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 sin ¼ x

Accepted Answer

Welcome back everyone. In this problem, we want to determine the amplitude period phase shift and vertical translation of the trigonometric function Y equals three plus five multiplied by the sine of 1/7 of X. For our answer choices, A says the amplitude is five, the period is 14 pi the phase shift is none and the vertical translation is three units up. B says the amplitude is half of five, the period is 14 pi the phase shift is none and the vertical translation is three units down. C says the amplitude is five. The period is two pi the phase shift is a half of pi units to the right and the vertical translation is three units up. And the D says the amplitude is five, the period is two pi the phase shift is none and the vertical translation is three units down. Now, we're trying to figure out these parameters here from our trigonometric function. But if we're going to figure it out, it helps for us to think about how the sine function is related to those. What do we know about the sine function? Well, recall, OK, recall that every sine function is written in the general form, Y equals a multiplied by the sine of B multiplied by X minus C. That expression plus D OK. Where A helps us to find the amplitude of our function B helps us to find a period of our function. Because the period of, as sine function equals two pi divided by the magnitude of B OK. C tells us the phase shift of our function, OK. That is how much it moves to the left or to the right. And the D tells us the vertical shift of our function, that is how much it moves up or down. So if we can get our trigonometric function to look like this one, then we will be able to compare and figure out those parameters. Now, let's take our function and recall that our function was Y equals three plus five multiplied by the sine of 1/7 of X. Now, if you look at both of these, probably you can tell that they are not exactly written in, in the same format in the same way. So let's see if we can rearrange our function to look more like the general function and then find those parameters. So if this is why, then we can rewrite our function as five sine of 1/7 of XK plus three. Now it's looking a little bit more like it. So now that we've had, we have it in the general form, let's compare, what do you notice, well, you probably realize that our amplitude A is the coefficient of the sine function. So A equals five, right? You probably realize that B here B would be the coefficient of what's inside the bracket here. We don't have any phase shift. OK. So it's as if we're saying we're working with BX, right? So if it's 1/7 of X, that means B equals 1/7. And as we said, we can tell that our phase shift C equals zero, right? And we can also tell that our vertical shift D equals positive three. So now that we have that information, we will be able to find our amplitude because no, this tells us that our amplitude which is the absolute value of our, of, of A. So that's five, we'll be able to find our period because remember our period equals two pi divided by B. So that's two pi divided by 1/7 and the two pi divided by 1/7 is the same thing as two pi multiplied by seven, which equals 14 pi. So our period is our period is 14 pi OK. We don't have a phase shift and our vertical translation is three units. So because it's positive, that tells us that our vertical shift is three units up. So that would be the amplitude period and vertical shift of our graph. If we look back at our answer choices, the correct answer choice would be answer choice. A 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 sin ¼ x

Key Concepts

Amplitude of a Trigonometric Function

Period of a Trigonometric Function

Vertical Translation and Phase Shift

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