An important concept in mathematics is that of direct proportion and inverse proportion. We frequently use the words direct proportion and inversely proportional in our daily lives. Direct proportion and inversely proportional are familiar to us and are concepts that everyone uses. The concept of direct proportionality and inversely proportionality are used in many situations.

When studying direct proportion and inversely proportion, we can’t understand it by looking at numbers. In mathematics, it is common to use graphs to learn about direct proportion and inverse proportion. Therefore, when learning direct proportion and inversely proportional, you must also understand coordinates.

Also, with direct proportion and inversely proportion, there is a plus or a minus. Depending on whether they are positive or negative, the shape of the graph will change.

Direct proportionality and inversely proportionality are also common sense for us. So learn the coordinates of the graph and try to understand the concept of direct proportionality and inversely proportionality.

Table of Contents

- 1 The Definition of the Function Is Whether the Numbers Are Determined
- 2 The Concept of Proportionality and Constant of Proportionality
- 3 Inverse Proportion: Formula, Tables, and Graphs Shape
- 4 Exercises: Word Problems and Graphs of Proportion and Inverse Proportion
- 5 Learn Equations and Graphs of Proportional and Inversely Proportional

## The Definition of the Function Is Whether the Numbers Are Determined

Most of the mathematics is represented by algebraic expressions. That is, it utilizes equations that use the alphabet. In equations that use letters, the term function is often used. When we study graphs in mathematics, we use the words linear functions and quadratic functions.

So what is a function? The definition of a function is as follows.

- Determining the value of $x$ determines the value of $y$.

When this condition is met, it is a function. For example, are the following two examples functions?

- A person with height $x$ cm is $y$ kg in weight.
- Walking for 3 hours at $x$ km per hour, the distance walked is $y$ km.

Height and weight vary from person to person. A person may be 170 cm tall and weigh 50 kg, while another person may weigh 70 kg. Even if we determine the value of $x$, we cannot determine the value of $y$. Therefore, it is not a function.

On the other hand, if the distance walked is $y$ km, then if we walked for 3 hours at $x$ km/h, we can make the following equation.

- $y=3x$

By determining the value of $x$, the value of $y$ is clearly determined. This is a function. Whether it is a function is determined by the ability to determine a specific number.

### The Concept of Coordinates: $x$ Axis, $y$ Axis, and Origin

After learning the definition of functions, to understand direct proportionality and inverse proportionality, we must understand the concept of coordinates. In a graph, it represents the position of a point and the shape of a line.

In a graph, the horizontal axis is called the $x$ axis. On the other hand, the vertical axis is called the $y$ axis. The intersection point of the $x$ and $y$ axes is called the origin.

When we use graphs in mathematics, we write points. It is the coordinates that indicate where the point is located in the graph. For example, in the following graph, the coordinate of the point P is $(3,-2)$.

Of the coordinates of $P(3,-2)$, 3 is called the $x$ coordinate and -2 is called the $y$ coordinate. In any case, the points that exist in the graph are called coordinates.

## The Concept of Proportionality and Constant of Proportionality

After you understand the definitions and concepts about functions and coordinates, you can learn about proportionality. Proportionality refers to a function that as the value of $x$ increases, the number of $y$ increases in the same proportion.

Let’s consider the following example, which is the same as the previous one.

- Walking for 3 hours at $x$ km per hour, the distance walked is $y$ km.

This function is $y = 3x$. Therefore, as the value of $x$ increases by 1, $y$ is increased by 3. Also, if the value of $x$ decreases, the value of $y$ decreases by the same ratio. The table looks like the following.

A change in the value of $x$ causes $y$ to increase (or decrease) by 3. This is direct proportionality. In this equation, $y=3x$, so the value of $y$ changes by 3. On the other hand, for example, if $y=5x$, then a change in the value of $x$ causes the value of $y$ to change by 5.

Therefore, in direct proportional coordinates, we use the following formula.

- $y=ax$

$a$ is called a constant of proportionality. The value of $a$ is different for each. In the previous equation, walking at $x$ km/h for 3 hours, resulting in $y=3x$. On the other hand, if you walk at $x$ km/h for 4 hours, $y=4x$. The constant of proportionality, $a$, changes in number depending on the question.

### How to Write a Graph of Direct Proportion: the Coordinates of $y=ax$

So what is the relationship between a function and a graph of direct proportionality? We have just described the table of $y=3x$ as an example.

