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 = -¼ cos (½ x + π/2)

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 negative 1/8 of the sine of a quarter of X plus a quarter of pi. For our answer choices. A says the amplitude is 1/8 the period is eight pi the phase shift is pi units to the lift and the vertical translation is none. B says the amplitude is 1/8. The period is four pi the phase shift is pi units to the left and the vertical translation is none. C says the amplitude is eight, the period is eight pi the phase shift is pi units to the right and the vertical translation is four units up. And D says the amplitude is eight, the period is four pi the phase shift is two pi units to the right and the vertical translation is none. Now, what are we trying to figure out here? We want to determine these parameters? OK. From or a trigonometric equation. So let's ask ourselves, how are these parameters related to the sine function? In other words, what do we know about sine functions? Well, recall that the general form of a sine function is that Y equals a multiplied by the sine of BX. Let me write that properly here, BX minus C plus D. In other words, this is what every sine function looks like, right? Where A helps us to find the amplitude. Because our amplitude of our function equal the magnitude of A, right B helps us to find a period. Because the period of our, of our sine function equals two pi divided by the magnitude of B C helps us to find our phase shift because our phase shift is equal to BC rather divided by B. And the D is what we call our vertica shift. OK. And all of these parameters can help us to graph our trigonometric function. Our amplitude A tells us how big our function is going to get. Our period tells us how long the wavelength will be. The phase shift tells us how much it moves horizontally. And of course, the vertical shift tells us how much it moves up or down. So if we can compare our equation to the general form, we'll be able to find the values of ABC and D and thus figure out what those parameters are. So given our function, OK, which we identified our function to be negative. One is of sign of the sign of a quarter of egg plus 1/4 of pie. OK. Then here if we compare both functions there or our equation is almost in the general form. The only issue here is that inside our bracket, it's adding, whereas in the general form, they're subtracting. So we need to rewrite it so that they are subtracting inside our angle for sign. How can we do that? Well, adding a quarter of pi is the same thing as subtracting its negative value. So what that means then is that we can rewrite our phone or rewrite inside the bracket. Kate as the sign of a quarter of X minus a negative fourth of pi, OK, know that we've done that inside the bracket. It's subtracting like in the general form and know that it's written in the general form, we can find the values of ABC and D because now this tells us then, right? Yeah, this is, this is us on those values because this law tells us that A is going to be equal to negative A negative A, right B, the coefficient of X is going to be equal to a quarter. See the value being subtracted inside the bracket is going to be negative quarter of pi and here there is no constant being added to the sine function. So D equals zero, now that we have the values of ABC and D, we will be able to find our parameters. So what this means, right. Therefore, since our amplitude is the magnitude of air, that means the amplitude is going to be equal to the magnitude of a negative eighth, which is just going to be an eighth. So our amplitude is 1/8 next our period OK, which is equal to two pi divided by the magnitude of B is going to be two pi divided by a quarter. Now, if we think about it, two pi divided by a quarter is the same thing as two pi multiplied by four and the two pi multiplied by four equals eight pi. So our period is eight pi finally, our fierce shift is going to be equal to C which is negative fourth of pi. OK. Let's write that pipe properly. So I have a negative fourth of pi divided by B which is a quarter. OK? And if we think about it again, we can rewrite this to as a negative fourth of pi multiplied by 44 cancers four. OK? And that leaves us with a negative pi and because it's negative, that means we're going to be moving to the left. So our first shift is pi units to the left. So based on our equation, our amplitude is an eighth, our period is eight pi and our phase shift is pun to the left and we have no vertical translation. If we look back on our answer choices, that would have been 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 = -¼ cos (½ x + π/2)

Key Concepts

Amplitude of a Trigonometric Function

Period of a Trigonometric Function

Phase Shift and Vertical Translation

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