Summary — What This Chapter Covers
- Direct proportion (y ∝ x): as x increases, y increases by the same factor. Equation: y = kx. Graph: a straight line through the origin.
- Direct proportion with powers/roots: y can be proportional to x², x³, √x, or ∛x — not just x itself.
- Inverse proportion (y ∝ 1/x): as x increases, y decreases by the same factor. Equation: y = k/x. Graph: a curve that never touches the axes.
- Inverse proportion with powers/roots: y can be proportional to 1/x², 1/x³, 1/√x, or 1/∛x.
- The constant of proportionality, k, is found by substituting one known pair of values into the general formula.
- The 4-step method — identify variables, form the equation with k, solve for k, rewrite and use — works for every single proportion question, direct or inverse.
- Variation models (linear, quadratic, cubic, reciprocal) let you match real data to the correct type of equation, and test/compare which model fits best.
1. Direct Proportion
What does "directly proportional" actually mean? Imagine you're paying for fuel. If 1 litre costs $2, then 2 litres cost $4, and 10 litres cost $20. Every time the amount of fuel doubles, the price doubles too. The ratio between the two quantities (price ÷ litres) never changes — it's always $2 per litre. That's direct proportion in a nutshell: two quantities that scale up and down together, always keeping the same ratio.
We write "y is directly proportional to x" using the special symbol ∝, like this: y ∝ x. This symbol is just shorthand — it doesn't give you a usable equation on its own. To actually calculate anything, you need to convert it into an equation using a constant, k:
Here's what that line looks like — notice it always passes through the origin, which makes sense: if x = 0 (you buy 0 litres of fuel), y must be 0 too (you pay $0).
y
| ●
| ●
| ●
| ●
|●________________________ x
0
A straight line through the origin.
Steeper line = bigger k.
Why does this matter?
Because k is CONSTANT, once you know it, you can predict y for ANY value of x — even ones you haven't been given. That's the whole power of proportion: one piece of data (one known x-y pair) unlocks the entire relationship.
1Direct Proportion with Powers and Roots
Not every direct proportion is as simple as y = kx. Sometimes y is proportional to a power or root of x instead of x itself. The idea is exactly the same — just swap out the "x" for whatever power/root you're told about:
- y proportional to the square of x → y ∝ x² → y = kx²
- y proportional to the square root of x → y ∝ √x → y = k√x
- y proportional to the cube of x → y ∝ x³ → y = kx³
- cube root
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