Shape of a Slope Function

Below you can see the graph of f(x) = a x² + b x + c and its tangent t through point P.

1. Try to find out how the coordinates of point M are connected with point P and the slope m of tangent t by dragging point P with the mouse.
2. Right click (Mac OS: ctrl-click) point M and select "Trace on". When you move point P now you get the graph of the slope function of f(x). What does the shape of the slope function look like?
3. Click the little icon in the right upper corner to reset the construction. Change the values of the sliders a, b, and c and turn on the trace of point M again. What does the shape of the slope function look like when you move point P now?