Question

If the graph of f(x) is given, with no equation, how do you draw the f'(x) graph, and vice versa.

Answer 1

generally you want to make the extrema and inflection point of f(x) have f'(x) at 0.

Answer 2

so:
f(x) ---> f'(x)
points of inflexion ---> turning points
turning points ---> x-intercepts

Answer 3

but what about x-intercepts? what happens to them?

Answer 4

THe x-intercepts have no special meaning for f'. It's just where the graph passes the x-axis. The slope of the graph could be anything there, so f' could be anything.