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 = cot (x/2 + 3π/4)

Accepted Answer

Welcome back. Everyone. In this problem, we want to find the amplitude period, vertical translation and phase shift for the function Y equals the core tangent of 1/4 of X minus 5/8 of pi. For our answer choices, all of them say that there is no amplitude and vertical shift. While A says the period is four pi and the phase shift is five halves of pi units to the left. B says the period is four pi and the phase shift is five halves of pi units to the right. C says the period is 1/4 of pi and the phase shift is five halves of pi units to the right. And the D says the period is four pi and the phase shift is 2/5 of pi units to the left. Now, here we're trying to find all of these things from the cotangent function. So first let's ask ourselves, what do we know about the cotangent function? We recall that like the tangent function, every cotangent function is written in the general form Y equals a multiplied by the core tangent of B multiplied by the expression X minus C plus D. OK. Where a, your function represents the vertical stretch of our cotangent or of our, of our cotangent graph B helps us to find a period of our graph because B tells us that the period equals pi divided by the magnitude of B or the absolute value of B C is what we refer to as our phase shift. That is how much the guy has shifted to the left or right and D represents our vertical shift. That is how much the graft has the graph has moved vertically. OK. So what that means is that if we compare our function to the general form of our cotangent function, then we should be able to figure out all these things to note is that the A represents the vertical stretch here for our co tangent graph. And that's not the same thing as the amplitude. The amplitude is not the same thing as the vertical stretch. So that means there is no amplitude for cotangent function. So let's go ahead and see if we can figure out the rest. So our function here was Y equals the core tangent of 1/4 of X. OK, minus 5/8 of pie. Now, if you compare both of them, you'll notice that it is not necessarily in the general form yet because here we have these two numbers, we have their difference, but we don't have a common coefficient for both. We can find that by factoring out a quarter from our expression. So if we factor out a quarter, it means we're going to divide both of them by a quarter and to divide by a quarter is the same thing as multiplying by four. So basically, we're going to multiply 1/4 of X by four and 5/8 of pi by four. So when we do that, then, so we have Y equals the co tangent of fourth factor out of fourth and then inside 1/4 into 1/4 of X leaves us with X and 1/4 into 5/8 of pi. That's basically the same thing as saying, four times, 5/8 of pi and four times 5/8 of pi 484 goes into it two times. So that leaves us with half of five pi. So in general form, our function is not the cotangent of a quarter of the expression X minus five halves of pi. No, it's in the general form. So let's look, let's let's compare starting with our period. Notice that B is equal to 1/4. Do you guys see that connection B is equal to 1/4? Right? So here, since B equals of worth then that means our period which equals pi divided by B is going to be equal to pi divided by 1/4. OK. And remember as we said earlier, dividing by 1/4 is the same thing as multiplying by four. So really our period is going to be four pi next, for our phase shift, for our phase shift, our phase shift equals our constant here, that's being subtracted. So in this case, if you notice our phase shift is five halves of pi, so since our field shift is five H of pi and it's positive, that means we're going five H of pi units to the right. If we look at our function, there's no constant that's being added to the core tangent. So that means there is no vertical shift. Therefore, our period is four pi and our field shift is five halves of pi units to the right. If we look back on our answer choices that would have been answer choice. B 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 = cot (x/2 + 3π/4)

Key Concepts

Amplitude of Trigonometric Functions

Period of Cotangent Function

Phase Shift and Vertical Translation

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