A graph of two real quantities
When a graph plots two measured quantities against each other, its features acquire physical meanings. Those meanings come entirely from the units, which is why the same shape says different things on different axes.
The gradient is the vertical unit divided by the horizontal unit. On a distance–time graph that is metres per second, a speed. On a speed–time graph it is metres per second per second, an acceleration. Reading the axes before interpreting anything is not a formality — it is the whole method.
| Graph | Gradient means | Area under means |
|---|---|---|
| distance–time | speed | nothing useful |
| speed–time | acceleration | distance travelled |
| cost–quantity | price per unit | nothing useful |
| volume–time | rate of flow | nothing useful |
| rate of flow–time | change in flow rate | total volume |
Switch between the two graphs with Gradient selected. A horizontal section means stopped on a distance–time graph and constant speed on a speed–time graph — the same shape, opposite meanings.
Distance–time graphs
Time runs along the horizontal axis and distance from the starting point up the vertical. The gradient is the speed, so a steeper line means faster.
Three shapes recur. A horizontal section means the distance is not changing — the object is stationary. A straight sloping section means constant speed. A curve means the speed is changing, getting steeper for speeding up and shallower for slowing down. A section sloping downward means returning towards the start.
The area under a distance–time graph means nothing
Multiplying metres by seconds gives metre-seconds, which is not a quantity anyone wants. Students trained on speed–time graphs sometimes calculate the area under a distance–time graph and present it as an answer. Ask what the units of the area would be before working it out — if they are meaningless, the area is too.
Speed–time graphs
Now the vertical axis is speed, so the gradient is the rate at which speed changes — the acceleration. A horizontal line no longer means stopped; it means moving at a constant speed. The object is only stationary where the graph touches zero.
And here the area does mean something: speed multiplied by time gives distance. The standard technique is to split the area into triangles and rectangles, find each, and add.
A car accelerates from rest to 20 m/s in 10 s, holds that speed for 30 s, then decelerates to rest in 20 s. Find the acceleration, the total distance and the average speed.
- Acceleration = gradient of the first section =
20 ÷ 10 = 2m/s².Change in speed divided by the time it took. - First section is a triangle:
½ × 10 × 20 = 100m.The area under a speed–time graph is the distance. - Middle section is a rectangle:
30 × 20 = 600m. - Last section is a triangle:
½ × 20 × 20 = 200m. Total= 900m.Split into standard shapes rather than trying to find the whole area at once. - Average speed
= 900 ÷ 60 = 15m/s.Total distance over total time — not the average of 0 and 20.
Acceleration 2 m/s²; distance 900 m; average speed 15 m/s
Conversion graphs and other practical graphs
A conversion graph is a straight line through the origin used to change between two units — kilometres and miles, rupees and dollars. Reading from either axis to the other converts in that direction, which makes it faster than arithmetic for repeated conversions and less accurate for any single one.
Other practical graphs follow the same principle. A water-filling graph of depth against time is steep where the container is narrow and shallow where it is wide, because the same volume raises the level less in a wider vessel. A graph of cost against quantity has a gradient equal to the unit price, and a vertical intercept equal to any fixed charge that applies regardless of quantity.
What a non-zero intercept tells you
A taxi fare graph that starts at Rs 200 when the distance is zero has a fixed charge of Rs 200 before any travelling. The gradient is then the rate per kilometre. Whenever a practical graph does not pass through the origin, the intercept is a standing cost or a starting value, and questions ask for its meaning in context rather than just its number.
Before you leave this chapter
- The gradient means vertical unit ÷ horizontal unit. Read the axes first.
- Distance–time: gradient is speed, horizontal means stopped, area means nothing.
- Speed–time: gradient is acceleration, horizontal means constant speed, area is distance.
- Find an area by splitting it into triangles and rectangles.
- A non-zero intercept on a practical graph is a fixed charge or a starting value.
Reading a graph back into a story
Half the marks in this topic come from describing a journey in words from its graph, or sketching a graph from a description. Both are done section by section: identify where the shape changes, then say what each section means in the language of the situation.
A good description names the quantity, the direction and the duration for every section — "travels away from home at a constant 40 km/h for 30 minutes", not "goes up".
Describe the journey shown by a distance–time graph that rises steeply for 10 minutes, is horizontal for 5, rises gently for 20, then falls steeply to zero over 15 minutes.
- First section: distance increasing quickly — travelling away from the start at a high constant speed for 10 minutes.Straight and steep means constant and fast.
- Second section: horizontal, so the distance is unchanged — stationary for 5 minutes.Not "moving slowly" — the distance is not changing at all.
- Third section: rising gently — continuing away from the start but more slowly, for 20 minutes.A shallower gradient is a lower speed, in the same direction.
- Fourth section: falling to zero — returning all the way to the starting point in 15 minutes, faster than either outward stage.Downward means coming back; steepness gives the speed.
Out quickly, a five-minute stop, out again more slowly, then a fast return to the start.