  Let $U$ be a set of polynomials of degree at most $k$ over $\mathbb{F}_q$,
the finite field of $q$ elements. Assume that $U$ is an intersecting family,
that is, the graphs of any two of the polynomials in $U$ share a common point.
Adriaensen proved that the size of $U$ is at most $q^k$ with equality if and
only if $U$ is the set of all polynomials of degree at most $k$ passing through
a common point. In this manuscript, using a different, polynomial approach, we
prove a stability version of this result, that is, the same conclusion holds if
$|U|>q^k-q^{k-1}$. We prove a stronger result when $k=2$.
  For our purposes, we also prove the following results. If the set of
directions determined by the graph of $f$ is contained in an additive subgroup
of $\mathbb{F}_q$, then the graph of $f$ is a line. If the set of directions
determined by at least $q-\sqrt{q}/2$ affine points is contained in the set of
squares/non-squares plus the common point of either the vertical or the
horizontal lines, then up to an affinity the point set is contained in the
graph of some polynomial of the form $\alpha x^{p^k}$.
