Under certain conditions, the number of predators in a predator-prey ecosystem can be modeled by a sinusoidal function, where X is the number of months elapsed and f(x) Is the number of predators in the ecosystem. Assume that the mountain lion population in a certain national park meets the conditions. If: - The number of mountain lions varies from the average by 22 over the course of a cycle The average number of mountain lions in the park is 67. One cycle of mountain lion population takes 12 months. From the beginning of the model, it takes 3 months for the number of mountain lions to reach its average Determine the appropriate periodic function to model the number of mountain lions in the park. Amplitude: square Period: square Phase Shift: square Vertical Shift: square f(x)=

To determine the appropriate periodic function to model the number of mountain lions in the park, we need to identify the amplitude, period, phase shift, and vertical shift of the sinusoidal function.<br /><br />Given information:<br />- The number of mountain lions varies from the average by 22 over the course of a cycle.<br />- The average number of mountain lions in the park is 67.<br />- One cycle of mountain lion population takes 12 months.<br />- From the beginning of the model, it takes 3 months for the number of mountain lions to reach its average.<br /><br />Step 1: Determine the amplitude.<br />The amplitude is the maximum deviation of the function from its average value. In this case, the amplitude is 22.<br /><br />Amplitude: 22<br /><br />Step 2: Determine the period.<br />The period is the time it takes for the function to complete one cycle. In this case, the period is 12 months.<br /><br />Period: 12<br /><br />Step 3: Determine the phase shift.<br />The phase shift is the horizontal shift of the function. Since it takes 3 months for the number of mountain lions to reach its average, the phase shift is 3 months.<br /><br />Phase Shift: 3<br /><br />Step 4: Determine the vertical shift.<br />The vertical shift is the average value of the function. In this case, the average number of mountain lions in the park is 67.<br /><br />Vertical Shift: 67<br /><br />Step 5: Write the periodic function.<br />The general form of a sinusoidal function is:<br />$f(x) = A \sin(B(x - C)) + D$<br /><br />Where:<br />- A is the amplitude<br />- B is the frequency (related to the period)<br />- C is the phase shift<br />- D is the vertical shift<br /><br />Substituting the values, we get:<br />$f(x) = 22 \sin\left(\frac{2\pi}{12}(x - 3)\right) + 67$<br /><br />Therefore, the appropriate periodic function to model the number of mountain lions in the park is:<br />$f(x) = 22 \sin\left(\frac{\pi}{6}(x - 3)\right) + 67$