 Question from the homework. Page 289-ish. Yo, 10. Love to. Okay? Often what they're going to do is they're going to give you part of a graph. So can you all look at number 10 and I think I like this question. Okay? You want to have in your mind, Victoria, this bad boy, there's sign and this one I always have a tougher time drawing. This bad boy, there's cosine. Okay? This is what I yelled at you last day. I said this is what we need to know because when I glance at this, I think they've given me a fraction of one trig wave, not a whole wave. And I think starting up high, I think it's cosine. So the first thing that I would be saying is, even if they hadn't said cos right here, I'd be going, it's cos. What's my amplitude? So I would say this for starters, 5 cos. Now I want to find B. Remember B is 2 pi over the period. And what I need to do is figure out what the period is. What fraction of a cosine wave have they given me here? I think half a wave, that much. See it? And that's pi by 9. Half the wave is pi by 9. How long is the whole wave? Don't overcomplicate it. Half the wave is 1 pi by 9. How long is the whole wave? 2 pi by 9. Is that okay? Yeah? So I'm going to say this. B is 2 pi over 2 pi by 9. How do I divide by a fraction? Flip it and multiply. Yeah. Dividing is not times. It's flipping and multiplying. So this is going to end up being 2 pi times 9 over 2 pi, which is quite convenient. The 2 pi's cancel. And in fact, B ends up being 9. I guess that's what it should say in the back. Okay? Similar for this one, by the way, for number 9, what fraction of the sine wave is this right here? I think half the wave. So I can get the period. Amplitude I can certainly get. Okay? Any others? Yep. 12 B. Any before 12? Earlier stuff went good. Did I sign 12 B? Yeah, I did. Okay. Can you all get out your calculators, please? And we want to make sure we're in radians. And you got 12 A, okay, yes? You said it was going to be equal to y equals 8 sine of. The period is 3. The period is 12 T. Now, remember I said that B is 2 pi over the period. What's the period here? If you're really lazy and you really want to do this quickly, you can just put a 2 pi over 12 there, because we're going to go to our calculator. Now, what is 2 pi over 12 in lowest terms? Okay. On your test, if this is multiple choice, the answer to pick from would be pi by 6. But I'm less interested in A. For part B then, here's what I'm going to do. I'm going to go to my calculator and I'm going to put a 4 in for T. I meant as why I said I'd get my calculator out. I'm going to go like this. 8 sine. Oh, it opened the bracket for me quite conveniently. 2 pi times T4 and then divided by 12 because there's a 12 in the bottom. That actually tells you how high the tide is. How high the sea level is. 6.9 meters. That's it. The other thing they can do is instead of giving you an X value, because this is sitting where our X normally is, they could give you a Y value. We'll talk about how you'll handle that. I think we'll be talking about that today if there wasn't one in the homework. Any others? So last day, I gave you kind of my little trick for sketching trig functions. I said I actually just draw it with the axis with no numbers and then I just adjust the scale, the numbers. Unfortunately, that starts to break down as a trick in today's lesson. Can you please turn to lesson 9, graphing trig functions or transformations of trig functions part 2. All we looked at yesterday was expansions and reflections. Expansions compressions. Stretches and shrinks and flips. And I said to you, by the way, when it comes to trig, we don't do horizontal flips very often at all and you'll see why in a while. In this lesson, we're going to consider graphs of sine and cosine that look like this. In particular, we're going to start to have slides, but we have to give them special names. So in the first part, we're going to concentrate on C and D. Describe how this graph here compares with this graph. How has this graph been moved? Troy, 30 degrees to the right. How has this, don't write that down. How has this graph been moved? Is the two inside next to the X? How would I show that it was inside next to the X if it was? Okay, we're going to put, we're going to be a little fussy, we have to be in brackets. It's not, so it's vertical. Two what? Two up. Okay. What about here? 60 degrees left, one down. What about here? This one first. 45 degrees right and not 45 degrees, but 45 up because it's next to the Y and everything's backwards. We're going to give these different names, which is why I said not to write that down. We call a horizontal slide a phase shift. We would actually say this has a phase shift of 30 degrees. Skip the second one for a second. The third one has a phase shift of negative 60 degrees or you could say 60 degrees to the left. I'll take either or. Or you could say 30 degrees to the right, but we're calling it a phase shift. This has a phase shift of 45 degrees to the right, so positive 45 degrees. Having said that, we're going to get tired of writing the words phase shift, I'm going to use PS as my abbreviation, if that's okay. What did you say this one was? Two up. Okay. When you move a trig function, a periodic function up or down, we call that a vertical displacement. This has a vertical displacement. I don't feel like writing vertical