## Exploration 11: Write-Up

Polar Equations:

Investigate:

To gain a better understanding of what this particular equation means as the variables vary, it is important to gain a grasp on the individual changes that occur when each variable is altered separately.

Let’s look at k first by letting a=0 and k=1.

Using Desmos, we can graph this equation and use the slider to let the value of k vary.

Try it Yourself on Desmos!

From this we can see that varying k creates pedals.

Pedals are a feature of a special polar equation, rose, of the form .

Below is a table of the k values from 0 to 5 and the corresponding number of petals.

 K # of rose petals 0 0 1 0 2 4 3 3 4 8 5 5 6 12 7 7 8 16

From the table we can see that:

• For all even values, the corresponding number of petals equals 2k.
• For all odd values, the corresponding number of petals equals k.

Thus, the value of k will determine the number of petals associated with the rose graph.

Now let’s take a look at a by letting the b=1 and k=1.

Using Desmos, we can graph this equation and use the slider to let the value of a vary.

Try it out Yourself!

From simply varying the value of a , it was difficult to make any significant observations. However, looking at the graph a=b=1, a heart-shaped graph was produced.

This led me to believe that there is a connection between the a and b values. To investigate, I looked at then cases when   and . By moving the sliders of the follows graph, both cases can be observed.

From this there were a few observations that I was able to make.

1. When , there is a loop that goes inward.

I also discovered that:

- The distance of the loop from the origin to the furthest point on the loop is equal to

- The distance from the origin to the farthest point of the graph, is

2. When , there is no loop .

Now let’s look at b. Let’s make a=0 and k=1.

Using Desmos, we can graph this equation and use the slider to let the value of b vary.

Try it out!

From this, I was able to confirm that the value of b alters the length of the diameter of the circle. The diameter is the distance between r=0 and r=b. Thus, the center of a circle with a diameter b in polar coordinates is . When b is negative, it is as if the circle is flipped across the line x=0 or  in polar coordinates.