The graph would look much like the current trace, would it not. At first, only a few people would raise their hands, then more, then more yet, until we reached a plateau when everybody who was awake had their hand in the air. The trace is linear below the reversal potential. This is because the cell is hyperpolarized, so all current is through leakage channels. Linearity means that slope is constant, which in turn means that conductance is not changing, because the slope of an IV plot is conductance.
Leakage channels have the same conductance at all potentials.
A little bit past the reversal potential, the slope begins to change. In fact, the slope becomes larger. This means that the conductance is increasing. It must be the case that we have crossed the threshold, and active potassium channels are opening, causing increased conductance. But then the curve becomes linear once again.