The concept of function traverses the mathematics studied at school and university and is an important transitional link between the two. We examine how an interval perspective on function may help with constructing graphical antiderivative functions. In doing so a number of important constructs needed to build such a graphical understanding are considered, along with how the students in the study linked them and built with them. In addition, some of the difficulties they faced and possible reasons for them are explained. The evidence presented shows that an interval perspective on function was important in being able to construct antiderivative functions graphically. Hence, we propose that this interval perspective on function may prove useful in helping students in transition to construct a local perspective on function. In turn we suggest what a potential path for thinking about graphical antiderivatives could look like and the kind of activities that could assist transition students along it.