For my final write-up, I have decided to return to Assignment 10 on Polar Equations and explore the first problem of that assignment.

1. Investigate

Note:

* When a and b are equal, and k is an integer, this is one textbook version of the " n-leaf rose."

* Compare with for various k.

* What if . . . cos( ) is replaced with sin( )?

We will begin this exploration by looking soley at the first equation, .

There are three variables that need to be defined in the equation above in order to produce a graph. Those variables are a, b, and k. Each of the three variables will change the graph in a different way and it is those changes that we will explore here.

We also have to define a range for theta, . Here it will be sufficient for theta to range from 0 to .

It is easiest to begin with the most simple values of the variables, when each variable equals 1.

Below is the graph when a = 1, b = 1, and k = 1.

Varying k for a = 1, b = 1: Positive Integers

Now that we have seen what we can call a base graph above, let's see what happens when a and b remain equal to 1, but k changes.

For k = 2, we get the following graph,

What is different between the original graph when k = 1 and the graph now when k = 2?

It looks like the graph has almost doubled - going from one leaf to two leaves.

Will the graph double again when k = 3 or will we get 3 leaves? What do you think?

Below is the graph for k = 3.

What happened? Now that k = 3, we see a graph with three leaves.

Do you detect a pattern here? When a = 1 and b = 1, it seems that we get a graph with k leaves.

Let's look at a few more values of k to verify our detected pattern.

k= 4:

4 leaves- check!

k = 5:

Yep, we see 5 leaves here.

k = 8:

Now we have 8 leaves.

k = 11:

Eleven leaves- just as predicted!

We can now safely conclude that when a = 1, b = 1 in the equation , the graph produced is a flower with k leaves. This is commonly known as the n- leaf rose.

Varying k for a = 1, b = 1: Non-Integer, Positive values

A good question to ask at this point would be what about values of k that are not integers. If we get a 3 leaf rose when k = 3, what will happen when k = 3.5? What about k = 1/2? Or k = 2.25? k = 1.6667?

What do you expect to happen here? Do you think the graph will change drastically now from what we saw above? Will the pattern still exist? Is that even possible? Can you picture a 1 and a half leaf rose? Let's see what happens!

Below is the graph for k = 3.5.

Whoa! It looks like we did get a graph with 3 and a half leaves. The graph did rotate from the graph earlier when k = 3.

Let's compare the graph of k = 3.5 with the graph of k = 3 and k = 4.

On the left below is the graph of k = 3.5 and k = 3. The graph on the right shows k = 3.5 and k = 4.

Above we can see the rotation that occurs as k increases. The flower seems to rotate to the right as k increases. Later, we will look at a movie of what happens to the rose with different k values.

Let's quickly look at some more graphs for non-integer k values.

k = .5

Below, we have k = 2.25 on the left, k = 2.5 in the center and k = 2.75 on the right:

Finally, we see k = 1. 3333 below on the left, k = 1.5 in the center, and k = 1.6667 on the right:

The following animation shows the rose graph as k increases from 0 to 10 and then decreases back to 0 again.

Varying k for a = 1, b = 1: Negative Numbers

What do you think will happen when k is a negative number? Is it possible for there to be -5 leaves on a flower? I don't think so.

If our pattern tells us we will have a k-leaf rose and we cannot have a negative number of leaves, what does that mean for our graph? Will the graph change into something alltogether different? Will there be no changes at all? Will there be no graph at all?

What do you think?

Let's see what happens when we graph for negative k values.

Below, k = -1.

Hmmm, the above graph looks suspiciously similar to the graph when k = 1. In fact, it looks exactly the same.

Let's look again at the graph when k = 1 and compare.

Yes, they appear to be the same graph.

Will this always happen? Let's look at a couple more examples.

k = 3 and k = -3:

k = 6 and k = -6:

It seems to be the case that when k is negative, the graph does not change at all from when k is positive.

Can this consistency be explained by looking at the equation?

Is there a difference between the following two equations: ?

