>Types of Functions>What Does Inversely Proportional Mean in Math?

What Does Inversely Proportional Mean in Math?

In any case where each value of $x$ , multiplied by each value of $y$ , gives a specific constant $k$ , this is known as inverse proportionality . We usually write this as in the rule box below: $y$ is equal to $k$ divided by $x$ . Our two explanations are exactly the same. Look here:

Example 1

A calculation shows that the expressions are the same:

\begin{array}{l} x \cdot y & = k \\ \frac{x \cdot y}{x} & = \frac{k}{x}, x \neq 0 \\ y & = \frac{k}{x} \end{array}

Rule

Two values, $x$ and $y$ , are inversely proportional if

y = \frac{k}{x},

where $k$ is a constant.

Below, in Example 2, you can see a case where $k = 1$ . Here are some rules for remembering what happens for different values of $k$ :

When $k > 0$ is positive, the graph slides outward from the first quadrant (with a positive $x$ -axis and $y$ -axis) and away from the origin.
When $k < 0$ is negative, the graph lies in the fourth quadrant (part of the coordinate system with a positive $x$ -axis and a negative $y$ -axis), but turned upside down. The shape of the graph is always the same.

Example 2

This graph shows $y = \frac{1}{x}$ , so $k = 1$ . Since the graph is inversely proportional, it means that all the coordinates on the graph are such that if you take the $x$ -coordinate and multiply them by the $y$ -coordinate, the answer is $k = 1$ .

Example of graph of inversely proportional function

Example 3

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

You can find this out with a few modifications:

\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}

\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 have now determined that

k = \frac{2}{3} \approx 0.67

, so the graph is inversely proportional.

Example 4

You are given the following points:

Do the points follow an inversely proportional function?

You know that if the points lie on the graph of an inversely proportional function, then you get the same answer when you multiply the $x$ -value by the $y$ -value for all the points:

\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}

It’s close, but since one answer is not the same as the others, you do not have inverse proportionality.

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