The ability to accurately interpret mathematical functions is crucial for understanding complex physical and mathematical concepts. Differentiating isothermal from adiabatic change of state in p-V diagrams respectively graphs of indirectly proportional functions from graphs of power functions in x-y diagrams presents a common challenge for students. This study examines the gaze behavior of 80 10th-grade students to identify the processing strategies they use when distinguishing these graphs of functions in mathematical and thermodynamic contexts. The cluster analysis of the gaze data revealed two main strategies: focusing on the value pairs, often resulting in a higher number of correct responses, and focusing on the graph itself, often resulting in misunderstandings. Interviews indicated that students who relied on graph-focused strategies often followed intuitive and less effective approaches. These findings underscore the need for adaptive support measures to address individual learning challenges and improve students' understanding of graphs of indirectly proportional functions. Beyond the specific focus on indirectly proportional and power functions, the methodology and results of this study offer a transferable framework for identifying learning difficulties across various function types and contexts, complementing existing research on students' understanding of linear functions.