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 = 3 cos (x + π/2)

Accepted Answer

Everyone. In this problem, we're asked to determine the amplitude period phase shift and vertical translation of the given trigonometric function. So we have Y is equal to six cosine of X plus pi divided by nine. We're given four answer choices A through D, each of them containing a different combination of these four values that we're looking for. For the amplitude, we can either have six or three for the period two pi or 18 pi for the phase shift pi divided by nine units to the left or to the right. And for the vertical translation, either no vertical translation or six units up. Yeah, those four answer choices are different combinations of those. So let's start. And when we have this equation, this cosine equation, we want to think about what the general equation looks like. And the general equation recall looks like why is equal to a cosign of BX minus C? What D? And when we think about comparing this equation to the equation, we have, what we can see is that this a value it is gonna be equal to six, that's the constant value outside multiplying our cosine. And if A is equal to six. Well, A corresponds to our amplitude. OK. So that means that we have an amplitude of six in this problem right now, if we take a look at our answer choices, option D has the amplitude being three. So we can eliminate that option already. A B and C have that amplitude of six. Nice. We've already eliminated some of those answer choices. Now, we can move along to the next value. And the next value we can look at is this B value and the B value is the coefficient on the X inside of the cosine function. And so it's that constant that's being multiplied out in front. And if we look at our equation, what we have is just X. So what that tells us is that B is just equal to one now B is equal to one. What does that tell us about this trigonometric function? What we call that B is related to the period of our function? And so the period of our function is gonna be equal to the regular period of the cosine function just cosine of X which we know is two pi divided by the B value. OK. Well, two pi divided by one is just two pi and so the period for our graph or our trigonometric function we were given is two pi again, taking a look at our answer choices A and B have this period of two pi C has a period of 18 pi so we can eliminate that. Now we move along to the next value in our function. And that is this C term and that is that shift inside of the cosine function, whatever is being added or subtracted. And what we can see is that we have positive pi divided by nine in the general equation, we're subtracting C OK. In our equation, we're adding this value and so this value is going to be negative. So we have that C is going to be equal to negative pi divided by nine. OK. What this tells us is that we have a face shift and the phase shift is going to be pi divided by nine units and the negative tells us it is true. So and so we have this phase shift of pi divided by nine units to the left. The last thing we wanna deal do is deal with this D. OK? So at the end, we are adding this value to everything else. OK? We don't have anything added to our cosine function in our equation. And so the value of D is just going to be zero. And what that tells us is that our equation or our function is not shifted vertically. OK? So it's not shifted up or down. So we have no vertical translation. OK? So let's take a look at those remaining answer choices. We had A and B left for option A. It has that phase shift being pi divided by nine units to the right. So that is not correct. We can eliminate option A and that leaves us with option B which is gonna be the correct answer. And we have that the amplitude is six, the period is two pi the phase shift is pi divided by nine units to the left and the vertical translation is none. Thanks everyone for watching. I hope this video helped see you in the next one.

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 3 cos (x + π/2)

Key Concepts

Amplitude

Period

Phase Shift

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