To choose graph axes, rewrite the physics equation in the form y = mx + c, then plot the quantity that plays the role of y against the one that plays the role of x. The gradient m and intercept c then carry physical meaning. This skill is part of physics investigations and explanations.
How do you turn an equation into a straight-line graph?
- Write the model in words or symbols, for example “pressure times volume is constant”.
- Rearrange it so that one measured quantity (or an expression made from it) stands alone, equal to a constant multiplied by another.
- Put the isolated quantity on the y-axis and the other on the x-axis.
- Say what the gradient represents and what it should equal.
If the relationship is inverse, the simple plot of one against the other gives a curve. Plotting against the reciprocal gives a line.
Worked example
Invented data, constant temperature, fixed mass of gas. The model is p × V = constant, written as p₁V₁ = p₂V₂ in the syllabus.
| V / cm³ | 20 | 25 | 40 | 50 | 80 |
|---|---|---|---|---|---|
| p / kPa | 100 | 80 | 50 | 40 | 25 |
Step 1: p × V = k, where k is a constant. Check: 20 × 100 = 2000, 25 × 80 = 2000, 40 × 50 = 2000, 50 × 40 = 2000, 80 × 25 = 2000. So k = 2000 kPa cm³.
Step 2: Rearrange: p = k × (1/V). This matches y = mx with y = p and x = 1/V.
Step 3: Work out 1/V for each reading: 0.0500, 0.0400, 0.0250, 0.0200, 0.0125 cm⁻³.
Step 4: Plot p (kPa) on the y-axis and 1/V (cm⁻³) on the x-axis. The points lie on a straight line through the origin.
Step 5: Gradient = (100 − 25) ÷ (0.0500 − 0.0125) = 75 ÷ 0.0375 = 2000 kPa cm³. This equals k, so it confirms the model.
The mistake to watch for
A student plots p against V, gets a curve through the points and draws a straight line “through the middle”.
Mistaken graph: p on the y-axis, V on the x-axis, straight line of best fit.
The line misses most points and the gradient has no meaning, because the relationship is not linear in V.
The correction is to ask “what is the equation, and what would make it look like y = mx?” before drawing any axes. Here p is proportional to 1/V, so the x-axis must be 1/V.
Check yourself
1. A trolley starts from rest with constant acceleration a, so s = ½ a t². What should be plotted to get a straight line, and what does the gradient equal?
Show answer
Plot s (y-axis) against t² (x-axis). Since s = (a/2) × t², the gradient equals a/2. Check with a = 2.0 m/s²: t = 1, 2, 3 s gives s = 1, 4, 9 m and t² = 1, 4, 9 s², so the gradient is 1.0 m/s², which is 2.0 ÷ 2.
2. For a resistor, which axes give a gradient equal to the resistance R?
Show answer
Since V = I × R, plot V on the y-axis and I on the x-axis. The gradient is V/I = R, in ohms. The line passes through the origin.
3. A student plots extension x on the y-axis and force F on the x-axis for a spring where F = kx. What does the gradient equal?
Show answer
x = F/k, so the gradient is 1/k, not k. To get k as the gradient, plot F on the y-axis against x on the x-axis.
Where this leads next
With the right axes, the next step is reading and interpreting the gradient with units. The graph-model and residual explorer lets you test how well a chosen line fits a small dataset, and the rate and energy graph interpreter helps with the meaning of gradients. Come back from naming the variables if the x and y roles still feel unclear.
Choosing axes is easier when someone asks you “why that one?” on every attempt. That is the kind of questioning you get in online one-to-one Physics tuition.