Remember back to Trigonometry. There were many trigonometric identities, including the following: .

The identity above tells us that for cosine, the negative does not affect the outcome.

Now we understand from our equation why the graph does not change when k is a negative number.

Varying a: positive integers

We have seen that the variable k affects the number of leaves in the graph, but what happens when the value of a changes?

To best see what happens to the graph as a changes, I will let k = 3 and b = 1 for the following exploration.

As a reminder, let's look at the graph when k = 3, b = 1, and a = 1:

Let's see what changes when a = 2. Below the red graph shows when a = 1 and the blue graph shows when a = 2:

What happened to the graph? It seems the leaves expanded a bit with a larger value of a.

When a = 3 we see the following:

The leaves have expanded even more, creating a less defined flower than we had before.

As the value of a increases, the shape of the graph grows and becomes less defined.

What happens if the value of a is not an integer?

Below is the graph when a = 1.5:

It appears that when a = 1.5, the graph lies between where it was when a = 1 and where it would be for a = 2.

Click here for a graph of different a values to see this incremental increase in a and growth of the graph.

Varying a: positive non-integers < 1

When a is a non-integer, the pattern noted above does not seem to change. When a is less than one, however, we do see a difference.

Below is the graph when a = .5 (in blue, compared to a = 1 in red):

Now, the leaves not only are smaller, but also now there are more leaves. It is almost as if the leaves folded in on themselves.

When a = .25, we get the following:

Again, we see three more smaller leaves in addition to the original leaves getting smaller and more compact.

Below is a comprehensive picture for values of a less than one.

The following animation shows the graph as a changes, with b = 1 and k = 3. In the animation, a goes from 0 to 4.

Varying a: negative values

We have seen what happens for different positive values of a, but what will happen when a is negative?

Below is the graph for a = -1:

The graph above is the exact same graph as the graph when a = 1.

Like we saw with the k-values, the sign of a does not affect the graph. The graph will be the exact same for -a as for a.

Varying b: positive integers

Now that we have seen what happens to the graph as the values of variables k and a change, we can finally look at the effect of the variable b on the graph.

We will continue to look at the case where k = 3. The value of a will be equal to 1 unless otherwise stated.

As a reminder, below is the graph of k = 3, a = 1, and b = 1.

Now let's see what happens when b = 2:

It appears the change in b to 2 has had a similiar affect to when a was less than one above. Now we have new smaller leaves inside the original leaves, which have expanded and grown from when b = 1.

Below we see that both the original and the new leaves expand further as b increases in value:

Varying b: positive non-integers > 1

When b equals a value greater than one that is a non-integer, the pattern we saw above continues. That is, the flower grows in size and has three more leaves inside the flower that also grow.

Below, b = 1.5:

For good measure, we'll show one more example. Below b = 2.75:

Below is a comprehensive graph for different values of b > 1:

Varying b: positive non-integers < 1

Next, we want to examine what happens when b is a positive number less than one. Before, we saw that as the value of b increased, the size of the flower increased and three inner leaves were added that also grew in size with b.

Do you think we will continue to see this pattern only with a decrease in size for values of b less than one? Or will the graph change shape again? Let's see!

Below is the graph for b = .5. Is it what you expected?

Now, with b < 1, it seems the graph of the flower has gotten smaller in scale, but the leaves have become less defined. The result of the leaves becoming less defined is similar to what we saw when the value of a was increased.

The leaves become less and less defined as b becomes smaller.

Below, b = .25:

The following image shows the graph for various b values less than one:

We can see that as b approaches zero, the graph itself becomes less defined.

Varying b: negative values

The only values of b left to investigate are negative values. Before, with k and a, we found that the sign of the value did not make any difference for the graph. Do you think this consistency will occur again with the variable b? Or will the sign of b make a difference in the graph? Let's see.

Below is the graph for b = -1. Is the graph the same or different than the graph for b = 1?

The graph is different than before!

Don't see the difference? Look at the following image to see the graphs of b = 1 and b = -1 together.

