# Linear Function

1

LinearFunction

Affiliation:

Linearfunctions are functions whose graphs are straight lines. Thesefunctions can be applied to explain variables that change in aconstant rate.

Westart on our learning of linear functions through investigating a fewlinear representations, where we will illustrate a systematicconversation of the modeling procedure, together with the concept ofdependent and independent variables. Then we will represent thenumbers using a graph.

Definition

Afunction *f*islinear if it can be expressed in the form

*f*(*x*)=*mx*+*b*

where:*m*and*b*areboth constants

*x*isan arbitrary member of the domain of *f*

Frequentlythe association between two variables (x and y) is a linear functionwritten as an equation:

*y*=*mx*+*b*

Ifone quantity changes with respect to a second quantity at constantrate, the relationship of the function is linear and the graph is aline.

Example

Emilywants to buy ice cream so she tells her brother, who is just about 20feet away from her. Emily then starts to leave her brother at aconstantrateof 4 feet per second. Now, represent the distance separating the twosiblings as a function of time.

Thefirst approach is graphical method. Let the variable *d*standsfor the distance (in feet) between the siblings and the variable *t*standsforthe quantity of time (in seconds) that has passed since Emily lefther brother. Because the distance that separates the siblings *depends*onthe quantity of time that has passed, then the distance, *d,*isthe *dependentvariable *andthe time, *t,*isthe *independentvariable*.

Inthe modeling procedure, position the independent variable, t, on thehorizontal axis and the dependent variable, d, on the vertical axis.Therefore, put distance on the vertical axis and time on thehorizontal axis, as revealed in Figure1.Observe that each axis was labeled with its corresponding variablesymbol including the units.

Figure1. Distance versus Time

Now*scale*everyaxis properly. Choose a scale for every axis with the subsequentview in mind. Emily is leaving his brother at a constant rate of 4feet per second. Allow every box on the vertical axis signify 4 feetand each two boxes on the horizontal axis signify 1 second as madeknown in Figure2.

Figure2. Scaling axes

Attime *t*=0, Emily 20 ft from her, *d*=20 feet. This match to the point (*t,d*)= (0*,*20)shown in Figure3(a).Next, Emily leaves her brother at a constant rate of 4 feet for everysecond. For each second of time that passes, the distance amid thesiblings increases by 4 feet.

Initiallyat the point (0*,*20)

After1 second (two boxes): move to the right and

4feet (1 box): move upward to the point (1*,*24),as shown in Figure3(b).

Figure 3(a). Initially 20 feet from his brother.

Figure 3(b). Walking away at 4 ft/s.

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.

References:

Shores, T. (2007). *Applied Linear Algebra and Matrix Analysis*. Springer: Undergraduate Texts in Mathematics.

Wagner, B. (2011).*Elemantary Algebra*. California, USA: Department of Mathematics, College of the Redwoods Press.

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