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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)

1 answer

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

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

    • This statement is true. The growth of the function \( a(t) \) is exponential, which means it grows slowly for small values of \( t \) and then accelerates as \( t \) increases. This is characteristic of exponential growth.

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

    • This statement is false. To find the y-intercept, we evaluate the function at \( t = 0 \): \[ a(0) = 2000(1.08)^0 = 2000 \cdot 1 = 2000. \] Therefore, the y-intercept is \( (0, 2000) \), not \( (0,8) \).

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

    • This statement is false. The function \( a(t) \) describes the amount of money in the account, which is always positive. So, the range of \( a(t) \) is \( (2000, \infty) \), not all real numbers.

Given the evaluations, the true statement about the graph of the function is Option #1: As \( t \) increases, \( a \) increases slowly at first and then quickly.