The gain control on a home audio subwoofer allows us to calibrate the bass levels produced by the subwoofer to match the output from the speakers in the system. This will ensure optimal system sound quality.

Is a gain control and volume control the same thing? No. While adjusting them can result in similar results (increase or decrease in volume) the gain control functions differently. A volume control adjusts the output levels. A gain control adjust the output levels relative to the input levels. That is an important distinction.

The gain control position is an indicator of how hard the subwoofer is working in the system---right?

Incorrect. The gain knob can be set very low and the subwoofer may be working at its maximum limits. The gain control can be set very high and the subwoofer may be coasting along and never even approach its maximum output limits. 

The phase control in a powered subwoofer allows the user to add electrical delay to the incoming signal. The phase control operates over a range of 0 to 180 degrees. Adding delay to the subwoofer signal can sometimes help the subwoofer to integrate better with the loudspeakers in the room. Setting the phase control to 180 degrees reverses the polarity of the subwoofer. In some application where the subwoofer is placed on a wall opposite from the speakers, reversing the subwoofer polarity can yield better in-room low-frequency response. When you reverse the polarity of a subwoofer, the subwoofer’s driver moves inward while all other speaker drivers in the system are moving outward. subwoofer feature a continuously variable 0-180 phase control which allows the most complete range of adjustment in small increments to achieve the best sound possible. 

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