# Constant function, Linear function and Affine function and their graphic representation

The functions that we will see in this lesson are very simple functions with only one variable, but they are essential to understanding the concept of numeric functions later.

## Constant Function

### Note:

A Constant Function is every function written in the form where is a known real number.

From the formula for the constant function, we can see that the value of is completely independent of the value of , in other words, the value of does not change if we change the value of .

As an example from reality, if we have a car traveling at a constant speed over a period of time (the variable is time t), then the expression of the function in this case is in which is the velocity of the car.

#### Let represent graphically the two defined functions and as follows:  • The number 3 is the intersection of the straight line that passes through the ordinate 3 (-axis) and is parallel to the axis of the abscissa (-axis).

• The number -2 is the intersection of the straight line that passes through the ordinate -2 (-axis) and is parallel to the axis of the abscissa (-axis).

## Linear Function

A Linear Function is any function written in the form where is a known real number and is called the slope of the line.

Through the formula of the linear function, we can see that the value of changes with the change of as follows:

The graphic representation of a linear function is a line that passes through the coordinates origin .

So how do we represent the line of a linear function in an orthonormal coordinate system? We said earlier that the graphic representation of a linear function is a straight line. So we need two different points to create this line. How will we get them?

• The first point is because the line passes through the origin of the coordinate plane.

• The second point we get by substituting into the coordinates.

That is, if we take , then:

Example1:

Let us represent graphically the linear function .

The line passes from and another point

Let's take for example , so .

We get the second point:

Let us represent graphically the linear function .

The line passes from and another point

Let's take for example , so .

We get the second point:  ## Affine Function

An Affine Function is any function written in the form where is a known real number and is called the slope of the line, and b is known real number called the y-intercept with the line.

Through the formula of the affine function, we can see that the value of changes with the change of as follows:

The graphic representation of the affine function intersects with the -axis. So how do we represent the line of an affine function in an orthonormal coordinate system?

The method is not very different from the linear function. First we represent on the -axis as the first point.

As for the second point, we will get it by substituting into the formula for the affine function as follows:

That is, if we take , then:

Example2:

Let us represent graphically the affine function .

We have , so, the first point is .  Let's take for example , therefore .

Hence, the second point is  Example3:

Let us represent graphically the affine function .

We have , so, the first point is .  Let's take for example , therefore .

Hence, the second point is  