Parabola and Derivative

The graph shows a parabola and its derivative function.
(1) Show algebraically the point of intersection E is at (2,4).

You can use V and H to slide the parabola around. Slide the vertex
up the y-axis until the parabola is TANGENT to the derivative line.
(2) Observe the result and show algebraically that it is correct.

It is possible to find other parabolas that are tangent to their derivative lines.
Slide the parabola around to find other solutions.

(3) Describe the set of parabolas that are tangent to their own derivative lines.

(4) Show algebraically that your general observation is correct.

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Dave Mulkey, Created with GeoGebra