Rates of change (Edexcel GCSE Maths): Revision Notes
Rates of change
What are rates of change?
Rate of change describes how quickly one quantity changes in relation to another. In maths, we often look at how distance, speed, or other measurements change over time using graphs.
A distance-time graph shows how distance changes as time passes. The steepness of the line (called the gradient) tells us the rate of change.
The gradient of a distance-time graph represents the speed of movement - the steeper the line, the faster the object is moving.
Distance-time graphs and pond filling
When water fills ponds of different shapes, the rate at which the water depth increases depends on the pond's shape:
- Wide sections: Water depth increases slowly
- Narrow sections: Water depth increases quickly
This creates different curve shapes on distance-time graphs.
Key Principle: The narrower the section of pond, the faster the water depth will increase.
This concept helps explain why distance-time graphs have different slopes for different situations.
Understanding rainfall graphs
Rainfall graphs show us how precipitation rates change over time, providing valuable insights into weather patterns.
Worked Example: Rainfall Analysis
Supraj recorded rainfall in mm per hour during one afternoon.
From the graph, we can analyse:
Step 1: Identify when rain started
- Rain started at 13:30 (when the line first goes up)
Step 2: Check the claim about 14:00
- The rain did not stop at 14:00 as claimed
- The rain was actually falling steadily from 14:00 to 16:00
Step 3: Determine when rain stopped
- It stopped raining at 17:30 (when the line reaches zero)
Key Insight: The graph shows the rate of rainfall - when the line is higher, rain is falling harder. When the line is flat at the top, rain is falling at a steady rate.
Velocity-time graphs
Velocity means speed in a particular direction. A velocity-time graph shows how velocity changes over time.
The gradient of a velocity-time graph tells you the acceleration (how quickly speed is changing).
Four key patterns to recognise
Essential Patterns on Velocity-Time Graphs
You need to identify these four situations:
-
Constant speed: Horizontal flat line
- Object moves at the same speed throughout
-
Accelerating: Line slopes upward
- Object is speeding up
-
Decelerating: Line slopes downward
- Object is slowing down
-
Not moving: Horizontal line at zero
- Object is stationary
Interpreting car journey graphs
When analysing a velocity-time graph for a car journey, look for these key indicators:
- Flat horizontal sections = constant speed
- Upward sloping sections = accelerating (speeding up)
- Downward sloping sections = decelerating (slowing down)
- Line at zero = not moving (stopped)
Always read graph axes carefully and identify the correct time periods for each phase of movement. Pay close attention to the velocity values at different time intervals.
Key Points to Remember:
- Rate of change shows how quickly one quantity changes compared to another
- Distance-time graphs use gradient to show speed - steeper lines mean faster movement
- Velocity-time graphs show four key patterns: constant speed, accelerating, decelerating, and not moving
- The gradient of a velocity-time graph represents acceleration
- Always read graph axes carefully and identify the correct time periods for each phase of movement