We report an investigation into the processes involved in a common graph-reading task using two types of Cartesian graph. We describe an experiment and eye movement study, the results of which show that optimal scan paths assumed in the task analysis approximate the detailed sequences of saccades made by individuals. The research demonstrates the computational inequivalence of two sets of informationally equivalent graphs and illustrates how the computational advantages of a representation outweigh factors such as user unfamiliarity. We describe two models, using the ACT rational perceptual motor (ACT-R/PM) cognitive architecture, that replicate the pattern of observed response latencies and the complex scan paths revealed by the eye movement study. Finally, we outline three guidelines for designers of visual displays: Designers should (a) consider how different quantities are encoded within any chosen representational format, (b) consider the full range of alternative varieties of a given task, and (c) balance the cost of familiarization with the computational advantages of less familiar representations. Actual or potential applications of this research include informing the design and selection of appropriate visual displays and illustrating the practice and utility of task analysis, eye tracking, and cognitive modeling for understanding interactive tasks with external representations.