Graphs in practical situations.
Read the journey in the shape: gradients show rates and speed–time areas show distances.
Draw conversion and travel graphs · interpret gradients and tangents · calculate speed–time areas
Read axes, scales and units first
A practical graph links two quantities from a situation. A point is meaningful only when you know what each axis measures, its units and the value represented by each grid interval.
Read the axis scale before reading a coordinate. A conversion graph gives two equivalent descriptions of the same quantity; a journey graph instead records how movement changes over time. Identify which kind you have so that the meaning of a slope or intercept is not guessed.
A graph connects quantities. Read the axis labels, scale and units before interpreting a coordinate or calculating a gradient.
Draw a conversion graph when 1 USD corresponds to 1.5 units of B.
Plot (0, 0), (10, 15), (20, 30) and (30, 45), then join with a straight line. The exchange rate here is an illustrative value provided for the problem.
Pause and explainWhy should you not assume one grid square means one unit?
The labelled scale determines the interval. A square might represent 5 minutes, 20 kilometres or another value.
Distance–time gradient gives speed
A distance–time graph tracks the distance travelled or distance from a reference point, as stated by the axis. Its slope compares a change in distance with the time taken for that change.
For a straight rising segment, equal time intervals give equal distance increases, representing constant speed. A horizontal segment means the plotted distance is unchanged. Read exactly what the distance axis represents before interpreting a downward segment as returning towards a starting point.
Gradient = change in distance ÷ change in time. A horizontal section means no change in distance, so the object is stationary. A steeper distance–time graph means greater speed.
Read the journey shown.
First section: 60 km ÷ 2 h = 30 km/h. Stop: 1 h. Last section: (120 − 60) km ÷ (6 − 3) h = 20 km/h.
Average speed for the whole journey = 120 ÷ 6 = 20 km/h.
For a curved distance–time graph, draw a tangent at the required instant. Its gradient estimates the speed at that instant.
Pause and explainWhy does a steeper distance–time segment represent greater speed?
More distance changes during the same time interval, so distance divided by time is larger.
Speed–time gradient gives acceleration
A speed–time graph puts speed, not distance, on the vertical axis. Its height tells you how fast the object is moving at a moment. Its gradient describes how speed changes with time.
Distance over a period comes from the area under the graph: constant speed gives a rectangle, and steadily changing speed can give a triangle or trapezium. Split the region into familiar shapes and check that the speed and time units combine into the required distance unit.
Find acceleration, deceleration and total distance for this graph.
First acceleration: (12 − 0) ÷ 4 = 3 m/s². Last acceleration: (0 − 12) ÷ 4 = −3 m/s²; deceleration magnitude is 3 m/s².
= 24 + 48 + 24 = 96 m
Split linear sections into triangles, rectangles or trapezia. For a trapezium of duration t with end speeds u and v, area = 12(u + v)t.
Pause and explainDoes a horizontal speed–time line mean the object is stopped?
Only if it is at speed zero. A horizontal line above zero represents a constant non-zero speed.
Your turn. Include units.
Name the graph type and the meaning of each axis before doing arithmetic. Decide whether the question needs a coordinate, gradient, interval or area.
Write units through the calculation. In a journey problem, connect the mathematical result back to the situation: moving, waiting, accelerating or completing a total distance.
A straight distance–time section joins (2 h, 40 km) and (5 h, 100 km). Find the speed.
Reveal worked answer
Speed rises uniformly from 5 m/s to 17 m/s in 6 s. Find acceleration and distance.
Reveal worked answer
What does a horizontal section on a speed–time graph mean?