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 [π/2 (x - ½)]

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 six multiplied by the cosine of the product of the expressions. A third of pi and X minus 1/5 for our answer choices. A says the amplitude is six, the period is six. The phase shift is 1/5 units to the left. And the vertical translation is none. B says the amplitude is six. The period is six. The phase shift is 1/5 units to the right and the vertical translation is none C says the amplitude is three, the period is two thirds. The phase shift is five units to the left. And the vertical translation is three units up. And D says the amplitude is three, the period is three halves. The phase shift is five units to the right and the vertical translation is three units up. Now, if we're going to find all these things for a trigonometric function, it helps to think about how our trigonometric function is related to the amplitude period, the phase shift and the translation, the vertical translation. Now, what do we know well recall that every cosine function is usually written in the form Y equals a multiplied by the cosine of B multiplied by the expression X minus C plus D. OK. We're in represents the amplitude of the graph B helps us to find the period of the graph. That is how far it is in between peaks where the period equals two pi divided by the magnitude of B. Right C helps us to figure out the phase shift of the graph. That is how much the graph moves to the left or right. And the D helps us to figure out the vertical shift of the graph. That is how much it moves up or down. So if we can compare our equation to the general equation for the cosine function, then we will be able to find those things. So let's write no, no function which we know that our function equals six multiplied by the cosine of a third of pie multiplied by X minus 1/5. OK. I don't know, let's look at both of these to see if we can recognize any similarities. Well here, by looking notice that the amplitude that is the coefficient of the cosine function is six. So A equals six B which is the coefficient of the expression inside the angle for the cosine function is a third of pi and the sea equal or C which is in our expression here equals 1/5. So A equals, let's write that down A equals six B equals a third of pi and C equals 1/5. So already we know our amplitude is six, our phase shift is 1/5. That means it moves 1/5 unit to the right now, we just need to use B to find our period. So here that means our period is going to be equal to two pi divided by a third of pi. OK. So let me write that properly. Two pi divided by a third of pi which we can rewrite as two pi multiplied by three, divided by pi. Here pi cancels pi and that leaves us with two multiplied by three which equals six. Therefore, the period is six. Let's highlight that the period is six. The phase shift is 1/5 units to the right and the amplitude is six. 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 helps.

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 3 cos [π/2 (x - ½)]

Key Concepts

Amplitude

Period

Phase Shift

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