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Hearts

There are many ways of writing equations or drawing figures that approximate the shape of a heart. Here are a few.

The ellipse.

Consider the following ellipse. The line y = x is shown for reference.

This ellipse has center at (0, 0) and it has rotational symmetry. Consider the reflection of the ellipse in the y-axis.

The reflection is accomplished the replacing the sign of the xy term. As can be observed, there are two heart shapes embeded in this figure. Note that the two ellipses are also relected in the x-axis.

The challenge is to use just one of the equations and modify it so that the values of the equation for negative x are reflected about the x-axis but the values for positive x are not changed. What if x was replaced by |x| in the equation?

If we vary the constant on the RHS, we can get multiple graphs like the following.

Vary the first coefficient:

Vary the second coefficient:

Vary the third coefficient:

Try a different equation. Using Polar Coordinates, let