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.
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:
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.
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.
, 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.
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.
