Graphing and linearization for NePhO

The authors of the source guide observe that graph-based questions in NePhO carry at least 10 marks out of 100. They also notice that, across recent papers, one skill keeps coming back: linearization. This is the authors' observation, not an official mark breakdown.

That makes graph work high-value practice. A clear table, a sensible scale and a well-read slope can earn marks even when the final number is not perfect.

Linearization means rewriting a relationship so that the data can be plotted as a straight line in the form y = mx + c. Once the points lie on a line, the slope and intercept can be matched to the unknown constants in the original equation.

If a relation involves only multiplication, division and powers, taking logarithms can turn it into a straight line. Sometimes taking roots or powers of both sides also helps. For example, if μ = a + b/λ², plotting μ against 1/λ² gives a straight line. The slope is b and the intercept is a.

A capacitor discharges so that V = α e^(−t/T). The measurements are shown below. The task is to find a straight-line form and use it to estimate α and T.

1. Take natural logarithms of both sides: ln V = −(1/T)t + ln α.
2. Compare with y = mx + c: use y = ln V and x = t. Then the slope is −1/T and the intercept is ln α.
3. Process the measured voltages into ln V values.
4. Plot ln V against t and draw a best-fit straight line.

With 2 squares = 1 s on the x-axis and 1 square = 0.14 on the y-axis, the points spread across the graph paper. A large triangle on the best-fit line gives a slope of −1.82 / 4.25 ≈ −0.43.

Switch between the raw measurements and the linearized values. The linearized view shows the best-fit straight line; the original view shows the fitted exponential curve.

The table in the worked example is the text alternative for the plotted values.

• Spread the points so the graph covers more than 80% of the paper.
• Choose a scale that goes slightly beyond the data range. For example, allow room for 7 s when the largest reading is 6 s.
• Prefer easy numbers. Use 2 squares = 1 s rather than 2.29 squares = 1 s.

• Cover more than 80% of the paper.
• Write the scale on the graph.
• Label both axes with the quantity and its unit. For powers of ten, put the multiplier in the axis label, for example “time (×10² s)”.
• Make each point easy to see, for example by circling it.
• Draw a best-fit line with roughly equal numbers of points above and below it.
• Find the slope using a large right-angled triangle and two far-apart points on the line.

These two prompts were written for this site.