Use the image to answer the question.

Question

An increasing curve with an arrow on the upper end is plotted on a coordinate plane. The curve begins at the point left parenthesis 0 comma 2,000 right parenthesis.

Nate’s parents invested $2,000 in a savings account when he was born. If the account has a growth rate of 8% per year, the situation can be modeled by the equation a(t)=2,000(1.08)t
, where a is the amount in the investment 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: The y-intercept of the function is (0,8)
.

Option #3: The range includes all real numbers.

(1 point)
Option #
is the true statement.

Answer 1

To analyze the function \( a(t) = 2000(1.08)^t \), let's evaluate the options provided:

1. Option #1: As t increases, a increases slowly at first and then quickly.
   This statement is true because the function is exponential. Exponential growth starts slowly, but as \( t \) increases, the amount \( a(t) \) grows more rapidly due to the nature of exponential functions.

2. Option #2: The y-intercept of the function is (0, 8).
   To find the y-intercept, substitute \( t = 0 \) into the function: \[ a(0) = 2000(1.08)^0 = 2000(1) = 2000. \] Therefore, the y-intercept is \( (0, 2000) \), not \( (0, 8) \). This statement is false.

3. Option #3: The range includes all real numbers.
   The range of the function \( a(t) \) is actually \( (2000, \infty) \) since the investment starts at $2000 and grows indefinitely, but does not include negative values or values less than $2000. Thus, this statement is false.

Based on the evaluations, Option #1 is the only true statement.

The correct answer is: Option #1 is the true statement.