Rules Of Transformations
Rules of transformations help in transforming the function f(x) to a new function f'(x), because of the change in its domain or the range values. The function can be transformed vertically, horizontally, or it can be stretched or compressed, with the help of these rules of transformation. The rules of transformations can also be represented graphically.
Let us learn the rules of transformation, its graphical representations, with the help of examples, FAQs.
1.  What Are The Rules of Transformations? 
2.  Graphical Representation of Rules of Transformations 
3.  Examples on Rules of Transformations 
4.  Practice Questions 
5.  FAQs on Rules of Transformations 
What Are The Rules Of Transformations?
Function transformation rules help in transforming the function based on the change of either the domain or the range of the function. For a function y = f(x), the domain is the x value and the range is the f(x) or the y value of the function. The transformation of a function to change the domain of the function is to replace the x value with a new value of x such as x + a, 5x, x  3, x/2. Similarly the change of the range of the function is possible by replacing f(x) with f(x), f(x) + 3, f(x)  2, f(x)/4.
Rules of transformation represent these changes in the x or y values of the function. The six important rules of transformation are as follows.
 Vertical Transformation : The function f(x) is shifted up by 'a' units upwards for the function f(x) + a. And the function f(x) is shifted vertically doward
 Horizontal Transformation: The function f(x) is shifted towards the right for the new function f(x + a). And it is transformed towards the left on replacing f(x) with f(x  a).
 Flipped Transformation about the Xaxis: The function f(x) is flipped about the xaxis by writing it as f(x).
 Mirror Transformation About the Yaxis: The function f(x) is changed to f(x) to obtain a mirror transformation about the yaxis.
 Stretched Vertical Transformation: The function f(x) is stretched vertically if a constant 'c' transforms it to cf(x). If c >1 then it is stretched vertically, and if 0 < c < 1 then it is compressed vertically.
 Stretched Horizontal Transformation: The function f(x) is stretched horizontally if a constant 'c' transforms it to f(cx). If c > 1 then it is stretched horizontally, and if 0 < c < 1 then it is compressed horizontally.
Graphical Representation of Rules of Transformation
The rules of transformation for functions can be represented graphically across the coordinate axis. The domain of the function  the x value can be represented along the xaxis, and the range of the function  the y value can be represented along the yaxis. The change in the domain or the range of the function, can be understood by the change in the xvalues and the yvalues. The function transformation rules can be shown as change in the graph of the function in the coordinate axis.
Vertical Transformation:
The function f(x) has been vertical shifted upwards by 'a' units for the function f(x) + a. The point (x, y) with reference to the function f(x), has been changed to (x, y + a) for the new function f(x) + a. Also the function f(x) has been vertically shifted downwards by 'a' units for the function f(x)  a. The point (x, y) with reference to f(x, y), has been changed to (x, y  a), for the new transformed function f(x, y  a).
As an example the function f(x) = x^{3}+ 2x^{2} has been vertically transformed by 4 units for f(x) = x^{3} + 2x^{2} + 4., and a point (x, y) with reference to the earlier function is not represented as (x, y + 4).
Horizontal Transformation
The function f(x) is horizontally shifted about the xaxis by writing x as x + a and the function is transformed as f(x + a). The point (x, y) with reference to the earlier function f(x), is now transformed to (x + a, y) with reference to the new function f(x + a). Also the function f(x) is horizontally shifted towards the left with reference to the xais by writing x as x  a, and the function is transformed as f(x  a). The point (x, y) with reference to the earlier function f(x) is now transformed to (x  a, y) with reference to the new function f (x  a).
Let us consider a function f(x) = 2x + 3, to be shifted horizontally about the xaxis, by 2 units and the new function would be f(x + 2) = 2(x + 2) + 3. And a point (x, y) with reference to the earlier function is now represented as (x + 2, y).
Flipped Transformation about the xaxis
The function f(x) is flipped transformed about the xaxis by writing it as f(x) and it is a mirror reflection of the function f(x) about the xaxis. The point (x, y) with reference to the function f(x) is now transformed to (x, y).
An an example the function f(x) = 3x +2 is transformed to is flipped across the xaxis and is represented as f(x) = (3x + 2). The point (x, y) has not been changed to (x, y).
Mirror Transformation about the yaxs
The function f(x) is mirror transformed or reflected about the yaxis by representing it as f(x). The point (x, y) with reference to the function f(x) is now changed to (x, y) with reference to the new function f(x). The function is observed to be reflected about the yaxis.