displacement. D period, D period. Okay. D period, D period of negative one. Or you could say justine one down. I'll take both. Vertical displacement of negative 45. Positive 45. Okay. So let's look at this little chart here. Let's do the horizontal phase shifts first. What's the phase shift here? It's a trick question. None as a number, please. Zero. None as a number is zero, right? What's the phase shift here? Pi by four. What that does is it takes your letter S, the sine curve, and moves it pi by four to the right. What's the phase shift here? Zero. What's the phase shift here? Negative three pi by two. And now we're going to try and generalize this. What's the phase shift here? See, what's the phase shift here? Still see. What you're going to have to watch out for again is an alarm bell if this B is not factored out. It's C as long as you have this lovely little set of brackets right there. If the B is not outside the brackets, we'll have to do our little alarm bell. What's the vertical displacement here? Zero. How about here? Zero. How about here? Five. How about here? Negative pi. How about here? D and D. That one I kind of like because I think vertical displacement D works for me. A for amplitude works just great for me. B for period. Well, B and the letter P kind of rhyme. Start. C for phase shift? I got nothing, but it's phase shift. Okay. It says, would you expect something similar for cosine? Yep. What about for tangent? Ugly cousin? Turn the page. So this page here is one of those lovely bookmark posted star asterisk pages. This is a summary of all of the graphing of sine and cosine. It says this, changing the parameter C on sine and cosine as long as it's all factored results in a horizontal phase shift to the right. If C is positive to the left, if C is negative. Changing the parameter D will do the following. A vertical displacement up if D is positive. A vertical displacement down if D is negative. You can also get the vertical displacement from a graph using this equation. Now this doesn't, this looks kind of confusing. You'll see it does make sense once you have a picture in front of you. Ah, here's the money page, right here. That's the one that summarizes all of our graphing. Let's see if we can understand all of this. First of all, for sine and cosine of Y equals A, sine or cosine of B bracket X minus C plus D, the amplitude we said highest minus lowest divided by 2. How far apart are they? Divide that by 2. That's your amplitude. Period. 360 over the absolute value of B or 2 pi over the absolute value of B. Why absolute value of B? Because we said that period is never what? Negative. Oh and amplitude. Why did I say absolute value of A? Because amplitude is never what? Negative. The phase shift is C and it's the same stuff as last unit. If you have a minus and then the positive number right there it's to the right. If you have a plus and a positive number right there it's to the left. The vertical displacement D up or down and you can find D by doing that. We'll talk about how that works in a second. We're gonna add one more thing by the way over here. If B, sorry, if the period is 2 pi over B what's B? 2 pi over the period. I always write that in because if they want me to find the equation they'll tell me the period and I need to find B. If they give me the equation they'll tell me B. I don't need to find the period. And both of those are totally fair game. Tangent, ugly cousin. Amplitude, all reels. A technically represents a vertical expansion or a vertical compression but we really aren't gonna deal with it all that much. Period is not 360 or 2 pi over B. It's 180 or pi over B. Period is pi. Yeah okay you can have a phase shift and a vertical displacement but we really don't deal with that much either. We will deal with this a bit because this here and a period change will move the asymptotes and the domain around instead of the gaps being at 90 pi by 2. If you slide it sideways the gaps move with it and that to me will be a good question. I might say hey where are the new asymptotes? Or what's the new domain? Remember for tangent the domain was what X couldn't be and the asymptotes were take the domain and instead of not equal to equal to or that was where the asymptotes were. Let's try some of these but this is a good page to you know keep track of. Example 3. Consider equations of that form there where A and B are 1 so we're not gonna add an amplitude or a period change yet. Write an equation which represents a cosine function having a horizontal phase shift 75 degrees to the right. Well Y equals what's the amplitude here? Oh they told me 1. I'm not gonna bother writing it any. What trig function do they want me to write? Let's write the word cos. What's B here? 1 so what's the period not 1? 2 pi okay bracket 75 degrees right X minus 75 degrees okay. And by the way nice review of transformations as well. A sine function having a horizontal phase shift of 3 pi by 5 radians okay. Y equals we said amplitude is 1 sine bracket. 3 pi by 5 radians left X plus 3 pi by 5 and then a vertical displacement 4 units up plus 4. So a good multiple choice question would be something like number 4 where I give you an equation and I ask you to tell me probably two of the four. I'll say tell me the phase shift and the vertical displacement or the phase shift and the period. I can make lots of nice wrong answers because there's four different numbers in there for you to pick from. Good little multiple choice question. So let's list everything here. It wants us to find the amplitude, the period, the phase shift and the vertical displacement. What's the amplitude? Sorry? 