Take the graph of a $f(x)$ which can be any arbitrary function, e.g. $x^2$ . We draw tangents to each and every point on it and colour them. If relative to a fixed points have $z$ times the number of tangents passing through it, it is said to be of degree $z$.
The interesting part is that for a general point $S(h,k)$ what is its degree? I think that I should consider easier cases first. Have you a solution/idea that how to solve the problem? (don't uphold or down vote the question will be edited and improved gradually) [for example in case of y=x. W.R.T. to (0,0) the degree of all other points is 1 on line; and 0 on others. I think that it is related to the multi-valued-ness and interjectivity of $f'(x)$.