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 - sin(3x - π/5)

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the amplitude period phase shift and vertical translation of the function Y equals nine minus the sine of eight X minus a third of pi. For our answer choices. A says that the amplitude is one, the period is 1/4 of pi the phase shift is a 24th of pi units to the left. And the vertical translation is nine units up. B says the amplitude is one, the period is 1/4 of pi the phase shift is a 24th of pi units to the right. And the vertical translation is nine units up. C says the amplitude is nine. The period is an eighth of pi the phase shift is a 24th of pi units to the left and the vertical translation is nine units down. And D says the amplitude is nine. The period is an eighth of pi the phase shift is a 24th of pi units to the right. And the vertical translation is one unit though now, if we're going to determine all of those things for our sine function, then it would help for us to think about how the sine function relates to them? That is the general form of the sine function. What do we know? Well, recall and recall that for every sine function is usually written in the form Y equals a multiplied by the sine by the sine. We write at N properly by the sine of B multiplied by the expression X minus C plus D OK. Where A represents our amplitude four function B can help us to find a period of our function because the period equals two pi divided by B. OK. C represents the phase shift of our function. That is how much it moves to the left or the right. And the D represents the vertical shift of our function. That is how much it moves up or down. So if we can compare our sine function, if we can get it looking like the general form and then compare the coefficients and the constants, then we should be able to find all of that information. So let's try to do that. So the function we have, I recall it was nine minus the sign of let's write that and properly the sine of eight X minus a third of pi. Now, here we can notice by looking it's not in the general form. So let's do a little work here to get it to look that way. OK? No, I'm going to write my sine function first. So that's negative sine eight X minus a third. Of pi plus nine. OK. So now I've, I've, I've got my constant there and no, all I need to do is to factor out the coefficient of X inside our parentheses here. No, it's eight X minus a third of pi. So we can factor it from both terms. And when we do that, we'll get Y to be equal to negative sine, OK? Of eight multiplied by eight into eight X goes X times and it into a third of pi goes or will leave us with pi divided by 24. Because if we think about it, we're really seeing a third of pi divided by eight, which is a third of pi multiplied by an eight. OK. So that's basically what we're seeing here. So that would be a 24 of pie. OK? And we'll have that plus nine. Now, when we look at our general form and our equation, they're both in the same form. So now we can tell the amplitude period phase shift and vertical shift because no, if we have a negative sign, we can imagine this is a negative one here. So now that tells us, then that tells us, then that A equals negative one B equals it. OK. C equals a 24th of pi positive and D equals nine. Now let's make sense of all of that here. If A is negative one, that means the amplitude, amplitude, the amplitude, which is the magnitude of A would be equal to one. OK. Next, if B is eight, that means the period is going to be equal to two pi divided by B. So that's two pi divided by eight, which equals 1/4 of pi, OK. So that's our period. Next C equals 1 24th of pi. So that means our field shift is a 24th of pi units to the right, because it's positive and finally D equals nine and because it's positive, that means our vertical shift is nine units to the right. Therefore, our amplitude is one, our period is 1/4 of pi or phase shift is a 24th of pi units to the right. And the D is nine units to the right or vertical shift. That is 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 - sin(3x - π/5)

Key Concepts

Amplitude of a Trigonometric Function

Period of a Sine Function

Vertical Translation and Phase Shift

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