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Math

Attribution

Some of the content of this guide was modeled after a guide originally created by the Openstax and has been adapted for the GPRC Learning Commons in March 2021. The graphs are generated using Desmos. This work is licensed under a Creative Commons BY 4.0 International License.

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:

 

3) f(cx), where c is a constant: c>1 compresses f(x) in the x direction and 0<c<1 stretches f(x). 
 
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.

 

4) cf(x), where c is a constant: c>1 stretches the graph in the y direction and 0<c<1 compresses the graph. 

 

Example: Sketch the graph of 
The original function is and the constant coefficient is which stretches the graph in the y direction.

 

5) -f(x): Reflects f(x) about the x axis.

 

Example: Sketch the graph of 

 

6) f(-x): Reflects f(x) about the y axis.

 

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

​​​​​​​

-3

0

2

-1

0

3

3

5

3

-1

4

-3

6

2

5

3

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