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1-30 AIM3 5.7 Sinusoidal Functions- Amplitude, Period, Phase Shift

Added Jan 30, 2024 •

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alright so let's complete our exploration of the sinusoidal functions by Taking Care all possible movements we can experience will be visiting amplitude. And phase shift and I'll be stressing a few things with you all right so first what is a sinusoidal function as if you've heard me say before freaking study something we better know what it is and here here's the definition basically Seinfeld function is a function cosine of something sign of something now we're going to look at it interpreted very special form and I want to emphasize that we are not going to use the way they've written this year not that it's a of course we're going to use a different form that's written it down below right here you can see it's very similar the difference is 1 things I change this this ax + B in here so if we think about it a x + b could be called a * x + B / a MLB over is just a number so okay and then I also if I want to call ex- negative B / a and that's more the former in do it except we're going to call this we're going to call this sea and so that's why I have this form here now number sinus signs or cosine so I put sign your bracket just put cosine the important thing is what are what information do we get from this a b c and d in one should identify that then we can say give information about the graph we can create the graph we can also use it to as we were on the next lesson when were given information we can then create this the appropriate function for the first thing I want to visit is the. First ice one of them will you tell me for sine or cosine what's the. So the. Would normally be too high for a sine x cosine X do we know can change so for example going to wax well that isn't just exiting more sleep. Is changed and we know that we should think of this as the sine of x over a half which means the. Is going to be half as much so that . would be a half of 2 pi on pretty would just be by okay go on the try to do is look for look for patterns so it's supposed I think of it this way just to really be deliberate about it called 1/2 * X price of keep that in mind and I got x / 2 so the period has been stretched by 2 so I. Is 2 * 2 pie until I gives us for pi the reason I wrote it 1/2 x just remind you can do it either way sometimes people for that way you'll see as we go through the problems what happens here after can I use but if you do can I use this coefficient years that's why I want to rewrite his 1/2 * X but I've got a cat crawling in front of me right now bill is back for a visit yes Bill good to see you can I use the numbers 2 and 1/2 and somehow use them on my initial basic function. Of 2pi to end up with a period of just pie and it when there's a two here and a. Of 4 Pi when there's a half so take a moment what could you do to two pie then involves a two or involves a half and ends up with the. We ended up with catsup hoping you end up with the realization that if we take the. Of 2pi then we divided it by this value 2 Pi / 28 x just pi to pi divided by a half of that would be for soap take two pie in / be right this value right here which is the coefficient of x right then that gives us the value of the. Also it's a two-way street if you knew be you I could figure out what the. Is but for the moment we'll just think of it this way so now I seen that if I can recognize the coefficient of x which is not too terribly difficult and I have insights into the. That's first bit of decoding of our sinuses sinusoidal form that we have let us look at some more big old cat head right in front of the screen now amplitude so amplitude I people sometimes misunderstand this amplitude is not the distance from the maximum to the minimum so let's just go out here minimum this is actually two amplitude amplitude is the distance from the median value up to a Max or down to admit it doesn't matter so that's one a but obviously if you have the maximum the minimum you have to weigh in if you had where you at 8 so we're going to be able to use that information to help us create our equations what is the. Well we know that. This time it takes to complete one cycle ride so for example for I will see if I can maybe go so kind of like a cosine we're going from the maximum to a maximum so that would be one. We would like to figure out what it is and we just talked about that fascia off a shift using sure that's not super rigorous but we're going to pretend that the sine and cosine functions start when X is zero Dawn the sine function we know we started a median and then we start going up if it was negative sign we saw comedian go down with cosine so we can pretend this is but we have to co-sign we know it starts at the maximum that goes down with negative cosine it would start to the minimum and then start heading up the fascia it's how far over you have to go do you have to go right now that's just ugly do you have to go write a certain distance or left a certain to get to our starting point in our starting point will be a median or mom or a minimum I usually like Max's minimum skills are easier to fine but but technically doesn't matter you'll have this discussion through up some of these videos and again let's just remember y equals a and what here cosine BX myself this is related to amplitude this is what see I hear I'll put a two there's information