Linear law turns a nonlinear relation into a straight line, so that an experiment’s results can be checked and its constants found from a graph. This page shows how the steps connect, the order to learn them in and where to start.
How do the steps connect?
Suppose variables x and y are related by y = ax² + b, where a and b are constants to find. The graph of y against x is a curve, which is awkward to read.
Plot y against x² instead. Then y = a(x²) + b has the form Y = mX + c, with Y = y, X = x², m = a and c = b.
Here is an original example with y = 3x² + 5.
| x | 1 | 2 | 3 |
|---|---|---|---|
| x² | 1 | 4 | 9 |
| y | 8 | 17 | 32 |
The gradient between the first two points is (17 − 8) ÷ (4 − 1) = 3, and between the last two it is (32 − 17) ÷ (9 − 4) = 3. The line crosses the vertical axis at 8 − 3 = 5. So a = 3 and b = 5.
What order should I study them in?
- Transforming a nonlinear relation into linear form builds the rearrangement skill.
- Choosing axes for a straight-line graph turns it into a plan for the graph.
- Finding constants from gradient and intercept reads the numbers off the line.
- Interpreting a linearised model in its original variables turns them back into the original relation.
Finish with the chapter practice set, which mixes the four steps.
Who should start where?
A student who cannot rearrange y = ax² + b into a line should start with step 1. A student who rearranges correctly but plots the wrong axes should start at step 2.
A student who draws the right graph but gets a and b wrong should practise step 3. A student whose constants are right but whose final equation loses marks should review step 4. The ideas also need logarithms, so a quick look at indices, surds and logarithms helps.
For a teacher to go through your graphs with you, see online one-to-one Additional Mathematics tuition. The wider SPM Additional Mathematics guide shows where linear law fits.