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Maths · Graphs

Graphing rates of change

The gradient of a graph is a rate of change. On a distance-time graph it is the speed; at Higher, on a velocity-time graph it is the acceleration, and the area underneath is the distance.

  • 6 key terms
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Last Lesson and Before

Answer each one, then check.

  1. 1

    Last lesson: what is the gradient between \((0, 0)\) and \((2, 60)\)?

    Show answerHide answer

    30

  2. 2

    A car travels 120 km in 2 hours. What is its average speed?

    Show answerHide answer

    60 km/h

  3. 3

    How many minutes are in 0.75 hours?

    Show answerHide answer

    45 minutes

  4. 4

    Area of a triangle with base 4 and height 12?

    Show answerHide answer

    24

Learning Objectives

  1. 1Read and draw distance-time graphs.
  2. 2Work out speed from the gradient of a distance-time graph.
  3. 3Work out average speed for a whole journey.
  4. 4(Higher) Use the gradient of a velocity-time graph for acceleration, and the area under it for distance.

Distance-Time Graphs

Distance goes up the side, time along the bottom.

  • Gradient is speed

    Speed \(= \dfrac{\text{distance}}{\text{time}}\) - the change up over the change across.

  • Horizontal line

    Not moving: the distance from the start is not changing.

  • Steeper

    Faster.

  • Minutes to hours

    For km/h, change minutes to hours first: 45 minutes \(= 0.75\) hours.

Speed from the Graph

On the journey in the diagram, work out (a) the speed on the way home, and (b) the average speed for the whole journey, including the stop.

Show the solutionHide the solution
  1. 1 (a) Distance and time on the way home 30 km in 45 minutes = 0.75 hours
  2. 2 Speed \(30 \div 0.75 = 40\) km/h
  3. 3 (b) Total distance and total time 60 km in 2 hours 15 minutes = 2.25 hours
  4. 4 Average speed \(60 \div 2.25 = 26.66\ldots\)

Answer(a) 40 km/h (b) 26.7 km/h

Other Rates of Change

Any graph of a quantity against time has a gradient that is a rate.

  • Water level against time

    Gradient = how fast the level rises (cm per minute).

  • Cost against number

    Gradient = cost of one more item.

  • Units of the gradient

    Units up the side "per" units along the bottom: km per hour, litres per minute.

  • Curved graphs

    A curve means the rate is changing: steeper parts change faster.

Velocity-Time Graphs

Now velocity (speed) is up the side.

  • Gradient is acceleration

    Acceleration \(= \dfrac{\text{change in velocity}}{\text{time}}\), in m/s².

  • Negative gradient

    Deceleration: slowing down.

  • Horizontal line

    Constant speed - NOT stopped.

  • Area is distance

    The area under the graph is the distance travelled. Split it into triangles, rectangles and trapeziums.

A Car Journey

A car accelerates uniformly from rest to 12 m/s in 4 seconds, travels at 12 m/s for 10 seconds, then slows uniformly to rest in 6 seconds. Work out (a) the acceleration in the first 4 seconds, and (b) the total distance travelled.

Show the solutionHide the solution
  1. 1 (a) Gradient of the first section \(12 \div 4 = 3\) m/s²
  2. 2 (b) Area: first triangle \(\frac{1}{2} \times 4 \times 12 = 24\)
  3. 3 Rectangle \(10 \times 12 = 120\)
  4. 4 Last triangle \(\frac{1}{2} \times 6 \times 12 = 36\)
  5. 5 Add \(24 + 120 + 36 = 180\)

Answer(a) 3 m/s² (b) 180 m

Don't Mix Them Up

Distance-time

  • Gradient = speed
  • Horizontal = stopped
  • Going down = coming back
  • Area means nothing useful

Velocity-time (Higher)

  • Gradient = acceleration
  • Horizontal = constant speed
  • Going down = slowing down
  • Area = distance travelled

Tell the Story

Draw a distance-time graph for this journey: Priya walks 2 km to a friend's house at 4 km/h, stays for 20 minutes, then they both cycle 6 km to the park in 20 minutes. Then write your own journey story for a partner to graph.

1. Work out each time first.

2. Plot the corners of the journey.

3. Join with straight lines.

A good answer shows: Walk: 2 km in 30 minutes. Flat for 20 minutes (to 50 minutes). Cycle: from 2 km to 8 km between 50 and 70 minutes - a steeper line, 18 km/h.

Can I...?