to be is related to the. She is related to the phase shift and this value of here after all if I had some function imagine a plus he wasn't there all this would just give me the Y value and now I'm adding to the Y value so this affects what they called vertical displacement mama give it another name in a sec while he'll do it right now call displacement we also called simply the shift or vertical shift hot spit too messy I put it to the side I know what I'll do so I'll say vertical that's the language you hear me you shift rather than displacement now look at the you know I'm a big fan of definitions but I'm also a big fan of understanding things rather than just memorizing so the vertical displacement of f is its average value that is its median right back to exchange colors here and do that it's average value in its average value is the median but how do we find it Well let's see what does it say it is the average of the maximum in of so if you have the average of the maximum men I will I know what average means average means add whatever you're talkin about his case we have two things we have the max come on now and we got Max and the men and then we divided by 2 I'm I'm so fine. There's nothing wrong with that amplitude what's the distance from its average value median right to the max or minimum all right I'm so fine we can do that but I want you to think about this how can I find the distance from here to here I mean if I gave you two points on a number line to find the distance between them what would do if I gave you two numbers and said what's the difference between the Mohegan do the key in there of course is always different so if we want to find the average value of the distance between them than what we would do if course is retake the Mack minus minus the minimum and / 2 now you can just brute-force memorize those but I think the picture can reinforce this we know that the amplitude is half of the distance from Max to men end to find distance you subtract so the picture I hope can inform us of this will we don't have to memorize the vertical displacement how far you go up what you've gone up not Yukon up is related to how much you moved up in the from the Max and how much you moved up the men from your basic sign extra cosine X they don't want to move it up twice yet you don't cuz you don't have both of them changing you want to go halfway between them until ex-policeman / 2 if you were going to come up with the equation for sinusoidal with keeping this stuff in mind and if you just need to keep it in mind if you want to spend the time understand why these are the way they are rather enjoy Brute Force memorizing cuz that doesn't lead understanding I just Brute Force memorizing that thinks that we're going to look at a couple question to be very closely related but that's okay alright so let's see here. Let's start with this one so we want to grab Alex graph up. Let's say if 12 sine X + 4 now one thing you're going to get in the habit of doing is identifying these five things what's the. What's the quarter. What's the amplitude what's the phase shift like the if any of course there may not be a face yet there may not be a vertical shift but obviously there's always a period of Court appeared in the amplitude all right so generally I like to start with the fascia much are the vertical shift if there is one number one that's very easy to identify it's what you've added or subtracted to your trick part and it's also very helpful to that because of the graphing aspect so let's say the vertical shift we can shoot Surplus were at one thing that's super important don't write for you need to ride right up for you can write the word up but don't just write for cuz then I don't know if you mean up or down you will not get credit right up for the reason this is important this is changed our median what are basic function like here Sinex we know the meeting is the x axis very good but now that we discovered that our vertical shift is for that means and I'll just ballpark this I'll call that for and so that means that up here at 4 I'm just put this over here y equals 4 that are medium that's where medium values and go and that's what we're going to build our amplitude off of and we can see the amplitude that's another easy I want to see if he'll let me just bump and then we'll go to a different color so the amplitude we can see is obviously 12 remember amplitude is a I'm so this was - 12 that would have to do with the shape of the graph of wind affect the amplitude and put his distance to Camping negative so if this was a question and it said negative 12 there and I asked me a tune you wrote - 12 map no good wrong answer since the amplitude is 12 and that's for then if I go 12 up and I'm not going to make this to scales don't think I have enough room so if I go well down from 4 well that would take me to - 8 if I go 12 up some for that would take us to it let's say 16 it's now I know my Max has a minimum so I taken care of Buerkle ship Napoli I usually do it in this order by the way but I really want to grab the attention of these okay so let's talk about the. Well it's just X so there's been no effect on it although technically of course there's no one there so my rule is 2 pi over the coefficient of x which is okay sir. Is 2 pi that's fine that means the quarter. Is Pi / 2 so I need to count by pi over 2 and I'm going to Ballpark it once it's one second quarter. Third quarter. Fourth quarter. hi of 253 Pi / 2 I think