Can’t tell from the screengrab, but that appears to be the position graph, and since you (or I) used “record position” there are keys at every frame. When you say you want to change the “start time of the particle moving across the curve” I assume you mean the start time of the emitter?
Let’s take a simple example. Add any simple point emitter to the stage at frame 1. Then jump to frame 30 and move the emitter so it should now take 30 frames to move in a straight line. With that stage emitter still selected, Copy it (using Edit menu, ctrl or cmd-c, or by r-clicking on it in the stage). Now make sure you’re at frame 30 and Paste using the Edit menu or ctrl or cmd-V — not the r-click method. The new emitter will act just like the original, but will start at frame 30.
Suppose we now want that one to start at frame 10 instead of 30, but leave everything else about it the same. With it selected use Cut (ctrl-X), move to frame 10, then Paste.
If you use the r-click Paste option, it repositions the emitter at the click point, but the other methods don’t change the position.
Hope this helps.
Alan.
………………..
Alan Lorence
Product Manager
GenArts, Inc.
http://www.genarts.com
http://www.wondertouch.com