A \(\lambda \) -translator in \(\mathbb {H}^2\times \mathbb {R}\) is a surface whose mean curvature H satisfies \(H= \langle N,\partial _z\rangle +\lambda \) , where N is the unit normal of the surface, \(\partial _z\) is the vertical Killing vector field and \(\lambda \in \mathbb {R}\) . In this paper, we study how the geometry of the boundary of a compact \(\lambda \) -translator affects the shape of the surface, asking under what conditions the symmetries of the boundary are inherited by the whole surface. Due to the product structure of \(\mathbb {H}^2\times \mathbb {R}\) and the geometry of \(\mathbb {H}^2\) , we distinguish between different notions of graphs and reflections. We provide conditions on the boundary curve of the surface to ensure that an embedded compact \(\lambda \) -translator is a graph. Finally, we present estimates for the area of a vertical graph \(\lambda \) -translator in terms of its height and volume.
