Reduction To A Linear Form

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Suppose you carried out an experiment, which gave the following results

t	x
2	8
4	11
6	15
8	18
10	20
12	24
14	27
16	31

Could you find an equation connecting x and t?

The first step would probably be to draw a graph of the data.

Since the data falls on a straight line, you can easily find an equation, by putting the gradient and intercept into the equation for straight lines,

We want to know how to find an equation connecting data that does not give a straight line.

Consider the following data

t	d
1	5
3	21
5	53
7	101
9	165
11	245
13	341
15	553

The graph of this data is

The graph appears to be a quadratic, suggesting that the data is linked by an equation in the form

The question is, how can we confirm that this equation really does fit the data and how can we determine the constants a and b.

The method we use is called reduction to a linear form. The idea is to plot the data in such a way that we get a straight line. In order to do this we have to write the equation, so that it mirrors the equation of a straight line.

We need to write our equation as

where Y & X are terms based on the variables, with no reference to the constant terms, and m &c are made up of the constant terms, with no reference to the variable terms.

In this case, the equation already looks like this

So, if we plot d against t², we ought to get a straight line

t²	d
1	5
9	21
25	53
49	101
81	165
121	245
169	341
225	453

You can see that we clearly have a straight line, so the data must be linked by an equation in the form we suggested. Further, we can measure the gradient,(2) and the intercept,(3) to find the constants a & b. The equation linking d & t is therefore

The whole process of reducing to a linear form can be a confusing one at first. The only way to see what is going on, is to look at a few examples.

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