2. What's the period? It's not 3. 2 pi over 3 remember period is 2 pi over B, B is 2 pi over the period. So 2 pi over 3 and if that reduced I would but I'm good with that. What's the phase shift? You can either say negative pi or you can say pi left. How do I know it's in radians? Not just because of the pi if I could be degrees how would I show it was pi degrees? There's no degrees symbol okay. You'll notice I was very fussy here I put this I put the degrees on the 75. Vertical displacement negative 4 or 4 down B amplitude period phase shift vertical displacement. What's the amplitude? 2 over 3 or negative 2 over 3? 2 over 3 almost certainly if I gave you a question for amplitude on the test I'll make it negative just to see if who will pick the negative one and I'll have that as an answer to pick from. What's the period? Well I got to do some arithmetic. The period is going to be 2 pi over a quarter. How do I divide by a fraction? 2 pi times 4 over 1 and now it's top times top bottom times bottom. I guess the period is 8 pi which makes sense to me because what we would have said last unit was horizontal expansion by 4 so instead of 2 pi 8 pi 4 times around the circle. Phase shift pi by 12 you can write to the right if you want to. Vertical displacement positive 3 or you could say 3 up okay. Now before we turn the page or move let's go back to this one here. What's my vertical displacement 4 down? So remember the cosine graph which started up high you would move it the whole thing 4 down. What's my amplitude 2 if I was sketching it from once I moved everything 4 down I'm not going to go one up one down one up one down what's my amplitude 2 up 2 down 2 up 2 down I would keep that in my mind as well. Phase shift I would move everything 2 pi by 3 to the right so instead of starting right on the y-axis up high I'd be starting 2 pi by 3 over. I'd have to do something with a period. We're going to talk about how you sketch these but we're going to do that mostly later on a bit this class mostly next class. Alright example 5 says find the amplitude the period the phase shift and the vertical displacement says y equals 2 sine bracket 3x plus pi minus 4. Alarm bell should go off. Matt why would alarm bell go off here not factored this I like I like this question you know what I love this question I love this question I guarantee on your test I'm going to give you sine or cosine or tangent unfactored and I'm going to say hey what's the phase shift and the first answer I'll put is negative pi. Phase shift is not negative pi I need to factor this 3 out I need to rewrite this which is why I left the space as 2 sine bracket factor the 3 out bracket bracket bracket minus 4. What the heck is going to go in here well every when I factor a 3 out of a 3x I'll get an x when I factor a positive out of a positive it'll stay positive. When you're factoring you're dividing pi over 3 so what's my amplitude 2 what's my period 2 pi over 3 2 pi over b what's my phase shift negative pi over 3 or pi over 3 to the left but certainly not pi vertical displacement negative 4. Is there an alarm bell in B? Do I need to yell? Try B on your own. Fractions? Don't panic. Are you becoming mediocre at fractions now? I like it. I got those. I'm willing to bet that at least five or more of you are just bursting to ask me where the heck that came from so if you're not sure you can ask I don't mind explaining it maybe everyone yes okay when you're factoring you're dividing so when I factor a 2 out of here and out of here it's the same as dividing by 2 which is why there's no more 2 and dividing by 2 dividing by 2 means putting an extra 2 in the denominator there's already a 2 down there you'll end up with a 4 down there so this has been a 3 6 it's been a 7 14 okay see we are becoming mediocre at fractions turn the page tougher much tougher much more challenging is writing the equation if they give you the graph if I give you the equation and say tell me about the graph all you have to do is memorize what the letters stand for if I give you the graph and say tell me the equation you really got to follow along and so I'm going to slow down I'm going to say if you're tired I did a yell a little while ago wake up think up yes area you too okay so here's what it says graphs a through d represent the same trig function and they do somebody I think it was in this class last day said something they said mr. duick I notice sign and cosine look the same you could slide sign over and it looks just like cosine as it turns out any sine graph I can write that as a cosine equation if I want to any cosine graph I can write that as a sine equation mathematically if I want to and not only that any positive sine graph I can write that as a negative sine graph if I want to and I mean negative cosine graph I can write it as a positive cosine graph any graph has at least four equations any trig function you can write it at least four different ways and that's without going all co-terminal like that's without going phase shift all code there's an infinite number of answers so this