As an example let us take a function f(x) = 5x + 1, which is reflected about the yaxis to obtain f(x) = 5(x) + 1 = 5x + 1. The point (x, y) is now changed to (x, y).
Stretched vertical Transformation
The function f(x) is stretched vertically by using a constant 'c' and it is now written as cf(x) to represent a new stretched function. The function cf(x) is stretched vertically if c > 1, and it is reduced if 0 < c < 1. The point (x, y) with reference to the original function is now changed to (x, cy).
The blue curve in the graph shows the stretched or compressed function, and the red curve shows the original function.
Stretched Horizontal Transformation
The function f(x) can be stretched horizontally by replacing x with cx in the function to obtain f(cx). The function is stretched horizontally if c > 1, and it is shrunk if 0 < c < 1. The point (x, y) with reference to the original function f(x) is now changed to (cx, y).
The blue curve shows the stretched or compressed function along the xaxis and the red curve shows the initial function in the graph.
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Examples on Rules Of Transformations

Example 1: What is the new function obtained on transforming the function f(x) = x^{2} + 5x + 3, by shifting the function by 3 units across the positive xaxis?
Solution:
The given function is f(x) = x^{2} + 5x + 3. The function is to be shifted by 3 unit across the positive xaxis.
Applying the rules of transformation, any point (x, y) on this given function transforms to (x + 3, y). Now we substitutes x = x + 3 in the function f(x), to obtain the new transformed function f'(x).
f'(x) = (x + 3)^{2} + 5(x + 3) + 3
f'(x) = x^{2} + 6x + 9 + 5x + 15 + 3
f'(x) = x^{2} + 11x 27
Therefore the new transformed function is f(x) = x^{2} + 11x 27.

Example 2: What is the new function obtained on transforming the function f(x) = 2x^{4}  5x^{3} + x^{2}  5x + 7, by stretching it horizontally by a factor of 0.3, along the positive xaxis?
Solution:
The given function is f(x) = 2x^{4}  5x^{3} + x^{2}  5x + 7. The objective is to stretch this function horizontally by a factor of 0.3.
Applying the rules of transformation, the point (x, y) for the given function f(x), transforms to (0.3x, y). Here we substitute x = 0.3x in the function f(x), to obtain the transformed function f'(x).
f'(x) = 2(0.3x)^{4}  5(0.3x)^{3} + (0.3x)^{2}  5(0.3x) + 7
f'(x) = 2(0.0081)x^{4}  5(0.027)x^{3} + (0.09)x^{2}  1.5x + 7
f'(x) = 0.162x^{4}  0.135x^{3} + 0.09x^{2}  1.5x + 7
Thus the transformed function is f'(x) = 0.162x^{4}  0.135x^{3} + 0.09x^{2}  1.5x + 7.
FAQs on Rules Of Transformations
What Are The Rules Of Transformations In Functions?
The rules of transformations are useful for transforming a given function f(x) into a new function f'(x). These transformations are a result of the change in the domain and range of the original function. The rules of transformation can be represented graphically to show change the shift in the curve of the function f(x).
What Are the Four Important Rules Of Transformations?
The four important rules of transformation are vertical transformation, horizontal transformation, stretched transformation, compressed transformation. The details of each of the transformations functions rules are as follows.
 Vertical Transformation: The function f(x) is transformed vertically to f(x) + a, or f(x)  a.
 Horizontal Transformation: The function f(x) is transformed horizontally to f(x + a), or f(x  a).
 Stretched Transformation: The function f(x) is transformed horizontally as f(cx), and vertically as cf(x). Here 'c' is a constant and c > 1.
 Compressed Transformation: The function f(x) is compressed horizontally as f(cx), and vertically as cf(x). Here 'c' is a constant such that 0 < c < 1.
What Are The Formulas For The Rules Of Transformations?
There are different formulas for different rules of transformation. For vertically transformation the function f(x) is transformed to f(x) + a or f(x)  a. For horizontal transformation the function f(x) is transformed to f(x + a) or f(x  a). Further for stretched or compressed transformation is it f(cx) or cf(x). For stretched transformation c > 1 and for compressed transformation 0 < c < 1.
How Do We Apply The Rules Of Transformations?
The rules of transformations are applicable by changing the coordinates. As an example for horizontal transformation of a function f(x) by 'a' units, towards the positive xaxis, the point (x, y) is replaced by (x + a, y). Similarly, the coordinates can be applied for different transformations.
What Are The Applications Of Rules Of Transformations?
The rules of transformation are applicable if the domain or the range of the functions are changed. These rules can be used to shit or change the graph of the given function in the coordinate axis.
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