To analyze the table and verify if a linear or exponential model best fits the data, let's examine the differences in values across each year.

The table shows:

Step 1: Calculate Differences

Let's check the differences between consecutive values to see if they are consistent (indicating a linear decrease).

1. $f(2) - f(1) = 23,755.48 - 28,780.48 = -5,025$
2. $f(3) - f(2) = 18,735.48 - 23,755.48 = -5,020$
3. $f(4) - f(3) = 13,715.48 - 18,735.48 = -5,020$

These differences are close to $-5,000$, indicating a roughly consistent decrease, which suggests a linear model.

Conclusion

A linear function best models the data. The rate of decrease (slope) is approximately $-5,000$, aligning with the answer shown in the image.

Would you like a deeper breakdown of this calculation, or have questions on linear versus exponential models?

1. How do we determine if a model is linear or exponential using data points?
2. What role does the slope play in identifying a linear model?
3. Could an exponential model ever fit this data?
4. How would we set up an equation for the linear model here?
5. What might cause small variations in the common difference?

Tip: To identify linear data, check if the change between consecutive values is approximately constant.