graph here we're going to write four different equations and we're going to use the instructions to kind of tell us our starting point says this it wants us to write this as a sine graph where a is positive so it wants us to write this as a positive sine graph now a positive sine graph over in the little margin right up here do this that's a positive sine graph except what I'm going to do is I'm going to erase that line there because now we're sliding graphs back and forth what I'm going to say is a positive sine graph is where this graph starts in the middle and goes up so I'm going to look along here and now it's in the middle in terms of its height right there but what direction is it heading down the first point that it's in the middle and starts going up is right there I'm going to put a little red dot right there that's where I'm going to sort of start my stopwatch as it were just that's where I'm going to start to count this graph from I could start all over the place which is why you can get so many different equations but since they wanted positive sine I know sine looks like this can you see it okay write it in this form y equals what's my amplitude okay total distance is 8 half of that for sine the period is 2 pi so B is 1 they didn't put a B here here's what I want you to notice if I want to draw this as if I want to write this as a positive sine graph what's my phase shift I gotta think how many squares make up pi count 6 so what's each square worth pi by 6 how many squares over to the right has this graph been moved x minus 2 pi by 6 and I'll draw a little arrow I'd write that as pi by 3 but just in our notes so we know if the heck we did we count that's that graph as a positive sine function what if they want me to write it in part B now in part B they still want me to write it as sine but Alex they want a to be negative they want me to write this is a negative sine function again I'm gonna do a little sketch negative sine looks like that and again I'm gonna get rid of this axis just to remind myself that's the shape that I'm looking for so I'm gonna look around and find where this graph is middle-high but heading downwards and I think the first point is that guy yeah y equals amplitude for oh heading downwards negative sine bracket x what's my phase shift now how many squares for left or right plus 4 pi by 6 is because that's what each square is worth by the way what I write 4 pi by 6 what would I write if this is on a test if I gave this to you as a multiple choice 2 pi by 3 I'll do a little arrow 2 pi by 3 okay Dylan here's what this really means if you graph that on your graphing calculator is y1 and you graph this as y2 only one graph would show up there same graph oh by the way back to here if I had wanted to I could have actually started counting right there is that also where it's in the middle and goes up so you could have Jesse phase shifted one two three four a bunch of more squares over let's decide from now on we're gonna try and keep our phase shift to the smallest easiest one I had a student about five years ago who delighted in phase shifting about 5,000 off the page every time and making me double-check that his mouth was right was Willie his math was right and after about three quizzes it was stopped being funny and just started being annoying because I had to go do all this arithmetic you can phase shift back and forth just so all of our answers look the same closest phase shift to zero please not only can I write this as positive sign not only can I write this as negative sign I can write this as a positive cosine equation write this as a cosine where a is positive now again I'm gonna do my little sketch positive cosine looks like that but I'm gonna get rid of this I'm looking for where this graph is up high it's up high in two locations here or here this is the closest one to zero isn't it I think justine I'm gonna put a big red dot there that's where I'm gonna start counting this graph from if I do that as my reference point it's gonna be y equals positive what's my amplitude still for cosine what's my period still to pie so what's be one bracket X what's my phase shift this time Tyler left or right so what am I gonna put here minus five pi by six could also have done plus seven pi by six and that would start right here same graph this is the weird thing about periodic functions Jesse's there's an infinite number of equations that can be drawn because they repeat themselves which makes it either should be an infinite number of equations that you can do because you could just keep moving over until it starts to repeat again and you kind of start your stopwatch right there so to speak hey I can write this as a negative cosine graph as well now a negative cosine graph instead of starting up high we'll start where yeah it's gonna look sort of like that so I'm gonna find somewhere you know what I think I'm gonna pick this one it's almost no phase shift at all and in fact what I'll teach you is whenever possible if you notice that right on the y-axis you're either up high or in the middle or down low use a phase shift of zero it's way easier and adjust your positive or negative sign or cosine to it this is gonna be y equals negative coasts amplitude still for cosine x plus pi by six all the same graph for different equations which ones right they all are usually I'll