Basics of Linear Functions

What exactly is a function, how do linear functions differ from other functions, and what are the characteristics of linear functions?

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What is a Function?

For example, let's say we have data on the number of cold drinks sold and the temperature for a certain store from June 1st to 11th. What kind of relationship might exist between the temperature and the number of sales?

Example) Number of cold drinks sold and temperature

If we plot this data on a graph with the vertical axis representing the number of sales and the horizontal axis representing the temperature, we can see that as the temperature rises, the number of sales also increases. If we could collect more detailed data, the number of points would increase. Try moving the blue slider on the graph below to the right to increase the number of points.

As we increase the number of points infinitely, we can see that they form a straight line. This straight line formed by connecting these infinite points represents the relationship between temperature and the number of sales.

When we denote the temperature as $x$ and the number of sales as $y$ , this relationship can be expressed mathematically as $y = 5x + 10$ . By doing this, we can represent a relationship where the number of sales $y$ is determined when the temperature $x$ is set.

What is a Linear Function?

Definition

When we express a function represented by a straight line like the one above in a general form, it becomes:

$$y = ax + b$$

(where $a$ and $b$ are constants). This type of function is called a linear function .

The term "linear" means that the degree of $x$ is 1. Functions where the degree of $x$ is 2 are called quadratic functions, and those where the degree of $x$ is 3 are called cubic functions.

Move the slider on the graph below to see the shapes of the graphs when $x$ is to the first, second, and third power. This will help you understand why we distinguish functions by their degree. The difference in shape means that the nature of the relationship between $x$ and $y$ is also different, so we give them different names to make them easier to distinguish.

Rate of Change

Let's consider a linear function $y=2x+1$ . If we express the relationship between $x$ and $y$ in a table, it would look like this:

From this table, we can see that $y$ changes twice as much as $x$ . The rate of change represents how much $y$ changes in relation to the increase or decrease in $x$ , and it's calculated by dividing the increase in $y$ by the increase in $x$ . In this table, $y$ increases by 2 when $x$ increases by 1, so the rate of change is 2.

$$Rate of Change = \frac{Increase in y}{Increase in x}$$

At this stage, you might not be able to imagine how the rate of change is used. When we draw the graph of a linear function later, you'll see how the rate of change is used, so don't worry and let's move on.