in this video you're going to learn about whether to use a s equation or a cosine equation when you're describing a sinusoidal graph so let's talk about some examples here this first example you can see they give us a sinusoidal graph what's a sinusoidal graph well it's basically you know this s-shaped graph and this question that students always ask is you know do I use a s equation or a cosine equation and I'm here to tell you that you can use either one now the next question will say well do I use a PO positive a value or a negative a value and again the question is uh the answer is you can use either one and I'll show you how to write these equations several different ways the first thing that I would do is I would sketch in this midline now I when I say midline what that means is you know it's the line that essentially splits this graph in half you've got half is above and half is below that'll help you a little bit when you're doing this problem because what this um midline tells us is what the vertical shift is and this case the graph is Shifting down too so no matter what equation that we write it's always going to have a k value or vertical shift of ne2 so we got that out of the way now the next thing is the amplitude and the amplitude is referring to you know how high the waves are but remember when you measure the amplitude you're always measuring from this midline okay so the amplitude itself is always positive if it's negative what that means it's reflecting it over the x-axis so you can see this amplitude here is just a height of one one so we know our a value is one now the question is do we want to use a s graph or a cosine graph well again that has to do with what you want to think of as your starting point you want to know your basic sign and cosine graphs so I'm just going to sketch them right here so you can see this s normally we think of as starting here at the origin and then it goes up and then back to zero and down and back up like that right so that's the basic sign graph whereas a cosine graph starts at the maximum goes through the zero the minimum back to the maximum like that they both have a period of 2 pi and what we want to do is we want to calculate what the period is for this particular graph now to get from let's say this point to where then it starts over and repeats again that's giving us a distance of 4 Pi okay 5 Pi minus 1 Pi 4 Pi till it repeats the other way to do it is you can measure from Peak to Peak or from value to Value so this is still giving us 4 Pi this is still giving us 4 Pi so in this case let's go ahead and calculate what this B value is and the formula for this is the period is equal to 2 Pi / B or you could say B is equal to 2 Pi / the period in this case the period we set is 4 Pi so if we reduce you can see that b is equal to 12 so in any of these equations that we're going to be writing here we're going to have the same basic format we're going to have amplitude is one B is 1/2 the vertical shift is -2 but then what we want to analyze is okay we're going to use a sign or a cosine let's maybe start with a cosine graph now you see this cosine graph down here if I was to reflect it over the xais then it would be over here right and it would look something like this dotted line that I'm drawing right now which looks a lot like the graph that we have here right so I'm going to make this a -1 cine 12 x okay now it hasn't shifted left or right so my H value is just going to be zero I'm just going to leave it like that the only thing that's differing between all these equations we're going to be writing is the a value whether it's positive or negative and the phase shift which is the left and right the horizontal shift okay so that's one possible answer here right the next one is let's maybe take a sign graph now so we're going to say s we still have the 1/2 we still have the -2 now remember sign starts here at the origin and it goes up and then down and back up so you can see this would be like a starting point but what's happening is is Shifting to the right Pi so when we shift to the right that's going to be minus Pi remember the one that's grouped with the X has the opposite effect on the graph minus Pi is actually shifting right pi plus pi is Shifting left Pi okay and then the other question is is that has it been reflected or not well this one hasn't been reflected you can see it's same shape as this graph here it's just being shifted right so that means the amplitude is just going to be one it's not being going to be multiplied by a negative okay let's take a look at another example let's maybe say for example we want to do a sign graph but we want to make it a negative amplitude this time okay so all this information here is staying the same the only thing that's changing is the fact that this is going to be a negative so it's reflecting over the x axis so where does that uh look like well that looks something like it's starting here now right okay and it's going down back up and like that so that means now it's being shifted to the left Pi which means that this is actually going to be the opposite plus pi okay remember the one group of X is the opposite effect on the graph and let's do maybe a a cosine graph and we'll just make this a positive one this time the B value is the same the vertical shift is the same but now let's see so if we do a positive cosine now cosine you can see starts over here at the max okay so we could either use this point here or we could use this point here doesn't really matter there's an infinite number of ways to write this equation for this graph it's just a matter of how far you want to shift it left and right whether you want to use S or cosine and whether you want it to be a multipli by negative 1 meaning it's reflecting so let's just go ahead and use this point over here so that means it's shifting to the left 2 pi which means that this is going to be plus 2 pi so any one of these would be the correct answer how do you know which one to use well if your teacher specifies it has to be a cosine graph you'd want to use one of those two if it's a sign graph one of those two but make it easy on yourself you don't want to be Shifting the graph way over here and then way over here try to pick something that's close ideally you might not have to shift it at all and that's what I did in this first example I thought that was the easiest was just to think of it as a CO sign graph that has only been shifted down but it was reflected over the x-axis first so I hope this helped you understand how to work with you know writing an equation uh s or cosine equation given a graph subscribe to the channel check out more math tutoring videos on my YouTube channel Mario's math tutoring and I look forward to helping you in the future videos I'll talk to you soon