Ratio, Proportion and Rates of Change in AlgebraAQA GCSE Maths: Revision notes
Section 1
How Do Graphs Model Real-Life Rates?
Graphs of real-life situations can be plotted and interpreted to find approximate solutions to problems involving distance, speed and acceleration — these are examples of kinematic problems.
The gradient of a straight-line graph represents a rate of change: for example, on a distance–time graph the gradient is speed, and on a speed–time graph the gradient is acceleration.
On a distance–time graph, a steeper line means a greater speed, because speed is distance time — the gradient.
Section 2
How Do Graphs Show Direct and Inverse Proportion?
Graphs can be used to recognise and interpret direct proportion and inverse proportion:
- A direct proportion graph is a straight line through the origin
- An inverse proportion graph is a curve that decreases as increases, never touching either axis
Section 3
How Do We Estimate Gradients and Areas on Non-Linear Graphs? (Higher)
For curved graphs (quadratic and other non-linear graphs), you can estimate the gradient at a point by drawing a tangent to the curve at that point and finding its gradient — this gives the instantaneous rate of change.
The area under a graph can be estimated (e.g. by splitting it into trapezia) and interpreted meaningfully — for example, the area under a speed–time graph represents distance travelled. These skills apply to distance–time graphs, velocity–time graphs and graphs in financial contexts.
An average rate of change between two points is found using the gradient of the chord connecting them, whereas the instantaneous rate of change at a single point uses the gradient of the tangent at that point.
Remember: chord = average rate of change over an interval; tangent = instantaneous rate of change at one point.
Must Know
- The gradient of a straight-line graph represents a rate of change (e.g. speed on a distance–time graph)
- A direct proportion graph is a straight line through the origin; an inverse proportion graph is a decreasing curve
- The gradient of a tangent at a point on a curve gives the instantaneous rate of change
- The gradient of a chord between two points gives the average rate of change over that interval
- The area under a graph can represent a real quantity, e.g. distance travelled under a speed–time graph
- These techniques apply to distance–time, velocity–time, and financial-context graphs
That's the notes covered.
Carry on to the next subtopic.