How to create custom easing function with Core Animation?

I am animating a CALayer along a CGPath (QuadCurve) quite nicely in iOS. But I'd like to use a more interesting easing function than the few provided by Apple (EaseIn/EaseOut etc). For instance, a bounce or elastic function.

These things are possible to do with MediaTimingFunction (bezier):

But I'd like to create timing functions that are more complex. Problem is that media timing seems to require a cubic bezier which is not powerful enough to create these effects:

_{(source: sparrow-framework.org)}

The code to create the above is simple enough in other frameworks, which makes this very frustrating. Note that the curves are mapping input time to output time (T-t curve) and not time-position curves. For instance, easeOutBounce(T) = t returns a new t. Then that t is used to plot the movement (or whatever property we should animate).

So, I'd like to create a complex custom CAMediaTimingFunction but I have no clue how to do that, or if it's even possible? Are there any alternatives?

EDIT:

Here is a concrete example in to steps. Very educational :)

1. I want to animate an object along a line from point a