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 the look as we're given, this graph looks like a periodic function that were 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 the answer choices that were given and there are four of them. So option A Y is equal to five sine of three X. Option by is equal to negative five sine of six X. Option cy is equal to five sign of six X and option DY is equal to six sign of five X. The first thing we wanna do is take a look at where this function crosses the Y axis. OK. So where is this function? When X is equal to zero? And in this case, that's the start of our graph. And when X is equal to zero, Y is also equal to zero. OK. So in this case, our function starts at zero and so we wanna use the sine equation, OK. Remember that the sine function also has a Y intercept of zero and it starts at 00. All right. So we're gonna use the sine function Y is equal to, we're gonna have plus or minus a sign of VX plus C plus D. No, let's start with C OK. And because our function starts at 00, we've chosen to use the sine equation by doing that, it saves us from having to do any sort of phase shift. And we're starting exactly where sine starts. So we have no phase shift and so C is going to be zero. All right. Now let's deal with DD is our vertical shift. And again, looking at our graph, OK. The vertical center of our graph is that X axis at Y equals zero. And so we don't have any vertical shift. OK. The other way we can look at this is the maximum is five, the minimum is negative five halfway between those two is zero. OK. So we have no vertical shift D is zero. So now that we've eliminated some of the complexity of our equation, we have Y is equal to plus or minus a sign of BX. So what we're gonna do is we're gonna go ahead and use our graph to find the value of a, use the graph to find the value of B. And then we're gonna talk about the sign is positive or negative when we do the A value as well. And so a recall is going to represent our amplitude. And when we're talking about our amplitude, what we're talking about is going from the maximum down to the X axis or alternatively from the minimum up to the X axis. And either of these are amplitude. In other words, it's the distance between the maximum and the minimum divided by two. OK. So in this case, the distance from the maximum which is five to the X axis Y equals zero is five. And so our amplitude A is going to be five. Awesome. So we have our amplitude of five, we know our A value. Now let's figure out the sign going on in front. In our graph, we're starting at that 0.00. And our graph is increasing up to the maximum before decreasing. When we think of just a regular sign curve, it does the exact same thing. It starts at 00 and increases. So we don't need to worry about reflecting this equation or having any sort of flipping. OK. Our graph is not reflected, it is just like the original sign curve. And so the coefficient, the leading coefficient is going to be positive. OK. All right. So we have determined the sign and the value up front. Now we have to figure out this B value and recall that B is related to the period. OK. So B is equal to two pi divided by the period T. Now the period is the, in this case, the a distance required to complete one cycle. So what we're gonna do is we're gonna choose a point to start at. So let's say we start at our starting 0.00. We're gonna follow this curl and our curve goes up to a maximum, comes back down, it crosses zero again, but this is not one full cycle. We haven't been down to our minimum value yet. So we have to go down to our minimum value and back to zero again before we've done one full cycle. And now we're at the same Y value of zero and we're headed in the same direction. Our curve is currently going upwards. OK. So that is one full cycle. And then we're just gonna take the distance in the X direction between these two points. Now we're starting at X equals zero, we're going to X equals pi divided by three. And so our period T is going to be pi divided by three. Now, we can calculate our B value two pi divided by pi divided by three. And when we work this out, the pies are going to divide out, we have two divided by one third which gives us a B value of six. OK. So we have an, a value of five A B value of six. We know that we have that positive term out in front. And so our equation is going to be Y is equal to five, died of six X. And that is the equation of the graph. We were given exactly like the question asked us to find if we compare this to the answer choices we were given, we can see that this corresponds with answer choice. C thanks everyone for watching. I hope this video help 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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