Investigations with polar equations


Write-up by

Blair T. Dietrich

EMAT 6680

Assignment #11 Investigation #2



How does the graph of change as a varies?



Here k = 0.1

Larger values of a tend to make the spirals grow in "width" and in "length."



How does the graph change as k varies?


Keeping all a values (and colors) as shown above, we can see the affect that the value of k has on the graph:


For k = 1…



For k = 2…


For k = 5…


For k = 10…


Try your own value of k here.




What about the graph of ?


For k = 0 we have nice concentric circles…                  For k = 1, the circles are nicely tangent at (0,0)…



For k = 2…


For k = 5…


For k = 8…

Try your own here.



How is the graph affected when a constant b is added to either of the previous equations?


Consider the graph of  for various values of b:



b = 0                                                                b = 3



b = 6                                                                            b = 9





Now consider the graph of   for various values of b (the color scheme is the same as for that above):


b = 0                                                                b = 3                                       



b = 5                                                                            b = 8


In each case it appears that adding the constant b yields a dilation in the graph.


Lastly, let us consider the equation .


Keeping the values of a, b, and c constant (here a=b=c=1), we can change k to see its affect on the graph. 

For k=0, the unit circle is graphed:


For k=1, the line y = -x + 1 is graphed.








As k increases in magnitude, the number of "branches" increases. 

Experiment with your own values of k here.


Now (while holding k=4) we will consider how changes in a, b, and c affect the graph.

c=2                                                                                          c=4


As c increases, the graph is dilated from the center and the graph intersects the axes at the c-values.


a=2                                                                                          a=4

As a increases, the graph is shrunk toward the center.  The intercepts appear to be 1/a on each axis.


b=2                                                                                          b=4


As b increases, the asymptotes to the curve get closer together.  The intercepts are unchanged.



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