Gradient as Rate of Change
Gradient as Rate of Change
The gradient of a straight-line graph measures how fast the vertical quantity changes relative to the horizontal quantity. This is a rate of change — and its units are always (units of y-axis) per (units of x-axis). Recognising this connection turns every linear graph into a rich source of real-world information.
Interpreting the Gradient
For any two distinct points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line:
\[ \text{gradient} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{change in } y}{\text{change in } x} \]In context, this equals the rate of change of the \(y\)-quantity with respect to the \(x\)-quantity.
| Graph | Gradient means |
|---|---|
| Distance–time | Speed (m/s, km/h) |
| Velocity–time | Acceleration (m/s²) |
| Cost–units | Unit cost (£ per item) |
| Temperature–time | Rate of cooling/heating (°C per minute) |
| Depth–time | Rate of filling/draining (cm per second) |
Direct and Inverse Proportion on Graphs
Direct proportion \( y = kx \): straight line through origin, gradient \( = k \). Doubling \( x \) doubles \( y \). The gradient is the constant of proportionality.
Inverse proportion \( y = k/x \): hyperbola — not a straight line. As \( x \) increases, \( y \) decreases; the graph approaches both axes asymptotically. The gradient changes at every point.
A linear graph that does not pass through the origin (\( y = mx + c \) with \( c \neq 0 \)) represents a linear relationship but not direct proportion.
Worked Examples
A cost–quantity graph passes through (0, 5) and (10, 35). Find the gradient and interpret it in context.
\[ \text{gradient} = \frac{35-5}{10-0} = \frac{30}{10} = 3 \]The gradient of 3 means the cost increases by £3 per item purchased. The y-intercept of £5 is a fixed charge regardless of quantity.
A graph of volume \( V \) (litres) against time \( t \) (minutes) passes through the origin with gradient 2.5. Write the equation and state what the gradient means.
\[ V = 2.5t \]This is direct proportion: \( V \propto t \). Gradient = 2.5 litres per minute — the rate at which water is filling the tank.
Explain how you would recognise an inverse proportion graph from its shape, and how it differs from a direct proportion graph.
Direct proportion: straight line through origin, constant gradient. Inverse proportion: decreasing curve in Q1 approaching both axes but never crossing them. Gradient is not constant — it changes at every point, becoming less steep as \( x \) increases.
Key Takeaways
- Gradient = rate of change = (change in \(y\)) ÷ (change in \(x\)). Units: y-units per x-unit.
- A straight line through the origin represents direct proportion; the gradient is the constant of proportionality \( k \).
- A straight line not through the origin is linear but not proportional — it has a non-zero y-intercept (fixed component).
- An inverse proportion graph is a hyperbola — a decreasing curve with asymptotes along both axes. It is not a straight line.
- Always read axis labels carefully before interpreting the gradient — units determine the physical meaning.