Bent-cable regression is an appealing statistical tool to model changepoint data. The model assumes a continuous function comprised of three segments connected together: two linear segments to describe the incoming and outgoing phases, joined by a quadratic function to model the transition zone. As demonstrated in this study, the assumption of a quadratic bend can lead to the starting and the end points of the transition to be either underestimated or overestimated. We propose a generalization of the bent-cable model to overcome this problem. The generalized model retains the interpretability of the regression coefficients, while improves the accuracy of locating the transition phase. We consider a comparison among the quadratic bent-cable, generalized bent-cable and piecewise linear models in analyzing both real-world and simulated data. This study suggests that the proposed generalization can be valuable in adequately describing different types of changepoint data.