Heart and Bell Curves

For this exploration, we are going to look at the polar equation

When n=5, the equation produces a graph that looks like a heart.

When n=-5, the equation produces a graph that looks like a bell.

Let’s look at the polar equation when n=-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5.

As n increases from -5 to -1, the bend in the “heart” gets closer to the origin.  As n increases from 1 to 5, the “bell” increases in size and the top gets farther away from the origin.

When n=0, the polar equation produces a 5-leaf rose.

To look further into n-leaf roses, see Exploration 11.

If we look at the 5-leaf rose, the two petals to the left of the y-axis look like a “heart” and the three petals to the right of the y-axis look like a “bell”.

As n increases from -5 to 0, the bend in the “heart” gets closer the origin to form the two petals to the left of the y-axis, and the three petals to the right of the y-axis begin to form.

As n increases from 0 to 5, the petals to the right of the y-axis grow farther away from the origin to form the “bell” shape, and the petals to the left of the y-axis get closer to the origin until they disappear.

What will the graphs look like if the polar equation was the sine function, instead of cosine?

Let’s look at the polar equation,  when n=-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5.

We can see that the graphs form a heart and bell just as the polar equation did with the cosine function.

When n=0, a 5-leaf rose is formed.  It is similar to the one made with the cosine function, except that it is rotated.  Three of the petals are on top of the x-axis, and two are below the axis.

When n increases from -5 to 0, the bend in the “heart” gets closer to the origin to form the two petals below the x-axis, and the three petals above the x-axis begin to form.

When n increases from 0 to 5, the three petals above the x-axis move away from the origin and form the “bell,” and the two petals below the x-axis get closer to the origin and disappear.