We have a discussion once a day on batteries, alternators, and starters. And once a day someone tells another person to disconnect the battery to check the alternator. While this was a good "test" with a pre 80's car, this can cause major problems with newer cars, as many other people have already stressed many times. If you want to check the alternator or voltage regulator, do like we did at Interstate Batteries. Invest in a cheap little voltmeter, connect it to the battery and watch for more than 13.8 volts....I'm sure someone knows exactly what a Toyota alternator puts out. If you have any large fluctuations then it could be the regulator. I'm betting that you need to take the battery out and give it a nice little charge overnight at 2amps works great. Allow the battery to stablize for 3-4 hrs if you want to get an accurate voltage reading. Otherwise stick it in, it should be fully charged.
Batteries are used to crank the starter, dampen the voltage coming from the alternator to prevent voltage spikes and we use it to power the car when the engine is off. Alternators power the whole electrical system with the engine running and recharge the battery, keep in mind it takes a long time to recharge not just a 20-30 min drive. I've said this before but 50-60% of the batteries we recieved as cores were indeed good batteries. Which in turn we sold as used batteries. One of the best tools to use when checking yoru battery is a 3 dollar hydrometer, and its simple to use.