use this concept in various situations, and the real-world applications of the rules of transformations are:  
 

• Art and crafts: Artists use the rules of transformations when they work with images and patterns. For example, to replicate a design, they can use reflections and translations to make similar copies of the design. 
   
• Engineering and construction: Engineers can use the transformations, such as translations and scaling, to shift and change the sizes of figures. For instance, if an engineer needs to resize the width or height of doors in a tall building, they can use the rules to alter the size. 
   
• Video gaming: Game developers use transformation rules to flip, move, and shift characters in a game. For example, in games like ‘Super Mario Bros,’ when a vertical translation happens, the character jumps. It can also move to the left or right depending on the horizontal translation.