Investigation of Polar Equations
Here weŐll be investigating .
LetŐs begin with a, b, and k =1.
Now lets investigate for different values of k, while a and b are held constant at 1.
We can see that the number of ŇleavesÓ correlates with the value of k, in this case that is k=2 so we have 2 leaves.
If we look at the x-axis we see that the x-coordinate is equal to a+b, or in this case 2.
And just because its prettyÉ and its always nice to check multiple values;
Here we can see that both the x and y-coordinates are equal to ± (a+b), which is still 2 and we can easily see why this is called the n-leaf rose.
When graphing for k=1/2 we have to increase the number of rotations (double) to get a complete graph. We can see there are 2 inverted leaves.
When graphing for k=1/4 we have to increase the number of rotations by 4 times to get a complete graph. We can see that there are 4 leaves getting progressively larger and superimposed on one another.
Now letŐs investigate situations where k and b are varied and a is held constant.
Here we have a double n-leaf rose with k=n smaller leaves and k=n larger leaves.
Notice that the leaf values on the x and y-axes still corresponds to ± (a+b).
As b increases the length of the second set of leaves gets longer.
For ał3, we get a graph with a radius of a, and center (b,0)
For a˛3 we get a starfish with k arms.
Here we will get a starfish with six arms.
What happens when we exchange sine for cosine?
The graphs are the same shape but the sine graph is rotated 900 which is logical if we consider the relationship of the sine and cosine functions.