Question

The estimated population of a certain city over time is given in the table below. Answer the questions below to determine what kind of function would best fit the data, linear or exponential. Number of Years Since Last Census, x 1 2 3 4 Estimated Population, f(x) 39,775 39,496 39,375 38,982 Answer function would best fit the data because as x increases, the y values change . The of this function is approximately .
and is it additively or multiplicatively? was it a common difference/ slope or a common ratio/ multiplier/base?

Answer 1

To determine whether a linear or exponential function fits the given population data, we need to analyze how the estimated population changes as the number of years since the last census increases.

Here’s the data provided:

| Years Since Last Census (x) | Estimated Population (f(x)) |
|------------------------------|------------------------------|
| 1 | 39,775 |
| 2 | 39,496 |
| 3 | 39,375 |
| 4 | 38,982 |

### Step 1: Analyze the Differences
1. Calculate the change in population between each consecutive year:

- From 1 to 2 years: \(39,496 - 39,775 = -279\)
- From 2 to 3 years: \(39,375 - 39,496 = -121\)
- From 3 to 4 years: \(38,982 - 39,375 = -393\)

The changes are:
- Year 1 to 2: -279
- Year 2 to 3: -121
- Year 3 to 4: -393

### Step 2: Determine If There's a Common Difference
The differences are not constant; they vary, so this indicates that the data might not fit a linear function (which would require a constant difference).

### Step 3: Analyze for Exponential Decrease
Next, let's look at the percentage decrease in population:

- From 39,775 to 39,496: \((39,496 / 39,775) \approx 0.993\) (approx. 0.7% decrease)
- From 39,496 to 39,375: \((39,375 / 39,496) \approx 0.997\) (approx. 0.3% decrease)
- From 39,375 to 38,982: \((38,982 / 39,375) \approx 0.993\) (approx. 1.0% decrease)

While the percentages also are not constant, they are relatively close, and indicate a consistent decrease in population.

### Conclusion
Based on the calculated differences and percentage changes:

- The function that would best fit the data is **exponential**, because the decreases are proportional to the population value at each step, even though they are not exactly the same.
- Since it does not have a constant change (common difference), it is best characterized by **multiplicative** behavior instead of additive.

Therefore, you would conclude that **an exponential function would best fit the data because as \(x\) increases, the \(y\) values change in a multiplicative manner, reflecting a common ratio rather than a common difference.**