A researcher observes and records the height of a weight moving up and down on the end of a spring. At the beginning of the observation the weight was at its highest point. From its resting position, it takes 12 seconds for the weight to reach its highest position, fall to its lowest position, and return to its resting position. The difference between the lowest and the highest points is 10 in. Assume the resting position is at y = 0.
Use the sine tool to graph the function. The first point must be on the mid-line and the second point must be a maximum or minimum value on the graph closest to the first point.
amplitude = 5
12 seconds seems to be a period
y = 5 sin (2 pi x/12) = 5 sin (pi x/6)
first point at 4 = 0
second point at x = 1/4 of 12 which is x = 3 seconds
y = 5 sin pi (3/6) = 5 sin pi/2 = 5
To graph the function using the sine tool, we need to determine the amplitude, period, and phase shift of the function based on the given information.
1. Amplitude (A):
The amplitude of a sine function represents the half of the vertical distance between the maximum and minimum values. In this case, the difference between the lowest and highest points is 10 in, so the amplitude is A = 10/2 = 5 in.
2. Period (T):
The period of a periodic function is the length of one complete cycle. In this case, the weight takes 12 seconds to complete one full cycle (from highest position, to lowest position, and back to the resting position). Therefore, the period is T = 12 seconds.
3. Phase Shift (C):
The phase shift is the horizontal displacement of the function. Since the observation starts with the weight at its highest point, the function is not horizontally displaced. Therefore, the phase shift is C = 0.
Based on these values, we can write the equation for the sine function as:
y = A * sin((2π/T) * (x - C)) + D
Substituting the known values:
y = 5 * sin((2π/12) * (x - 0)) + 0
Simplifying further:
y = 5 * sin((π/6) * x)
To graph the function using the sine tool, you can plot points by substituting different x values into the equation and calculating the corresponding y values. The first point should be on the midline, and the second point should be a maximum or minimum value on the graph closest to the first point.
To graph the given function using the sine tool, we need to understand the properties of a sine function and how they relate to the given information.
1. Period: The period of a sine function is the length of one complete cycle. In this case, the time it takes for the weight to complete one cycle (from highest point to lowest point and back) is 12 seconds. Therefore, the period of the function is 12 seconds.
2. Amplitude: The amplitude of a sine function represents half of the distance between the highest and lowest values of the function. In this case, the difference between the lowest and highest points is given as 10 inches. Since the amplitude represents half of this distance, the amplitude is 5 inches.
3. Horizontal Shift: Since the weight starts at its highest point, we know that the mid-line of the graph (the horizontal axis) represents the resting position of y = 0. Therefore, there is no horizontal shift in this case.
4. Vertical Shift: Since the resting position is at y = 0, there is no vertical shift in this case.
With these properties in mind, we can now graph the function using the sine tool:
1. Start by plotting the first point on the mid-line, which is (0, 0).
2. The next point must be a maximum or minimum value on the graph closest to the first point. Since the weight starts at its highest point, the next point should be a maxima. This will help us determine the phase shift as well.
3. The phase shift (horizontal shift) is calculated using the formula:
phase shift = (time taken for the first maxima or minima to occur) / (period) * 2π
In this case, the time taken for the first maxima to occur is 12 seconds (same as the period). Therefore, the phase shift is:
phase shift = (12 / 12) * 2π = 2π
4. Now, we can plot the second point. Since we know the amplitude is 5 inches and the phase shift is 2π, the second point will be located at:
(phase shift, amplitude) = (2π, 5).
5. Using these two points, you can draw the sine curve through them, making sure to repeat the pattern every 12 seconds.
By following these steps, you will be able to accurately graph the function using the sine tool.