 Mark your own. Here we go. You guys okay back there? What's the matter? There's a leftover fingernail on the desk. Would you like to grab a paper towel from behind you and gently... Okay. You realize that's going on the internet for posterity and we have no idea how it gets there. Teacher-scholars for generations will wonder at my classroom. I'm gonna guess somebody was stressed and was biting their nails. Maybe a calculus student. Ready? Here's your trigger phrase that you know you're not supposed to be using your calculator to go shift sign or shift coast or shift tan. You're supposed to use special triangles and the unit circle and all that. Solve. Do I know the angle? It's meant to be really obvious questions, Benz. Do I know the angle? No. Because I'm gonna give you both questions on your cast and YouTube to distinguish between the two of them. Sometimes you're going to be given the angle and I want the root 3 over 2 answer or the 1 over root 2 negative or positive quadrant. Sometimes I'm gonna tell you the answer and you have to rationalize what two angles it came from. So we start out with our good old cast rule. Which trig function did they give me, Benz? Negative or positive? Negative or positive? Gotta be there and there. Then I say to myself, self, I need the reference angle. It would be helpful if I could find the reference angle. I need a triangle with a root 3 and a 2 in it. I think it's the 1, 2, root 3 triangle where that's 90 degrees or pi by 2 radians. Which of these two angles has a sign of root 3 over 2? The bottom one or the top one? Justin. Ah, how big is this top one? Which means that this angle right here is pi by 3 and this angle right here is pi by 3. Oh, and this is 3 pi by 3 and this is 6 pi by 3. So I'm pretty sure my first answer, theta 1 is 4 pi by 3, yes. And theta 2 is 5 pi by 3. Late comers, don't think I'm gonna forget. I'll get you later. Two marks each. In all honesty, on your test, I'll probably give you a half mark if you get the correct quadrant, a half mark for getting the correct triangle and then a half mark for the angle and a half mark for the angle. But here, I'll just go one mark for each angle because it's easier to mark. But as long as you got them both right, you're gonna get full marks. By the way, you could have, remember that big ugly circle that I said you could memorize, remember that diagram? So you could have just memorized that and just written down the answers that I have to take it. But that's a lot of memorizing. B. Once again, we do our cast rule. This looks an awful lot like tangent is negative, which means we're there and there. Oh, I need a triangle with a root 3 in it. Why? I'm pretty sure that's a job for the 1, 2, root 3 triangle. By the way, sandally, root 3 over what? What's underneath there? It's invisible. Yes, really, I'm looking for a triangle with a root 3 and a 1 in it. I got a clue in. Let's see. Oh, I want a tangent of... I think this guy has a tangent of opposite over adjacent of root 3. And once again, I think my reference angle happens to be pi by 3. Oh, and once again, this is 3 pi by 3 and 6 pi by 3. I think theta 1 is 2 pi by 3. Theta 2 is... Whoops, Mr. Duke. Not 4 pi by 3. 5 pi by 3. You're lucky, late person number 4. I'm in a grumpy mood. I'll get the 3 people in a second. The 4 people in a second. Did I mention how I'd mark this one? I probably did this one the same way. Okay, now they gave me reciprocals. Now, if this was not an exact value question, first thing I would do would be flip both sides. In other words, if this was like 2 over 5, do I have a triangle with a 2 over 5 in it? No, I would flip both sides and I would say, I don't deal with secant. I'd deal with cosine as 5 over 2. But because it's an exact value, I know for a while I said, how do I deal with secant and cotangent? We actually can now for exact value questions. I can still start out by drawing the cast rule. Secant goes with cosine. Cosine is negative here and here. Which triangle? Again, it's the 1, 2, root 3 triangle, because I got a 2 and a root 3 in it. Mr. Newark, aren't they always that one though? Sometimes they'll be a root 2, and you know it's the 1, 1, root 2. You only got 2 triangles to pick from. So, Ellen, it's going to be pretty repetitive. Sorry. Oh, that's 90 degrees right there. I want a secant of 2 over root 3. Secant goes with cosine. Cosine is adjacent over hypotenuse. I want a hypotenuse over adjacent. Oh, it's this guy