**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.**

**return**