To determine whether the data represents a linear or exponential function, let’s analyze the changes in the number of zombies over time based on the given values.
Here is the data provided:
| Hours (x) | Number of Zombies (f(x)) | |-----------|---------------------------| | 1 | 33 | | 2 | 50 | | 3 | 67 | | 4 | 85 |
Step 1: Check for linearity
To see if the data follows a linear trend, we can calculate the differences between consecutive values of \( f(x) \):
-
From \( x = 1 \) to \( x = 2 \): \( 50 - 33 = 17 \)
-
From \( x = 2 \) to \( x = 3 \): \( 67 - 50 = 17 \)
-
From \( x = 3 \) to \( x = 4 \): \( 85 - 67 = 18 \)
While the differences are close (17, 17, and 18), they are not consistent, which suggests it is not a perfect linear function.
Step 2: Check for exponential growth
An exponential function would show a multiplicative rate of growth rather than additive. To see if there is a percentage increase:
-
From \( x = 1 \) to \( x = 2 \): \((50 - 33) / 33 \approx 0.515 \) or about 51.5%
-
From \( x = 2 \) to \( x = 3 \): \((67 - 50) / 50 \approx 0.34 \) or about 34%
-
From \( x = 3 \) to \( x = 4 \): \((85 - 67) / 67 \approx 0.269 \) or about 26.9%
The percentage changes also show a decrease, which isn't characteristic of exponential growth.
Conclusion:
Since the differences in the number of zombies are not consistent, we cannot classify the data perfectly as linear, but since the changes are relatively steady and could resemble a linear relationship for this small dataset, we might consider it approximately linear in this context.
To summarize:
- The data suggests a linear format because of the incremental increases being relatively uniform (though not exactly consistent).
- Thus, it is reasonable to say that a linear function best fits the data.
Final Answer:
A linear function would best fit the data because as \( x \) increases, the \( y \) values change in a quasi-consistent manner. Rounded to the nearest .5, the slope (the average change) of this function is approximately \( 17 \) zombies per hour at first glance, or \( \textit{(approximately 17)} \) when averaged out.