GCSE · Maths · AQA · Spec 8300 · Foundation
Gradient as rate of change; proportion graphs
Pounds per kilogram, miles per hour, litres per minute: behind every 'per' is a straight-line graph, and its gradient is that 'per'.
Lift the line: the gradient is the price per kilogram
This line shows cost = gradient × mass, so it starts at (0, 0). Lift the point and the whole line tilts: a steeper line means each kilogram costs more. The gradient is the rise for a run of 1 kg — the price per kilogram. At the starting setting, 4 kg cost £10 and 8 kg cost £20: 10 ÷ 4 = 2.5 and 20 ÷ 8 = 2.5. Whatever the setting, cost ÷ mass equals the gradient at every point on this line, so the ratio cost : mass never changes along it — 2.5 : 1 at the start.
Direct, inverse — or neither?
Put each graph in its category before you read why. Two of them are traps.
Still to sort
Direct proportion (0)
A straight line through the origin: y = kx.
Where the line is: Straight is not enough — the line must pass through (0, 0), so that doubling x doubles y.
Inverse proportion (0)
A curve y = k/x: doubling x halves y.
Where the line is: Falling is not enough — a falling straight line is not inverse proportion.
Neither (0)
Fails both tests, even if it looks tidy.
Where the line is: A straight line that misses the origin is linear, but not proportional.
One direct, one inverse — solved both ways
Problem
(a) y is directly proportional to x, and y = 15 when x = 6. Find y when x = 10. (b) y is inversely proportional to x, and y = 8 when x = 3. Find y when x = 6, and find x when y = 2.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Every 'per' is a gradient — and a straight line through the origin is direct proportion
What you need to know
- Gradient = change in y ÷ change in x. As a rate, it is how much y changes for each 1 unit increase in x.
- The units of a gradient are y-units per x-unit, such as £ per hour or cm per minute.
- A straight line has the same gradient everywhere, so it shows a constant rate of change.
- Direct proportion: y = kx, a straight line through the origin with gradient k.
- Inverse proportion: y = k/x. For positive x and k it is a curve that falls as x increases and never meets either axis.
- If y : x is always a : b, then y = (a/b)x — the fraction a/b is the gradient of a line through the origin.
The big picture
The gradient of a straight-line graph is a rate of change: the change in y for each 1 unit of x, with units such as £ per kg. On a straight line it is the same everywhere. Direct proportion is a straight line through the origin, y = kx, where k is the gradient and the constant value of y ÷ x. Inverse proportion is the curve y = k/x, where x × y is constant. Proportion problems are solved by finding k from one known pair, then using the equation or reading the graph.
Key points
Worked example
Problem
A straight-line graph shows the fuel left in a car's tank (litres, up) against the distance driven (km, across). It passes through (0, 45) and (600, 0). Find the gradient and explain what it means, with units.
⚠ Watch out
Dividing one point's y-value by its x-value to find the gradient of a line that does not pass through (0, 0), or calling any falling graph 'inverse proportion'.
Memory hook
Gradient means 'per'. Direct: straight through zero, double x doubles y. Inverse: a curve, double x halves y.
Check yourself
A straight line through the origin passes through (4, 14). What is its gradient, what is its equation, and what is y when x = 10?
Flashcards
(9)What does the gradient of a straight-line graph tell you?
What are the units of a gradient?
What does a graph of direct proportion look like?
What does a graph of inverse proportion look like?
In y = kx, what is k on the graph?
When does y ÷ x at a single point give the gradient?
How do you find k when y is inversely proportional to x?
What does a negative gradient mean as a rate?
If y : x is always 3 : 2, what linear function links them?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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