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Inversely Proportional Functions

Two variables $x$ and $y$ are inversely proportional if their product is constant. That means you always get the same answer when you multiply $x$ and $y$ together.

Theory

Inverse Proportionality

Two variables $x$ and $y$ are inversely proportional if

y = \frac{k}{x}

where $k$ is a constant.

Example 1

Saying that $y = \frac{k}{x}$ is the same as saying that $x \cdot y = k$ : $\begin{array}{l} x \cdot y & = k, & x \neq 0 \\ \frac{x \cdot y}{x} & = \frac{k}{x}, & x \neq 0 \\ y & = \frac{k}{x}, & x \neq 0 \end{array}$

Example 2

This graph shows $y = \frac{1}{x}$ , that is, $k = 1$ . The graph is inversely proportional, so for all points on the graph, if you multiply the $x$ -coordinate by the $y$ -coordinate, the answer is $k = 1$ .

Example 3

Is the graph $y = \frac{2}{3 x}$ inversely proportional?

You can find that out by doing a few conversions:

\begin{array}{l} y & = \frac{2}{3 x} \\ = \frac{2 \cdot 1}{3 \cdot x} \\ = \frac{2}{3} \cdot \frac{1}{x} \\ \approx 0.67 \cdot \frac{1}{x} \\ \approx \frac{0.67}{x} \end{array}

You’ve found that $k = \frac{2}{3} \approx 0.67$ , so the graph is inversely proportional.

Example 4

You have been given the following points:

$x$ -values 1 2 3 4 5
$y$ -values 20 10 7 5 4

Do the points follow an inversely proportional function?

From the theory, you know that if multiply the $x$ -value by the $y$ -value and the answer is the same for all points, then the points follow an inversely proportional function. You check the points from the table:

\begin{array}{l} 1 \cdot 20 & = 20 \\ 2 \cdot 10 & = 20 \\ 3 \cdot 7 & = 21 \\ 4 \cdot 5 & = 20 \\ 5 \cdot 4 & = 20 \end{array}

Because one of the answers is not the same as the others, they do not follow an inversely proportional function.

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