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The approximate number of zombies in 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 Hours Since Zombies First Spotted, x 1 2 3 4 Approx. Number of Zombies, f(x) 33 50 67 85 Answer function would best fit the data because as x increases, the y values change . Rounded to the nearest .5, the of this function is approximately

and is it additively or multiplicatively

2 answers

To determine whether the data suggests a linear or exponential function, let's take a look at the values given in the table:

| Hours since zombies first spotted, \( x \) | Approx. number of zombies, \( f(x) \) | |--------------------------------------------|-------------------------------------| | 1 | 33 | | 2 | 50 | | 3 | 67 | | 4 | 85 |

Step 1: Check for linearity

First, we will check if the data suggests a linear pattern by calculating the differences between the \( f(x) \) values:

  • From \( f(1) \) to \( f(2) \): \( 50 - 33 = 17 \)
  • From \( f(2) \) to \( f(3) \): \( 67 - 50 = 17 \)
  • From \( f(3) \) to \( f(4) \): \( 85 - 67 = 18 \)

The differences are approximately constant (17 and a change of 1 at the end). This suggests that the data could likely be modeled by a linear function because the changes in the number of zombies are relatively consistent.

Step 2: Check for exponential growth

Next, we will check for exponential growth by calculating the ratios of consecutive values:

  • Ratio from \( f(1) \) to \( f(2) \): \( \frac{50}{33} \approx 1.515 \)
  • Ratio from \( f(2) \) to \( f(3) \): \( \frac{67}{50} = 1.34 \)
  • Ratio from \( f(3) \) to \( f(4) \): \( \frac{85}{67} \approx 1.265 \)

The ratios are not consistent, which further suggests that an exponential function is not the best fit.

Conclusion

Since the changes in the values are more consistent with linear growth than with exponential growth, we conclude that a linear function would best fit the data.

Characteristic of the function

As \( x \) increases, the \( y \) values change additively (by a constant amount), rather than multiplicatively (which would be characteristic of an exponential function).

Summary

  • Type of function: Linear
  • Change: Additively (the increase in zombies is approximately constant at about 17 per hour for \( x = 1, 2, 3 \))

If we need to find the specific linear function, we can derive it from the points:

  1. Slope (\( m \)):

    • Using the consistent increase, \( m = 17 \).

  2. Starting point: Plugging in one of the points to find the y-intercept (\( b \)):

    • Using point (1, 33): \( 33 = 17(1) + b \), leads to \( b = 16 \).

Thus, the linear equation could be approximately \( f(x) = 17x + 16 \).

was it a common difference/ slope or a common ratio/ multiplier/base?