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In this study, I identified important differences in how students conceive points and lines. For example, students envisioned points as a circular dot that represents two quantities’ measures and envisioned that points on a graph (e.g., a line) do not exist until they are physically and visually plotted. Therefore, they conceived the line as representing a direction of movement of a dot on a coordinate plane. On the contrary, students’ meaning of a line included a record of two covarying quantities with the result of the trace consisting of infinitely many points, each of which represents both quantities’ measures. I illustrate the role of these particular meanings for lines and points in students’ construction and interpretation of continuous images of covariation in graphical contexts. I also provide implications by drawing attention to nuances in students’ conceptions of points and lines, including how to incorporate these conceptions into the teaching and learning of mathematics.