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Using Algebra and Discrete Mathematics to Investigate Population Changes in a Trout Pond
Part Three - Graphical Analysis

Table Graph  Equation
  Make a Conjecture Numerical Analysis Graphical Analysis Symbolic Analysis

In the previous part of this i-Math investigation, you found the population sizes, A(n), that correspond with different years, n. It is often helpful to draw a graph. Examine a graph of n versus A(n) below.
View a Graph
Graph

Years "n" versus Population "A(n)"
for n versus A(n).

Analyze the Graph

Describe characteristics of the graph, and describe what information the characteristics give you about the trout population.

How is the long-term population shown in this graph?

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What if ... (revisited)

Now use the Interactive Graphing Tool to investigate the questions from the previous part. That is, consider the following three questions.

(a) If the initial population doubles, what will happen to the long-term population?
(b) If the annual restocking amount doubles, what will happen to the long-term population?
(c) If the annual population decrease rate doubles, what will happen to the long-term population?

Draw graphs for each of these three situations. Compare the graphs to the numerical results you got before.

 

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A "Recursive" Graph

In a recursive situation like this, there is another type of graph that can provide a useful visual representation of the situation. This type of graph is called a "cobweb graph" or "stairstep graph". Click on Plot Graph, below, to draw a stairstep graph. The graph will plot slowly. Watch carefully.


This graph is sometimes called a "cobweb graph" or a "stairstep graph".

 

Analyze the Graph

Explain what is being graphed and what information is shown by the graph.

How is the long-term population shown in this graph?

If you'd like a hint to help you understand this graph, and watch graph above.

Using the hint, explain what information is shown by the stairstep graph.

 


Next Step

In addition to a graphical and numerical analysis it is helpful to do a "symbolic" analysis.
.

Go to Symbolic Analysis

 
 
Graphical Analysis
© 2000 - National Council of Teachers of Mathematics


CD Version last updated: August 8, 2000