A Circle or Not Quite A Circle:
This write-up is for the first year Algebra student. The write-up is for students who are looking to learn the graphical behaviors of the equation , when is a positive integer.
If one was to graph the equations , and , one would see that , is an equation of a circle with y-intercept at ±1 and x- intercepts at ±1. Intercepts are points where the graphs touches the x or y axis.
The intercepts of the equation , are at one for the y-axis and at one for the x-axis.
When graphed simultaneously, see graph, the two graphs intercept with each other at one on both the y-axis and the x-axis.
This type of behavior extends to , , , and etc.
What is unique about the equations is that when is an even exponent the graph is the expansion of the circle. Each graph have intercepts at and , but the values of the domains increase with every graph. See figure
On the other hand, the equations with odd integers for all intercept at and , but instead of expanding outward from ,,they move to the left of the equation between and . Then the graphs of the equations start to merge at about 3 and −3.
From this observation it can be expected that the graph of , to continue to expand outward from , and have x and y intercepts at ±1. See graph
Also, as seen in the picture, it can be expected that the graph of , to move to the left of , in the second and fourth quadrant. It can further be expected that the graph will intercept at the x-axis and at the y-axis at one. Then merge in the second and fourth quadrant after the domain of ±3.
From the above exploration it can be assumed that graphs where is even integer will behave as stated and the graphs where is an odd integer will behave as stated above.