Let \(f:D\rightarrow Y\) be a function from a nonempty convex subset \(D\) of a vector space \(X\) into a Banach space \(Y\) and \(K\) be a nonempty closed convex cone in \(Y\). One says that \(f\) is \(K-\)quasiconvex if for all \(y\in Y\) the level set \(f^{-1}(y-K)\) is convex. Assuming the directedness condition \((y_1+K) \cap (y_2+K) \neq \emptyset \) for all \(y_1,y_2\in Y\) and that the polar cone \(K^+\) coincides with the weak-star closed convex hull of the set \(\text{extd }K^+\) of its extreme directions, the authors prove that \(f\) is \(K-\)quasiconvex if and only if the scalar function \(l\circ f\) is quasiconvex for every \(l\in \text{extd} K^+\). This result extends an earlier characterization of \(K-\)quasiconvexity obtained by \textit{Dinh The Luc} [``Theory of vector optimization'' (Lect. Notes Econ. Math. Sci. 319, Springer, Berlin) (1988; Zbl 0654.90082)] in the particular case when \(K\) is the polyhedral cone generated by \(n\) linearly independent vectors of an \(n-\)dimensional Euclidean space.