  The linear orbit of a degree d hypersurface in $\mathbb{P}^n$ is its orbit
under the natural action of PGL(n+1), in the projective space of dimension $N
=\binom{n+d}{d} - 1$ parameterizing such hypersurfaces. This action restricted
to a specific hypersurface $X$ extends to a rational map from the
projectivization of the space of matrices to $\mathbb{P}^N$. The class of the
graph of this map is the predegree polynomial of its corresponding
hypersurface. The objective of this paper is threefold. First, we formally
define the predegree polynomial of a hypersurface in $\mathbb{P}^n$, introduced
in the case of plane curves by Aluffi and Faber, and prove some results in the
general case. A key result in the general setting is that a partial resolution
of said rational map can contain enough information to compute the predegree
polynomial of a hypersurface. Second, we compute the leading term of the
predegree polynomial of a smooth quadric in $\mathbb{P}^n$ over an
algebraically closed field with characteristic 0, and compute the other
coefficients in the specific case n = 3. In analogy to Aluffi and Faber's work,
the tool for computing this invariant is producing a (partial) resolution of
the previously mentioned rational map which contains enough information to
obtain the invariant. Third, we provide a complete resolution of the rational
map, which in principle could be used to compute more refined invariants.
