Linear Function

What is a linear function?

Linear functions are functions whose graphs are straight lines. These functions can be applied to explain variables that change in a constant rate.

We start on our learning of linear functions through investigating a few linear representations, where we will illustrate a systematic conversation of the modeling procedure, together with the concept of dependent and independent variables. Then we will represent the numbers using a graph.

Definition:

A function f is linear if it can be expressed in the form

f (x) = mx + b

Where: m and b are both constants

            x is an arbitrary member of the domain of f

Frequently the association between two variables (x and y) is a linear function written as an equation:   

y = mx + b

If one quantity changes with respect to a second quantity at constant rate, the relationship of the function is linear and the graph is a line.

Example:

Vanessa wants to buy ice cream so she tells her brother, who is just about 20 feet away from her. Vanessa then starts to leave her brother at a constant rate of 4 feet per second. Now, represent the distance separating the two siblings as a function of time.

The first approach is graphical method. Let the variable d stands for the distance (in feet) between the siblings and the variable t stands for the quantity of time (in seconds) that has passed since Vanessa left her brother. Because the distance that separates the siblings depends on the quantity of time that has passed, then the distance, d, is the dependent variable and the time, t, is the independent variable.

In the modeling procedure, position the independent variable, t, on the horizontal axis and the dependent variable, d, on the vertical axis. Therefore, put distance on the vertical axis and time on the horizontal axis, as revealed in Figure 1. Observe that each axis was labeled with its corresponding variable symbol including the units.

Figure 1. Distance versus Time

Now scale every axis properly.  Choose a scale for every axis with the subsequent view in mind. Vanessa is leaving her brother at a constant rate of 4 feet per second. Allow every box on the vertical axis signifies 4 feet and each two boxes on the horizontal axis signify 1 second as made known in Figure 2.

Figure 2. Scaling axes

At time t = 0, Vanessa 20 ft from her, d = 20 feet. This match to the point (t, d) = (0, 20) shown in Figure 3(a). Next, Vanessa leaves her brother at a constant rate of 4 feet for every second. For each second of time that passes, the distance amid the siblings increases by 4 feet.

Initially at the point (0, 20)

After 1 second (two boxes): move to the right and

4 feet (1 box): move upward to the point (1, 24), as shown in Figure 3(b).

                             

The rate of partition is a constant 4 feet per second. Continuing indefinitely will create the linear relationship between distance and time. Assuming that the distance is an unbroken function of time, the distance is increasing incessantly at a constant rate of 4 feet for every second, and then the graph will be

Figure 4. Discreet and Continuous Model

The graph can also be use for predictions.

Example:

Find out the distance amid the siblings after 8 seconds.

First, locate 8 seconds on the time axis; sketch a perpendicular arrow to the line, then a parallel arrow to the distance axis.

Figure 5. Predicting the distance between

Determining the pattern that explains the distance d between the siblings as a function of time t, note that:

• At t = 0 seconds, the distance among siblings is d = 20 feet.

• At t = 1 second, the distance among siblings is d = 24 feet.

• At t = 2 seconds, the distance among siblings is d = 28 feet.

• At t = 3 seconds, the distance among siblings is d = 32 feet.

Table 1. Determining a model Equation

Table 1 summarizes the result. It disclose a relationship between distance d and time t that can be expressed by the equation

d = 20 +4t

Furthermore, this equation can be used to calculate the distance between the siblings at any time, for example 2 minutes. First, change t = 2 minutes to t = 120 seconds, then replace this number in the model equation.

d = 20 + 4(120) = 500

Therefore, the distance between siblings after 2 minutes is d = 500 feet.