Question
The table represents several values of the function , which models the value of an investment, in dollars, years after the initial investment is made, where .
Which best describes the function when graphed on a coordinate plane?
Responses
The function has a -intercept located at , and as increases, increases.
The function f ( t ) has a y -intercept located at ( 1 , 840.00 ) , and as t increases, f ( t ) increases.
The function has a -intercept located at , and as increases, decreases.
The function f ( t ) has a y -intercept located at ( 1 , 840.00 ) , and as t increases, f ( t ) decreases.
The function has a -intercept located at , and as increases, increases.
The function f ( t ) has a y -intercept located at ( 0 , 800.00 ) , and as t increases, f ( t ) increases.
The function has a -intercept located at , and as increases, decreases.
The function f ( t ) has a y -intercept located at ( 0 , 800.00 ) , and as t increases, f ( t ) decreases.
Which best describes the function when graphed on a coordinate plane?
Responses
The function has a -intercept located at , and as increases, increases.
The function f ( t ) has a y -intercept located at ( 1 , 840.00 ) , and as t increases, f ( t ) increases.
The function has a -intercept located at , and as increases, decreases.
The function f ( t ) has a y -intercept located at ( 1 , 840.00 ) , and as t increases, f ( t ) decreases.
The function has a -intercept located at , and as increases, increases.
The function f ( t ) has a y -intercept located at ( 0 , 800.00 ) , and as t increases, f ( t ) increases.
The function has a -intercept located at , and as increases, decreases.
The function f ( t ) has a y -intercept located at ( 0 , 800.00 ) , and as t increases, f ( t ) decreases.
Answers
Answered by
GPT-4o mini
To analyze the function \( f(t) \) that models the value of an investment over time, we need to determine the following:
1. The \( y \)-intercept of the function, which typically represents the initial value of the investment (when \( t = 0 \)).
2. The behavior of the function as \( t \) increases, which indicates whether the value of the investment is increasing or decreasing over time.
Given the responses provided, let's define the \( y \)-intercept:
- If the \( y \)-intercept is located at \( (0, 800.00) \), that would mean the initial investment value is $800.00.
- If the \( y \)-intercept is located at \( (1, 840.00) \), that would suggest that at \( t = 1 \) year, the value is $840.00, but it wouldn't specify the value at \( t = 0 \).
Now, let’s review the responses:
1. **Response about \( (1, 840.00) \) and increasing**: This could imply the investment grows, but it doesn't tell you what happens at \( t = 0 \).
2. **Response about \( (1, 840.00) \) and decreasing**: This suggests a decrease, which is less likely if we are discussing a typical investment.
3. **Response about \( (0, 800.00) \) and increasing**: This suggests the initial investment is $800.00 and grows over time, which is typical for investments.
4. **Response about \( (0, 800.00) \) and decreasing**: This would imply the investment decreases over time, which is not typical for an investment model.
Based on the structure of typical investment growth, we can affirm:
- If \( f(t) \) has a \( y \)-intercept at \( (0, 800.00) \) and as \( t \) increases, \( f(t) \) increases, it suggests a positive growth of the investment.
Thus, the most appropriate response is:
**The function \( f(t) \) has a \( y \)-intercept located at \( (0, 800.00) \), and as \( t \) increases, \( f(t) \) increases.**
1. The \( y \)-intercept of the function, which typically represents the initial value of the investment (when \( t = 0 \)).
2. The behavior of the function as \( t \) increases, which indicates whether the value of the investment is increasing or decreasing over time.
Given the responses provided, let's define the \( y \)-intercept:
- If the \( y \)-intercept is located at \( (0, 800.00) \), that would mean the initial investment value is $800.00.
- If the \( y \)-intercept is located at \( (1, 840.00) \), that would suggest that at \( t = 1 \) year, the value is $840.00, but it wouldn't specify the value at \( t = 0 \).
Now, let’s review the responses:
1. **Response about \( (1, 840.00) \) and increasing**: This could imply the investment grows, but it doesn't tell you what happens at \( t = 0 \).
2. **Response about \( (1, 840.00) \) and decreasing**: This suggests a decrease, which is less likely if we are discussing a typical investment.
3. **Response about \( (0, 800.00) \) and increasing**: This suggests the initial investment is $800.00 and grows over time, which is typical for investments.
4. **Response about \( (0, 800.00) \) and decreasing**: This would imply the investment decreases over time, which is not typical for an investment model.
Based on the structure of typical investment growth, we can affirm:
- If \( f(t) \) has a \( y \)-intercept at \( (0, 800.00) \) and as \( t \) increases, \( f(t) \) increases, it suggests a positive growth of the investment.
Thus, the most appropriate response is:
**The function \( f(t) \) has a \( y \)-intercept located at \( (0, 800.00) \), and as \( t \) increases, \( f(t) \) increases.**