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 xaxis by a factor of 3
 Transformation 2: Stretching in the yaxis 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
