Testing and graphing inverse proportion

If \(y=\dfrac{k}{x}\) with \(k>0\) and \(x>0\), the graph of \(y\) against \(x\) is a curve called a hyperbola. It falls as \(x\) increases and never touches either axis. The graph of \(y\) against \(\dfrac{1}{x}\) is a straight line through \(O\) with gradient \(k\).

Why: \(y=\dfrac{12}{x}\) is never \(0\) and \(x\) cannot be \(0\), so the curve approaches the axes without meeting them. Writing \(y=12\left(\dfrac{1}{x}\right)\) shows that \(y\) is directly proportional to \(\dfrac{1}{x}\), so that graph is a straight line through \(O\).

Method

1. To test a table, multiply each pair of values. If every product is the same, the data are consistent with inverse proportion and the product is \(k\).
2. A table in which \(y\) decreases as \(x\) increases is not enough; check the products.
3. To draw \(y\) against \(\dfrac{1}{x}\), work out \(\dfrac{1}{x}\) for each value, plot the points \(\left(\dfrac{1}{x},y\right)\) and join them with a ruler towards \(O\).

Example

\(2\times20=5\times8=8\times5=40\). The products are equal, so \(y=\dfrac{40}{x}\).

For the pairs \((1,9)\), \((2,7)\) and \((3,5)\), \(y\) also decreases, but the products are \(9\), \(14\) and \(15\). These values are not in inverse proportion; they fit the straight line \(y=11-2x\).

Check: a straight line sloping downwards is never inverse proportion, and a hyperbola for positive values never crosses an axis.