In the present study, we ask whether and how online assessment can inform teaching about students' understanding of advanced concepts. Our main goal is to illustrate how we study design of tasks that support reliable online formative assessment by automatically analyzing the objects and relations that characterize the students' submissions. We aim to develop design principles for e-tasks that have the potential to support the creation of rich and varied response spaces that reflect convincingly the students' perceptions. We focus on studying the design principles of such an interactive rich task, and the representations and tools that support automatic real-time analysis of submitted answers, primarily in the form of free-hand sketches. Using a design research methodology, we describe a two-cycle study focusing on one e-task concerning tangency to the graph of a function. We characterize the mathematical attributes of examples constructed by students in an interactive environment. Checking the correctness of answers is only one of the functions of the STEP platform that we use. That platform can identify submissions that contain partial information (e.g., missing or misplaced tangency points), can categorize these answers, and allow the teacher to analyze them further. The reported analysis demonstrates how the design of the tasks and the characterization of mathematical attributes can help to expand our understanding of different ways of conceptualizing.