here. Pi by 6. This angle is pi by 6. This angle is pi by 6. And right there is 6 pi by 6. So my first theta 1 is yonk. 5 pi by 6. And my second theta 2 is yonk. 7 pi by 6, yeah? B, cotangent. Oh, that goes with tangent. That one I can remember easily, Brett. Tangent is negative, cotangent is negative, here and here again. Same quadrants. I got a root 3 again, so it's going to be the 1, 2, root 3 triangle again. Mr. Dewey, how come you keep drawing the 1, 2, root 3 triangle when you got it right there? Is it worth me practicing memorizing it by drawing each time so that I don't stupidly forget it on a test? Yeah, I think the one second it takes is worth it. Oh, cotangent. Tangent is opposite over adjacent, so cotangent is adjacent over opposite. I need an adjacent over opposite of root 3 over 1. Oh, here again. Pi by 6. I think theta 1 is 5 pi by 6. Theta 2 is 11 pi by 6. And you'll notice I took a sneaky shortcut. I didn't even write the 6 pi by 6 and the 12 pi by 6. I think I'm good enough now that I can just remember that those are there and that can count a little bit. Number 3. Emily says exact values. That means special triangles and all that. Can you read A to me, Emily? Ah, the alarm bell goes off, right? Here, now I hear. Oh, and here. Oh, and this will be a square root both. So these are both a little bit tricky. Now, cosecant is undefined. Undefined is that negative or positive? It doesn't help me. Castrol really won't get me anywhere. Now I have to fall back on the very definition of what cosecant was. Cosecant goes with sine. Cosecant was... because sine was y over r. What they're really saying is r over y is undefined. When will this be undefined? When y is 0. Where is y 0 on my little unit circle analogy? Am I 0 high right here? No, no. 0 high right there and right there. There are two answers. The first answer is 0 radians. The second answer is pi radians. Now, the workbook that we're using always has an or equal to right here and that would suggest that 2 pi would be a third answer. And if you put that on the quiz, I won't take marks off, but that's actually not in the domain. I always think the workbook is a bit silly and one extra angle. I think just go around the circle once and quit. Sine theta equals negative 1. The alarm bell would be going off because although sine is negative, which suggests it's in one of two quadrants, it's going to be right on one of my four arms. It's not going to be in an actual quadrant. There are two ways I can do this one. The first one is to say to myself, hmm, this is the unit circle. This is y over r, sine is 1 in my unit circle. Sine is just my y coordinate. And I can do a lovely little sketch of a circle where the radius is 1. And then I can say to myself, self, if the radius is 1 and y is 1, when am I 1 high? Right there. And I can get the only answer, which is pi by 2. That's one way to do it. Yeah, am I wrong? Oh, negative 1, Mr. Dewick. Negative 1, Mr. Dewick. And I can get my only answer, which is, of course, as I said, 3 pi by 2. There's a second way you can handle these ones. You could have just done this. What's that a sketch of? How high? How high? What do I put here? 2 pi pi pi by 2 3 pi by 2. And you may notice that I am negative 1 high right at 3 pi by 2. You can get your negative ones and your zero sandaly either from the unit circle or from the graph. So now you got two methods. Secant is negative 1. That's when r over x, because secant goes with cosine is negative 1. That means that the radius is equal to my x coordinate but negative. If I visualize my unit circle over here, or my circle over here, when is the radius the same as the x coordinate but negative? I think only right there. Because this distance is x, this distance is also the radius, but this distance is negative x. The only theta that works is pi. d quadratic square root both sides. And I'll get cosecant theta equals plus or minus 2 over root 3. Oh, I'm going to be in the 1, 2, root 3 triangle. It's plus or minus. I'm still going to do the cast rule because I want to get into my regular groove, my regular habit. cosecant goes with sine. Sine is positive there and there. It's negative there and there. That's going to be four answers. The triangle is going to be the 1, 2, root 3 triangle. cosecant goes with sine which is opposite over hypotenuse. So I want a hypotenuse over opposite that has 2 over root 3. Justin, I'm pretty sure that's this guy up here. I'm pretty sure my