For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = (1/2)csc (2x - π/4)
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Identify the amplitude: For the cosecant function, amplitude is not typically defined as it is for sine and cosine. However, the coefficient in front of the csc function, , affects the vertical stretch or compression.
Determine the period: The period of the cosecant function is determined by the coefficient of inside the function. The formula for the period of is . Here, , so the period is .
Find the phase shift: The phase shift is determined by the horizontal translation inside the function. The formula is where is the constant added or subtracted from . Here, and , so the phase shift is .
Identify the vertical translation: There is no constant added or subtracted outside the function, so the vertical translation is 0.
Summarize the transformations: The function has a vertical stretch by a factor of , a period of , a phase shift of to the right, and no vertical translation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude
Amplitude refers to the maximum distance a wave reaches from its central axis. In the context of trigonometric functions, it is typically associated with sine and cosine functions. However, for cosecant functions, which are the reciprocal of sine, the amplitude is not defined in the same way, as cosecant can take on values from negative to positive infinity.
The period of a trigonometric function is the length of one complete cycle of the wave. For the cosecant function, the period can be determined from the coefficient of x in the argument of the function. In this case, the period is calculated as 2π divided by the coefficient of x, which is 2, resulting in a period of π.
Phase shift refers to the horizontal shift of a trigonometric function along the x-axis. It is determined by the constant added or subtracted from the x variable in the function's argument. For the function y = (1/2)csc(2x - π/4), the phase shift can be found by setting the inside of the function equal to zero, leading to a shift of π/8 to the right.