At an ocean depth of 20 meters, a buoy bobs up and then down 2 meters from the ocean's depth. Four seconds pass from the time the buoy is at its highest point to when it is at its lowest point. Assume at x = 0, the buoy is at normal ocean depth.
Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.
up2 and down 2 -- amplitude = 2
center line = -20
period = 4 so 2π/k = 4
y = -20 + 2sin(π/2 t)
To graph the function using the sine tool, we need to identify the amplitude, period, phase shift, and vertical shift of the function.
First, let's identify the amplitude. The buoy bobs up and down 2 meters from the ocean's depth. The amplitude of a sine function is half the vertical distance between the maximum and minimum values. In this case, the amplitude is 2 meters.
Next, let's determine the period of the function. The period is the length of one complete cycle of the function. We are given that it takes four seconds for the buoy to go from its highest point to its lowest point and back to the starting position. Therefore, the period is 4 seconds.
Now, let's find the phase shift of the function. The phase shift determines the horizontal shift of the function. We are given that at x = 0, the buoy is at the normal ocean depth. The starting position of the buoy corresponds to the midline of the sine function. Since the midline is the horizontal line halfway between the maximum and minimum values, the phase shift is 0.
Lastly, let's determine the vertical shift of the function. The vertical shift is the vertical displacement of the function. In this case, there is no vertical shift since the starting position is at the midline.
Now that we have all the necessary information, we can write the equation of the sine function. The general form of a sine function is y = A*sin(B(x - C)) + D, where A represents the amplitude, B represents the reciprocal of the period, C represents the phase shift, and D represents the vertical shift.
In our case, the equation of the function is y = 2*sin((2π/4)(x - 0))+ 0, which simplifies to y = 2*sin((π/2)x).
To graph the function using the sine tool, you can plot points by substituting various values of x into the equation and calculating the corresponding y-values. In this case, the first point on the midline is (0,0). The second point, which must be a maximum or minimum value closest to the first point, can be found by substituting x = 1 into the equation. Evaluating this, we get y = 2*sin((π/2)(1)) = 2*sin(π/2) = 2*1 = 2. Therefore, the second point is (1,2).
Now you can plot these two points on a graph and continue to plot additional points to complete one full cycle of the sine function. Once you have enough points, you can connect them smoothly to create the graph of the function.