Gradients, Rates of Change, and Proportion
Graphs are a powerful way to see how things change. This guide shows you how the steepness (gradient) of a graph represents a rate of change, and how to recognise direct and inverse proportion from real-world graphs.
The Core Concepts
Gradient as a Rate of Change
The gradient of a line tells you how quickly the $y$-value changes compared to the $x$-value. In real-world graphs this represents a rate of change.
Example: On a distance-time graph, gradient = change in distance ÷ change in time = speed.
Direct vs. Inverse Proportion
Direct proportion: As one quantity increases, the other increases at the same rate. The graph is a straight line through the origin.
Inverse proportion: As one increases, the other decreases. The graph is a curve that never touches either axis.
Worked Example: Distance-Time Graph
A cyclist’s journey is shown on the graph. Calculate their speed during segment A.
Calculate Speed for Segment A
- Start: $(0 \text{ min},\ 0 \text{ km})$. End: $(30 \text{ min},\ 10 \text{ km})$.
- Rise (distance): $10 – 0 = 10 \text{ km}$.
- Run (time): $30 – 0 = 30 \text{ min}$.
- Gradient (speed): $\dfrac{10}{30} = \dfrac{1}{3} \text{ km/min}$.
Convert to km/h: $\dfrac{1}{3} \times 60 = \mathbf{20 \text{ km/h}}$.
Tutor Insights
🤔 Common Misunderstandings
- Forgetting what the gradient means. Students can calculate a gradient but struggle to explain “a gradient of 20 represents a speed of 20 km/h”.
- Confusing direct and inverse graphs. Direct = straight line through the origin. Inverse = curve that never touches either axis.
📝 Common Exam Mistakes
- Mixing up rise and run when calculating the gradient (doing run ÷ rise instead of rise ÷ run).
- Incorrect or missing units (e.g., forgetting km/h or writing the wrong ones).
- Not checking the origin for direct proportion. A straight line is not enough — it must pass through (0, 0).
Practice Questions
Q1. The graph shows the volume of water in a tank over time. What does the gradient represent? Calculate the rate, including units.
Q2. A car travels at constant speed. Distance $d$ is directly proportional to time $t$. Which equation describes this?
a) $d = \dfrac{k}{t}$ b) $d = kt$ c) $d = k + t$
Show Answers
Q1. The gradient represents the rate at which the tank drains (flow rate).
Using points $(0, 25)$ and $(5, 0)$: gradient $= \dfrac{0-25}{5-0} = -5$.
Answer: 5 litres per minute (the negative sign indicates draining).
Q2. Answer: b) $d = kt$. This is the only equation representing a straight line through the origin.
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