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 as we're given, this graph looks like a periodic function. We're gonna come back and kind of look at the details of this graph in just a minute. But let's first look at our answer choices. So there are four of them. Option A Y is equal to five sine of three X. Option by is equal to negative five S of three X. Option cy is equal to 10 sine of three X and option DY is equal to negative 10 sine of three X. All right. So let's take a look at a graph. And the first thing we're gonna look at is where this graph starts. OK? When X is equal to zero, where is this graph? And when X is equal to zero, our function is at zero as well. So because we have our graph starting at X equals zero at zero, we're gonna use the sine function. OK? Because recall that the sine function starts there as well. All right. So if we're using the sine function, then our equation is going to be Y is equal to a multiplied by sign of B multiplied by X plus C four D. All right now, so it's a little complicated. We have four unknown values in this equation. So it's a lot to figure out, but we can simplify this down right off the bat. OK. Now we chose the sine function because our function starts right at zero at X equals zero. OK. So because that function starts right at zero, we don't have to worry about any phase shift. OK. The sine function starts at 00, our function starts at 00. And so this phase shift C is just going to be zero. Now, for D, they recall that D corresponds to a vertical shift. If we look at our graph, the maximum value we have is at five, the minimum value we have is at negative five. Our function is centered vertically around X equals zero. OK. So we don't need to worry about a vertical shift. OK. This graph has not been shifted upwards or downwards. And so D is going to be zero as well. And so once we factor that in, we're left with just Y is equal to a multiplied by S of BX. And let me just put a plus or minus in front of this equation. OK. And that's gonna be something that we have to figure out as well. So let's start with a, we need to figure out A and A is our amplitude. Now, if we go to our graph, we're gonna find our maximum value and we're gonna look at the distance between that maximum to the vertical center. OK. Which is the line X equals zero or sorry, Y equals zero. OK. So along Y equals zero, we have the vertical center, we're going from our maximum down to that line right down to our X axis. And that's gonna be our amplitude. We could also look from the minimum if you wanted. So from our minimum at negative five all the way up to that X axis, that's also our amplitude A, OK. It is not the total distance from maximum to minimum. OK. We're going from the max or the min to the center. Now, in this case, maximum is at five, our center is at zero. And so the amplitude is just going to be five. All right. So we figured out the a value. Now, along with that a value, we need to figure out the correct sign. OK. The sign of our leading term. Now let's think about our regular sine function. OK. If I were to draw our regular sine function, we know it starts at 00 and it increases at first before decreasing. OK. So increases to its maximum and then decreases. The function we have starts at that same 0.00 but it starts by decreasing and then increasing. So this function has been flipped compared to the regular sine function. And so the leading term needs to be negative. OK. So the sign we're gonna have is negative. So we're gonna have that Y is equal to negative five, multiplied by side of BX. All right. Now that we figured that out, we're left with determining what B is. And let's recall that B is related to the period of our graph B is equal to two pi divided by the period T. Now how can we figure out T or recall that the period is going to be the time to complete one full cycle. So we're gonna go to a graph, we're gonna choose a point. So let's choose our starting point at 00. We're gonna follow the graph until we complete one full cycle. So we're going downwards to the minimum, we're going back up, we're not going to stop yet. Even though we're at the point Y equals zero, we haven't completed one full cycle. OK? So we're gonna keep going to our maximum and then when we come back down to zero again, that is when we've completed one full cycle. OK? We're at the same Y value and we're headed in the same direction and we're headed downwards. And so the period is going to be that X distance between the two points. OK? Now, in this case, we're starting at X equals zero, we're ending at X equals two pi divided by three. And so our period T is just going to be two pi divided by three and that distance between powerful cycle. Now we have our period T we can calculate this B value KB is going to be two pi divided by that period. Two pi divided by three, the two pi are going to divide out. We're gonna be left with B is equal to three. So if we put this all together, we get that Y is equal to negative five sign of three. Uh And that is the function that represents the graph we were given, comparing this to our answer choices. We can see that this corresponds with answer choice by is equal to negative five sign of three X. Thanks everyone for watching. I hope this video helped 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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