Bouncing Ball Pattern

I too would like to join the Dan Meyer fan club, for his perspective and helpfulness, but this time for his "Will the ball hit the can" activity.

His post got me thinking about the bouncing ball pattern:

And I wondered how the parabolas were related, in particular, how are the equations of the quadratics related?

So, back to geogebra (using Dan Meyer's work as a template), and in short I was able to change:


Reflections:

• On the construct:

1. Neither the presentation of this problem, nor the geogebra file are student ready. This was just a question I had. [I'm still learning geogebra and this was a way for me to get more comfortable with the tool. I now know a way to *add a picture as a background and *create a slider.]
2. The picture is not mine and therefore is a bit forced as an 'application' problem. I wanted to try the geogebra aspect and so just used a pre-existing pic this time.
3. I feel that the whole set-up of the factored form is too helpful to be used as-is.

• On the problem:
  

1. In this set-up, the parabolas have very similar leading coefficients and I think its worth testing to see if that's always true. This observation was not my original assumption.
   
2. The next question that came to my mind was what I'm calling the basketball fractal (forgive my touchpad-sketching capabilities). The question is, if basketball is shot, hits the rim and is continuously bounced back into the air (with no new force added), then, how will the 'nested parabolas' be related?