How does the degree relate to the extrema?

I'm assuming this is for apolynomial function. The question of whether a degreee is odd or even changes the look of a graph. An even-numbered degree forms a parabola, where (in the most basic form), the one minimum point (extrema) just touches the origin. An odd-numbered degree, in its most basic form, doesn't touch a point, it crosses it. It expands infinitely without extrema.

Let's assume you're just talking about quadratic functions (or [even] parabolic functions, to be more general), in which case something like x^2 (the simplest quadratic equation) and x^50 would have the same extreme minimum point.