Construct a direct or inverse proportional model

By UK GCSE Mathematics · independent study · myAudience

Original myAudience mathematics teaching illustration; labels, relationships and lesson calculations independently checked. · Original project artwork; no exam-board figures or textbook illustrations reused.

Construct a direct or inverse proportional model

To construct an equation, first choose the relationship justified by the statement. Direct proportion to a quantity gives a constant times that quantity. Inverse proportion gives a constant divided by it; the constant is fitted from a supplied matching pair.

If y is directly proportional to x², write y = kx² and find k using y/x². If y is inversely proportional to x², use y = k/x² and find k from yx². The phrase identifies the proportional quantity, not necessarily x itself.

After finding the constant, check the supplied data and state excluded inputs. Do not fit a model from one pair without an explicit proportionality assumption: many different rules can pass through that pair.

Worked example: y is directly proportional to x²; y = 18 when x = 3. Find y when x = 5.

Write y = kx².

Use 18 = 9k, so k = 2.

Substitute x = 5.

Answer: y = 2x²; required value 50.

Check your understanding: If y = 12/x², what happens when a positive x doubles?

Answer: y becomes one quarter as large.

Why: The squared denominator becomes four times as large.