## 7.2.3 Graphing the Situation

Grade:
9-12
Standards:
Math Content:
Algebra

An interactive graphical analysis provides a visual interpretation of the results.

The interactive figure calculates the amount of medicine in a person's body immediately after taking a dose. In this scenario, the individual takes an initial dose of medicine followed by recurring doses, taken faithfully at fixed intervals of time. The interactive figure allows the initial dose to be different from the recurring doses.

The simulation requires three inputs:

• The initial dose—the amount of medicine given for the initial dose
• The elimination rate—the percent of medicine (given as a decimal) that the kidneys remove from the system between doses
• The recurring dose—the amount of the medicine to be given at fixed intervals

Click Calculate.

The points graphed are (n, A(n)), with A(n) representing the amount of medicine in the body immediately after taking the nth dose of medicine.

Placing the cursor over a data point will display the value of A(n) for that point.

The horizontal scroll bar at the bottom of the applet will allow you to observe the graph for larger values of n.

### Task

So far you have investigated this situation primarily numerically, by looking for patterns in the values of A(n) generated by the interactive figure. To get a better understanding of the patterns and why they are occurring, it is helpful to do a graphical analysis.

• Enter appropriate values for the parameters in the graphing tool below to get a graph of the original situation. Use the graph to explain what is happening to the level of the drug.
• Describe the characteristics of the graph, and explain what information the characteristics give you about the amount of medicine in the body over time. How is the stabilization level shown in this graph?
• Explore other values for the parameters. How does the shape of the graph change? What parameter seems to affect the steepness of the curve?
• In many of the results generated in this example, a final stabilization level appears to have been reached. Are these values really reached, mathematically? How is this situation shown in the graphs?
• Think of another applied situation that could be modeled and analyzed using methods and equations similar to those used in this example.

### Discussion

In this example, students have used multiple representations to analyze a real-world situation. They have used equations, tables, and graphs. By analyzing the problem using all three representations and seeing the connections among the representations, students develop a richer understanding of the problem, its solution, and the important mathematics involved.

This example also illustrates the use and power of recursion. A recursive point of view is used to generate the equations and tables. This approach makes this problem accessible to more students. The equation NEXT = 0.4 NOW + 440 (start at 440), described in the Algebra Standard, is easy for students to generate and understand. This leads naturally to the more formal equation A(n+1) = 0.4 A(n) + 440, A(0) = 440. Using these equations, spreadsheet tables and graphs can be generated and analyzed.

The non-recursive equation (called the explicit formula or closed-form equation) for this situation is A(n) = –293.3333(0.4)n + 733.3333. This is a more difficult equation for students to generate and work with. In fact, students are able to use this approach only when and if they get to more advanced high school mathematics. In contrast, the recursive approach can be undertaken early in the high school years. At an appropriate time, the closed-form equation should also be brought into the analysis of problems like this.

Even though students may not be able to write an explicit formula for A(n), they should realize that A is a function of n. The graph displays the fact that A(n) has a horizontal asymptote at 733.33. If the initial dose is greater than 733.33, then A(n) decreases to the asymptote, and if the initial dose is less than 733.33, then the function increases to the asymptote.

Finally, problems like this are important to study and teach for several reasons. They provide a rich environment in which to use important processes of mathematics. This example helps students develop skill in problem solving, mathematical modeling, communication, reasoning, finding connections, and using multiple representations. Such problems also provide experience with important mathematics content. The basic equation used in this example, expressed in several different formats, is equivalent to A(n) = r • A(n – 1) + b. If r = 1, then this equation represents arithmetic sequences and linear change. If b = 0, then this equation represents geometric sequences and exponential change. In applications, equations like this can be used to model and analyze many situations that involve sequential change, like the growth of money in an investment program, year-to-year population growth, or daily change in the chlorine concentration in a swimming pool.

### Tale Time to Reflect

•  How can the recursive point of view be used to enrich your understanding of linear and exponential functions? For example:
•  Find an equation using the words NOW and NEXT that corresponds to the linear equation y = 3x + 4. How does the slope appear in the NOW-NEXT equation?
•  Do the same for the exponential equation y = 3x. How does the base of the exponential function show up in the other NOW-NEXT equation?
•  Could you model the medicine problem using either a linear or an exponential function?
•  How might this situation lead to an initial discussion of asymptotes and limits?
•  Do you think that the multiple-representation approach used here is an effective way to build students' understanding

### Reference

National Research Council. High School Mathematics at Work: Essays and Examples for the Education of All Students. Washington, D.C.: National Academy Press, 1998.