A carefree child is at recess swinging on her swingset. When her height is plotted as a function of time, we obtain a sinusoldal function where x is time in seconds and f(x) Is her height in feet. If: The child's height varies by 2.4 feet above and below the average height while she is swinging It will take 2 .03 seconds for the child to fall to her minimum height and rise again to the average height. The child's average height is 4.2 feet off the ground. It takes 2.7 seconds for the child to swing back and forth once. Determine the best function to model the child' height as she swings. Amplitude: Period: Phase Shift: Vertical Shift: f(x)=

To determine the best function to model the child's height as she swings, we need to consider the given information and the properties of sinusoidal functions.<br /><br />1. Amplitude:<br />The amplitude of a sinusoidal function represents the maximum deviation from the average value. In this case, the child's height varies by 2.4 feet above and below the average height, so the amplitude is 2.4 feet.<br /><br />2. Period:<br />The period of a sinusoidal function represents the time it takes for the function to complete one full cycle. In this case, it takes 2.7 seconds for the child to swing back and forth once, so the period is 2.7 seconds.<br /><br />3. Phase Shift:<br />The phase shift of a sinusoidal function represents the horizontal shift of the graph. In this case, it will take 2.03 seconds for the child to fall to her minimum height and rise again to the average height. This means that the graph will start at the minimum height at x = 2.03 seconds, so the phase shift is -2.03 seconds.<br /><br />4. Vertical Shift:<br />The vertical shift of a sinusoidal function represents the vertical shift of the graph. In this case, the child's average height is 4.2 feet off the ground, so the vertical shift is 4.2 feet.<br /><br />Based on the above information, the best function to model the child's height as she swings is:<br /><br />$f(x) = 2.4 \sin\left(\frac{2\pi}{2.7}x - 2.03\pi\right) + 4.2$<br /><br />This function has an amplitude of 2.4, a period of 2.7 seconds, a phase shift of -2.03 seconds, and a vertical shift of 4.2 feet.