Identify a constant ratio and its proportionality coefficient

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Identify a constant ratio and its proportionality coefficient

If y is directly proportional to x, write y = kx. The constant k is found from a corresponding nonzero input by y/x. Doubling x doubles y; multiplying x by any permitted factor multiplies y by the same factor.

The graph is a straight line through the origin when the relevant domain includes zero. A model restricted to positive inputs can show only part of that line. The constant’s units depend on the two quantities being compared.

A straight line with a fixed initial charge is not direct proportion. Check the constant ratio rather than judging a relationship only by a straight appearance on a diagram.

Worked example: y is directly proportional to x; y = 12 when x = 4. Find y at x = 7.

Calculate k = 12/4 = 3.

Use y = 3x.

Substitute x = 7.

Answer: 21.

Check your understanding: Does doubling x always add 3 to y in this model?

Answer: No; it doubles y.

Why: Direct proportion preserves a ratio, not a fixed increment.