The feed-forward-loop (FFL), a network motif in genetic regulatory networks, involves two transcription factors (TFs): one regulates the expression of the second, and both TFs regulate the expression of an effector gene. Analysis of FFL design principles has been initiated, but the functional significance of the FFL is still unclear. In theoretical studies so far, the TFs are assumed to interact with different signals, which is common. However, we have found examples of FFLs in Escherichia coli in which both TFs interact with the same signal. These examples belong to the type 2 incoherent class of FFLs, in which each TF acts exclusively as a repressor of transcription. Here, we analyze mathematical models of this class of circuits, examining a comprehensive array of subclasses that differ in the way a signal modulates the activities of the TFs. Through parameter variation, we characterize statistically how input/output (I/O) behavior and temporal responsiveness are predicted to depend on the wiring of signal interactions in a circuit. We find that circuits can exhibit any of 13 qualitatively distinct steady-state I/O patterns, including inducible and repressible patterns. Some subclasses exhibit as many as six patterns. Transient pulses are also possible, and the response of a circuit to a signal may be either faster or slower than that of a gene circuit in which there is only one TF. Our results provide a catalog of functions for a class of FFL circuits, whose subclasses have different breadths of possible behaviors and different typical behaviors.