What Makes a Function Proportional?

You often come across two quantities whose relationship is such that if you double one of them, the other one doubles too. For example, if you are baking, and you need $1$ kg flour to make 24 buns, it is easy to figure out how much flour you need to make twice as many buns. You need $2$ kg flour to make 48 buns. Doubling one quantity leads to the other one doubling!

The mathematical word for this concept is proportionality. If all values of $y$ divided by all corresponding values of $x$ give you a certain constant $k$ , that means $x$ and $y$ are proportional quantities. We call this constant $k$ the proportionality constant. We usually write this as shown in the rule box below— $y$ is $k$ multiplied by $x$ . The two explanations below for why are exactly the same:

Rule

A calculation shows that the expressions are equal:

\begin{array}{l} \frac{y}{x} & = k \\ x \cdot \frac{y}{x} & = k \cdot x \\ y & = k x \end{array}

Rule

Two entities $x$ and $y$ are proportional if

y = k x,

where $k$ is a constant.

You can see in Example 2 below what happens when $k = 1$ . Here are some rules for remembering what happens to different values of $k$ :

If the value of $k$ is higher than $1$ , then the line will be steeper.
If the value of $k$ is lower than $1$ , then the line will be more gradual.
If $k$ is a negative number, then the graph will slope downwards from left to right and eventually cross under the $x$ -axis.

A proportional function is a special case of the linear function $y = a x + b$ . If you set $a = k$ and $b = 0$ , then you get the formula of proportionality. The graph is always a straight line through the origin at $(0, 0)$ . The ratio between $y$ and $x$ is always equal to the slope $k$ .

Example 1

This graph shows

y = x

, which means that

k = 1

. The function is proportional, so if you take all the

y

-coordinates and divide it by the

x

-coordinates, the answer will always be

k = 1

Example of a graph of a proportional function

Example 2

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

You can find out by doing a few modifications:

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

y = \frac{2 x}{3} = \frac{2 \cdot x}{3 \cdot 1} = \frac{2}{3} \cdot \frac{x}{1} \approx 0.67 \cdot x \approx 0.67 x

You determined that

k \approx 0.67

k = \frac{2}{3}

, so the function is proportional.

Example 3

You are given the following coordinates:

Do the points follow a proportional function?

You know that if the points follow a proportional function, you’ll get the same answer when you divide the value of $y$ by the value of $x$ for all the coordinates:

\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}

Since all the answers are equal, you have proportionality.

Note! Functions in the form $y = \frac{k}{x}$ are called inversely proportional.

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