  1. 1Read a distance-time graph.
  2. 2Draw a distance-time graph.
  3. 3Work out speed from a gradient.
  4. 4Work out an average speed.
  5. 5(Higher) Work out acceleration from a velocity-time graph.
  6. 6(Higher) Work out distance from the area under a velocity-time graph.

Summary & Exam Focus

  • Distance-time: gradient = speed; flat = stopped.
  • Average speed = total distance ÷ total time.
  • (Higher) Velocity-time: gradient = acceleration; area = distance.

Exam focus

The distance-time graph shows Tom's cycle ride. Work out Tom's speed on his way home, in km/h. (2 marks) (2 marks)

Read the scales carefully - small squares are often 5 or 10 minutes, not 1. Convert minutes to hours before dividing to get km/h.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Speed
Distance travelled per unit of time.
Average speed
Total distance divided by total time, including stops.
Rate of change
How fast one quantity changes compared with another; the gradient.
Velocity
Speed in a given direction.
Acceleration
Rate of change of velocity, in m/s².
Deceleration
Slowing down; a negative acceleration.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 4 marks

    The distance-time graph shows Tom's cycle ride from his home to a lake and back. (a) At what time did Tom arrive at the lake? (b) How long did Tom stay at the lake? (c) Work out Tom's speed on his way home, in km/h.

    A distance-time graph of Tom's ride: 30 km out by 10:00, stopped until 10:30, and home by 11:15.
    Show answerHide answer

    Model answer

    (a) 10:00 (b) 30 minutes (c) 30 km in 45 minutes: \(30 \div 0.75 = 40\) km/h

    Mark scheme

    • (a) 10:00 — B1
    • (b) 30 minutes — B1
    • (c) 30 km and 45 minutes, or \(30 \div 0.75\) — M1
    • (c) 40 km/h — A1
  2. Question 2 Calculator 2 marks

    For the whole of his ride, including the stop, Tom travelled 60 km in 2 hours 15 minutes. Work out his average speed. Give your answer to 1 decimal place.

    Show answerHide answer

    Model answer

    \(60 \div 2.25 = 26.66\ldots = 26.7\) km/h

    Mark scheme

    • \(60 \div 2.25\) — M1
    • 26.7 km/h — A1
  3. Question 3 Non-calculator 2 marks

    Water is poured into a tank at a constant rate. After 4 minutes the depth is 10 cm. After 10 minutes the depth is 25 cm. Work out the rate at which the depth increases. Give the units of your answer.

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    Model answer

    \(\dfrac{25 - 10}{10 - 4} = \dfrac{15}{6} = 2.5\) cm per minute

    Mark scheme

    • \(\dfrac{25 - 10}{10 - 4}\) — M1
    • 2.5 cm per minute — A1
  4. Question 4 Calculator · Higher 4 marks

    A car accelerates uniformly from rest to 12 m/s in 4 seconds. It travels at 12 m/s for 10 seconds, then slows down uniformly and stops after another 6 seconds. (a) Work out the acceleration in the first 4 seconds. (b) Work out the total distance the car travels.

    Show answerHide answer

    Model answer

    (a) \(12 \div 4 = 3\) m/s² (b) \(\frac{1}{2} \times 4 \times 12 + 10 \times 12 + \frac{1}{2} \times 6 \times 12 = 24 + 120 + 36 = 180\) m

    Mark scheme

    • (a) 3 m/s² — B1
    • (b) Splitting the area into parts, or using the trapezium rule with parallel sides 20 and 10 — M1
    • (b) One correct part: 24, 120 or 36 — M1
    • (b) 180 m — A1

Quick check

  1. What does a horizontal line on a distance-time graph mean?

    1. AConstant speed
    2. BSpeeding up
    3. CComing back
    4. DStopped
    Show answerHide answer

    D: Stopped

    The distance from the start is not changing, so the object is not moving.

  2. A distance-time graph rises 12 km in 20 minutes. What is the speed?

    1. A0.6 km/h
    2. B36 km/h
    3. C24 km/h
    4. D240 km/h
    Show answerHide answer

    B: 36 km/h

    20 minutes is \(\frac{1}{3}\) of an hour, so \(12 \times 3 = 36\) km/h.

  3. (Higher) What does the area under a velocity-time graph give?

    1. ADistance travelled
    2. BAcceleration
    3. CAverage speed
    4. DTime taken
    Show answerHide answer

    A: Distance travelled

    Area = velocity × time = distance.

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