Determine an equation of the form y = a cos bx or y = a sin bx...

Question

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.

Accepted Answer

Hey, everyone in this problem, we're asked to find the equation of the graph given below. OK. So we're shown this graph and it looks like a periodic type function. We're gonna come back to that in just a minute. But we have four answer choices. Option A Y is equal to four cosine of one third X. Option by is equal to eight cosine of one third X. Option cy is equal to one quarter cosine of one third X. And option DY is equal to four cosine of three X. OK? All right. So let's take a look at our graph. And the first thing we wanna notice or point out is that this graph starts at a, its maximum. OK. So it starts at its maximum value and then it decreases. So when we're thinking about writing an equation for this graph, we're gonna use the cosine function. OK? Because the cosine recall also start at its maximum value. OK. So if we use cosine, we don't have to do any phase shifting or anything like that. OK? It's gonna be a little bit simpler and that's good. All of those answer choices have a cosine function So we're on the right track. Now, in general, we're using cosine, the equation is gonna be Y is equal to a cosign A B multiplied by X plus five plus D. OK. Now, that looks a little complicated. There's a lot to figure out there, but we can simplify this right off the bat. We've already said there's no phase shift and there's no phase shift because we're starting at our maximum value just like the cosine function does. And so by the phase shift is going to be zero, we don't need to worry about that. Now, in terms of D, our vertical shift, we're gonna be looking at where the center is in the vertical direction of this function. OK. So the maximum is at four, the minimum is at negative four. So this graph is centered at Y equals zero or centered around Y equals zero. So the vertical shift is going to be zero as well. We don't have to shift this thing up or down at all. It is already centered at zero. OK. So D that vertical shift is going to be zero as well. So with that, our equation simplifies to Y is equal to a cosine of BX. And so we have two things to figure, we have our amplitude A and we have our B value. Let's start with our amplitude A. OK. So again, A is going to correspond to our amplitude. And if we take a look at our graph and we wanna find the amplitude, we're gonna go to the maximum value and we're gonna look at the distance between that maximum value in the vertical center. So in this case, that's Y equals zero. OK. So we're looking at the distance from that maximum down to the X axis or we could look from the minimum upwards, we're gonna get the same value. OK? So either way you wanna do this, that's gonna be our amplitude. And the key here is that it's from the maximum or the minimum to the center, but not from the maximum to the minimum. OK? It's not the entire distance, it's half of that distance. All right. So our amplitude, in this case, we've said that our maximum is four. OK? So the distance from Y equals four to Y equals zero is going to be four. OK? So our amplitude is four. Now, our a value is going to remain positive. OK? We always have to figure out this sign, let's put a plus or minus out in front here. And we're choosing positive because our graph starts at the maximum and decreases. And that's exactly what the cosine function does. It starts at the maximum and decreases. So we don't need to flip this graph, it already matches cosine. And so we're good to go taking that positive out in front. So we've got a, now we can move on to figuring out B and recall that our B value is related to the period through the following. OK. B is going to be two pi divided by the period T, OK. So in order to determine B, we need to find the period teeth and we can do that using our graph. So go, let's go to our graph and recall that the period is gonna be the time it takes the complete one cycle. OK. So we wanna identify one full cycle in our graph and we can take say the maximum values. So we start at the first maximum that occurs at X equals zero. We follow along our curve until we get to the next maximum value. OK. And the X distance between those two is going to be our period T, OK. So we're starting at X equals zero, we're going up to X equals two pi divided by three. And so our period T is going to be equal to two pi divided by three. Again, the X distance or the time taken to complete that full cycle. Now, if we want to calculate our B value, we have that B is going to be two pi divided by the period. We just found two pi divided by three. And this gives us a B value of three. OK. The two pi will divide it. So if we're writing our function, like we want to, for our answer, we have Y is equal to A which we found was four, multiplied by cosine of V which we found as three X. And that is the equation that represents this graph. If we compare this to our answer choices, we can see that this corresponds with answer choice. Dy is equal to four cosine of three X. Thanks everyone for watching. I hope this video helped see you in the next one.

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.
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Key Concepts

Amplitude

Period

Phase Shift

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