## 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, graphed at the right.

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 y-axis but the values for positive x are not changed.

What if x was replaced by |x| in the equation?

This is equivalent to

Consider

How is it different?

You will probably want to experiment with Graphing Calculator or some other graphing tool to experiment with these and other equations.

If we vary the constant on the RHS, we can get multiple graphs like at th right.

If we vary the first coefficient graphs similar to those at the right result.

If we vary the second coefficient graphs similar to those at the right result.

If we vary the third coefficient graphs similar to those at the right result.

Try some different equations.

Using Polar Coordinates,  let

Begin with simpler equations for the ellipse:

Try experimenting with this equation: