Lincoln’s parents invested $5,000 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 equation a(t)=5,000(1.12)t

Question

𝑎
(
𝑡
)
=
5
,
000
(
1.12
)
𝑡
, where a is the amount after t years. Which one of the following statements is true about the graph of the function?

Option #1: As t increases, a increases slowly at first and then quickly.

Option $2: As t increases, a increases quickly at first and then slowly.

Option #3: As t increases, a decreases slowly at first and then quickly.

Option #4: As t increases, a decreases quickly at first and then slowly.

Answer 1

The function provided is \( a(t) = 5,000(1.12)^t \), which represents exponential growth since the base of the exponent (1.12) is greater than 1.

In an exponential growth function, the growth is slow at first and becomes more rapid as time increases because the amount increases multiplicatively. Initially, as \( t \) starts at 0 and increases, the amount grows slowly because the base raised to a small power (e.g., \( (1.12)^1 \), \( (1.12)^2 \), etc.) results in relatively small increases. However, as \( t \) increases, the exponent makes the amount grow much larger very quickly.

Therefore, the correct statement regarding the graph of the function is:

Option #1: As \( t \) increases, \( a \) increases slowly at first and then quickly.