If you flag down a vehicle, especially a taxi, you wave at it as a signal for the driver to stop. They flagged down a passing family who stopped to help them. [
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Look for the light – if the roof light on top of the cab is lit, the taxi is available for service. Raise your arm to “flag” the cab, and do your best to make eye contact with the driver. Be prepared to let the driver know where you're headed.
Taxi and For-Hire Vehicle (FHV) relief stands allow drivers to park and leave their vehicles for up to one hour. NYC DOT maintains and installs taxi and FHV relief stands.
The rule derives its name from the tradition by which a hackney carriage driver at the head of a queue of taxicabs is obliged to take the first passenger requesting a ride.
Report the incident. Even if it's been a bad day for the cabbie driver, or he was just reacting to circumstances, rudeness should never be tolerated (credit miller). Contact the cab company right away and let them know what happened.
From Driver Rule 54-20 (in PDF): It is against the law to refuse a person based on race, disability, or a destination in New York City. A taxicab driver is required to drive a passenger to any destination in the five boroughs. You can make a refusal complaint by calling 3-1-1.
Tips are big business in New York, as they are in the rest of the US. But sadly you'll be expected to pay over the odds in the Big Apple. 20% of the fee is the recommended amount. If you're paying with card instead of cold hard cash, the amount of gratuity will automatically be added to the charge.