Question

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.

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).