Question

The amount of money in a savings account 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 Account Opened, x 1 2 3 4 Money in account ($), f(x) 178. 39 213.86 257.14 307.95 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

Answer 1

To determine whether the data in the table represents a linear or exponential function, let’s analyze the changes in the values of \( f(x) \) as \( x \) increases.

The data provided is:

| Years (x) | Money in Account (f(x)) |
|-----------|--------------------------|
| 1 | 178.39 |
| 2 | 213.86 |
| 3 | 257.14 |
| 4 | 307.95 |

### Step 1: Calculate the Differences
To check if it's linear, we calculate the first differences (the change in \( f(x) \) as \( x \) increases by 1).

- \( f(2) - f(1) = 213.86 - 178.39 = 35.47 \)
- \( f(3) - f(2) = 257.14 - 213.86 = 43.28 \)
- \( f(4) - f(3) = 307.95 - 257.14 = 50.81 \)

### Summary of Changes
- First difference from year 1 to 2: 35.47
- First difference from year 2 to 3: 43.28
- First difference from year 3 to 4: 50.81

The first differences are not constant, which indicates that the function does not follow a linear pattern.

### Step 2: Calculate the Ratios
Next, let’s check for exponential growth by calculating the ratios of successive values.

- \( \frac{f(2)}{f(1)} = \frac{213.86}{178.39} \approx 1.197 \)
- \( \frac{f(3)}{f(2)} = \frac{257.14}{213.86} \approx 1.202 \)
- \( \frac{f(4)}{f(3)} = \frac{307.95}{257.14} \approx 1.195 \)

### Summary of Ratios
The ratios are close to 1.2, which suggests that the function increases by a constant factor, indicating that it could be exponential.

### Conclusion
Given that the first differences are not constant and the ratios are relatively consistent, the function that best fits the data is **exponential**.

### Summary Statement
The function would best fit the data because as \( x \) increases, the \( f(x) \) values change **multiplicatively** (by a consistent factor).

Thus, the appropriate conclusion is:

- The amount of money in the savings account is modeled by an **exponential function**.
- The change in the function is approximately **multiplicative**.