Proportional Functions

Two variables $x$ and $y$ are proportional if the ratio between them is constant. That means you always get the same answer when you divide $y$ by $x$ , regardless of where you are along the two graphs.

Theory

Proportional Variables

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

y = k x

where $k$ is a constant.

Example 1

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

Proportional functions are actually just a special case of linear functions, $f (x) = a x + b$ . What has happened is that $b = 0$ , so $b$ is gone. This means that the graph intersects the $y$ -axis at the origin each time. In addition, the slope $a$ is replaced by the proportionality constant $k$ . Here are a few examples:

Example 2

This graph shows $y = x$ , that is, $k = 1$ . Since the graph is proportional, for all the coordinates on the graph, if you divide the $y$ -coordinate by the $x$ -coordinate, the answer is $k = 1$ .

Example 3

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

You can find that out by doing a few conversions:

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

You have determined that $k \approx 0.67$ or $k = \frac{2}{3}$ , so the graph is proportional.

Example 4

You have been given the following points:

$x$ -values 1 2 3 4 5
$y$ -values 7 14 21 28 35

Are these points part of a proportional function?

From the theory, you know that if you divide the $y$ -value by the $x$ -value, and the answer is the same each time, all points belong to a proportional function. You check the points from the table:

\begin{array}{l} \frac{7}{1} & = 7 \\ \frac{14}{2} & = 7 \\ \frac{21}{3} & = 7 \\ \frac{28}{4} & = 7 \\ \frac{35}{5} & = 7 \end{array}

Because all the answers are the same, you are working with a proportional function. The function is $f (x) = 7 x$ .

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Activities 4