For each function, give the amplitude, period, vertical transl...

Question

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = (1/2)csc (2x - π/4)

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine the amplitude period phase shift. And the vertical translation of the given trigonometric function of Y equals a half of the cosecant of four X minus 1/12 of pi. For our answer choices, A says the amplitude is none. B says the period is a half of pi C says the phase shift is a 48th of pi unit to the left. And D says there's no vertical translation. B says the amplitude is a half, the period is two pi the phase shift is a 48th of pi units to the right and no vertical translation C says there's no amplitude. The period is a half of pi the phase shift is a 48th of pi units to the right and no vertical translation. While B says the amplitude, there is no amplitude, the period is two pi the phase shift is a 48th of pi units to the right. And the vertical translation is 1/12 units up. Now, what do we, what do we, what do we know here? Well, we're talking about the Coi kind function. OK. And for the core and function. We want to think about how it is related to the amplitude period phase shift and vertical translation. So what do we know about the co and function? Well, recall that generally the form, the sequent function is written in the form Y equals the cosecant of B minus C plus D. OK? Where our period, OK. Our period B helps us defend our period because the period equals two pi divided by B OK. Or feel shift, our fears shift is equal to C divided by B. So that's what C can help us to find. And our vertical shift is given by D so D is our vertical shift. So what this tells us then is that if we can figure out if we can figure out the values of BC and D, then we should be able to find a period phase shift. And a vertical shift. To note is that the co equant function does not have a amplitude. OK. So we know already that the amplitude is not. So let's take our co and function. So remember that our function was why equals a half of the co second of a second of four X minus 1/12 of pi OK. So here, if we compare both of these, it's already in the general form because we have our exterm here with its coefficient B, we have a phase shift, but we don't have any vertical shift. So that means here from our equation, we can tell that B equals four by comparison. And we can tell that C is going to be equal to 1/12 of pi. Now that we have those, we can use them to find our period and phase shift. Because that note tells us therefore, our period OK is going to be equal to two pi divided by B, the magnitude of B which is four, which is going to be equal to a half of pi. Likewise, the phase shift is going to be equal to C divided by B. So our phase shift, the phase shift is going to be equal to 1/12 of pi divided by four and 1/12 of pi divided by four is the same thing as 1/12 of PI multiplied by a quarter and 1/10 of pi multiplied by a quarter equals a 48th of pi. Therefore, we don't have any amplitude, any vertical translation, but our period is a half of pi and our phase shift is a 48th of pi. Notice too that our phase shift is positive. So it's a 48th of pi units to the right when we look back on our answer choices, that would have been answer choice. C thanks a lot for watching everyone. I hope this video helped.

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = (1/2)csc (2x - π/4)

Key Concepts

Amplitude of Trigonometric Functions

Period of Trigonometric Functions

Phase Shift and Vertical Translation

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