Section 1: Linear Models

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 3.1 Linear Models 219 Version: Fall 2007 3.1 Linear Models Sebastian waves good-bye to his brother, who is talking to a group of his friends ap- proximately 20 feet away. Sebastian then begins to walk away from his brother at a constant rate of 4 feet per second. Let’s model the distance separating the two brothers as a function of time. Our first approach will be graphical. We will let the variable d represent the distance (in feet) between the brothers and the variable t represent the amount of time (in seconds) that has passed since Sebastian waved good-bye to his brother. Because the distance separating the brothers depends on the amount of time that has passed, we will say that the distance d is the dependent variable and the time t is the independent variable . It is somewhat traditional in the modeling process to place the independent variable on the horizontal axis and the dependent variable on the vertical axis. This is not a hard and fast rule, more a matter of personal taste, but we will follow this rule in our example nonetheless. Thus, we will place distance on the vertical axis and time on the horizontal axis, as shown in Figure 1 . Notice that we’ve labeled each axis with its variable representation and included the units, an important practice. t (s) d (ft) Figure 1. Distance depends upon time. Warning 1. The label on the horizontal axis, t (s), might look like function notation to some readers. This is not the case. Rather, the variable t represents time, and the (s) in parentheses that follows represents seconds, a standard ab- breviation in physics. Similar comments are in order for the label d (ft). The variable d represents distance, and the (ft) in parentheses that follows represent feet, another standard abbreviation in physics. There are a number of different ways that you can label the axes of your graph with units appropriate for the problem at hand. For example, consider the technique presented in Figure 2 , where the labels are placed to the left of the vertical axis and Copyrighted material. See: 1
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220 Chapter 3 Linear Functions Version: Fall 2007 underneath the horizontal axis. Another difference is the fact that the unit abbrevia- tions in Figure 1 are spelled out in their entirety in Figure 2 . t (seconds) d (feet) Figure 2. Distance depends upon time. Some instructors prefer that you rotate the distance label on the vertical axis ninety degrees, so that it appears sideways. Others prefer that you label the ends of each axis with the variable, as we have done in Figure 1 , but spell out the units in their entirety alongside each axis as we’ve done in Figure 2 . The list of preferences goes on and on. Tip 2. It is important to have a conversation with your instructor in order to determine what your instructor’s expectations are when it comes to labeling the axes and indicating the units on your graphs.
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