## 7.2.3 Graphing the Situation

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 *n*th 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**

**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 atWork: Essays and Examples for the Education of All Students.Washington, D.C.: National Academy Press, 1998.