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