We provide a matrix factorization model for the derived internal Hom (continuous), in the homotopy category of k-linear dg-categories, between categories of graded matrix factorizations. This description is used to calculate the derived natural transformations between twists functors on categories of graded matrix factorizations. Furthermore, we combine our model with a theorem of Orlov to establish a geometric picture related to Kontsevich's Homological Mirror Symmetry Conjecture. As applications, we obtain new cases of a conjecture of Orlov concerning the Rouquier dimension of the bounded derived category of coherent sheaves on a smooth variety and a proof of the Hodge conjecture for n-fold products of a K3 surface closely related to the Fermat cubic fourfold.