Now we can clearly see that the negative value of b has rotated the graph what looks like 60 degrees.

Let's look at more examples of negative versus positive values of b to see this pattern continue.

Below, b = 2 and b = -2:

The following graph shows b = .5 and b = .-5:

Unlike for variables a and k, the sign of b does affect the graph. A negative value of b rotates the graph.

The following animation shows the changes in the graph for the range of values of b from 0 to 2.5

New Equation

Next, we want to look at the following equation and compare it with the first equation we have thus far examined.

compared to our first equation,

We will begin the same as we have before, looking at the equations when a = 1, b = 1, and k = 3. The original equation is in red, while the new equation is graphed in blue.

It appears that the new equation gives a smaller scale graph than the first equation. Does this make sense looking at the equation? Yes, because the only difference in the equations is that the original equation has "+ a" added to the b cos (ktheta) part of the equation. It makes sense that in this case, the "+a" removed would scale down the graph.

Let's see if this scaled down graph continues with different values of a, b, and k.

Below is the graph for a = 2, b = 1, and k = 3.

Above, we do not see a scaled down version of the original equation in the graph of the new equation. What do we see? The original equation acts as we would suspect after our above investigation, while the new equation remains unchanged from the first time we graphed it. Does this make sense? Yes, because the only change we made in the first and second images here was to change the value of a from 1 to 2. Since there is no variable a in the new equation, it makes sense that the graph would go unchanged.

If the value of a does not change our new equation, let's look at what happens when we change the value of b.

The values in the following image are a = 1, b = 2, and k = 3.

What do we see above? The change in the value of b does not affect the new equation in the same way it affects the original equation. As we noted earlier, the increase in the value of b expands the graph of the original equation and adds three new leaves to the graph. For the new equation, the growth in b expands the original graph, but does not add any new leaves.

To continue, I will show the graphs of the two equations for different values, without explanation. As you view each image, think about the differences between the two graphs and how these differences could be explained by the equations themselves.

Below, a = 1, b = 2, and k = 2.

0

Below, a = 1, b = .5, and k = 2.

Below, a = 1, b = -2, and k = 3.

It appears that there is no exact pattern for the differences between the graphs of the two equations, even though the equations appear to be very similar. The differences in the graphs are explainable, they are just not consistent for all changes in the variables.

In a regular, non-parametric equation, the addition of "+a" would lead to a vertical shift in the graph. We have seen above how in parametric equations, however, the addition of "+a" to an equation can change the graph drastically, in a hard to predict manner.

Sine instead of Cosine

To end this exploration, I will briefly look at what happens when we replace the cosine function in the original equation with the sine function.

Our new equation is .

The image below shows the graph of the original equation (with cosine) and the new equation (with sine) for a = 1, b = 1, and k = 3.

Above, we see a rotation between the two graphs, similar to the earlier rotation seen with the sign change in the variable b.

Will the graph with sine always be a rotation of the cosine graph?

Below, we see the graphs for both equations when a = 1.5, b = 2, and k = 4.

Below, a = 2, b = -2, and k = 2:

From the above examples, we can see that the difference between the cosine equation and the sine equation is a rotation of the graphs.

Final Remarks

In the investigation above, we have explored a small set of parametric equations. First we looked the changes in the graph produced by different values of a, b, and k. We saw the differences depended on whether or not the variable was an integer and if it was greater than or less than one. With the variable b, the sign of the value also affected the graph. We then looked at a new equation that appeared to be similar to the orginal equation, but produced graphs very different than the original equation's graphs. Parametric equations do not act the same as non-parametric equations in the way that we can predict shifts, rotations, etc based on changes in the equation. Finally, we briefly examined the effect of changing the function in the equation from cosine to sine. We found a rotation between the two graphs that were otherwise kept the same.

Parametric equations can be very interesting to study, as we have found here. Sometimes the changes are predictable and other times they can be very surprising. It is important to realize that there exists a whole other class of equations and graphs outside of the typical x, y functions we see most often.

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