EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Graphing rates of change
Graphs · Lesson 3 of 8
Last Lesson and Before
Answer each one, then check.
1. Last lesson: what is the gradient between (0, 0) and (2, 60)?
30
2. A car travels 120 km in 2 hours. What is its average speed?
60 km/h
3. How many minutes are in 0.75 hours?
45 minutes
4. Area of a triangle with base 4 and height 12?
24
Learning Objectives
1. Read and draw distance-time graphs.
2. Work out speed from the gradient of a distance-time graph.
3. Work out average speed for a whole journey.
4. (Higher) Use the gradient of a velocity-time graph for acceleration, and the area under it for distance.
Reading a Distance-Time Graph
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On a distance-time graph the gradient is the speed. The first section rises 30 km in 1 hour: 30 km/h. The flat section is a 30-minute stop. The return covers 30 km in 45 minutes: 30 ÷ 0.75 = 40 km/h. |
Steeper means faster; flat means stopped; going down means coming back. |
Distance-Time Graphs
Distance goes up the side, time along the bottom.
▸ Gradient is speed. Speed = distance/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
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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. |
1. (a) Distance and time on the way home
30 km in 45 minutes = 0.75 hours
2. Speed
30 ÷ 0.75 = 40 km/h
3. (b) Total distance and total time
60 km in 2 hours 15 minutes = 2.25 hours
4. Average speed
60 ÷ 2.25 = 26.66…
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.
PART TWO · HIGHER
Velocity-Time Graphs
Gradient is acceleration; area is distance.
Velocity-Time Graphs
Now velocity (speed) is up the side.
▸ Gradient is acceleration. Acceleration = (change in velocity)/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
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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. |
1. (a) Gradient of the first section
12 ÷ 4 = 3 m/s²
2. (b) Area: first triangle
½ × 4 × 12 = 24
3. Rectangle
10 × 12 = 120
4. Last triangle
½ × 6 × 12 = 36
5. Add
24 + 120 + 36 = 180
Answer: (a) 3 m/s² (b) 180 m
Don't Mix Them Up
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DISTANCE-TIME |
VELOCITY-TIME (HIGHER) |
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▸ Gradient = speed ▸ Horizontal = stopped ▸ Going down = coming back ▸ Area means nothing useful |
▸ Gradient = acceleration ▸ Horizontal = constant speed ▸ Going down = slowing down ▸ Area = distance travelled |
Key Terms
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Speed Distance travelled per unit of time. |
Average speed Total distance divided by total time, including stops. |
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Rate of change How fast one quantity changes compared with another; the gradient. |
Velocity Speed in a given direction. |
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Acceleration Rate of change of velocity, in m/s². |
Deceleration Slowing down; a negative acceleration. |
Your Task: Tell the Story
10 minutes
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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...?
☐ Read a distance-time graph.
☐ Draw a distance-time graph.
☐ Work out speed from a gradient.
☐ Work out an average speed.
☐ (Higher) Work out acceleration from a velocity-time graph.
☐ (Higher) Work out distance from the area under a velocity-time graph.
Summary
✓ Distance-time: gradient = speed; flat = stopped.
✓ Average speed = total distance ÷ total time.
✓ (Higher) Velocity-time: gradient = acceleration; area = distance.
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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) Read the scales carefully - small squares are often 5 or 10 minutes, not 1. Convert minutes to hours before dividing to get km/h. |