reference angle is pi by 3. And that's 3 pi by 3 and that's 6 pi by 3. So it looks like for a half mark theta 1 is going to be pi by 3. For other half mark theta 2 is going to be 2 pi by 3. For yet another half mark theta 3 is going to be 4 pi by 3. And if you want all of the half marks theta 4 ends up being 5 pi by 3. Am I correct? I hope, yeah. This will be the bulk of the written section of your test. The bulk of the written section of your test will be probably 3 or 4 trig equations to solve. I guarantee you'll have either a 1 or a 0 or an undefined. I guarantee there will be a reciprocal. There will be one quadratic up. And then the very last question on your test and we only got by the way one more lesson so we're nearly at. Your test is not going to be next week it's going to be towards the end of the week after probably. I'm going to do a domain change. I'm going to do a domain change. Which is actually far easier than you realize. You see the difference here? Eric, they haven't gone from 0 to 2 pi. Eric, they haven't gone from 0 to 2 pi. See the difference here? Haven't gone from 0 to 2 pi. No worries. I'll just go with what I know. I'll pretend I'm going from 0 to 2 pi first. I'm going to start out same old, same old. Last one. Sign is positive here and here. I want the 1, 2, root 3 triangle. I think that angle has a sign of 1 half. So I think my reference angle is pi by 6, pi by 6. So the two answers that I get right now. Theta 1 pi by 6 Theta 2 pi by 6. Now let's double check. Is this number here between negative pi and positive pi? Is this number here between negative pi and positive pi? Oh, then I'm done. Wait a minute. Let's double check. Maybe there's some negative answers that would fit. What I could do is I could check is there a theta 3? Now if there is, it would be minus 2 pi because that's how we found coterminal angles except that would be a dumb way to write it. Cara, what's my denominator? 5 take away 12 is negative 7 pi by 6. Is that between negative pi and positive? Okay, you know what? That's not an extra angle to find. I'll do the same thing with this one just to double check. I'll say let's see. Pi by 6, subtract 12 pi by 6. Is negative 11 pi by 6? Is that between negative pi and positive pi? Oh, you know what? Even though they changed the domain on this one, it didn't actually change my answers at all. Which, by the way, was a mistake on my part when I made up this question because I meant it to change the answers. I should have made sign negative. Oh well. I will next year. Oh, no I won't. I'm not teaching this course again. See, Kant is undefined between negative pi by 2 and ignore temporary domain change. See, Kant undefined. Alarm bell, Emily. I'll be going. Okay. See, Kant. R over X is undefined. Which really means I think that X is zero. When is X zero here or here? So originally I would say theta 1 is yonk pi by 2 and theta 2 is yonk 3 pi by 2 is pi by 2 in this domain. Yeah, is 3 pi by 2 in this domain. Ooh, not quite. Sneaky people then. So I need to find a co-terminal angle. I would either add or subtract 2pi except since I'm too big I'm going to subtract 2pi except that's a silly way to do it. Eric, what's my denominator here? Why don't I do myself a favor and actually just subtract and I'll get 3 take away 4. I'll get negative 1 pi by 2. Is that in my domain? Oh yeah, it is or equal to there. Ah, Alex, how do I handle a domain change? I don't. I ignore it. At the very end I just look at my answers and I add or subtract if I'm too big subtract 2pi if I'm too small until I end up in the domain that I want and I toss any answers that aren't in the domain. That's it. One mark for each of those. One mark for each of those. Give yourself a lovely little score on your quiz out of count to Holy smokes 20. And then pass them inwards please. Hey, can you open your books please to the homework from last day? Finally, did all of our transformations of trig functions and you'll notice I haven't done much on tangent. Haven't looked at all at secant, cosecant, and cotangent. We will, but all of those behave strangely. You gonna be okay? You sure? I'd be happy to scare you to try and get those, but you know. Page 296 Any questions from the homework that you would like me to go over? Yo! 1f Did you get the amplitude okay of 7, I hope? Did you get the vertical displacement okay of negative 1, I hope? Because those are the easy ones. Oh, period. Now the period is going to be 2pi