Two movable points on parabola

I am trying to re-create part of slide 2 of this activity:

I want to have a parabola with two movable points, like the black one in the activity: 1 point that translates the parabola (I can do this), and 1 point that stretches the parabola. I am having trouble with the second point. I can create each one individually, but when I try to have both points on the same parabola it breaks down. Does anyone know how to re-create the black parabola with the two movable points?

Might not be the most elegant solution but something like:

might do what you’re looking for.

Thank you, this is perfect!

Funny, I never think of using regressions for stuff like this… here’s how I did it.