Quadratic Battlefield

Back in the day, when my parents sent me on a summer camp, I was taught this Korean game involving pebbles and a patch of hard dirt. A square with side lengths of two metres was drawn, and in each of the corners, players marked their "base" by drawing in a small square. The object of the game was to move a pebble out of and back into your territory with three flicks. The area carved by the pebble's path was added to your existing territory, unless you were unsucessful, in which case no territory was added.

Now I know some people here have a fetish for parabolas, (along with ninjas, custard and other such items), so I'm proposing the same Korean game, except that the area which the pebble carves is now replaced by a quadratic equation, the coefficients of which are randomly drawn and pieced together by the players in a given time limit. Players draw coeffecients from a bag Scrabble-style, and must create parabolas that open and close within their respective quadrants on a 20x20 cartesian plane. The player with the most area after x turns wins.

Requires visualization and quick calculation to predict whether or not your parabola will make it back. Would also require the use of technology as a graphing aid, so this is best played on a tablet PC or PDA.