Question

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

answer attempt 1 out of 2
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 \boxed{}.

Explanation:

Step1: Check differences (linear)

Calculate the differences between consecutive \( f(x) \) values:
\( 56 - 24 = 32 \)
\( 88 - 56 = 32 \)
\( 122 - 88 = 34 \) (close to 32, likely rounding)

Step2: Check ratios (exponential)

Calculate the ratios of consecutive \( f(x) \) values:
\( \frac{56}{24} \approx 2.33 \)
\( \frac{88}{56} \approx 1.57 \)
Ratios are not constant, but differences are nearly constant (linear behavior).

Step3: Find slope (rate of change)

Use two points, e.g., \( (1, 24) \) and \( (2, 56) \).
Slope \( m = \frac{56 - 24}{2 - 1} = 32 \). Check with \( (2, 56) \) and \( (3, 88) \): \( \frac{88 - 56}{3 - 2} = 32 \). With \( (3, 88) \) and \( (4, 122) \): \( \frac{122 - 88}{4 - 3} = 34 \), rounded to nearest 0.5 is 32.0 or 32.5 (34 is close to 32.5? Wait, 32, 32, 34: average ~32.67, nearest 0.5 is 32.5 or 32? Wait, 32, 32, 34: the first two differences are 32, third is 34. The slope (rate of change) for linear is approximately 32 (or 32.5 when rounding 34 to nearest 0.5? Wait, 34 to nearest 0.5 is 34.0? No, 34 is integer, nearest 0.5 is 34.0? Wait, no: 32, 32, 34. The average of differences: \( \frac{32 + 32 + 34}{3} \approx 32.67 \), nearest 0.5 is 32.5 or 33? Wait, 32.67 is closer to 32.5? No, 32.67 - 32.5 = 0.17, 33 - 32.67 = 0.33. Wait, no, 32.67 is closer to 32.5? Wait, 0.17 < 0.33, so 32.5? Wait, but first two differences are 32, third is 34. Maybe the problem expects using the first two differences (32) or average. But the question says "rounded to the nearest.5", so 32 (or 32.5? Wait, 32 is already a multiple of 0.5? 32.0 is nearest 0.5. Wait, 32, 32, 34: the rate of change (slope) for linear is approximately 32 (since first two differences are 32, third is 34, maybe the data has a slight round, so slope ~32, or 32.5? Wait, 32 is exact for first two, third is 34, so average is (32+32+34)/3 = 98/3 ≈ 32.67, which rounds to 32.5 (nearest 0.5) or 33? Wait, 32.67 is 32 + 0.67, so 0.67 is closer to 0.5 or 1.0? 0.67 - 0.5 = 0.17, 1.0 - 0.67 = 0.33. So closer to 32.5? Wait, no: 32.67 is between 32.5 and 33.0. The distance to 32.5 is 0.17, to 33.0 is 0.33. So nearest 0.5 is 32.5? Wait, but 32 is a possible answer. Wait, the first difference is 32, second 32, third 34. So the rate of change (slope) is approximately 32 (linear function, since differences are nearly constant).

Answer:

linear function would best fit the data because as \( x \) increases, the \( y \) values change at a nearly constant rate. Rounded to the nearest \( 0.5 \), the rate of change (slope) of this function is approximately \( 32.5 \) (or \( 32 \), but based on the third difference, \( 34 \) is close to \( 32 \), average ~32.67 → nearest 0.5 is \( 32.5 \) or \( 32 \); likely \( 32 \) or \( 32.5 \), but first two differences are 32, so \( 32 \) is more accurate, but 34 is 2 more than 32, so average is ~32.67 → 32.5 when rounded to nearest 0.5).

(Note: Depending on interpretation, the slope is approximately \( 32 \) or \( 32.5 \). Given the third difference is 34 (close to 32), the answer is likely \( 32 \) or \( 32.5 \).)

Final answer for the rate of change (slope): \( \boldsymbol{32.5} \) (or \( 32 \), but 32.67 rounds to 32.5).