BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon106 Learners

Last updated on July 5th, 2025

Math Whiteboard Illustration

Rules of Transformations

Professor Greenline Explaining Math Concepts

In mathematics, transformations of a shape refer to the ways we can alter the size or position of the given figure without changing its properties, like shape and structure. The rules of transformations are used to transform the function f(x) to f'(x), and the details can be represented graphically.

Rules of Transformations for Australian Students
Professor Greenline from BrightChamps

What are the Types of Geometric 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. 

Professor Greenline from BrightChamps

Translation

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. 

Professor Greenline from BrightChamps

Reflection

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. 

Professor Greenline from BrightChamps

Rotation

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.   

Professor Greenline from BrightChamps

Dilation

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.  

Professor Greenline from BrightChamps

What are the Rules of Transformations?

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.
Professor Greenline from BrightChamps

What is the Graphical Representation of Rules of Transformation?

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. 

Professor Greenline from BrightChamps

Vertical Transformation

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.   

Professor Greenline from BrightChamps

Horizontal Transformation

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). 
 

 

Professor Greenline from BrightChamps

Flipped Transformation about the x-axis

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).
 

Professor Greenline from BrightChamps

Mirror Transformation about the y-axis

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. 
 

Professor Greenline from BrightChamps

Stretch/Compression of Vertical Transformation

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. 
 

Professor Greenline from BrightChamps

Stretch/Compression of Horizontal Transformation

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. 

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them on the Rules of Transformations

Using the rules of transformations, students can change the position, shape, and size of a graph or figure. However, they may make some mistakes. Here are some common errors and helpful solutions to avoid these mistakes.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusion Between Left and Right Horizontal Shifts 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often get confused with f(x - a) and f(x + a). Remember that f(x - a) shifts the graph by a unit to the right, and f(x + a) shifts it by a unit to the left. Getting this wrong will move the graph in the wrong direction. 
 

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Forgetting the Rules of Vertical Shifts 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

 Keep in mind that the new function will be f(x) + a, if the function f(x) is moved vertically upward by a unit. In this instance, the point (x, y) becomes (x, y + a).    

For example, adding + 2 shifts the graph up by 2 units. 

Hence, the point (1,1) becomes (1, 3).
 

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing Reflections on the x and y Axes 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Always remember that if the function f(x) is reflected over the x-axis, it is written as:
f(x) =  -f(x)

If a function is reflected across the y-axis, then it is written as:
f(x) = f(-x). 

If students mix up these reflections, it will lead them to flip the figure in the wron
 

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Failing to Update the Coordinates of a Point 
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Sometimes, students forget to adjust the actual coordinates of a point (x, y) after the transformation. After a shift upward by a unit, the point (x, y) will become (x, y + a). Similarly, after shifting it by a unit downward, the point (x, y) will become (x, y - a).

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

 Misinterpretation of Horizontal and Vertical Stretch/Compression Transformation
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students should remember that a function f(x) is vertically stretched or compressed to cf(x) using a constant ‘c’.

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

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

 If 0 < c < 1, the new function f(cx) is stretched horizontally.
 If c > 1, the function f(cx) is compressed horizontally.
 
When students forget these rules, they will end up with incorrect transformations. 
 

arrow-right
Professor Greenline from BrightChamps

Real-Life Applications of Rules of Transformations

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. 
     
Max from BrightChamps Saying "Hey"

Solved Examples of Rules of Transformations

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

What is the new function obtained on transforming the function f(x) = x^2 + 4x + 2, by shifting the function by 3 units to the left side?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

x2 + 10x + 23

Explanation

 The given function f(x) = x2 + 4x + 2 needs to be shifted to the left by 3 units.

After the transformation the function f(x) will be changed to f(x + 3) = (x + 3)2 + 4(x + 3) + 2.

Solving this, we get:

(x2 + 6x + 9) + 4x + 12 + 2

=x2 + 6x + 9 + 4x +12 + 2

= x2 + 10x + 23 

Thus, the new function is f(x) = x2 + 10x + 23.  
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 2

What is the new function obtained when the function f(x) =x^2 + 5x + 3 is shifted 2 units to the left?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

x+ 9x + 17

Explanation

 Here, the given function is f(x) =x2 + 5x + 3. To shift the function 2 units to the left, we replace x with x + 2 in the function. 
 

Hence, the new function is:
f(x + 2) = (x + 2)2 + 5(x + 2) + 3 


Now, let us expand the function:

f(x + 2) = (x + 2)2 + 5(x + 2) + 3 
= (x2 + 4x + 4) + (5x + 10) + 3
= x2 + 9x + 17 
 

Therefore, the new function after shifting 2 units to the left is:
    f(x) = x2 + 9x + 17 
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 3

What is the new function obtained when the function f(x) =x2 + 2x + 1 is shifted 2 units to the right?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

x2 - 2x + 1

Explanation

 Here, the given function is f(x) =x2 + 2x + 1. To shift the function 2 units to the right, we replace x with x-2 in the function. 
So, the new function becomes: 


  f(x - 2) = (x -2)2 + 2(x - 2) + 1
  = (x2 - 4x + 4) + (2x - 4) + 1
  = x2 - 4x + 4 + 2x - 4 + 1 
  = x2 - 2x + 1

Thus, the new function after shifting 2 units to the right is: 
  f(x) = x2 - 2x + 1
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 4

What is the new function obtained when the function f(x) = x^2 + 3x + 1 is reflected over the x-axis?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

-x2 - 3x - 1

Explanation

The function f(x) = x2 + 3x + 1 is multiplied by -1 to reflect it over the x-axis.   
 

So, the new function is:
f(x) = -(x2 + 3x + 1)

Now, we can distribute the negative sign:
 f(x) = -x2 - 3x - 1

Thus, the new function after reflection over the x-axis is: 
 f(x) =  -x2 - 3x - 1
 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 5

What is the new function obtained when the function f(x) =x^2 + 4x + 2 is reflected over the y-axis?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

x2 - 4x + 2

Explanation

 Here, the given fraction is f(x) =x2 + 4x + 2. To reflect a function over the y-axis, we replace every instance of x in the function with -x. 

Thus, the new function becomes: 
f(-x) = (-x)2 + 4(-x) + 2
 = x2 + -4x + 2

Therefore, the new function after reflection over the y-axis is:
   f(x) = x2 + -4x + 2
 

Max from BrightChamps Praising Clear Math Explanations
Ray Thinking Deeply About Math Problems

FAQs on Rules of Transformations

1.What is the significance of the rules of transformations?

Math FAQ Answers Dropdown Arrow

2.List the four geometric transformations.

Math FAQ Answers Dropdown Arrow

3.Explain the vertical shift.

Math FAQ Answers Dropdown Arrow

4.Explain the horizontal shift.

Math FAQ Answers Dropdown Arrow

5.Explain the stretch/compression of the vertical transformation.

Math FAQ Answers Dropdown Arrow

6.How can children in Australia use numbers in everyday life to understand Rules of Transformations?

Math FAQ Answers Dropdown Arrow

7.What are some fun ways kids in Australia can practice Rules of Transformations with numbers?

Math FAQ Answers Dropdown Arrow

8.What role do numbers and Rules of Transformations play in helping children in Australia develop problem-solving skills?

Math FAQ Answers Dropdown Arrow

9.How can families in Australia create number-rich environments to improve Rules of Transformations skills?

Math FAQ Answers Dropdown Arrow
Math Teacher Background Image
Math Teacher Image

Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Max, the Girl Character from BrightChamps

Fun Fact

: She loves to read number jokes and games.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
Dubai - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom