A population of coyotes is known to show periodic behavior over time. The table shows the population of coyotes in a specific area over time.

Question

Time (years) 0 1 2 3 4 5 6 7 8 9 10 11
Population 25 40 62 75 66 35 26 38 64 74 61 32

Which equation best models the behavior of the data?

(1 point)
Responses

f left parenthesis x right parenthesis equals 25 cosine left parenthesis Start Fraction pi over 3 End Fraction x right parenthesis plus 50
Image with alt text: f left parenthesis x right parenthesis equals 25 cosine left parenthesis Start Fraction pi over 3 End Fraction x right parenthesis plus 50

f left parenthesis x right parenthesis equals negative 25 cosine left parenthesis Start Fraction pi over 3 End Fraction x right parenthesis plus 50
Image with alt text: f left parenthesis x right parenthesis equals negative 25 cosine left parenthesis Start Fraction pi over 3 End Fraction x right parenthesis plus 50

f left parenthesis x right parenthesis equals negative 50 cosine left parenthesis Start Fraction pi over 3 End Fraction x right parenthesis plus 25
Image with alt text: f left parenthesis x right parenthesis equals negative 50 cosine left parenthesis Start Fraction pi over 3 End Fraction x right parenthesis plus 25

f left parenthesis x right parenthesis equals 50 cosine left parenthesis Start Fraction pi over 3 End Fraction x right parenthesis plus 25

Answer 1

To determine which equation best models the population of coyotes over time based on the provided data, we need to analyze the periodic behavior of the population values.

1. Identifying Key Characteristics:

   • The maximum population values appear to be around 75, while the minimum values seem to drop to around 25.
   • The population shows an initial increase, peaks, then decreases and fluctuates periodically.

2. Finding the Midline:

   • The midline (average of max and min) can be calculated as follows:
     • Midline = (Max + Min) / 2 = (75 + 25) / 2 = 50.

3. Determining Amplitude:

   • Amplitude is half the distance between the max and min values.
     • Amplitude = (Max - Min) / 2 = (75 - 25) / 2 = 25.

4. Determining the Period:

   • Count the number of years it takes to complete one cycle. Observing the data, it seems there is a peak every ~6 years (from years 3 to 9).
   • This suggests a period of approximately 6.

5. Identifying the Frequency:

   • The frequency can be determined from the period: Frequency = 2π / Period = 2π / 6 = π / 3.

6. Choosing the Cosine Function:

   • Since the population starts at a relatively lower value and increases, the cosine function is appropriate.

7. Equations:

   • The general form of a cosine function is given by: \[ f(x) = A \cos(B(x - C)) + D \] where
     • A is the amplitude,
     • B affects the period,
     • C is horizontal shift (phase shift, which we can ignore for this analysis since we start from a known position),
     • D is the midline.

Based on our findings:

• The midline is 50.
• The amplitude is 25.
• The function should start at its maximum, which would imply a negative cosine function because it starts lower than the midline.

Thus, the best fit based on the analysis is: \[ f(x) = -25 \cos\left(\frac{\pi}{3} x \right) + 50 \]

However, rearranging gives: \[ f(x) = 50 - 25 \cos\left(\frac{\pi}{3} x \right) \]

This corresponds to the second option given (the negative cosine explicitly):

Answer: \( f(x) = -25 \cos\left(\frac{\pi}{3} x\right) + 50 \) (which is the second option).