Why is the signal inverted on a VDB transistor?
Your schematic shows a typical common emitter single transistor amplifier. To understand why the input signal is inverted at the output consider the following:  The input signal voltage appears across the emitter resistor.  As it increases in voltage, it increases the emitter current. This also increases the collector current since, in an active transistor with a reasonable value of beta (current gain), the collector current is the same as the emitter current.  However, note that the output voltage at the collector is equal to the supply voltage minus the product of the collector current and the collector resistor.  Since the actual output voltage is AC coupled through a capacitor, the output voltage does not include the supply voltage but is equal to just the product of the collector current and collector resistor. However, it is inverted with respect to the input signal voltage because of that minus sign.