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 = 1/3 tan (3x - π/3)

Accepted Answer

Welcome back, everyone in this problem, we want to find the amplitude period, vertical translation and phase shift for the trigonometric function Y equals 1/5 of the tangent of five X minus 1/5 of pi. For our answer choices, all of them say that there is no amplitude and vertical shift. But A says the period is 1/5 of pi while the phase shift is 1/5 of pi units to the right. B says the period is 1 25th of pi and the phase shift is 1/5 of pi units to the left. C says the period is 1/5 of pi and the phase shift is a 25th of pi units to the right. And D says the period is pi and the phase shift is a 25th of pi units to the right now, let's try to make sense of what's really going on here. So if we're going to find all of these things, we have to think about the nature of the tangent function. So let's ask ourselves, what do we know about that function? Will recall that generally every tangent function can be written in the form Y equals a multiplied by the tangent of B multiplied by X minus C plus D. OK. Where A represents the vertical stretch of our tangent. OK. B is a part of how we can find a period for a tangent because the period is going to be equal to pi divided by the absolute value. Let me write that pi properly pi divided by the absolute value of B OK. C equals our first shift. And the D is our vertical shift. Let's put D over here D is our vertical shift. So that means if we compare our equation with the general form, then we should be able to find all of these things. Now notice here we said that the coefficient of the tangent A is the vertical structure of the tangent graph and it is not equal to the amplitude. So it's, it's not, we, we don't necessarily need to find the amplitude of a tangent graph because a tangent graph doesn't have an amplitude. So let's focus on the period, the phase shift and the vertical shift. So given our tangent function, which we know that it said Y equals 1/5 of the tangent of five X minus 1/5 of pi we can try to rewrite it in the general form. And then compare now notice that in the general form there is a coefficient of X minus C but here we only have a coefficient of X. So we need to factor out that term from X in what we have here for our angle five, X minus 1/5 of pi. So that means we need to factor 05. OK. So if we come down a bit, then that means why is going to be equal to 1/5 multiplied by the tangent of five multiplied by five X goes X times. OK? And five into 1/5 of pi five into 1/5 of pi goes 25 times. OK. So here we have no red meat in the general farm. So what does this mean then? Well, no here notice that if we to write it above five is B OK, or a period of 21 25th of pi is C but there is no vertical shift. So now that we have B and C, we can find our period and our phase shift. So that means then our period is going to be equal to pi divided by the absolute value of B which is five. So that's 1/5 of pi and our face shift is going to be equal to or value here which is a 25th of pi. Now it's positive. So that means you're going to be going to the right. Therefore, the period of our graph is 1/5 of Pi and its sphere shift is 1 25th 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 helps.

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 1/3 tan (3x - π/3)

Key Concepts

Amplitude of Trigonometric Functions

Period of the Tangent Function

Phase Shift and Vertical Translation

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