Let f be a function from a finite field Fp with a prime number p of elements, to Fp. In this article we consider those functions f(X) for which there is a positive integer n>2p−1−411 with the property that f(X)i, when considered as an element of Fp[X]/(Xp−X), has degree at most p−2−n+i, for all i=1,…,n. We prove that every line is incident with at most t−1 points of the graph of f, or at least n+4−t points, where t is a positive integer satisfying n>(p−1)/t+t−3 if n is even and n>(p−3)/t+t−2 if n is odd. With the additional hypothesis that there are t−1 lines that are incident with at least t points of the graph of f, we prove that the graph of f is contained in these t−1 lines. We conjecture that the graph of f is contained in an algebraic curve of degree t−1 and prove the conjecture for t=2 and t=3. These results apply to functions that determine less than p−2p−1+411 directions. In particular, the proof of the conjecture for t=2 and t=3 gives new proofs of the result of Lov\'asz and Schrijver \cite{LS1981} and the result in \cite{Gacs2003} respectively, which classify all functions which determine at most 2(p−1)/3 directions