Mr Kidman's Classroom
Revise the key ideas
Direct and inverse relationships
- Direct proportion means y changes by the same factor as x: y = kx for constant k. The graph is a straight line through the origin. A straight line with a non-zero intercept is not direct proportion.
Direct and inverse proportion - Worked example: Five notebooks cost £12.50 at a constant unit price. One costs £2.50, so eight cost £20. Here cost = 2.5 × number of notebooks.
- For direct proportion, is constant. Doubling x doubles y; tripling x triples y. Test ratios rather than differences.
- Inverse proportion means y = , so xy is constant. For positive quantities, doubling x halves y. Its graph is a reciprocal curve, not a straight line.
- Worked example: Four equally productive workers take 9 hours for a fixed job. Six workers take (4 × 9) ÷ 6 = 6 hours, assuming work is shared perfectly and each worker's rate is unchanged.
- For a fixed distance, time is inversely proportional to speed. This needs a constant journey length; a general time-versus-speed situation may not be inverse proportion.
Using equations
- To use a proportional equation, find the constant from known values, then substitute the new input. Keep the complete relation, not just the value of k.
- Worked example: y = . When x = 3, y = 8; when x = 8, y = 3. x = 0 is not allowed.
- Read the context carefully: a fixed starting fee, changing productivity or a changing total can invalidate a proportional model.
Higher — powers and constructing models
- If y is directly proportional to x2, write y = kx2. If y is inversely proportional to x2, write y = . State which power the question specifies.
- Worked example: y ∝ x2, and y = 18 when x = 3. Then 18 = 9k, so k = 2 and y = 2x2. At x = 5, y = 50.
The symbol ∝ means “is proportional to”. The equation with k states the constant factor for this particular relationship.
- Worked example: y ∝ , and y = 12 when x = 2. Then k = 48, so y = . At x = 4, y = 3.
When x = 2, x2 = 4. Multiplying y = k ÷ 4 by 4 gives k = 12 × 4 = 48.
- If y ∝ √x and y = 15 at x = 9, then k = 5 and y = 5√x. To find x when y = 20, √x = 4, so x = 16.
- Plotting y against x2 gives a straight line through the origin for y = kx2. Its gradient is k. Plotting against does the same for inverse proportion.
For the first graph, calculate x2 for each input and use that as the horizontal coordinate; y stays on the vertical axis. Do not label that horizontal axis simply x.
