You don't need to search "verlets" or "verlet". For anyone else reading this: verlet is a mostly outdated type of integration in modern physical simulation. This sort of thing can be achieved in two dimensions whith custom (almost gameplay specific) code. The exact details of doing this sort of thing don't have to be very fancy (not that verlet is fancy at all).
I would imagine that rendering the rope would done separately from any physical simulation. Perhaps one way to fake the trampoline would be to apply a constant acceleration (or force) to the ball if the ball is "colliding" with the trampoline. The acceleration can be aimed towards the average of the two anchor points, or in other words the direction from the ball to the average of the two anchor points (a vector).
The collision detection and rendering of the trampoline should be harder than the manipulation of forces.