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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

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

$$y=kx$$ |

where $k$ is a constant.

Example 1

Saying that $y=kx$ is the same as saying that $\frac{y}{x}=k$: $$\begin{array}{llll}\hfill \frac{y}{x}& =k\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x\cdot \frac{y}{x}& =k\cdot x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =kx\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

Proportional functions are actually just a special case of linear functions, $f(x)=ax+b$. 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 $y=\frac{2x}{3}$ proportional? **

You can find that out by doing a few conversions: $$\begin{array}{llll}\hfill y& =\frac{2x}{3}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{2\times x}{3\times 1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{2}{3}\times \frac{x}{1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx 0.67\times x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =0.67x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

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: $$\begin{array}{llll}\hfill \frac{7}{1}& =7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{14}{2}& =7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{21}{3}& =7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{28}{4}& =7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{35}{5}& =7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

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