Learning outcomes
- Use interpolation and extrapolation correctly.
- Read graph values to suitable precision.
- Construct a large gradient triangle.
- Calculate gradient with units.
- Estimate uncertainty in graph-derived values.
10.1 Interpolation
Interpolation estimates a value inside the measured data range using the best-fit line or curve. Draw a thin guide line from the known axis value to the trend, then across to the other axis.
Because the estimate lies between measured points, it is generally more reliable than extrapolation, though it still depends on the quality of the data and trend.
10.2 Extrapolation
Extrapolation extends the trend beyond the measured range. It is less certain because the relationship may change outside the tested conditions. Extend only as far as needed and state the limitation if asked to evaluate.
A straight line within the data range does not guarantee the same relation indefinitely. A spring may leave the proportional region, a wire may heat, or a cooling curve may flatten toward room temperature.
10.3 Reading values
Read graph values to about half of one smallest grid square. Show guide lines where they make the method clear. Do not report more digits than the graph can support.
If the scale is 0.2 units per small square, a value such as 4.37 is unjustified unless the graph allows that precision.
10.4 Gradient
Choose two points on the best-fit line, not necessarily original data points. Draw a right-angled triangle whose hypotenuse spans at least half the length of the line. Calculate gradient = change in y divided by change in x.
Write coordinate differences with units. The gradient unit is y-unit divided by x-unit. A negative gradient is expected when y decreases as x increases.
10.5 Curves and tangents
For a gradient at one point on a curve, draw a tangent touching the curve locally and calculate the tangent gradient using a large triangle. The tangent should reflect the local direction, not cut across the curve arbitrarily.
Graph-derived intercepts or gradients may later be used in a physical formula. Preserve suitable significant figures, normally two or three.
Worked examples
Gradient with units
If a force–extension graph has ΔF = 3.6 N and Δx = 0.072 m, gradient = 3.6/0.072 = 50 N/m.
Interpolation
To find the temperature at 150 s, move vertically from 150 s to the cooling curve, then horizontally to the temperature axis and read to half a small square.
Practical focus
Investigation or training activity
Use a printed graph to obtain one interpolated value, one extrapolated value, one straight-line gradient and one tangent gradient. Mark every construction line.
Examination guidance
- Use points on the best-fit line for gradient.
- Make the triangle large.
- Include gradient units.
- Do not overstate graph-reading precision.
- State that extrapolation is less reliable beyond the measured range.
Check your understanding
- Why is interpolation usually more reliable than extrapolation?
- Where should gradient points be chosen?
- How large should a gradient triangle be?
- What is the unit of gradient on V against I?
Answers
- It lies within the tested data range.
- On the best-fit line.
- Its hypotenuse should span at least half the line.
- V/A, equivalent to ohms.
