9. Linearisation

Our analysis must begin with a scatterplot, and recognition of the type of model appropriate for the situation. But we have seen several times that it is not possible to identify the type of function from its graph (e.g. see Unit 11).

The only graph that one can be reasonably sure about is the straight line with linear formula y = ax + b. Most models (except the trigonometric functions) can be made linear by transforming the data in some way.

In this activity we will briefly work through some pure, exact mathematical examples to illustrate the principles of linearisation. Open the Scatterplot applet (it includes some help tips) and move it so that you can work in the applet and here at the same time.

Example 1
In the applet, click on Load Data File to load Example 1.
You should see from the table that the formula for the data simply is y = x² and the graph is a parabola. The scatterplot also shows the linear regression line and its parameters. Linearisation means that we want to adapt the scatterplot, so that we can use the linear regression line to help us to formulate a formula for the data. To do that, we want the scatterplot to be a straight line, so that the regression line is a good fit for the data, so that we can use the parameters of the regression line to deduce a formula for the data.

Now in the plot box, type X^2 and press ENTER or click Update Display. The graph is now a straight line! Can you explain why?

Example 2
In the applet, scroll down the drop-down list, select Example 2, and click Load Data File and make sure that the plot shows X vs. Y. This graph also looks like a parabola, but it may also be a cubic! Linearisation can help us decide. Again type X^2 vs. Y.

The resulting graph of X^2 vs. Y is a straight line with slope 2 and y-intercept 0, so its equation is y = 2x, and from this we can deduce that the equation of X vs. Y is y = 2x². You should check in the table that you agree that this is correct, and make sure that you understand why the graph of X^2 vs. Y is a straight line and why the equation of X vs. Y is y = 2x²!

Note: This is all that is necessary, but you can be really smart and get the equation directly from the applet by making transformations until you get y = x, like in Example 1: Plotting 2*X^2 vs. Y is the straight line y = x, so the exact formula of the data is y = 2x²!

Example 3
In the applet, scroll down the drop-down list, select Example 3, and click Load Data File and make sure that the plot shows X vs. Y. This graph also looks like a parabola, but it may also be a cubic! Linearisation can help us decide. Again type X^2 vs. Y.

The resulting graph of X^2 vs. Y is a straight line with slope 1 and y-intercept 5, so its equation is y = x + 5, so we can deduce that the equation of X vs. Y is y = x² + 5. You should check in the table that you agree that this is correct, and make sure that you understand why the graph of X^2 vs. Y is a straight line and why the equation of X vs. Y is y = x² + 5!

Note: The graph of X^2 vs. Y – 5 is the straight line y = x, so the exact formula of the data is y – 5 = x²!

Example 4
In the applet, select Example 4 from the drop-down list, click Load Data File and make sure to reset the plot to X vs. Y. It is not easy at all to see the formula from the table, and the graph may be some quadratic or cubic or exponential function. One simply has to try some transformations and see which gives a straight line.

Plotting X^2 vs Y gives a straight line with equation y = 0,5x + 3. Try to understand how we can now deduce that the required equation of X vs. Y is y = 0,5x² + 3 (check in the table).

An explanation
To understand the algebraic transformations underlying the graphical linearisation, study the following animation:

Now find the best-fit equations of Examples 5 to 8 in the applet through linearisation …
Make sure you reset the plots to X vs. Y after each example.
Check your formulae against the data in the tables.

Check your answers here: