Reading graphs: linear, proportional or inverse-square?

A graph’s appearance is evidence only after you know how it was drawn. Two plots of the same measurements can look very different if one axis starts above zero or uses logarithmic spacing.

Read the labels before the line

Identify the horizontal and vertical quantities, their units and any multipliers such as ×10³. Then check whether equal tick spacing represents equal differences or equal ratios.

On ordinary linear axes, slope is Δy/Δx and carries units of the vertical quantity divided by the horizontal quantity. Calculate it from labelled coordinates, not the angle of the line on your screen. Changing the drawing’s aspect ratio changes that angle without changing the underlying data.

For motion-specific meanings of slope and area, use the separate speed, velocity and acceleration guide.

A straight line need not be proportional

Compare y = 3x with y = 3x + 2. Both have slope 3. Only the first has a constant y/x ratio and passes through the origin.

For the second relationship, increasing x from 2 to 4 changes y from 8 to 14. Doubling x has not doubled y because the offset remains 2. This distinction follows from the slope-intercept form of a linear function.

A cropped graph may not display the origin. Do not assume its intercept is zero simply because the visible segment is straight.

Use a doubling test to distinguish models

For positive x, nonzero constant k and the models below:

Model	Quantity that stays constant	Effect of doubling x
y = kx	y/x	y doubles
y = kx²	y/x²	y quadruples
y = k/x	xy	y halves
y = k/x²	x²y	y becomes one quarter

These tests compare candidate models; they do not establish that a real system follows one exactly. Other variables must be held fixed, and measurement uncertainty matters. OpenStax introduces direct and inverse variation.

Worked data: recognise an inverse square

Consider this original, idealised data table:

x	y	xy	x²y
1	36	36	36
2	9	18	36
3	4	12	36

The ordinary product xy changes, but x²y stays at 36. The data are consistent with y = 36/x². That model predicts y = 2.25 at x = 4.

The prediction extends beyond the supplied x values, so it is an extrapolation. In a real experiment, treat that extension as a model prediction to test, not an additional measurement.

Logarithmic axes change the interpretation

On a base-10 logarithmic axis, equally spaced ticks might read 1, 10 and 100. The spacing represents multiplication by ten rather than addition of a fixed amount. Zero cannot appear on an ordinary logarithmic axis.

Taking logarithms explains useful straight-line transformations:

• If y = kxⁿ for positive x, y and k, then log y = log k + n log x. A log–log plot has slope n.
• If y = Abˣ with positive A and b, then log y = log A + x log b. Plotting log y against x gives a straight line.

These statements concern transformed coordinates. Use the printed scale to interpret a plot rather than treating every straight trace as y = mx + c in the original variables. The logarithm rules also underlie the pH scale.

Before choosing an answer

Ask whether you used the actual coordinates, checked the intercept and accounted for the scale. A downward curve alone does not distinguish inverse from inverse-square behaviour. Likewise, a small-looking change on a logarithmic plot may represent a large concentration ratio.