Problem:
Through graphical means, investigate the effects of allowing the variable a to range in the closed interval [0,2] on the following polar equation:

This function is called the Cartioid.


Visualization:
Using Derive:

We will start with a = 0 and then use ascending values.

  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 = 1 + cos(0* +)
  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.

The graph you now see is called the Cartioid

Now we allow a to take increasing values in the interval (0, 2]

  1. Next press to return to the algebra window.
  2. Press for Author.
  3. Enter the function
    r = 1 + cos(.25* +)
  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.
    That graph appears as follows:

Note the clockwise rotation.
  1. Next press to return to the algebra window.
  2. Press for Author.
  3. Enter the function
    r = 1 + cos(.5* +)
  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 = 1 + cos(.75* +)
  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 = 1 + cos(1* +)
  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.
  16. Next press to return to the algebra window.
  17. Press for Author.
  18. Enter the function
    r = 1 + cos(1.25* +)
  19. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  20. Now press and then to see the new graph drawn.
  21. Next press to return to the algebra window.
  22. Press for Author.
  23. Enter the function
    r = 1 + cos(1.5* +)
  24. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  25. Now press and then to see the new graph drawn.
  26. Next press to return to the algebra window.
  27. Press for Author.
  28. Enter the function
    r = 1 + cos(1.75* +)
  29. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  30. Now press and then to see the new graph drawn.
  31. Next press to return to the algebra window.
  32. Press for Author.
  33. Enter the function
    r = 1 + cos(2* +)
  34. Press to get the plot window.
    Derive will automatically redraw any old graphs; so don't be surprised when the old graph(s) appear.
  35. Now press and then to see the new graph drawn.
    Now we have the following picture:

Extra Credit or Bonus Task

Problem: Determine what values of a and b are necessary to obtain the following picture, using the equation listed below:

Hints
  1. The picture requires multiple equations, i.e. multiple values of a and b.
  2. If you need more help see Polar Coordinates - 2.
    This will help you learn how to use the a values.
Click here for the solution.

This document was originally written by Richard Rupp.