 Previously in lectures 10 and the earlier parts of lecture 11 in our lecture series we've been learning how to take a trigonometric functions formula like sine and cosine and then graph that function in this last video for Lecture 11 we want to reverse the process if we're given the trigonometric graph can we come up with a Equation that represents that trigonometric function now We've already seen as we've been graphing these things and also as one studies trigonometric identities Then in fact the same graph can actually be represented by more than one trigonometric function We saw we had examples where we're graphing a cosine function But in the end because of a phase shift the graph actually looked like a sine wave and vice versa We are graphing a sine and in the end it looked like a cosine So when it comes to trying to find an equation to Model the graphs were given it turns out there are multiple solutions But our goal is to then choose the least complicated Possibility what's the simplest equation that will give us this graph? And so what I mean by that is when you look at this first graph, right? Is it a sine or is a cosine? It's hard to tell maybe at first. I like to first identify the midline The midline would be the middle of our sinusoidal wave, which we see right here The midline is y equals 2 this represents. There's going to be a horse a vertical shift to the graph We get our k value is equal to 2 now looking at the Looking at the midline. Does our graph? Does it look like a sine or a cosine? So focusing on the y-axis over here. We start on the midline We then go up then we return to the midline then we go down then we return to it This right here is one complete cycle and there's a lot of information we get from this So first of all the fact that it goes up then down and it starts on the midline This makes me think that this is a sine function. Yes, you could do cosine But we want to avoid horizontal shifts as much as possible. So I'm going to base this around a sign It's a sign that has no reflection because it starts off by increasing It's a sign with no reflection our basic function is supposed to look like y equals K plus a sine of B times x minus h like so So we've already identified that there is a shift of some kind So we're gonna end up with y is equal to 2 plus the next thing we want to identify is the Amplitude how far above the midline do we get so the midline is at 2 the tip top of this thing is at 5 That's a difference of 3 which is going to give us the amplitude. So we have y equals 2 plus 3 Sine of well then what goes inside here? The B will determine the period change, right? And so the period which we can see on the graph, right? We went from 0 to pi So one single period turned out to be pi now the relationship between the period and this coefficient B is that P is equal to 2 pi over B like so Or in particular it probably a little bit easier to use is B is equal to 2 pi over P Which tells us B is equal to 2 pi over pi You cancel the pies you end up with a 2 so we're gonna put a coefficient of 2 in there And you'll notice that since we started on the y-axis There was no shift to the left or right that should the convenience of using sine is that no shifting was necessary So we just end up with a 2x right here and this gives us the simplest Least complicated trigonometric function that gives us this graph y equals 2 plus 3 sine of 2x Let's consider another example Looking at this trigonometric graph. The first thing I want to identify is the midline The middle of the graph is going to be right here. It's again gonna be at y equals 2 Does this look like sine or cosine? Well, it starts on the midline So if you start on the midline with respect to the y-axis, that's gonna be a sign, right? But this time we're decreasing. So this one is gonna be a sign with a reflection With a reflection on it. So our basic model we're working with here is again gonna be sine So we get y equals k plus a sine of B times x minus h like so that's what we're looking for The shift we've already identified because that's the location of the midline So we get y equals 2 plus. Let's try to find the amplitude the amplitude is how big these bumps are, right? So how far above the midline do we go? This one goes all the way up to 7 the midline that the midline is at 2 so the amplitude is gonna be 5 They're different 7 minus 2 but because it's been reflected we need to have a negative sign incorporated in there So in fact, we have a we have y equals 2 minus 7 not 7 5 the difference between the top And the midline so 7 minus 2 is 5 times sine So now we have to adjust for any Any period changes are shifting the advantage that's looking at the y-axis and since we start on the y-axis There's no shifting going on here. So we just need to figure out the period change We see that one period is displayed on the screen starting at zero and ending at 2 pi thirds This is one cycle So we get that the period is going to equal 2 pi thirds, which means B is going to equal 2 pi divided by 2 pi thirds Which to make life a little bit easier I'm going to times top and bottom by 3 So these 3's can't allow the 2 pies are going to cancel out in the end. He doesn't end up with a 3 like so and so then finishing up our equation here we get y equals 2 minus 5 sine of 3x and This then gives us the correct equation For this sine wave again, there's more than one equation you could use But this is going to be the simplest one. We want to avoid Horizontal shifts as much as possible. Let's consider one last example here now when I look at this one Well, the midline is a little bit harder to see it's tempting to say it's the x-axis Is it the x-axis? Well, if you look at the very bottom of the graph with respect to the Y-axis it does actually start on the y-axis here So this already tells me that oh the graph I could probably get away with a cosign there the lowest value is negative 5 The largest value is here at 3 So what's the midpoint? Negative 5 plus 3 is a negative 2 divide that by 2 is negative 1 So the midline is going to be at negative 1 It might it kind of looked like it could be the x-axis, but the correct midline is going to be here at negative 1 like so All right, so our basic function. We're looking for is y equals k plus a Times cosine of b times x minus h. We're using a cosine model for this one right here So can we identify the shift the shift is where the midline is located at so we end up with y is equal to negative 1 Now is this cosine reflected the standard cosine it starts off at 1 it comes down it goes up something like this So ours is in fact reflected notice how we start off by increasing not decreasing So we're gonna put a negative sign right here. What's the amplitude? Well, the amplitude is how far above the midline are we so if the midline's at negative 1 it goes all the way up to 3 The difference there 3 minus negative 1 is 4 the amplitude is going to be a 4 Which is what we need to put here So we have y equals negative 1 minus 4 times cosine of we need to identify a period Which we have exactly one period listed we can go from 0 to 6 Right here the period is 6 which if the period is 6 that means b is equal to 2 pi over 6 2 goes into 6 3 times so we end up with pi Thirds as our coefficient b and so then we end up putting that in there So we're gonna get pi thirds x and this then gives us the simplest equation for our cosine wave right here and Again, we've made all of our decisions to avoid the phase shift as much as possible We don't like horizontal shifts because it definitely complicates things so if at all possible if the y axis coincides with a Intersection of the midline or a maximum or a minimum of the graph Then you don't have to do a phase shift and life becomes so much easier for us So that then ends lecture 11 which is going to conclude our discussion of transforming sine and cosine and lecture 12 We're gonna learn about we're gonna continue the transformations We've learned in the last couple lectures, but apply them to the other trigonometric functions tangent cotangent secant and coat Co-secant as well, so I hope you'll stay tuned for those If you've learned things in these videos Please give it a like if you'd like to see more videos like this in the future subscribe to the channel And as always if you have any questions feel free to post them in the comments of this video or any of the videos You watch and I'll be glad to answer them at my suitonist capability