Tools and Inquiry — IB Physics skills

T.3.8 Graphing in Physics — IB Physics HL Revision Notes & Topical Questions

5 min read 11 parts Lesson 13 of 22 1 quiz

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Lesson summary

T.3.8 Graphing in Physics — IB PHYSICS SL notes: Construct, plot and interpret graphs, find gradients, intercepts and areas, and linearise relationships including logarithmic graphs.Sketch graphs show a qualitative trend — axes are labelled but need no scale or data points.Linear…

Linear graph
Independent variable on the x-axis
Plot data points accurately as small crosses
Plot all points within the graph area
A line of best fit may be straight or curved
A straight line is drawn with a ruler

Part 1 of 9Sketch graphs and relationships

1Sketch graphs show a qualitative trend

Sketch graphs show a qualitative trend axes are labelled but need no scale or data points.

2Linear graph

Linear graph a straight line means the rate of change is constant.
Directly proportionaly against x is a straight line through the origin and xy is not constant but y/x is.
All directly proportional graphs are linear — but not all linear graphs are directly proportional.

3Non-linear graph

Non-linear graph a curve means the rate of change is not constant.
Inversely proportionaly against x is a curve that crosses neither axis, y against 1/x is a straight line through the origin, and xy is constant.

Boyle's law: P against 1/V is a straight line through the origin, P against V is a curve, and PV against P is a horizontal line.

Part 2 of 9Types of chart

Chart	Data	Use
Bar chart	Qualitative or discrete	Columns whose height shows group size
Pie chart	Qualitative or discrete	Slices show proportions
Histogram	Quantitative, continuous	Columns with no gaps
Scatter graph	Continuous variables	Points with a line of best fit, e.g. force against extension
Line or curve graph	Functional relationship	Points joined in sequence, e.g. distance against time

Part 3 of 9Plotting a graph

Plotting a graph

Label both axes with quantity and unit.

Independent variable on the x-axis

Independent variable on the x-axis dependent variable on the y-axis.

Use appropriate linear scales.

Points should fill at least half the sheet;
- if you can double the scale and still fit the points, the scale is too small.

Plot data points accurately as small crosses

Plot data points accurately as small crosses in pencil so corrections are easy.

Plot all points within the graph area

Plot all points within the graph area and include error bars where uncertainties are known.

Part 4 of 9Lines of best fit

1A line of best fit may be straight or curved

A line of best fit may be straight or curved depending on the trend of the data.

2A straight line is drawn with a ruler

A straight line is drawn with a ruler a curve is drawn smoothly.

It has a balance of points on either side.

It passes through the origin only if the data and trend allow.

Part 5 of 9Gradient, intercept, maxima and areas

Gradient, intercept, maxima and areas

Gradient of a straight line

Δy / Δx.

Draw a large triangle and read values from points on the line, not from scattered data points.
The unit of the gradient is the unit of y divided by the unit of x, e.g. N m^-1 for force against extension.

Gradient of a curve at a point

Gradient of a curve at a point draw a tangent with a ruler, then find its gradient.

Straight line y

Straight line y mx + c — m is the gradient and c the y-intercept (the value of y at x = 0).

Maximum

Maximum gradient goes from positive through zero to negative; minimum — negative through zero to positive.

Area under a graph often represents a physical quantity

Area under a graph often represents a physical quantity the area under a velocity–time graph is displacement.
Use base × height for a rectangle and ½ × base × height for a triangle.
For a curve, divide into rectangles and triangles and count the remaining squares.

Part 6 of 9Extrapolation and interpolation

Extrapolation and interpolation

Interpolation

Interpolation using the line of best fit to estimate values between data points.

Extrapolation

Extrapolation extending the line of best fit to estimate values beyond the data.

Extrapolation is less reliable

Extrapolation is less reliable the trend may not continue outside the measured range.

Part 7 of 9Linearising graphs

Linearising graphs

Linearising rearranges a non-linear relationship to fit y

Linearising rearranges a non-linear relationship to fit y mx + c — straight lines are easier to interpret.

