Math

Transformations of Functions

A general function of x is defined as y=f(x). There are some basic transformations of functions which are explained below.

 

1) f(x)+c, where c is a constant:  where c>0, moves  f(x) c units upward and c<0  moves f(x) c units downward.

 

Example: Sketch the graph of 

We start with the graph of and shift the graph 4 units upward: 

 

2) f(x+c), where c is a constant: c>0 shifts f(x) c units to the left and c<0 shifts f(x) c units to the right. 

 

Example: Sketch the graph of 

We start with the original function which is and shift it 6 units to the right:

 

 

Example: Sketch the graph of  

The original function is and the function after transformation is . The constant number in the transformation is which means that it stretches the original function along the x axis.

 

 

Example: Sketch the graph of 

The original function is and the constant coefficient is which stretches the graph in the y direction.

 

 

Example: Sketch the graph of 

 

 

Example: Sketch the graph of 

 

Example: Sketch the graph of 

This example includes a series of transformations. The original function is and we have the following transformations:

• The function is multiplied by -2, which reflects the function about the x axis and stretches the function in the y direction.

• The original function is shifted 2 units to the right.

The following graph shows the original function and all the transformations that results in the function given in this example:

 

Example: Sketch the graph of 

The original function is  and we have the following transformations:

• The original function is shifted 7 units to the left.

• The original function is compressed in the y direction.

The original function and the transformations are shown in the diagram below:

Transformation using points on a graph

As the transformations are combined, a graph can be transformed point by point for simpler graphs.  For the transformation of the following form, the individual points of the graph can be transformed accordingly

In this example, take the graph above and apply the following transformations

*Equations will be replaced to match above at a later date*

• Transformation 1: Compression in the x-axis by a factor of 3
  • 
• Transformation 2: Stretching in the y-axis by a factor of 2
  • 
• Transformation 3: Translation of 3 units to the right and 1 unit down
  • 

This final form of the transformation written out in terms of initial x and y values to the transformed x and y values is as follows with the below graph showing the results of the transformation of each point

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-3	0	2	-1
0	3	3	5
3	-1	4	-3
6	2	5	3