Is there a relationship between the degree of a polynomial and how "steep" it is on the left and right edges? If so, what is it

The larger the degree, the steeper the graph's branches towards the right and left edges.

Step-by-step explanation:

Yes, there is a relationship between the degree of a polynomial and how steep its branches are at their end behavior (for large positive values of x, and to the other end: towards very negative values of x).

This is called the "end behavior" of the polynomial function, and is dominated by the leading term of the polynomial, since at very large positive or very negative values of the variable "x" it is the term with the largest degree in the polynomial (the leading term) the one that dominates in magnitude over the others.

Therefore, larger degrees (value of the exponent of x) correspond to steeper branches associated with the geometrical behavior of "power functions" (functions of the form:

which have characteristic end behavior according to even or odd values of the positive integer "n").

Recalling the behavior of such power functions, the larger the power (the degree), the steeper the graph.