In this study, we investigate students' ways of understanding graphing tasks involving quantitative relationships in which time functions as an implicit variable. Through task-based interviews of students ages 14-16 in a summer mathematics program, we observe a variety of ways of understanding, including thematic or visual association, pointwise thinking, and reasoning parametrically about changes in the two variables to be graphed. We argue that, rather than comprising a hierarchy, these ways of understanding complement one another in helping students discover an invariant relationship between two dynamically varying quantities, and develop a graph of the relationship that captures this invariance. From these ways of understanding, we conjecture several mathematical meanings for graphing that may account for students' behavior when graphing quantitative relationships.