they almost like trigonometry he's hanging out and looking at things to Pi OK and then we haven't talked off a shift yet but we see the risk want it's just exo-k none don't don't leave it blank you got to answer the question sometimes the answer is there is no answer right now we can start plotting points and since we're talking about it sign remember well I had on the previous Facebook sign from 0 to 2 pi quote-unquote starts at the median so I will and there's no face shift tonight that means I'm right here and I'll change colors to make it stand out sending I'm going to snow work out the median a quarter. In since it's a positive sign I need to be going up to a quarter. In I'm at the maximum quarter. Later median quarter. Later minimum which is about there little bit messy McWhorter. Later I'm back to the median and then I can do a semi-decent oh and I did want to mention also let me see how things down here just the other way of time about save vertical shift you work all vertical shift the meeting is Max Plus monceaux you shortly or especially if I'm going to ask her if we can see here suppose you had the Max and Men well that we said you add so 16 + -8 / 2 well that's 8 / 2 and there's 4 but we recognize it the amplitude well that's only 16 - - 8 / 2 which is 24 / 22 and there's the 12th so whether using our how we approach it logically Leesville formulas or we have an equation than we can recognize them so let's look some supposed to unbutton this page is just a reminder that we're going to use this kind of notation. This time I put sign here and I want to remind you also with our transformational stuff since its other form x- see if she is positive we're going to go right if she is negative going to go left and I'm going to explore that on this next page so you really want to make sure you're comfortable with how I got all this information how I start with my vertical shift because there's a vertical shift cuz there was one this time and then I'd be stop the vertical shift you going to find that much more convenient when graphic so we're going to grab something that's very similar to this year going to see if you're ready if not go back and look and listen to it again I choked look at you except this time can you see I've got a phase shift so let's go through and identify all the bits and pieces so since it's just acts just one extra that. Is that, it's in the wrong spot. . is 2 pi to the quarter. Is pi over 2 amplitude is clearly 12 now the fascia pits of x minus positive private Uso it's only right Pi / 2 go shift is up for so now how do we craft listened as I said last time start by using your vertical shift so let me change that back all right Bill you're in the way. To go I know I was going to sit on my papers very good feel very good. So let's say this is for thanks for somewhere to last time the only real difference is we've got a phase shift and what they are a was 12 so let's just say you know up 12 so there's 16 down 12-0 that -8 hope that looks reasonable build off my meat find facie case 01234 quarter. Pi over 2 pi 3 pi over 2 Tupac but here's the thing folks because we have a phase shift that's where you start you are knocking started zero when there's a face if you're going to start where the phase shift takes you and since it's a positive sign I know that means I'm going to start it and meet and then go up at least if I go to the right so pi over 2 which of course take me to hear this is where I'm going to make my first point and since again it's a positive sign that means I start at the median now that's the first point I plot do not start at 0 if there is a phase shift right positive sign means I'm going up to a quarter. Later I should be at the maximum there's one quarter. Quarter. Later quarter. Later minimum and oh dear I don't have another Market well that's because I made my first four and I forgot that there was sufficient oh silly me okay no problem I'll just go back in and let not you say it's about there so to be 5 pi over 2 and now I can add my fifth point right five-point technique so this take me back to the median and now I can graph. and you know I can't stand that of course I'm just doing it for free hand you got to have a certain amount of wiggle room when your crap that's like graph paper so much nicer if I needed to graph a second. Well no big deal I would either keep adding pi over 2 to the right or pie over to the left and do four more of them and there we go we would have graft dysfunctional attacking. who wanted shop again you must I expect to see this see these parts we take advantage of vertical shift and phase shift the quarter. Then we plot are five point then we go in and make a reasonable sketch and the next video start looking at data and using the data to come up with the equation rather than already having the equation and plotting it but it's me based still on recognizing so I could have left it there on recognizing these important values and how we can find them John what we understand about I'll take this picture house we have the maximum in we have access to the amplitude if we have the maximum in we have access to Vertical shift and other information but you'll see as we go through it all right I was at work for you if not prepare your questions we will talk about them tomorrow take care

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1-30 AIM3 5.7 Sinusoidal Functions- Amplitude, Period, Phase Shift

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