Charlie Conway 7/27/2009

**Polar Equations**

For this exploration, we will investigate equations of the type

When **a** and **b** are equal and **k** is an integer, then this formula is commonly known as the "n-leaf rose." This is a fairly obvious name, as you will see when we investigate the graph.

When **a** and **b** and **k** are all 1, then the graph looks like a single leaf with x-intercepts of 0 and 2 and y-intercepts of 1, 0, and -1.

If we change the values of **a** and** b**, still making them equal, and leaving the value of **k=1, **the shape of the graph is the same, only bigger. The y-intercepts move to the positive and negative of the values of **a** and **b**. Also, part of the graph always comes to a point at the origin. The other x-intercept increases to cross the x-axis at the sum of **a** and **b**. If the values of **a **and **b** are negative, the graph is simply reflected across the y-axis.

What if **a** and **b** are different, still leaving **k=1**? Perhaps the simplest form of this graph would be when **a=1** and** b=2**, so our equation would be

As you can see, the y-intercepts are 0, 1, and -1, reflecting the positive and negative value of **a** and the crossing at the origin. The x-intercepts other than the origin are 1 and 3, which interestingly correspond to **b+a** and **b-a**. Perhaps a better way to think about this fact is with respect to the distance from the origin along the x-axis of each loop. The big loop extends 3 units or **b+a** units, and the smaller loop extends** b-a=1** unit.

This relationship also holds when **a** is larger than **b**. In the case where **a=3** and **b=2**, the graph looks like this

Again, the graph has y-intercepts of -3 and 3, which correlate the positive and negative values of **a. **The x-intercepts are 5 and -1, which correlate to **b+a** and **b-a**.

Now that we have investigated many of the properties of our equation with **k=1**, let's change the value of **k**. As we said earlier, when **a** and **b** are equal and **k **is an integer, then the graph is known as an "n-leaf rose."

for **k** ranging from 0 to 10 to 0

As its name suggests, the graph "grows" a new leaf at every integer **k**.

The graph as **k** changes while **a=b** is interesting but fairly simple. The graph as **k** changes with **a** and ** b** not equal is a bit more complex but not something that is necessarily shocking. Here are two examples of equations with **a** not equal to **b**, with different values for **k**.

In the first graph, in which **k=5** (an odd integer), there are 5x2=10 pedals, five larger pedals and five smaller pedals. Each of the smaller pedals is centered inside one of the larger pedals. The bisectors of two pedals is on the x-axis, with one reaching its maximum x-value at 4 and the smaller one reaching its maximum x-value at 2. This is consistent with what we have discovered thus far in that the bigger "loop" corresponds to **b+a** and the smaller loop corresponds to **b-a**. These distances are the same for all of the pedals, not just the pedals on the x-axis. The graphs of all the equations with odd values of **k** have similar properties, with the smaller pedals being inside of the larger pedals and one pedal being centered on the positive x-axis.

In the second graph, we have **k=4**, an even integer, and there are as you might expect, 4x2=8 pedals. The difference, as you can see, is that with even values of **k**, the smaller pedals are centered in between the larger pedals. However, the length of the pedals still correspond to the values of **a+b** and **a-b**.