give you a hint I'll say write it as a sine function or write it as a cosine function not always usually one will be way easier I'll be honest if I had a choice I might have picked D because it's a nice small phase shift I find the mistake kids make the most is they count squares wrong because they're in a rush that's me why on the page okay let's put the whole package together ready here is a trig function I can write this as sine or as cosine or as negative sine or as negative cosine it's all gonna depend if I wanted to write it as sine where I start my stopwatch or as cosine or as negative sine or as negative coast but we're gonna have to have some things in common what's the amplitude let's see how high does this graph go one how low negative five total distance six amplitude three what's the period I find the easiest way to count most of these is from either a peak to a peak or a trough to a trough yes those are the actual fancy words how many squares apart is that count carefully 12 is it 12 yeah what's one square worth look at your scale how'd you get that how many squares make up pi three of them so one square is pi by three so if I hear you the period is 12 pi by threes oh this one I will tidy up what is 12 pi by three and lowest terms 4 pi okay I don't want the period right next to the period in the margin here what's B because that's what it's gonna show up in our equation what's B what was the expression for B 2 pi over okay let's write that down 2 pi over the period which is gonna be 2 pi over 4 pi oh what's B I'm gonna write down here B equals one half that's what's gonna show up in my way now we're gonna come back to the phase shift in a second what's my vertical displacement now the vertical displacement is where the center vertically of this graph is how high are we one how low are we negative five find halfway between them now while you're thinking that look up that's what this meant find halfway between the top and the bottom that's how you find actually there's an easier way there's an easier way did we already get the amplitude Alex then go three down from there or go three up from there but that's your middle what is the vertical displacement not to negative to you know what all of you right now put a little dotted line right there I always do that the vertical displacement is negative to and the reason that's so important is that's where the y-axis the x-axis has been moved to that's the middle of my graph that's where sign is gonna begin sign starts in the middle and positive sign starts in the middle and goes up that's the smallest possible left right phase shift for positive sign what is the phase shift negative pi by three or pi by three to the left now I can write the equation but I need to write big so I can't fit it in here I'm gonna write it right here y equals three amplitude which trig function did we commit to here sign be a half bracket x phase shift is pi by three to the left or to the right plus pi by three close bracket close bracket minus two that's that equation B sorry nope B is positive periods always positive B is gonna be positive that's why I said we're not gonna deal with the horizontal reflection very much at all hey what if I wanted to write it as a negative sign graph I'm gonna change colors I'm gonna go blue negative sign still starts in the middle but instead of heading up which way is ahead you know what right there folks the equation's gonna be nearly identical look up just watch for a second the y is not gonna change the amplitude is not gonna change the sign is not gonna change the B because the period is not gonna change and the vertical displacement is not gonna change what is gonna change two things negative in my phase shift how many squares to the right how many squares to the right always counting from the y-axis right from zero one two three four to the right minus five pi by threes sorry what is that a question see is this the same equation same is the same graph as the previous one thread it is cosine everything but the phase shift is gonna be the same amplitude three period four pi B one half vertical displacement negative two now positive cosine starts where up high remember the little sketch that we did right there yes you could start over here but it wants us to do the smallest phase shift you could start over it wants us to do the smallest phase shift how many squares to pi by threes so let's write it as a positive cosine y equals equals what three cosine B is a half bracket x we're to the right minus two pi by three minus two what if I want to write this in part D as a negative cosine graph well okay y equals negative three coasts that's not gonna change I'll put a negative in front of the three the one half the period is not gonna change the x is not gonna change the minus two way over here is not gonna change where would negative cosine start bottom right there yes Miguel or right there but that's further how many squares Miguel left or right plus four pi by threes if you were to graph all four of those equations very very carefully you'd get one graph okay almost done on the page says consider sine and cosine and there's a b c and d a b c and d which of these parameters a b c or d if you change them affect the domain well what is the domain of sine and cosine all reels hey all you guys like this because you need to