Lincoln's parents invested $5000 in a college savings account when he was born. If the account has a growth rate of 12%, the situation can be modeled by the function a(t)=5000(1.12)t

Question

, where a is the amount after t years. Which one of the following statements is true about the graph of the function?

(1 point)
Responses

As t increases, a increases slowly at first and then quickly.
As t increases, a increases slowly at first and then quickly.

As t increases, a increases quickly at first and then slowly.
As t increases, a increases quickly at first and then slowly.

As t increases, a decreases slowly at first and then quickly.
As t increases, a decreases slowly at first and then quickly.

As t increases, a decreases quickly at first and then slowly.

Answer 1

The function given is \( a(t) = 5000(1.12)^t \), which models exponential growth, as the amount \( a \) increases over time with a growth rate of 12% per year.

In exponential growth functions, as \( t \) increases, the output \( a \) increases slowly at first (for small values of \( t \)) and then begins to increase much more rapidly as \( t \) becomes larger. This is characteristic of exponential growth, where the increase is not linear; instead, it accelerates as the base of the exponential (in this case, \( 1.12 \)) is raised to higher powers.

Therefore, the correct statement is:

As \( t \) increases, \( a \) increases slowly at first and then quickly.