MOS Capacitor Band Edges
The y-axis of a band diagram represent the energy of an electron. If you plot the energy of a positive elementary charge, it will be the mirror image of this about x-axis. So increasing voltage, which corresponds to potential energy of a positive charge, will reduce the energy of electron. That is why the energy level is shifted down on applying a positive voltage and vice versa. 

See the diagrams below. First one shows positive gate voltage and second one shows negative gate voltage applied with respect to the substrate (Semiconductor). Some portion of the applied voltage will be dropped across semiconductor and remaining across the oxide. Because of this, the voltage at the left side and right of the oxide will be different. The more positive side will be lower than the other. That is why there is an upward slope for positive gate voltage and vice versa.

Since there are no charge inside the oxide (ideal MOS cap assumed), the electric field has to be uniform inside (read Poisson's equation) and hence potential vary linearly. And conduction band follows the potential variation. That's why the band edges are drawn linearly.