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As we can see from our initial equation:
The graph makes a standard circle with radius 1 and center at (0,0). The even power of the exponents makes this graph symetric both on the x and y axis.
On the other hand, our equation that cubes both x and y has orgin symmetry. This seems to be the pattern for all odd powers of x and y.
Let's now explore just the even exponents by looking at the range from 2 -- > 24. Click here to watch the transformation.
As you can see, there is a shift change from a circle to a square. We can make an assumption that if 'n' continues to increase, the end result will simply be a finer and finer square. It will never become a perfect square, but it will approach it.
So ... as 'n' approaches infinity, the graph will come closer and closer to resembling the following:
On the flipside, if 'n' is odd and has a range of 3 --> 25, the results looks more like this.
We can observe that the graph is very similiar to y = -x away from the orgin, but around it, it strives to cross the coordinates (-1,1), (1,1), and (1, -1) but never quite reaches them.
In fact, even as 'n' approaches infinity, it will never reach those coordinates, but it will look very similiar to the following graph:
