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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