In order to work with non-linear relationships in Further Mathematics, data transformation is applied. This involves changing the scale of one of the variables so that it is compressed or stretched to create a straight line. In this course, three main transformations are regarded.

## Types of Transformations

• $x^2$
• decreases values less than 1
• increases values greater than 1
• large values are increased the most
• stretches the values

### Logarithm

• $\log{x}$
• reduces all values
• values between 0 and 1 become negative
• large values are reduced much more than small values
• compresses values
• note: can only be applied to values greater than 0

### Reciprocal

• $\dfrac{1}{x}$
• reduces all values greater than one
• large values are reduced much more than small values
• compresses values (more than log)

## Transforming the x-axis

Above is a representation of how the data points are stretched or compressed. The graphs show a linear line being drawn by transforming the x-axis. Notice how the x-axis has a different scale.

### Application

The following data from Table 1 was used to construct two separate graphs (Graph 1 & Graph 2). By observing the two graphs, it is obvious that the data points represent a more linear pattern when a reciprocal transformation has been applied to the x-axis.

Table 1

Graph 1Graph 2

## Transforming the y-axis

In the same way, data can be transformed in the y-axis. Notice how, by changing the scale on the y-axis, the data values were able to demonstrate a linear relationship.

### Application

The data in Table 2 shows the values that were applied when creating Graph 3, 4 and 5. Displayed is also the $r^2$ value in order to show which transformation was a better fit. The value of $r^2$ is the greatest with the 1/y transformation which signifies that this is the best transformation to apply.

Table 2

Graph 3

Graph 4

Graph 5

## Which transformation?

By comparing the shape of the data points to those in this table, the possible transformations can be experimented with. After applying all of these transformations, the coefficient of determination is observed in order to determine which of the transformations is most appropriate.

Essential Further Mathematics Units 3 & 4, 4th Edition Enhanced, pg 190.