To find a rate from a straight-line graph, divide the change up the vertical axis by the change along the horizontal axis. The answer is the gradient, and its unit is the vertical unit per horizontal unit.
This skill appears in physics, chemistry and biology questions alike, so it sits at the centre of physical reasoning in mixed tasks. The data below is invented for teaching.
What is a rate, and where is it hiding in the graph?
A rate says how much something changes in one unit of another quantity. Speed is metres per second. Cooling rate is degrees Celsius per minute, and reaction rate and growth rate follow the same pattern.
Each time, the rate is the steepness of the line. A steeper line means a faster change. A flat line means no change.
How do I calculate it, step by step?
- Name the two axes and their units. The unit of your answer comes from these.
- Choose two points exactly on the line, well apart, where the line crosses grid intersections if possible.
- Read both coordinates carefully, checking the scale of each square.
- Subtract to get the change in y and the change in x.
- Divide change in y by change in x, then write the unit.
Worked example
A trolley moves along a straight track. The line on its distance-time graph passes exactly through (2 s, 3 m) and (10 s, 19 m). Find its speed.
Step 1, axes: distance in metres on the vertical axis, time in seconds on the horizontal axis. So the unit is m/s.
Step 2, points: (2, 3) and (10, 19).
Step 3, changes: change in distance = 19 − 3 = 16 m. Change in time = 10 − 2 = 8 s.
Step 4, divide: speed = 16 ÷ 8 = 2.0 m/s.
Check by reading forward: in 8 s at 2.0 m/s the trolley covers 2.0 × 8 = 16 m, which matches the change in distance.
What mistake should I watch for?
A common slip is to divide one point’s y-value by its x-value, ignoring where the line started.
Mistaken working: speed = 19 ÷ 10 = 1.9 m/s
This treats the line as if it began at the origin. The trolley was already 3 m along the track at 2 s.
The correction is to use the change between two points, not a single reading. Only a line that passes through (0, 0) lets you divide a single point, and the safer habit is to subtract every time.
Check yourself
Work these without a calculator, then open each answer.
1. A straight line passes through (0 s, 0 m) and (6 s, 15 m). Find the speed.
Show answer
Change in distance = 15 m, change in time = 6 s. Speed = 15 ÷ 6 = 2.5 m/s.
2. The temperature of a cooling liquid falls in a straight line from 80 °C at 2 min to 50 °C at 8 min. Find the cooling rate.
Show answer
Change in temperature = 80 − 50 = 30 °C. Change in time = 8 − 2 = 6 min. Rate = 30 ÷ 6 = 5 °C/min.
3. On a speed-time graph a straight line passes through (2 s, 4 m/s) and (5 s, 16 m/s). What does the gradient represent, and what is it?
Show answer
The gradient of a speed-time graph is acceleration. Change in speed = 16 − 4 = 12 m/s. Change in time = 5 − 2 = 3 s. Acceleration = 12 ÷ 3 = 4 m/s².
Where this leads next
Rates link naturally to energy, so continue with connecting a circuit observation with energy transfer. The scientific investigation critic helps you judge whether a graph in an investigation supports the rate you calculated.
Some students can calculate a gradient in a worksheet but lose it inside a long mixed question. That is where our teachers work with you in online one-to-one Combined Science tuition.