When scientists measure ozone pollution, one thing they have to take into account is the fact that It varies hourly throughout the day.One major component of that variation can be modeled with a sinusoidal function. If: The pollution varies from average by 0.01 parts per million (ppm) over the course of the day. The cycle takes 7 hours from midnight to reach average pollution levels. The cycle takes 24 hours to return to its start state. The average pollution throughout the day is 0.02 parts per million (ppm) Determine the approprlate periodic function to model the concentration of air pollution during the day. Amplitude: Period: Phase Shift: square Vertical Shift: square f(x)=

To determine the appropriate periodic function to model the concentration of air pollution during the day, we need to consider the given information and use it to identify the amplitude, period, phase shift, and vertical shift of the sinusoidal function.<br /><br />Amplitude:<br />The amplitude of the sinusoidal function represents the maximum deviation from the average pollution level. In this case, the pollution varies from the average by 0.01 ppm over the course of the day. Therefore, the amplitude is 0.01 ppm.<br /><br />Period:<br />The period of the sinusoidal function represents the time it takes for the function to complete one full cycle. In this case, the cycle takes 24 hours to return to its start state. Therefore, the period is 24 hours.<br /><br />Phase Shift:<br />The phase shift of the sinusoidal function represents the horizontal shift or displacement of the function. In this case, the cycle takes 7 hours from midnight to reach average pollution levels. Therefore, the phase shift is 7 hours.<br /><br />Vertical Shift:<br />The vertical shift of the sinusoidal function represents the vertical displacement of the function. In this case, the average pollution throughout the day is 0.02 ppm. Therefore, the vertical shift is 0.02 ppm.<br /><br />Based on the above information, the appropriate periodic function to model the concentration of air pollution during the day is:<br /><br />$f(x) = A \sin(B(x - C)) + D$<br /><br />Where:<br />A = Amplitude = 0.01 ppm<br />B = $\frac{2\pi}{\text{Period}} = \frac{2\pi}{24}$<br />C = Phase Shift = 7 hours<br />D = Vertical Shift = 0.02 ppm<br /><br />Substituting the values, we get:<br /><br />$f(x) = 0.01 \sin\left(\frac{2\pi}{24}(x - 7)\right) + 0.02$<br /><br />Therefore, the appropriate periodic function to model the concentration of air pollution during the day is:<br /><br />$f(x) = 0.01 \sin\left(\frac{2\pi}{24}(x - 7)\right) + 0.02$