Gradient as a rate of change
- The gradient of a graph shows how fast one quantity changes compared with another.
- On a distance-time graph the gradient is the speed. On a velocity-time graph the gradient is the acceleration.
- A straight line shows a constant rate of change. A horizontal line on a velocity-time graph shows a constant velocity.
Average and instantaneous rates of change
- The average rate of change between two points is the gradient of the chord (straight line) joining them. A car that travels 120 m in 8 seconds has an average speed of 15 m/s.
- The instantaneous rate of change at one point is the gradient of the tangent to the curve at that point.
- To find the gradient of a tangent, pick two points on it that are far apart, then work out change in y ÷ change in x.
Area under a graph
- The area under a velocity-time graph is the distance travelled.
- Split the area into triangles, rectangles and trapeziums. A car accelerating from 0 to 20 m/s in 10 seconds travels ½ × 10 × 20 = 100 m.
- For a curve, split the area into strips (trapeziums) to estimate it.
Key terms
- Rate of change
- How fast one quantity changes compared with another.
- Gradient
- Change in y ÷ change in x.
- Chord
- A straight line joining two points on a curve.
- Tangent
- A straight line that touches a curve at one point.
- Instantaneous rate of change
- The rate of change at one moment: the gradient of the tangent.
- Acceleration
- The rate of change of velocity: the gradient of a velocity-time graph.