by Oktay Mercimek
Let's start to investigate with first one.
for a=1 and k=1 , equation is which is a circle equation in polar coordinates.
Making a bigger doesn't change that much , it just changes the size of circle.
If we set a as negative number, it changes the direction of circle
For k>1 , graph is changing slightly
If a=1 and k=2, then graph looks like
And again changing a is changes the size of curve
Let's make k=3 and see what is going on
Isn't is little different than what you expected?
make it k=4 and k=5
Can we say it is different for odd and even values for k ?
For k=10 , I expect 20 leaves
And for k=11, I expect 11 leaves
To understand the behavior of this function, I put k=n and make n variable between 0 and 10.
So it is for , Click here to view Graphing Calculator File.
Click play button to see how it changes while n varies and you will be surprised when you see the difference between odd and even values for k. (you can download Graphing Calculator Viewer (FREEWARE) from http://www.pacifict.com/FreeStuff.html in order to open a GCF file)
has similar to , for k=1 circle is on the x-axis instead of y-axis.
For other k values, it is again similar to sin function but rotating with certain angle
To understand better you can look HERE where a varies (ASSGN11 A vary) or HERE for k varies.
More amazing things is happening with
Let's look for couple b values,
Click HERE to GCF (COS B VARY) file for other b values for function .
For sin version click HERE (SIN B VARY).
As a last investigation let's look this function
Let's make c=10, a=1 , b=1 and k=4
Isn't it nice?
click Here for different c values ( LAST C VARY)
For c=10, a=1 , b=1 and k=5.5
click Here for different k values ( LAST K VARY)
click Here for different a values ( LAST A VARY)
click Here for different b values ( LAST B VARY)
You can see that changing k is related with the expanding lines and curves, changing c is related with magnifying picture of graph, and changing a and b is related with rotating the picture of graph.