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Dynamic Graphs with Geometers’ Sketch Pad

 

            We begin now a look at the mathematical notion of function,  which we can think of as a process which takes any element of a set A and transforms it into a unique element of a set B.   You have probably already studied some common functions:  linear, quadratic, polynomial.  For this lab activity, it is helpful to think about a function as a set of inputs with corresponding outputs.

 

           It is difficult to understand the dynamic nature of functions by looking at a formula, a graph, or even a table of values.   In this activity, we will look at the nature of change of the output (value of the dependent variable) as the input (value of the independent variable) changes.   Specifically, we will look at the nature of change in linear and  exponential functions.

 

Activity 1

 

Open the GSP file entitled “dyna x,” and play with the different points on the axes there.  You might find it helpful to know the following: 

 
  • What is O and what happens when you move it?
  •  What is U and what happens when you move it?
  • What is Z and what happens when you move it?

 

Now place your cursor on the point marked “A” and play with it a bit. (Be sure to place your cursor just on the black point or you may end up moving things you don’t want to move!)   Jot down any interesting observations.

 

The following questions offer some guidelines for your investigation (but please feel free to just play around for a while and notice anything interesting on your own.)

 
  1. Does it appear that the output changes more quickly or more slowly than the input as you drag point A around?

 
  1. What is the initial value of the output, i.e. what is the value of the output when the input = 0?

 
  1. Is the value of the output ever negative?  Why do you think this is so?

 
  1. How many output values are there for each input?

 
  1. Does every possible value on the input axis have an output?

 
  1. Compare the change in the output for equal changes in input.  For example, on the input line, place point A on the origin.  Then move point A one unit to the right and note how much the output changed.  Repeat several more times, until the output moves off the second axis.  Jot down your observations.

 
  1. Place the point A back on the origin again.  Move it to the right one unit, then to the left one unit.  How do the outputs compare?  Now move A to the right 2 units, then left 2 units.  How do the outputs compare?

           When will this pattern occur? Why does it occur?

 

Activity 2

 
  1. Open the GSP file marked “dyna x triple.”   Note that you now have three points on the same input axis.  Play with them for a while.

 
  1. Is the process connecting the input to the output the same for all three points?  Explain.

 
  1. Use these three points to answer questions 4 and 5 in Activity 1.

 
  1. For what values are the inputs and outputs equal to each other?

 

Repeat Activity 1 for each of the following:

 

a)  dyna x+3

b) dyna x-1

c) dyna 2x

 

 

Activity 3

            Open the GSP file entitled “dyna 2^x,” and play with the different points on the axes there.    Please answer questions 1-7 of Activity 1 for this new function, paying extra close attention to question #6.

 

 

Activity 4

 

Open the GSP file marked “dyna 2^x triple” and answer the questions in Activity 2.  

 

 

Activity 5

 
  1. Open the GSP file entitled “dyna guess” and play around with it for a while.

 

     2.  Formulate an expression of the process from the input axis to the output

          axis and test your conjecture with several different values for the input.  Then click    

          with your mouse on “show function” to see if you were correct.

 

Exercises

 

For the input/output processes represented in each of the GSP files below, do the following.

 

A.  Open the GSP file and play around with it for a while.

 

B.   Is the process connecting the inputs to the outputs similar to the linear functions

     above or the exponential?   Explain.

 

            These questions might guide your investigation.

 

           i)  Is the output ever negative?

          ii)  Do all possible input values have an output?

         iii)  Do any 2 different values of the output have the same input

              values?

                       

         iv)  What is the initial value?

         v)  What is the nature of change in the output as the input changes?   

             Perform the same type of experiment as we did in #6 of Activity 1.

 

C.  Formulate an expression of the process from the input axis to the output

     axis and test your conjecture with several different values for the input.

 
  1. dyna2.1 guess1  
  1. dyna2.1 guess2  
  1. dyna2.1 guess3  
  1. dyna2.1 guess4  
  1. dyna2.1 guess5           
  1. dyna2.1 guess6  

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