There are actually several levels of interaction in the graph editor. By default, tangents start in a very locked-down state, which makes it easier to deal with raw keyframes in bulk.
When you want to get finer control, select 'curves->weighted tangents' in the graph editor menu. Then, to fully unlock the tangent of a particular keyframe, hit the 'unlock tangent' button ... it's the one that shows the tangent handles as empty circles rather than filled circles. Now you should be able to drag your tangent anywhere you wish.