Pendulum: T

2π√(L/g),

so T² = (4π²/g)L — a graph of T² against L has gradient 4π²/g.
What the symbols mean: T = period (s) · l = length of the pendulum (m) · g = 9.81 m s⁻²

Common linearisations.

Exponential decay N = N₀e^-λt gives ln N against t, gradient −λ.
Power law y = ax^b gives log y against log x, gradient b.
Lenses 1/v against 1/u.

Part 8 of 9Logarithmic graphs

1On a logarithmic scale intervals increase exponentially

On a logarithmic scale intervals increase exponentially it handles values spanning several orders of magnitude.

2Semi-log graph

Semi-log graph only the y-axis is logarithmic; exponentials y = ab^x become straight lines.
log y = x log b + log a: gradient is log b and intercept is log a.

3Log–log graph

Log–log graph both axes are logarithmic; power functions y = ax^b become straight lines.
log y = b log x + log a: gradient is b and intercept is log a.
Kepler's law T² ∝ r³ gives a straight line of log T against log r, with gradient 3/2.

4Check the base

Check the base log x = k gives x = 10^k, and ln x = k gives x = e^k.

Part 9 of 9More from the specification

1Gradient of distance-time graph

Gradient of distance-time graph calculating the instantaneous slope on a distance-versus-time plot directly yields the scalar speed of the body.

2Logarithmic scale

Logarithmic scale each equal step multiplies the value by a constant factor, such as 10, so it fits data spanning many orders of magnitude.

3Linearising

Linearising plotting a transformed quantity, such as y against 1/x or ln y against x, turns a curve into a straight line.

4Gradient uncertainty

Gradient uncertainty draw best-fit and worst-fit lines;
- half the difference between the steepest and shallowest gradients gives the uncertainty.

5Logarithmic scale

Logarithmic scale each equal step multiplies the value by a constant factor, such as 10, so it fits data spanning many orders of magnitude.

6Linearising

Linearising plotting a transformed quantity, such as y against 1/x or ln y against x, turns a curve into a straight line.

7Gradient uncertainty

Gradient uncertainty draw best-fit and worst-fit lines;
- half the difference between the steepest and shallowest gradients gives the uncertainty.

8Derived unit

A unit formed by multiplying or dividing SI base units.

9Scale drawing

Accurate drawing with a chosen scale.

10Parallelogram method

Vector addition using a parallelogram of vectors.

11Vector

Quantity with magnitude and direction.

12Resultant

Vector sum of two or more vectors.

13Scalar

Quantity with magnitude only.

14Significant figures

Digits carrying meaningful information in a number.

15Resultant vector

Single vector equivalent to a combination of vectors.

16Percentage uncertainty

The absolute uncertainty divided by the value, times 100%.

17Absolute uncertainty

The range ±Δx in the same unit as the measurement.

18Percentage error

Difference from the accepted value as a percentage of it.

19a

− ω2x

What the symbols mean: a = acceleration (m s⁻²) · ω = angular frequency (rad s⁻¹) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre

20Logarithmic scale

Logarithmic scale each equal step multiplies the value by a constant factor, such as 10, so it fits data spanning many orders of magnitude.

21Linearising

Linearising plotting a transformed quantity, such as y against 1/x or ln y against x, turns a curve into a straight line.

22Gradient uncertainty

Gradient uncertainty draw best-fit and worst-fit lines;
- half the difference between the steepest and shallowest gradients gives the uncertainty.

Key terms

Term	Meaning
Gradient	Change in y divided by change in x.
Line of best fit	Smooth line or curve with balanced points on both sides.
Tangent	Straight line touching a curve at a point; its gradient is the gradient of the curve there.
Interpolation	Estimating values between data points.
Extrapolation	Extending a line to estimate values beyond the data.
Linearising	Rearranging a relationship into y = mx + c.
Logarithmic scale	Scale whose intervals increase by powers of ten.

Avoid these mistakes

Describing any rising straight line as directly proportional — it must pass through the origin.

Using a small gradient triangle — a large one reduces the uncertainty.

Reading the gradient from data points that lie away from the line of best fit.

Forgetting units on the axes or on the gradient.

Mixing log base 10 and natural logs when reading a logarithmic scale.

Last updated: 18 October 2026