stretch anyways everybody like this don't hit someone and stretch while you're at it feels good here's my domain if you change the amplitude is that gonna change your domain we do this is that gonna change the domain at all no what about what's the next one be if I change the period now that might make it happen more often but will that still make it all reels even if I squish it or stretch it okay so that's not gonna change the domain what about the phase shift if I move it left and right will that change the domain no what about the vertical displacement if I move it up and down will that change the domain you know what the domain of sine and cosine always is all reels I probably won't ask you that boring range what's the range normally between sine and cosine how low how high okay so everybody all of you put one hand like this there's negative one there's positive one there's your range okay if I change the amplitude will that change the range ah okay let's write that down hey okay back to our normal range if I change the period that's horizontal isn't it I don't think that's gonna change my range okay what about if I change the phase shift if I move this back and forth is that changing the distance between them no oh D vertical displacement if I move this whole thing up or down will that change the range yeah because instead of between negative one and positive one the distance will be the same but you might be between negative two and zero okay so D does three is more obvious which one changes the amplitude a which one changes the period me which one changes the zeros the what the x intercepts okay now I got to think a little bit more here when in doubt because it takes all of one second okay if I change the amplitude will that change the zeros what about if I change the period by making this not as long or not as short will that move these three points yeah I think the period will so I'm gonna add say B what about the phase shift if I move that whole graph back and forth yep what if I move that thing up or down yeah in fact I can move it so high that it has no zeros um oh what if I moved it so high that it had no zeros but then put a huge amplitude in front of it could it get the zeros again turns out actually the amplitude if there's a phase shift can also change the zeros no phase shift amplitude won't do a thing sorry I said patient vertical displacement vertical displacement if there's a vertical displacement well then the amplitude can also affect the zeros because you could be so high that you have none put an amplitude of 10,000 in front of it suddenly you do have a zero it'll reach down that far state the highest and lowest values of the function in terms of a b c and d so they want high values that's x that's x I'm not even gonna look at those they're talking about vertical here what they're really saying is tell me the range okay for sign and for cosine here's the range the first thing you do is you move it up or down the vertical displacement so let's pretend the vertical displacement boom to there d high first and then from there you go amplitude down and amplitude up you say that again say that again to find the range Dylan algebraically and I probably might give you an algebraic one with no numbers there's your vertical displacement that's the center of your graph and from there you go up amplitude and you go down amplitude the lowest you get is start d high and go down that far less than or equal to y less than or equal to start d high and go up from there that's your range the algebraic one is tougher I'm just thinking right now what I should have done is skip b gone to c first and then d b would have made more sense but let's look at c it wants us to find the range let's see if we can do this in our head what's the vertical displacement negative four so the whole graph has moved four down look up four down we're right here negative four what's my amplitude that means from negative four I'm gonna go three up and three down I go three down from negative four what's my lowest point gonna be negative seven three up from negative four and now that we did that maybe that makes a bit more algebraic sense through a lot at you today okay hopefully it ties in sort of to the transformation stuff that we've learned already except stacy because these graphs are periodic phase shift vertical displacement and the fact that there's multiple answers for an equation because they repeat themselves number one is good number one good good good because that's not in brackets alarm bell good alarm bell okay and then we're gonna practice getting the equations so number three or Victoria this is very similar to the one that you asked me this is only a fraction of a wave okay it wants you to graph it as a positive cosine where does positive cosine start you'll have to figure out how long one whole wave is but again Victoria I think they gave you a half wave because I think the whole cosine graph looks like that so I think half the period double it there's your whole period and now you can figure out B okay five is good which graph haven't I talked about ugly cousin I'm gonna assign six with the expectation I'll probably have to talk about it but see you can get it okay remember what I taught you about tangent seven I'm gonna say how about a and D eight nine no ten there you go graphing trig functions