Gradients, intercepts and linearisation
Transforming an equation into straight-line form lets a graph test a model and determine constants.
Transform the proposed law to y = mx + c so gradient and intercept reveal constants.
By the end of this lesson, you can:
- compare an equation with y = mx + c;
- select transformed axes;
- calculate a gradient using a large triangle;
- use gradient and intercept to find constants.
Linear form
Rewrite the theoretical relationship so the plotted quantities correspond to y and x. The physical constants then correspond to gradient m or intercept c.
Calculating gradient
Choose two far-separated points on the best-fit line, not necessarily measured points. Use gradient = Δy/Δx and include units derived from the axes.
Power laws
For y = axⁿ, take logarithms: lg y = n lg x + lg a. A plot of lg y against lg x has gradient n and intercept lg a.
WORKED EXAMPLEReveal method
For V = IR, plotting V against I gives gradient R. For T = k√l, plotting T against √l gives gradient k.
Try another worked problem
Attempt the question yourself, then reveal the examiner-style solution when you are ready.
Linearisation
If y=kx², what graph has gradient k?
CHECK YOUR ANSWERReveal solution
Plot y against x²; the gradient is k and expected intercept is zero.
Mastery check
For lg y = n lg x + lg a, the graph gradient is: