Gradient as rate of change
When a graph models a real situation, the gradient is the RATE at which the y-quantity changes per unit of the x-quantity.
Units of the gradient
The units come straight from the axes: y-units per x-unit.
- Distance (m) vs time (s) → m/s (speed).
- Cost (£) vs quantity (kg) → £/kg (price per kg).
- Volume (litres) vs time (s) → litres/s (flow rate).
- Population vs years → people/year (growth rate).
Reading rate from a straight-line graph
m = (y₂ − y₁) / (x₂ − x₁) using any two distinct points on the line.
Reading rate from a curve
For a curve, the rate changes at each point. Two techniques:
- Average rate over an interval — gradient of the chord.
- Instantaneous rate at a point — gradient of the tangent.
Proportion graphs
If y is directly proportional to x, the graph is a STRAIGHT LINE THROUGH THE ORIGIN. The constant of proportionality k is the gradient: y = kx.
If y is inversely proportional to x, the graph is a hyperbola (y = k/x), NOT a straight line.
If you plot y against 1/x for an inversely-proportional relationship, the graph IS a straight line through the origin (with gradient k).
Detecting proportion from a graph
Three checks:
- Straight line? Yes for direct proportion.
- Passes through (0, 0)? Required.
- Gradient k > 0? Required.
A line of the form y = mx + c with c ≠ 0 is NOT direct proportion (even though it's straight).
✦Worked example— Worked example — flow rate
A tank's volume (litres) vs time (s) gives a straight line from (0, 50) to (60, 230).
- Gradient = (230 − 50) / 60 = 3.
- Units: litres/s.
- Interpretation: water enters at 3 litres per second.
WJEC exam tip
When asked to "interpret" the gradient, always state TWO things: the value AND the meaning in context. "3 litres per second — the tank fills at this rate." A bare numerical answer loses the interpretation mark.
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