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 given statements about the investment growth model \( a(t) = 2000(1.08)^t \):

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

   • This statement is true. The function is an exponential growth function, which means that as \( t \) increases, the rate of growth becomes faster over time. Initially, the growth is slow, but it accelerates exponentially.

2. Option #2: The y-intercept of the function is (0, 8).

   • This statement is false. The y-intercept of the function occurs when \( t = 0 \):
     \[ a(0) = 2000(1.08)^0 = 2000 \times 1 = 2000 \]
     So the y-intercept is (0, 2000), not (0, 8).

3. Option #3: The range includes all real numbers.

   • This statement is false. The amount \( a(t) \) is always positive and greater than zero, so the range of the function is \( (2000, \infty) \) and does not include all real numbers.

Based on the analysis, the true statement is:

Option #1 is the true statement.