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Match each equation to its graph. Be prepared to explain how you know which graph belongs with each equation.
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Suppose a windmill has a radius of 1 meter, and the center of the windmill is at on a coordinate grid.
Suppose a bike wheel has a radius of 1 foot and we want to determine the height of a point, , on the wheel as it spins in a counterclockwise direction. The height, , in feet of point can be modeled by the equation , where is the angle of rotation of the wheel. As the wheel spins in a counterclockwise direction, the point first reaches a maximum height of 2 feet when it is at the top of the wheel, and then a minimum height of 0 feet when it is at the bottom.
The graph of the height of looks just like the graph of the sine function but it has been raised by 1 unit:
The horizontal line , shown here as a dashed line, is called the midline of the graph.
What if the wheel had a radius of 11 inches instead? How would that affect the height, , in inches, of point over time?
This wheel can also be modeled by a sine function, , where is the angle of rotation of the wheel. The graph of this function has the same wavelike shape as the sine function, but its midline is at and its amplitude is different:
The amplitude of the function is the length from the midline to the maximum value, shown here with a dashed line, or, since they are the same, the length from the minimum value to the midline. For the graph of , the midline value is 11 and the maximum is 22. This means that the amplitude is 11 since .
For the graph of , the midline value is 11 and the maximum is 22. This means that the amplitude is 11 since .
The maximum distance of the values of a periodic function above or below the midline.
The value halfway between the maximum and minimum values of a period function. Also the horizontal line whose -coordinate is that value.