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9.7 More Real-life Graphs

GCSE Maths Foundation · Edexcel · Chapter 9: Graphs

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: gradient as a rate of change, filling containers, speed-time graphs and interpreting curved graphs. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Interpret the gradient of a real-life graph as a rate of change, with units
  • Match the shape of a container to the graph of depth against time as it fills
  • Read a speed-time graph and work out an acceleration from its gradient
  • Compare rates on different parts of a real-life graph

Key words

  • Rate of change: How fast one quantity changes compared with another, e.g. litres per minute.
  • Constant rate: Changing by the same amount every unit of time. The graph is a straight line.
  • Speed-time graph: A graph with time across and speed up the side.
  • Acceleration: How quickly speed increases, e.g. in metres per second per second (m/s²).
  • Deceleration: Slowing down: a negative acceleration.

Gradient as a rate of change

On any real-life graph, the gradient tells you how fast the quantity on the vertical axis is changing. Its units are “vertical units per horizontal unit”.

Worked example: filling a tank

The volume goes from 20 litres to 140 litres in 8 minutes.
Rate = change in volume ÷ change in time = 120 ÷ 8 = 15 litres per minute.
The line crosses the axis at 20, so there were already 20 litres in the tank at the start.

Answer: 15 litres per minute

Key fact: A negative gradient means the quantity is going down, e.g. water draining from a bath at 6 litres per minute.

Filling containers

Water flows into each container at a constant rate. Think about how fast the depth rises. Where the container is narrow the depth rises quickly (steep graph). Where it is wide the depth rises slowly (shallow graph).

Speed-time graphs

On a speed-time graph a line sloping up means speeding up, a horizontal line means moving at a constant speed (not stopped), and a line sloping down means slowing down.

Worked example: acceleration

From 0 to 10 s the speed rises from 0 to 20 m/s. Acceleration = 20 ÷ 10 = 2 m/s².
From 10 s to 22 s the car travels at a constant 20 m/s for 12 seconds.
From 22 s to 30 s the speed falls by 20 m/s in 8 s. Deceleration = 20 ÷ 8 = 2.5 m/s².

Answer: acceleration 2 m/s², deceleration 2.5 m/s²

Comparing rates on a graph

When a graph is made of different straight sections, work out the rate for each section and compare. The steeper section has the greater rate.

Worked example: a phone battery

0 to 30 minutes: 80 − 20 = 60% in 30 minutes = 2% per minute.
30 to 70 minutes: 100 − 80 = 20% in 40 minutes = 0.5% per minute.
From 70 minutes the graph is flat: the battery is full.

Answer: it charges 4 times as fast at first

Common mistakes

  • Giving a rate without units. Say “15 litres per minute”, not just 15.
  • Thinking a horizontal line on a speed-time graph means stopped. It means constant speed. Stopped is speed 0.
  • Mixing up container shapes: a wide part fills slowly, so the depth graph is less steep there.
  • Reading the change in y as the final value. Rate = (end value − start value) ÷ time taken.

Revision questions

Try these, then check your answers below each one.

Question 1. The volume of water in a tank rises steadily from 30 litres to 150 litres in 8 minutes. At what rate does it fill, in litres per minute?

Answer: 15 litres per minute

Working: (150 − 30) ÷ 8 = 120 ÷ 8 = 15 litres per minute.

Question 2. Water is poured at a constant rate into a cylinder with straight vertical sides. Which describes the graph of depth against time? (Choose from: A straight line, A curve that gets steeper, A curve that gets less steep, A horizontal line.)

Answer: A straight line

Working: The width does not change, so the depth rises by the same amount each second: a straight line.

Question 3. Water is poured at a constant rate into a vase that gets wider towards the top. Which describes the graph of depth against time? (Choose from: A curve that gets less steep, A curve that gets steeper, A straight line, A horizontal line.)

Answer: A curve that gets less steep

Working: As the vase widens the depth rises more and more slowly, so the graph gets less steep.

Related lessons

Continue in the full course

This lesson is section 9.7 (EGCMF0907) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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