Exponential ^ Exploration

What does the graph of x^2+y^2=1 look like?

What about x^3+y^3=1?

In general, when n is an integer, what does x^n+y^n=1 look like as n increases?

As most of you know, the graph of x^2+y^2=1 is a circle centered at the origin with a radius of 1.

The graph of x^3+x^3=1 has the same intercepts as our circle; however, instead of coming together to create a circle, this curve turns up in the second quadrant and heads toward infinity as x decreases. And again in the fourth quadrant, the curve turns away from the origin and approaches negative infinity as x increases. The curve in the first quadrant is very similar to that of circular graph of x^2+y^2=1, with the only difference being a slightly sharper corner.

As the exponents increase, it becomes clear that the equations with even indexes resemble the graph of x^2+y^2=1, but with increasingly sharper corners approaching what appears to be a square. The equations with odd indexes also resemble the first equation with odd index, x^3+y^3=1. But again, as the exponent increases, the corner in the first quadrant becomes more squared off. Also, the curve in the second and fourth quadrant moves further along the x-axis (backward in the second quadrant and forward in the fourth quadrant) before making the its turn away from the origin, making for sharper angles.

Here is an animation that represents the situation.

(It might be more helpful to move the cursor, as opposed to letting it play so fast)

As you can see, the curves begin very round, then become very sharp as the exponent get larger. They then round back off as the exponent decreases again.