Last updated on July 5th, 2025
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.
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.
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:
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.
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.
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).
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).
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.
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.
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:
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?
x2 + 10x + 23
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.
What is the new function obtained when the function f(x) =x^2 + 5x + 3 is shifted 2 units to the left?
x2 + 9x + 17
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
What is the new function obtained when the function f(x) =x2 + 2x + 1 is shifted 2 units to the right?
x2 - 2x + 1
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
What is the new function obtained when the function f(x) = x^2 + 3x + 1 is reflected over the x-axis?
-x2 - 3x - 1
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
What is the new function obtained when the function f(x) =x^2 + 4x + 2 is reflected over the y-axis?
x2 - 4x + 2
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
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.
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