To determine what kind of function best fits the given data, let's first observe the values provided in the table:
| Hours (x) | Zombies (f(x)) | |-----------|----------------| | 11 | 3333 | | 22 | 5050 | | 33 | 6767 | | 44 | 8585 |
Step 1: Check for Linear Growth
We can check whether the change in the number of zombies is constant, indicating a linear relationship.
- Calculate the change in zombies for each interval:
- From 11 to 22 hours: \( 5050 - 3333 = 1717 \)
- From 22 to 33 hours: \( 6767 - 5050 = 1717 \)
- From 33 to 44 hours: \( 8585 - 6767 = 1718 \) (rounded difference)
The difference is approximately constant (around 1717), suggesting a linear relationship.
Step 2: Check for Exponential Growth
Next, we can check if the ratio of the changes indicates an exponential growth pattern:
- Calculate the ratios of the changes:
- From 11 to 22 hours: \( \frac{5050}{3333} \approx 1.515 \)
- From 22 to 33 hours: \( \frac{6767}{5050} \approx 1.339 \)
- From 33 to 44 hours: \( \frac{8585}{6767} \approx 1.270 \)
The ratios are not constant, indicating that the relationship isn't exponential.
Conclusion
The function that best fits the data is linear because the number of zombies increases consistently over equal intervals of x (hours).
Step 3: Determine Linear Function
To find a linear function, we can use the format \( f(x) = mx + b \), where:
- \( m \) is the slope
- \( b \) is the y-intercept
Using two points, for example, (11, 3333) and (22, 5050), we can determine the slope (\( m \)): \[ m = \frac{(5050 - 3333)}{(22 - 11)} = \frac{1717}{11} \approx 156.09 \]
Now to find \( b \) using \( f(11) = 3333 \): \[ 3333 = 156.09 \cdot 11 + b \] \[ 3333 = 1717 + b \] \[ b \approx 1616 \]
Final Linear Function
So, the approximate linear function is: \[ f(x) \approx 156.09x + 1616 \]
Thus, the function would best fit the data because as \( x \) increases, the \( y \) values change linearly. Rounded to the nearest .5, the slope of this function is approximately 156.1.