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. So we're given this graph, we have a periodic function shown on it. 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 we have four of them. Option A Y is equal to six cosine of four X. Option by is equal to negative six cosine of four X. Option cy is equal to four cosine of six X or option DY is equal to negative four cosine of six X. Now let's take a look at our graph. And the first thing we wanna look at is the Y intercept. Where does this function cross the Y axis? When we look, it's going to be at its minimum value. OK. And because this function crosses the Y axis at its minimum value, we're gonna use cosine to represent our graph. OK. This will save us from having to do any sort of phase shifting. OK. So we have that our equation is going to look something like Y is equal to plus or minus. A cosign of B X plus C plus D. Now, like we had just mentioned, we're choosing to use the cosine function because we are starting at a minute. OK. We're starting at a maximum or a minimum. We can use the cosine function and we don't have to worry about a phase shift because cosine starts at its maximum. OK. And so the C value is going to be zero. Now, what about this devalue? Well, this is a vertical shift. So we wanna look at where the vertical center is. Now, if we look at our maximum value, it's at four, the minimum value is at negative four, halfway between four and negative four is zero. And so Y equals zero or that X axis is going to be our vertical center. That means that our graph has not been shifted up or down at all. And D is just going to be zero. We don't need to worry about it. OK. So our function is gonna simplify down to Y is equal to plus or minus a cosine of BX. OK. So we need to figure out the sign of the leading term. We need to figure out A and we need to figure out B let's start with A and recall that A is going to represent our amplitude. Now, if we go to our graph, we're gonna look at a maximum value. OK. So the maximum occurs three times in what we're given. We're just gonna choose the first one, it has a wide value of four. And what we're gonna do is we're gonna take the distance from that maximum down to our X axis or down to the vertical center that we just talked about. That's our amplitude. Alternatively, we could do it from the minimum. So we could go from the minimum up to the X axis. OK. That's our amplitude. One other way we could do it is we could look at the total distance from the maximum to the minimum and divide it by two. OK. So in this case, the distance between our maximum and the X axis is four. OK. Distance between the minimum and the X axis it's four. So our amplitude is going to be equal to four. Now our leading coefficient should it be positive or should it be negative? Well, let's think about what happens with a regular cosine curve. If we draw out a regular cosine curve at X equals zero, our cosine curve is at its maximum value and it's gonna decrease to its minimum before coming back up again. In our case, we're starting at the minimum value in increasing. OK. So the graph that we're shown here is actually a reflection of the regular cosine graph. And so we need a negative outfront of our equation in order to represent that reflection vertically. All right. So we are gonna choose a negative sign there. Now, the last thing we need to do is figure out this B value and recall that B is related to the period of our graph. So B is gonna be two pi divided by the period T. Now T in this case is going to be the X distance to complete one full cycle. So we're gonna go to our graph and we're gonna choose a starting point and we're gonna choose the point X equals zero and Y is equal to negative four. And we're gonna follow our curve all the way around up to the maximum and back to our minimum until we've completed one full cycle. And now we're gonna look at the distance in the X direction between those two points. And so this X distance between this full cycle and we're starting at X equals zero, we're going up to X equals pi divided by three. And so that period T is going to be pi divided by three. Coming back to our B value, we have B is going to be two P divided by pi divided by three. The pies are gonna divide out, we get two divided by one third, which gives us a B value of six. So now to write the entire equation out with all of the information we figured out we have that Y is going to be equal to negative four cosine of six. And this is the equation that represents the graph we were given if we compare this to our answer choices. We can see that this corresponds 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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