A good graph does something a table never can: it makes a relationship visible. The shape of a line tells you whether two quantities are proportional, the gradient often is a physical quantity, and the area underneath can be another one entirely. Learn to plot cleanly, fit a line fairly, and read a graph properly, and you’ll unlock some of the most reliable marks in the whole course — especially in your Internal Assessment.
📘 What you need to know
Put the independent variable on the x-axis, the dependent one on the y-axis, both labelled with units
Choose sensible scales so the points fill at least half the grid
A line of best fit may be straight or curved, with points balanced on either side
The gradient of y against x is Δy ÷ Δx, found with a large triangle
The y-intercept is the value of y where the line crosses the axis (x = 0), giving y = mx + c
The area under a graph often represents a physical quantity (e.g. area under velocity–time = displacement)
Linearising a curved relationship turns it into a straight line you can analyse
Plotting a graph well
Before any physics comes out of a graph, the plotting has to be right. The independent variable — the one you controlled — goes across the bottom; the dependent one goes up the side. Label both axes with units, pick linear scales that spread the data across the grid, and mark each point with a small, sharp cross.
🎨 Plotting checklist
Label both axes with the quantity and its unit
Choose scales that make the points fill more than half the grid
Plot accurately with small crosses, in pencil, so mistakes can be fixed
Draw the line of best fit — straight with a ruler, or a smooth freehand curve
Lines of best fit
A line of best fit shows the underlying trend, ignoring the scatter of individual points. It can be straight or a smooth curve, whichever the data suggests — the key is that points sit balanced on both sides. It’s only drawn through the origin if the data genuinely supports that.
A trap worth naming: “line of best fit” doesn’t mean “straight line”. If the points curve, force a ruler through them and you’ll misrepresent the physics. Let the data choose the shape — and if it’s a curve, draw it smoothly in one confident sweep rather than joining the dots.
Reading the gradient
On a straight-line graph the gradient is constant, and it usually carries real meaning — the gradient of a resistance-against-length graph, for instance, tells you resistance per metre. To find it, draw a large triangle on the line (large triangles shrink the reading error), take the values off the axes, and divide.
Gradientm = Δy ÷ Δx = (y2 − y1) ÷ (x2 − x1)
Draw the gradient triangle as large as the line allows, then read Δy and Δx off the axes.
Intercepts and the equation of a line
Every straight-line graph obeys y = mx + c, where m is the gradient and c is the y-intercept — the value of y where the line crosses the vertical axis at x = 0. Reading off both lets you write the full relationship between the variables.
Straight-line equationy = mx + c
Area under a graph
The region between a line and the x-axis often stands for a physical quantity. The classic case: the area under a velocity–time graph is the displacement. For straight-line graphs you find it by splitting the region into triangles and rectangles; for curves, you count squares or divide it into strips.
Split the area under a straight-line graph into a triangle and a rectangle; their sum is the displacement.
Linearising a curve
Straight lines are far easier to read than curves, so physicists often linearise — rearrange a relationship until plotting the right combination of variables gives a straight line. The pendulum is the perfect example. Its period is T = 2π√(L ÷ g), a curve when you plot T against L. Square both sides and it straightens out.
Linearising the pendulumT² = (4π² ÷ g) L
Plotting T² against L turns the curve into a straight line whose gradient is 4π² ÷ g.
Quick recap: plot with labelled axes and sensible scales; fit a balanced line (straight or curved); gradient = Δy ÷ Δx from a large triangle; intercept gives c in y = mx + c; area often means a physical quantity; and linearise curves to analyse them.
WE 1
A best-fit line on a graph of resistance against length passes through the points (0.20 m, 4.0 Ω) and (1.80 m, 28.0 Ω). Find the gradient and state what it represents.
Use two points that lie on the line (not raw data points) and apply the gradient formula.
Gradient
m = Δy ÷ Δx = (28.0 − 4.0) ÷ (1.80 − 0.20)
= 24.0 ÷ 1.60 = 15 Ω m⁻¹gradient = 15 Ω m⁻¹The units (Ω per m) reveal the meaning: this is the resistance per unit length of the wire.
WE 2
A student plots T² against L for a simple pendulum and measures a gradient of 4.0 s² m⁻¹. Given T² = (4π² ÷ g)L, find a value for g.
Compare the graph to y = mx: the gradient equals 4π² ÷ g, so rearrange for g.
Match to y = mx
gradient = 4π² ÷ g
Rearrange for g
g = 4π² ÷ gradient
g = (4 × π²) ÷ 4.0g = 9.9 m s⁻²Reassuringly close to the accepted 9.81 m s⁻² — the linearised graph gives g straight from its gradient.
💡 Top tips
Take gradient points off the line, not the data — the line already averages out the scatter
Make the gradient triangle big — it shrinks the percentage error in your reading
Check the units of a gradient — they usually tell you exactly what it represents
When linearising, compare with y = mx + c to see what the gradient and intercept mean
⚠ Common mistakes
Forcing a straight line through points that clearly curve — let the data pick the shape
Reading gradient values from scattered data points instead of from the best-fit line
Drawing a tiny gradient triangle, magnifying the reading error
Assuming a straight line through the origin means directly proportional — check it actually passes through (0, 0)
Up next: Determining Uncertainties from Graphs — we’ll add error bars to those points and use best and worst-fit lines to put an uncertainty on the gradient and intercept, closing the loop between graphing and uncertainty.
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