As the value of $x$ changes, the value of $y$ also changes. Note that the values of $x$ and $y$ are coordinates. In short, we have the following.

Let’s type these coordinates into a graph. And if we connect the coordinates of each, we get the following.

In a direct proportional graph, it will always be a straight line. Also, a line always passes through the origin. We can create a graph from the formula $y=ax$.

Conversely, you can also get the formula $y=ax$ from the graph. For example, if $y$ is increasing by two for each $x$ increase, we know that the formula is $y=2x$.

### The Shape of the Graph Differs Whether the Constant of Proportionality Is Positive or Negative

There’s something else that’s important when considering the proportional formula. That is, whether the constant of proportionality $a$ is a positive number or a negative number.

So far, we have considered the graph of proportionality by assuming that $a$ is a positive number. On the other hand, if $a$ is a negative number, what does a linear graph of direct proportionality look like? The idea is the same as before: consider the graph of direct proportionality when $a$ is negative. For example, suppose we have the following equation.

- $y=-2x$

In this case, for every $x$ increase by one, $y$ increases by -2. The table looks like the following.

A graph of this table would look like the following.

If $a$ is a positive number, the graph of the line will go up to the right. On the other hand, if $a$ is a negative number, the graph of the line will be downward to the right. Depending on whether the value of $a$ is positive or negative, the shape of the graph will change.

## Inverse Proportion: Formula, Tables, and Graphs Shape

After learning the functions of direct proportionality, one thing you must study is inverse proportionality. Compared to direct proportionality, inverse proportionality is a little more difficult to understand.

If $a$ is a positive number, then in direct proportionality, as the value of $x$ increases, the value of $y$ increases by the same ratio. In contrast, in the inversely proportional case, if $a$ is positive, as the value of $x$ increases, the value of $y$ decreases. For direct proportionality, inversely proportional is the opposite.

Whereas $y=ax$ is a formula of direct proportionality, in inverse proportionality we have the following formula.

- $y=\displaystyle\frac{a}{x}$

When does it become a function of inverse proportion? For example, if you multiply speed by time, you get the distance. If you walk 6 km at $x$ km/h and $y$ hours, you have the following equation.

- $x×y=6$
- $y=\displaystyle\frac{6}{x}$

There is a fixed distance of 6 km to walk. Therefore, if you increase $x$ km/h (walking speed), the value of time to arrive ($y$) will be smaller. On the other hand, if you decrease the speed of $x$ km/h, the value of time to arrive ($y$) will increase. If one value is larger and the other value is smaller, there is an inverse proportional relationship.

For the equation of inverse proportionality just mentioned, the table looks like the following.

If the value of $x$ is doubled, the value of $y$ will be $\displaystyle\frac{1}{2}$. And if the value of $x$ is tripled, the value of $y$ is $\displaystyle\frac{1}{3}$. In the case of inversely proportional, it is the opposite of proportional.

A graph of this table is shown below.

A graph of direct proportionality is always a straight line. On the other hand, a graph of inverse proportionality is always a curve. The inversely proportional curve is called hyperbola. Proportional and inversely proportional graphs have quite different shapes.

### Curve Graphs Differ in Positive and Negative

In direct proportionality, we explained that the shape of the graph depends on whether the constant of proportionality, $a$, is positive or negative. The same can be said for graphs of inverse proportionality.

We described earlier what the inversely proportional graph looks like if $a>0$. On the other hand, if $a<0$, the graph of inversely proportional graphs will look like this.

As $x$ doubles, triples or quadruples in number, the value of $y$ becomes $\displaystyle\frac{1}{2}, \displaystyle\frac{1}{3}, \displaystyle\frac{1}{4}$, the same way. However, it is important to understand that the graphs shape is different depending on whether the constant of proportionality is a positive or negative number.

## Exercises: Word Problems and Graphs of Proportion and Inverse Proportion

**Q1.** Do the following calculations.

You buy a metal for 40g for \$100.

- How much is the price for 1 gram of metal?
- If the amount to be paid is $y$ dollars and the weight of the metal is $x$ g, create an equation that expresses $y$ as $x$.
- How much does 26g of metal cost?

