Can I control the resolution of fit lines?

Yes. Use the Resolution box on the Fitting dialog (accessible by double-clicking on the fitted line). Once a fitted function has been determined, the fit is approximated with segments on the x-axis. The Normal fit line is composed of 200 segments, and the number of segments increases in exponential fashion as you select Medium, High, Very High, or Perfect. Note that this option is only beneficial for fits with high curvature (i.e., a straight line fit will not be improved by this option, but a high-level polynomial fit will be). Selecting a higher number of points will result in a smoother appearance of the fitted function in the graph; however, selecting a higher number of points will also slightly slow down the graphing procedure.