Final answer:
The modern vice president is particularly influential as the presiding officer of the Senate, with the power to cast tie-breaking votes.
Step-by-step explanation:
In modern American politics, a vice president is particularly influential as the presiding officer of the Senate. The Constitution grants the vice president this role, notably providing a tie-breaking vote in the event of a split within the Senate. While the vice president's influence has varied throughout history due to the variable roles presidents have carved out for them, this position in the Senate bestows a significant legislative function, particularly when it comes to the confirmation of key appointments and the passage of legislation.