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. And the graph we're given has this periodic looking function shown in blue. And we're gonna come back to the details of this graph as we work through the problem. But first, let's take a look at our answer choices. So there are four of them. Option A Y is equal to negative four cosine of 1/5 X. Option by is equal to negative two cosine of five X. Option cy is equal to two cosine of 1/5 X. And option DY is equal to negative two cosine of 1/5 X. So let's take a look at our graph. And the first thing we're gonna point out is that this graph starts at a minimum value. OK. So it's starting at its minimum value. So we're gonna use the cosine function in our equation. OK. This does not start at zero. So in order to save us from having to do any sort of phase shifting, we're gonna use cosine for our equation. And when we do that, our equation is gonna be Y is equal to plus or minus A multiplied by cosine of B multiplied by XC plus D. OK. So in this equation, we have four values that we need ABC and D. And then we have to also sort out this plus or minus thing at the beginning. What's our leading sign? Yeah, it looks like a lot, but we can narrow this down pretty quickly. So we've just said that this starts at a, a minimum value and because our graph starts at a minimum value, we don't need to phase shift from our regular cosine function. And so the phase shift parameter zero is just, or C sorry is just going to be zero. OK. So C is zero, we don't have to worry about that. Now, D represents our vertical shift. OK? And if we look at our graph, if we look at the distance from the X axis upwards, it's two. If we look at the distance from the X axis downwards to the minimum value, it's also two. OK. So our function is centered vertically around the X axis at Y equals zero. So we have no vertical shift, OK? We don't need to move this function or this function hasn't been shifted upwards or downwards. And so D is going to be zero as well. And so when we get all of that together, we just have that our function simplifies. So Y is equal to plus or minus a cosine of BX, OK. So there's two big things we have to figure out the amplitude A in our B value. And then we also need to sort out the sign. OK. So let's start with A and A is going to represent our amplitude. OK. So let's go over to our function and we're gonna pick out the maximum value. OK. And we have a maximum value of two that occurs at five pi and we're gonna look at the distance from that maximum down to our vertical center, which we've said is the X axis. OK. That's gonna be our amplitude. Now, alternatively, we could do this from the minimum. So we have a minimum of negative two at 10 pi and we could look at the distance from that minimum up to the X axis for that vertical center. OK? Either of those are our amplitude. But what's really important, what sometimes can catch us is it's not the total distance from the maximum to the minimum. OK. We're only going from the maximum or the minimum to the center. Hi. And so in this case, our maximum is two, our center is at Y equals zero. And so our amplitude, the total distance between those two is two. All right. So we have our a value along with our a value, we need to figure out the sign. So our leading term in this case should be negative. Why is that? Well, if we think about our regular cosine function, let's draw it out here. It starts at its maximum value and decreases to its minimum before increasing again, in the function or the graph that we have, we start at the minimum value and we increase. OK. So this is like the flipped or reflected version of the regular cosine graph. And so because we're starting at that minimum value and increasing, we have this reflection, we need a negative out in front of our equation, right? All right. So we figured out that we want a negative sign, we know what A is. Well, let's move to B and recall that B is related to the period of our graph. So B is equal to two pi divided by the period T. Now the period T, the time to complete one full cycle. So what we're gonna do is we're gonna go to a graph, we're gonna choose a point and let's choose the starting point. OK. So we start at X equals zero and Y equals negative two. And we're gonna follow our graph until we reach one full cycle. So we're going up to our maximum all the way back down to our minimum until we get back to negative two. And on the Y axis and we are at 10 pi on the X axis. That is one full cycle of this graph. And the X distance between these two points is going to be our period T. Now the first X value is zero. The final X value is 10 pi So the distance between the two is 10 pi and so our period in this case T is going to be 10 P. All right. So now we can calculate the B value and the B value. OK? Two pi divided by our period 10 pi is going to give us 1/5. OK. The pies will divide out, we can divide the top and bottom numerator and denominator by two to get 1/5. So we have our, a value, we know the sign of the leading term. We have our B value so we can write our equation. It's gonna be Y is equal to negative too multiplied by cosine of 1/5 X. OK? And that is that final equation. The, the question was asking us to find, we can compare this to our answer choices and we see that this matches with answer choice. D 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

General Form of Sine and Cosine Functions

Amplitude and Period of Trigonometric Functions

Phase Shift and Choosing Between Sine and Cosine

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