Real-life Graphs — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers graphs that describe practical situations: journeys, conversions between units, filling containers, and the changing cost of a service. By the end of this guide you should be able to read values off such a graph, describe what each section shows, and calculate rates from gradients.
You should also be able to tell a distance–time graph from a speed–time graph and say what each of their features means, work out a distance from the area under a speed–time graph, and recognise when a graph is telling a story that is not about motion at all.
The organising idea is that every feature of the graph is a statement about the situation, and translating between the two is the whole skill. A steep section is a fast rate. A flat section is no change in whatever the vertical axis measures. A kink is a moment when something altered. Read the axis labels, and each shape on the page turns into a sentence about what happened.
Key terms and definitions
Distance–time graph — distance on the vertical axis against time on the horizontal. Its gradient is speed.
Speed–time graph — speed on the vertical axis against time. Its gradient is acceleration and the area beneath it is distance.
Gradient — the steepness of a line: the change in the vertical quantity divided by the change in the horizontal one.
Conversion graph — a straight line through the origin used to change one unit into another.
Average speed — total distance divided by total time, which is not generally the average of the individual speeds.
Rate — how quickly one quantity changes with respect to another.
Core concepts
Distance–time graphs
The vertical axis records how far the object is from its starting point, so the gradient is distance divided by time — the speed.
A steeper line means a faster speed. A straight sloping line means the speed is constant. A horizontal line means the distance is not changing, so the object is stationary. A line sloping back down towards zero means the object is returning to where it started.
A curved line means the speed itself is changing, and the speed at any instant is then the gradient of a tangent at that point.
Calculating speed from a journey graph
Take the change in distance and divide by the change in time for that section.
A car covering 150 km in 3 hours has an average speed of 150 ÷ 3 = 50 km/h. The units come straight from the axes: kilometres divided by hours gives km/h.
For a journey in several stages, calculate each section separately. The average speed for the whole journey is the total distance divided by the total time — not the average of the section speeds, because the stages usually last for different lengths of time.
Speed–time graphs
Here the vertical axis records speed, so the gradient is speed divided by time — the acceleration.
A positive gradient means speeding up, a negative gradient means slowing down, and a horizontal line means the speed is constant. That last point is the one most often confused: a flat line here means steady movement, whereas a flat line on a distance–time graph means no movement at all. Check the vertical axis label before saying anything about a horizontal section.
The area beneath a speed–time graph is speed multiplied by time, which is the distance travelled. Split the region into rectangles, triangles and trapeziums, work each one out, and add them.
Conversion graphs
A conversion graph converts between two units — pounds and dollars, miles and kilometres, litres and gallons. Because doubling one quantity doubles the other, the graph is always a straight line through the origin.
To use one, find the known value on its axis, go across or up to the line, and read off the matching value on the other axis. Extending the line is fine, but it becomes less reliable a long way beyond the plotted range.
Other real-life graphs
Not every real-life graph is about motion, and the same reading skills apply.
A container-filling graph shows depth against time. A narrow container fills quickly, giving a steep line; a wide one fills slowly, giving a shallow one. A container that widens as it goes up produces a curve that becomes less steep.
A cost graph may start at a value above zero, showing a fixed charge before any usage, and then rise at a rate given by its gradient — the cost per unit.
In every case the story is the same: read the axis labels first, then translate the shapes.
Worked examples
Example 1: Describing a journey
A journey graph plots distance in kilometres against time in hours. It rises steadily from the origin to 60 km over the first hour, stays level at 60 km for the next half hour, then falls steadily back to 0 km over the following hour and a half.
The first section is a straight rise, so the speed is constant: 60 ÷ 1 = 60 km/h.
The level section means the distance is not changing, so the object is stationary for 30 minutes.
The falling section means it is travelling back towards the start, covering 60 km in 1.5 hours, so its speed is 60 ÷ 1.5 = 40 km/h — slower than the outward leg, which matches the shallower slope.
The total distance travelled is 120 km over 3 hours, so the average speed is 40 km/h.
Example 2: Distance from a speed–time graph
A train's speed rises steadily from rest to 30 m/s over 20 seconds, then stays at 30 m/s for 60 seconds. How far does it travel altogether?
The first stage is a triangle: ½ × 20 × 30 = 300 m.
The second stage is a rectangle: 30 × 60 = 1800 m.
The total distance is 300 + 1800 = 2100 m.
The acceleration in the first stage is the gradient there: 30 ÷ 20 = 1.5 m/s².
Example 3: A conversion graph
A conversion graph passes through the origin and shows that 5 miles is equivalent to 8 kilometres. Convert 20 miles into kilometres.
Because the line passes through the origin, the two quantities are in a fixed ratio. Twenty miles is four times five miles, so the answer is four times eight kilometres: 32 km.
Reading off the graph directly would give the same result, and either method is acceptable — but the ratio method stays exact where a reading from a grid is only as good as the scale allows.
Common mistakes and how to avoid them
Mixing up the two motion graphs. Read the vertical axis label first. A horizontal line means stationary on a distance–time graph but constant speed on a speed–time graph.
Averaging the section speeds. Average speed is total distance divided by total time.
Taking the area under a distance–time graph. It has no meaning. Only the gradient matters on that graph.
Reading a value off the wrong axis. Trace carefully across and up, and state the units with the answer.
Assuming every curve means acceleration. On a container-filling graph a curve means the container's width is changing, which has nothing to do with speed.
Forgetting the units of a gradient. They are always the vertical unit divided by the horizontal unit.
Describing a section without saying what it means. "It goes up" earns nothing; "the car travels at a constant 60 km/h" earns the mark.
Exam technique for "Real-life Graphs"
Label what each axis measures before you start, and keep the units in front of you. Most errors in this topic are errors of interpretation, not arithmetic.
When describing a journey, work through it section by section in order, and say what each section means in terms of the situation rather than the shape.
Show the subtraction when calculating a gradient — for example "60 ÷ 1.5" — because the method mark sits in that fraction.
For an area question, draw the split lines on the graph and give each shape's area separately before adding them.
Always attach units, and match the units the axes use. A speed read from a graph in kilometres and hours is in km/h, not m/s, unless you convert.
Answer "average speed" questions with total distance over total time, and say so explicitly in the working.
Quick revision summary
Read the axis labels first. They decide what the gradient and the area mean.
On a distance–time graph the gradient is the speed: steeper means faster, horizontal means stationary, and a line falling back to zero means returning to the start. A curve means the speed is changing, and a tangent gives the speed at an instant.
On a speed–time graph the gradient is the acceleration and the area beneath is the distance travelled. A horizontal line here means constant speed, not a stationary object.
Average speed = total distance ÷ total time, never the average of the separate speeds.
A conversion graph is a straight line through the origin, so the two quantities stay in a fixed ratio and can be scaled up or down directly.
Other real-life graphs follow the same reading: a container-filling graph shows depth against time, where a steeper line means a narrower container, and a cost graph may begin above zero to show a fixed charge.
Give units with every answer, taken from the axes.