**A1.** Answers.

Direct proportionality is frequently used in everyday life. When shopping, the calculation of how many items to buy and how much to spend is proportional. If you can’t do this calculation, you’ll have trouble in shopping.

**(a)**

Since we don’t know the price of a gram of metal, let’s take this number as $a$. 1g of metal costs $a$ dollars, so when you buy 40g of metal that costs you \$100. Therefore, the formula is as follows.

$40×a=100$

$40×a\textcolor{red}{×\displaystyle\frac{1}{40}}=100\textcolor{red}{×\displaystyle\frac{1}{40}}$

$a=2.5$

**(b)**

The price of 1 gram of metal is 2.5 dollars. If the amount to be paid is $y$ dollars and the weight of the metal is $x$ g, we can build the following proportional equation.

- $y=2.5x$

**(c)**

Once we make a proportional equation, we can also get the value of $y$ by substituting the value of $x$. If we buy 26 grams of metal, we get the following amounts.

$y=2.5×26$

$y=65$

**Q2.** Do the following calculations.

The distance from your house to the library is 3000m. Walk at a speed of $x$ m/min and it will take you $y$ minutes to arrive.

- Represent $y$ in the equation $x$.
- If you walk at a speed of 50 m/min, how many minutes will it take to arrive?
- How many meters per minute should you run if you want to arrive in 30 minutes?

**A2.** Answers.

**(a)**

Multiplying the speed and time result in distance. Therefore, you will have the following inversely proportional equation.

$x×y=3000$

$y=\displaystyle\frac{3000}{x}$

**(b)**

Substituting a number into the equation of inversely proportional, we can get the answer.

$y=\displaystyle\frac{3000}{50}$

$y=60$

**(c)**

In calculating a function, we don’t necessarily substitute $x$. Sometimes we substitute $y$ to get the answer. Since $y=30$, we have the following.

$30=\displaystyle\frac{3000}{x}$

$30\textcolor{red}{×\displaystyle\frac{x}{30}}=\displaystyle\frac{3000}{x}\textcolor{red}{×\displaystyle\frac{x}{30}}$

$x=100$

**Q3.** Answer the following questions.

- For the line in (1), find the constant of proportionality for $y=ax$
- For the hyperbola of (2), find the constant of proportionality for $y=\displaystyle\frac{a}{x}$
- Find the coordinates of P

**A3.** Answers.

In proportional and inversely proportional problems, it is important to be able to read and solve graphs as well as word problems.

(a)

In a proportional graph, we can substitute the coordinates to $y=ax$ to get a constant of proportionality. The graph of a straight line has coordinates of $(6,-3)$, so let’s substitute them.

$-3=6a$

$6a\textcolor{red}{×\displaystyle\frac{1}{6}}=-3\textcolor{red}{×\displaystyle\frac{1}{6}}$

$a=-\displaystyle\frac{1}{2}$

**(b)**

Likewise, we can assign the coordinates to yield a constant of proportionality. The graph of the hyperbola has coordinates of $(-2,4)$, so let’s substitute them.

$4=\displaystyle\frac{a}{-2}$

$\displaystyle\frac{a}{-2}\textcolor{red}{×(-2)}=4\textcolor{red}{×(-2)}$

$a=-8$

**(c)**

The $x$ coordinate of the point P is 1. We can also calculate the $y$ coordinate of the point P by using the answer to (b). The result is as follows.

- $y=-\displaystyle\frac{8}{x}$ (substitute $x=1$)
- $y=-\displaystyle\frac{8}{1}$
- $y=-8$

The coordinates of the point P is $(1,-8)$.

## Learn Equations and Graphs of Proportional and Inversely Proportional

In math functions, we learn about direct proportionality and inverse proportionality. Proportional and inversely proportional are familiar and many people use them in their daily lives, such as when shopping. If you can’t calculate direct proportionality and inverse proportionality, you will be in trouble in everyday life. So, let’s understand the concept of proportional and inversely proportional.

Proportional is that when the value is doubled, the number you want to find is also doubled. On the other hand, it is inversely proportional if the value is doubled and the amount you want to find is $\displaystyle\frac{1}{2}$.

We also learn about graphs in proportional and inversely proportional at the same time. So understand the concept of coordinates and be able to create proportional and inversely proportional expressions from the graph.

Distinguish between proportional and inversely proportional from the problem sentence. Then, find the constant of proportionality and calculate it to find the answer.