Problem:
Use graphing to determine the effects of changing the a and b values in the following polar equation:

Visualization:
Using Derive:

    First we will examine the effects of changing the a value

  1. Prepare the program for graphics.
  2. If the first menu item is Algebra then press .
  3. Press for Author.
  4. Enter the function
    r=sin
  5. Press to get the plot window.

    Now we need to prepare Derive to graph in polar coordinates.

  6. Press to enter the options menu.
  7. Now press to enter the state menu.
  8. Finally, press and then to tell Derive to graph using continuous polar coordinates.
    Note: Derive will continue to graph in polar coordinates for the remainder
    of the session or until the state is chaged back to rectangular.


  9. Now press and then to see the graph.
  1. Next press to return to the algebra window.
  2. Press for Author.
  3. Enter the function
    r=.5 sin
  4. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  5. Now press and then to see the new graph drawn.
  6. Next press to return to the algebra window.
  7. Press for Author.
  8. Enter the function
    r=2 sin
  9. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  10. Now press and then to see the new graph drawn.
  11. Next press to return to the algebra window.
  12. Press for Author.
  13. Enter the function
    r=3 sin
  14. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  15. Now press and then to see the new graph drawn.
    At this point our graph will be like the following:
n.b. this picture is the first few iterations of a topological object known as the Hawaiian Earring.

Now, we examine the effects of changing the b value

  1. First we have to clear the plot window by pressing and then
  2. Next press to return to the algebra window.
  3. Press for Author.
  4. Enter the function
    r=3 sin
  5. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  6. Now press and then to see the new graph drawn.
    As expected from before, the graph looks like the following:
  1. Next press to return to the algebra window.
  2. Press for Author.
  3. Enter the function
    r=3 sin (2)
  4. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  5. Now press and then to see the new graph drawn.
  6. Next press to return to the algebra window.
  7. Press for Author.
  8. Enter the function
    r=3 sin (4)
  9. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  10. Now press and then to see the new graph drawn.
  11. Next press to return to the algebra window.
  12. Press for Author.
  13. Enter the function
    r=3 sin (8)
  14. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  15. Now press and then to see the new graph drawn.
    At this point our graph will be like the following:

Feel free to try any additional experiments and combinations of your own.

This document was originally written by Richard Rupp.