Graphs are often used in real life to show how one quantity changes compared to another — for example, how far someone has travelled over time, or how to change a value from one unit into another. In this topic, you will learn how to read and understand these graphs, and how to draw your own graph from a table of data.
1. Travel Graphs (Distance-Time Graphs) #
On a travel graph, time is shown on the horizontal axis (x-axis) and the distance from the starting point is shown on the vertical axis (y-axis).
The gradient (steepness) of a travel graph tells you the speed:
$$\text{speed} = \text{gradient} = \frac{\text{change in distance}}{\text{change in time}}$$- A steeper line means a faster speed
- A flat (horizontal) line means the object is not moving (it is stationary)
- A line sloping back down means the object is returning towards the starting point
The graph below shows Amir’s journey from home to a shop 60 km away, and back again.
Fig 1: Amir’s journey from home to a shop 60 km away, and back
You can read values directly from the graph. For example, reading up from 4 hours on the time axis to the line, then across to the distance axis: after 4 hours, Amir was 30 km from home (on section C, on his way back).
Formula:
$$\text{speed} = \frac{\text{change in distance}}{\text{change in time}}$$| Section | Time interval | Change in distance | Speed |
|---|---|---|---|
| A | 0 h to 2 h | 60 km (away from home) | $\frac{60}{2} = 30$ km/h |
| B | 2 h to 3 h | 0 km | $\frac{0}{1} = 0$ km/h (resting) |
| C | 3 h to 5 h | 60 km (back towards home) | $\frac{60}{2} = 30$ km/h (returning) |
2. Conversion Graphs #
To use a conversion graph:
- Find the known value on one axis
- Draw a line straight up (or across) until it meets the graph line
- From that point, draw a line across (or down) to the other axis
- Read off the converted value
The graph below converts between miles and kilometres, using the approximation that 5 miles is about 8 km.
Fig 2: Conversion graph between miles and kilometres
Question: use the graph to convert 30 miles into kilometres.
- Find 30 on the miles axis.
- Draw a line straight up until it meets the graph line (shown in orange above).
- Draw a line across from that point to the km axis.
- Read off the value: 30 miles ≈ 48 km
A conversion graph is a straight line, so the gradient (the rate of change between the two quantities) stays the same everywhere on the line. This means you can use the same graph to convert any value, not just the one shown in the example.
3. Drawing a Graph from Given Data #
Sometimes you are given a table of data and asked to draw the graph yourself.
Steps to draw a graph from a table of data:
- Choose a sensible scale for each axis, so the graph uses the space well
- Label both axes with the quantity and its units
- Plot each point from the table accurately
- Join the points in order. In practical graphs, points are usually joined with straight lines, since the speed is constant between each pair of given points
The table shows Priya’s cycle ride to her friend’s house and back to the point where she stopped again.
| Time (minutes) | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| Distance from home (km) | 0 | 2 | 2 | 5 | 5 |
Fig 3: Graph plotted from Priya’s data table above
Each point from the table has been plotted, then joined in order with straight lines. The flat sections (10 to 20 minutes, and 30 to 40 minutes) show Priya was stationary at those times.
- A travel graph (distance-time graph) shows distance from a starting point against time.
- The gradient of a travel graph is the speed: $\text{speed} = \frac{\text{change in distance}}{\text{change in time}}$
- A flat section means stationary (not moving); a line sloping back down means returning to the start.
- A conversion graph is a straight line used to change a value from one unit to another — draw a line to the graph, then across to read the converted value.
- To draw a graph from data: choose a sensible scale, label the axes, plot the points accurately, and join them in order.
