Year 9 Maths The Thai Guy

Lesson 12.5 · 25 min

Real-Life Graphs

Distance–time and speed–time graphs: what the gradient means when the axes are not just x and y.

In this lesson you will learn to…

  • Read distance, time and speed from a distance–time graph
  • Work out a speed from the gradient of a section
  • Interpret a flat section, a steep section and a return journey
  • Find acceleration and deceleration from a speed–time graph
  • Calculate an average speed over a whole journey

The same maths, different labels

You already know how to find a gradient. Real-life graphs simply put meaningful quantities on the axes, and then the gradient stops being an abstract number and becomes something you can describe in words.

Vertical axisHorizontal axisThe gradient means
DistanceTimeSpeed
SpeedTimeAcceleration
CostNumber of itemsPrice per item
VolumeTimeFlow rate

That is the whole idea of the topic. Look at the axes, and the gradient tells you the rate.

Reading a distance–time graph

  • A steep line — covering a lot of distance in little time, so a high speed.
  • A gentle line — slower.
  • A flat line — the distance is not changing at all, so the object is stationary. This is the single most commonly misread feature: a flat line does not mean “constant speed”.
  • A line coming back down — the distance from the starting point is decreasing, so the object is returning.

Average speed is not the average of the speeds

If you drive 60 km/h for one hour and 20 km/h for three hours, the average speed is not 40 km/h. Work it out properly:

average speed=total distancetotal time=60+604=30 km/h\text{average speed} = \frac{\text{total distance}}{\text{total time}} = \frac{60 + 60}{4} = 30 \text{ km/h}

Any time spent stopped counts in the total time. This is exactly why the answer is lower than you might guess.

Speed–time graphs

Swap the vertical axis to speed and everything shifts meaning:

  • A flat line now means a constant speed — still moving, just not changing.
  • An upward slope means accelerating.
  • A downward slope means decelerating, and the gradient comes out negative.

Because the two graph types look so similar, always read the vertical axis label before saying anything about a flat section. Getting that wrong turns a correct calculation into a wrong answer.

Units follow the axes. Distance in km and time in hours gives km/h. Distance in metres and time in seconds gives m/s. Speed in m/s divided by time in seconds gives m/s2\text{m/s}^2. Write the units down as you go and they will look after themselves.

Key rules and formulas

  • Speed from a distance–time graph

    A steeper section means a faster speed. A flat section means distance is not changing — the object is stationary.

  • Acceleration from a speed–time graph

    This is the gradient again, but of a speed–time graph. A downward slope is a deceleration and comes out negative.

  • Average speed

    Use the whole journey, including any time spent stopped. It is almost never the average of the individual speeds.

  • Read the axes first

    A flat line on a distance–time graph means not moving. A flat line on a speed–time graph means moving at a steady speed. Same shape, opposite meaning — always check the vertical axis label.
Try it

Gradient as a rate

Equation
y = 3x + 0
Crosses the y-axis at
(0, 0)
For every 1 across
3 up

Imagine the horizontal axis is time in hours and the vertical axis is distance in km. Then the gradient is the speed. Set in your head: 60 km every hour. A flat line () means stopped.

Worked examples

Worked example 1

Speed from a section

On a distance–time graph, a car travels from 0 km to 40 km in the first hour. Find its speed.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

A journey with a stop

A cyclist rides 40 km in 1 hour, rests for 1.5 hours, then rides a further 15 km in 2.5 hours. Find the average speed for the whole journey.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Deceleration

On a speed–time graph, a car slows from 25 m/s to 0 m/s over 12 seconds. Find the deceleration.

Try it yourself first, then reveal the solution one step at a time.

Practice: real-life graphs

9 questions

Nine questions, including three where you click straight on a distance–time graph, and a timed speed drill.
  1. Question 1 of 9Click the diagram
    This is a distance–time graph for a journey. Click the section where the speed is greatest.

    Click the right part of the diagram. You can also tab to each part and press Enter.

  2. Question 2 of 9Click the diagram
    On the same graph, click the section where the object is not moving.

    Click the right part of the diagram. You can also tab to each part and press Enter.

  3. Question 3 of 9Type the answerFoundation
    A car travels 40 km in the first hour of its journey. What is its speed, in km/h?
    km/h
  4. Question 4 of 9Multiple choice
    On a speed–time graph, what does a horizontal (flat) line mean?
  5. Question 5 of 9Step by step
    Find an average speed for a whole journey.
    A cyclist rides 40 km in 1 hour, rests for 1.5 hours, then rides a further 15 km in 2.5 hours.
    1. What is the total distance travelled, in km?

  6. Question 6 of 9Type the answerChallenge
    A car slows from m/s to m/s over 12 seconds. Find the size of its deceleration, in . Give your answer to 1 decimal place.
    m/s²
  7. Question 7 of 9Match the pairs
    Match each pair of axes to what its gradient measures.

    Pick an answer, then choose the row it belongs to. You can also drag it.

    • Distance against time
    • Speed against time
    • Cost against number of items
    • Volume against time

  8. Question 8 of 9True or falseChallenge
    Driving at 60 km/h for 1 hour and then 20 km/h for 3 hours gives an average speed of 40 km/h.
  9. Question 9 of 9Quick-fire drill
    Quick-fire: speed, distance and time. Give numbers only.

    8 questions · 90 seconds · answer as many as you can.