## Equations of the form:

#### by: BJ Jackson

This section increases the exponent from by one degree. We
now have third degree equations which will allow the graph to
be in any of the quadrants. So, the question is what will this
new degree do to the origal graph which was a circle. By increasing
the degree, the graph folds the curves of the circle inward between
the inflection points at 0, 90, 180, and 270 degrees and looks
like this.

Now, based on the previos explorations. One would make the
conjecture that when a and b increase, but remain equal, the inflection
points will equal a and b. The following graph shows that this
is exactly what happens when a and b are increased to two and
three.

Now, based on the previous explorations with these parametric
equations, we can conclude that when a < b that the shape will
remain the same but that the x-axis will now be a major axis (longer
than the y-axis) and that the inflection points will equal a on
the x-axis and b on the y-axis. The graphs below use values of
a from 1 to 4, and values of b less than a to support the conclusion.

Finally, based on the previous, one would conclude that the
scenario when a > b would work exactly like when a < b except
for the fact the major axis and minor axis would be reversed.
Another way to think of it would be as a 90 degree rotation of
the figure above which is exactly what happens as the following
picture shows.

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