parabola and triangle area (hint2)

Arthur Lee
This page continues an exploration discussed in another page:
The line and parabola are moved upward without changing their shape. We can now construct another triangle below it, on the x-axis for comparison. A'and B' on the x-axis, directly below A and B, form the base of a triangle. The height C'D' is the same as the segment CD. You may drag C to check how the two triangles vary together. Can you explain why they should have the same area? Note carefully how D' changes and reaches its maximum height. You may right click point D' to turn on 'show trace' in the menu. Drag point C again and you will then see more clearly the path of D'. How would you describe the path of D'?
In order to describe more precisely the path of D', you may refer to the functions of the line and parabola, which are f(x) and g(x). By creating another function g(x)-f(x), you should obtain a graph that gives that the path of D'.