Reply To: Animation of Bezier

Mesh might be easier, but if you want to keep it a curve then I have 4 methods I can think of: Note: You can change the radius with alt+s hotkey. 

1.  Shape keys are the easiest way to do it.
2.  Use a lattice object along with a lattice modifier.
3.  You can use Keyframe animation. 
3A. Driver method: create custom properties. then right click and copy as new driver. Then in edit mode of the curve select the control point and on the n-panel item tab right click and paste driver on the radius.
3B. While in edit mode keyframe the radius. Either right click on radius and click insert keyframe or with mouse cursor over radius field press the I key. Me personally I’d just go thru and create the first keyframe for all of the control points. Then in dopesheet click view menu and check show sliders. This way you can adjust the values thru the channels of the dopesheet. As soon as you make a change in the slider it will change an existing keyframe or add a keyframe if there isn’t one on the frame you’re on. 
4. Use another curve as a tapper object. works best if it has the same number of control points. If I remember correctly you just have the move the control points of the taper object along the Y axis. this can be done with shapekeys or using hooks. select control point and press Ctrl+h->New object. it will create an empty and hook the control point to that empty. then just animate the empty.