Let's evaluate each statement based on the provided results:
1. The function that best represents the data is [tex]\( f(x) = 24512 \cdot (0.755)^x \)[/tex].
- This statement asserts that the exponential function [tex]\( f(x) = 24512 \cdot (0.755)^x \)[/tex] closely matches the data in the table. The analysis determined that this function does indeed fit the data points well.
This statement is true.
2. The function that best represents the data is [tex]\( f(x) = 554x^2 - 5439x + 24600 \)[/tex].
- This statement suggests that the polynomial function [tex]\( f(x) = 554x^2 - 5439x + 24600 \)[/tex] best matches the data. However, the analysis showed that this function does not fit the data points as closely as the exponential function does.
This statement is false.
3. The function decreases indefinitely.
- Both the exponential function [tex]\( f(x) = 24512 \cdot (0.755)^x \)[/tex] and the polynomial function [tex]\( f(x) = 554x^2 - 5439x + 24600 \)[/tex] indeed decrease indefinitely over time. The exponential function will get closer and closer to 0 but never quite reach it, and the polynomial function will continue to decrease as [tex]\( x \)[/tex] increases.
This statement is true.
4. It is reasonable to interpolate to the value of the car at 4.5 years.
- Interpolation is the process of estimating unknown values that fall within the range of known data points. Since 4.5 years falls between the existing data points of 4 and 5 years, it is reasonable to interpolate to estimate the car's value at this point.
This statement is true.
5. It is reasonable to extrapolate to 40 years.
- Extrapolation involves estimating values outside the range of known data points. Extrapolating to 40 years is not reasonable because factors affecting the car’s depreciation may change significantly over such a long time period. The function may not accurately predict the car's value that far into the future.
This statement is false.
Thus, the valid statements are:
- The function that best represents the data is [tex]\( f(x) = 24512 \cdot (0.755)^x \)[/tex].
- The function decreases indefinitely.
- It is reasonable to interpolate to the value of the car at 4.5 years.
This corresponds to the results: [tex]\((\text{True, False, True, True, False})\)[/tex].
