IB Physics: Linearising Graphs Explained

Linearising a graph means algebraically rearranging a non-linear equation so a plot of two chosen quantities gives a straight line, y = mx + c. Instead of eyeballing a curve of best fit, you get a gradient and intercept you can actually calculate a physical constant from — that's the entire point.

Gradients and intercepts of straight lines can be measured with real precision; curves can't. A curved best-fit line hides the constant you're after inside its shape. Rearrange the equation first, plot the transformed variables, and the gradient — or intercept — becomes a direct, calculable value, usually the exact quantity the experiment set out to measure.

For radioactive decay, N = N₀e^(−λt) isn't linear, so you take natural logs of both sides: ln N = ln N₀ − λt. Plotting ln N against t gives a straight line with gradient −λ and y-intercept ln N₀, letting you read the decay constant straight off the graph.

For a pendulum, T = 2π√(L/g) becomes linear once you square both sides: T² = (4π²/g)L. Plotting T² against L gives a straight line through the origin with gradient 4π²/g, so g = 4π² ÷ gradient — a classic power-law linearisation IB examiners expect you to recognise instantly.

Most marks disappear at the algebra stage, not the plotting stage. Students often plot a variable against a partly-transformed version of another — say, T against L² instead of T² against L — choose axes that don't actually produce a straight line, or never write out the rearranged linear equation the mark scheme is looking for.

Draw two extra lines through the steepest and shallowest paths your error bars allow — a 'maximum' and 'minimum' gradient line — then take the uncertainty as half the difference between those two gradients. Quoting a bare gradient with no uncertainty is one of the most common ways marks are lost on data-based questions.

The gradient's units come from dividing the y-axis units by the x-axis units — never assume it's dimensionless. For T² (s²) plotted against L (m), the gradient carries units of s²m⁻¹, and getting this wrong usually costs a mark even when the numerical value itself is correct.

Yes — if your raw data had uncertainties, they don't disappear when you transform the axes, they change size. Taking a logarithm of a value with a 5% uncertainty, for instance, turns that percentage error into a fixed absolute uncertainty in ln(y), and your error bars need to reflect that transformation.

Linearising graphs mainly shows up in the data-based, structured questions within Paper 2, where you're given a table of results and asked to process, plot and interpret them. According to the IB, the 2025 DP Sciences guide removed the separate Paper 3, so this practical data-analysis skill is now examined entirely inside Paper 2.

In the Physics internal assessment, linearising is central to the data analysis criterion. Examiners want to see you recognise a non-linear relationship, derive the correct linear form, and use its gradient or intercept to calculate an experimental constant — ideally compared against a literature value with a stated percentage difference.

Use a semi-log plot (ln y against x) for exponential relationships like radioactive decay or capacitor discharge, where the gradient gives a rate constant directly. Use a log-log plot (log y against log x) for power laws such as y = kxⁿ, where the gradient itself equals the unknown power, n.

Software like LoggerPro or a graphical calculator will happily plot ln y against t for you, but examiners still want the algebra: the rearranged linear equation, labelled axes, and a stated gradient with units and uncertainty. Skipping that working, even with a perfect computer-generated graph, loses the analysis marks the mark scheme is actually awarding.

The most useful thing you can do is sit with a past paper data-based question and ask your child to explain, out loud, why they chose those two axes — not just check the final gradient. If they can't explain the algebra that got them to a straight line, that's the gap worth fixing before the next mock.

Look for resources that walk through the full worked method — rearranging the equation, choosing axes, extracting units and uncertainty — rather than just supplying final answers. On RevisionPrep, the Physics Topical Worksheets and Revision Notes cover linearisation alongside the rest of the data-analysis and IA skills students need across both SL and HL.