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 = 2 sec(πx - 2π)

Accepted Answer

Welcome back everyone. In this problem, we want to determine the amplitude period phase shift and vertical translation of the trigonometric function of Y equals two multiplied by the second of a half of pi X minus four pi. For our answer choices. A says there's no amplitude. The period is four pi the phase shift is a units to the right. And the vertical translation is four pi units up B says there is no amplitude nor vertical translation but the period is four and the phase shift is eight units to the right. C says the amplitude is two, the period is four, the phase shift is eight units to the left. And the D says there is no amplitude nor vertical translation but that the period is four and the phase shift is four pi units to the right. Now, if we're trying to figure out how these parameters relate to the second function, it would be good for us to think about what we know when it comes to the second function. So what do we know? Well, recall recall that the second function is generally written in the form Y equals the second of BX minus C plus D, OK. Where we can use B to help us find the period of our function. Because the period of our function equals two pi divided by B, we can use C to help us find the phase shift of our function. Because phase shift equals C divided by B and D is our vertical shift. OK. D is the vertical shift of our graph. Now, to note when it comes to the second function, there is no amplitude. The second function has no amplitude. So if we can find the values of BC and D from our equation that we're given, then we should be able to solve for the period, the phase shift and the vertical shift. So let's compare our equations. So for our equation which said that Y equals two multiplied by the second of, let me write it this way, a half of pi E, OK, a half of pi X minus four pi if we compare both of those equations, right, notice that essentially both of them are in the general form. Our equation is already written in the general form. So by comparison, we can tell that B equals a half of pie. OK. And C equals four pi there is no constant here. So that tells us then that our vertical shift to D equals zero. So there's no vertical shift. But we can use B and C to solve for the P and phase shift serving for the period. That means the period is going to be equal to two pi divided by B. So that's two pi divided by a half of pi, which if we rewrite it is the same as two pi multiplied by two divided by pi pi cancels pi and two multiplied by two equals four. Next, the fi shift, this shift of our graph then is going to be equal to C which is four pi divided by B which is a half of pi, which if we add a thought that's going to be four pi multiplied by two divided by pi pi cancels pi four multiplied by two equals eight. So for our graph, its phase shift is eight, it's positive. So that means it's eight units to the right. OK. Its period is four and it has no amplitude nor vertical translation. When 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 = 2 sec(πx - 2π)

Key Concepts

Amplitude of Trigonometric Functions

Period of Secant Function

Phase Shift and Vertical Translation

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