Question
Use the image to answer the 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.
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.
Answers
GPT-4o mini
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.
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.