Assignment 11

POLAR EQUATIONS


In this investigation, we will look at the equation r = a + bcos(kx).

In this graph, a and b both equal 1 and k is the integer 1. This is one version of the n-leaf rose. Let's now compare this to

r = bcos(kx).


Purple:

Red:

Notice that when the a is 0, we have a circle with center (.5,0) and rdius .5.


It is interesting to see what will happen when we replace cosine with sine.

Purple:

Red:

Notice that the graph has the same characteristics, it just rotates around the origin.


Now we will see what happens when a,b,c, and k are varied.

In this graph, k was increased to a value of 2. Everything else remained the same. This resulted in the graph splitting into two loops.


For k = 3:

This results in three loops.


For k=4:

This results in 4 loops.


To see the pattern from k=1 through 10 click HERE!


What if a and b remain equal, but they both increase? Click here to check it out.


Next I will investigate quite a few different equations.


Click HERE to see for varying a's.

Click HERE to see for varying a's.

**Notice that this changes the size of the graphs and there is no loop.


Click HERE to see for varying a's.

Click HERE to see for varying a's.

**Notice that this changes the size of the graph.


Now I will look at .

This first graph shows when a,b,c, and k are 1.


Now I will let a and b vary at the same rate.

Click HERE to see what happens to the graph.


Now I will let c vary.

Click HERE to see the results.


Now I will let k vary.

Click HERE to see the results.

The k seems to be the most interesting change because it changes the number of lines.


 

 

 

 

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