over a quarter. It's 2pi over b. Did you get the period of 8pi? Ah, sorry. Not 2pi, Mr. Dewick. It's in degrees, good gosh. 360 divided by a quarter. How do I buy a fraction? Flip it and multiply. It's going to be 360 times 4, 720 times 4. Do you get 1,440 degrees as your period? Oh. Because what, you put a 2pi in there? How do you divide by a fraction? Flip it and multiply, right? It's degrees. Oh, and you're going to have to, because this is an alarm bell as well, you're going to have to factor out that one quarter. You're going to have to rewrite this as the sign of one quarter bracket. You'll get an x, you'll get a plus. When you factor out a one quarter here, it's the same as dividing by one quarter. It's actually a phase shift of 80 degrees left. It's 20 over a quarter. Those might be on internet posterity too. Yeah. Number four? Excellent. First of all, they told me which trig function they want me to graph. Cos. Oh, positive cos. Which means where does positive cos start up high? I'm not going to say one and negative one anymore because often the amplitude is not going to be one. I'm just going to say positive cos up high, negative cos at the bottom. Sign in the middle, positive heading upwards. Sign in the middle, negative heading downwards. Positive cos, you know what? Right there, baby. Let's fill in what we can. First of all, what's the amplitude? Well, how high does this graph go, Vlad? No, how high does this graph go? 9. How low? Negative 3. Total distance was what you said, 12 amplitude, half of that 6. It goes 6 up, 6 down, 6 up, 6 down, 6 up, 6 down. Cos. What did I say the amplitude was? So, 6 down from the top will give me the middle, or 6 up from the bottom will give me the middle, which is my vertical displacement. I always do that next, 1, 2, 3, 4, 5, 6. My vertical displacement is that high, right there. My vertical displacement is positive 3. I always do those two first because I find those are by far the oh, and you know what? I can even tell you my phase shift. 1, 2, 3, 4, 5 squares to the right. X minus, what's the square worth? 5 pi by 4s. The tricky part is going to be what goes here. Well, what fraction of a wave have they given me? Half a wave. How many squares is half a wave count? 1, 2, 3, 4, 5, 6, 7, 8. So, a whole wave is 16 squares. What did you say 1 square was worth? 16 pi by 4s. Which is 4pi, so B is going to be 2pi over the period. The pi's cancel in what's 2 over 4 in lowest terms. That's what your final answer would look like. Is that what it says in the back? I hope it ain't. Are there other answers? Oh, there's an infinite number of answers. Although they did say minimum possible phase shift because I could have gone back here if I wanted to do a positive coast. Could have gone way over here if I wanted to do a positive coast because it does throw up over here again. Could have done this as a positive sign graph. That would stay the same. That would stay the same. That would stay the same. This would be minus pi by 4. Could have done it as a negative cosign. Put a negative there. Put a plus sign there and move 1, 2, plus 3pi by 4. I could have done it as a negative sign, but why the heck would I want to? That's not even showing up on my graph paper. I wouldn't say all of those are right except for the fact that they demanded they wanted me to do this as a positive coast. Is that all right? Any others? I'm away on Monday. Monday is going to be a review period and I have for you two huge trig reviews. Unfortunately, Tyler, the trig reviews cover the entire trig section which is two separate units which means you can't do all the questions. I'm going to give you the reviews and then I'm going to tell you for this upcoming test which questions are fair game and that's what you're going to work on on Monday. I've already put the answer keys online. So here they come. Take a look at this one, first of all, please. And if you want to on the very, very bottom here on the back page or on the top or somewhere I'm just going to type questions that are fair game. Somewhere you've got to write down which questions you've worked on on Monday. I already showed which one it was to get with the program. Yes. Okay. One is fair game. Two is fair game. Three, since you can't do that, one without a calculator is not fair game but you should know how to go from radians to degrees. Four. Five is a word problem that comes later. Eleven. Twenty. Twenty-nine. Thirty-five. Fifty-eight. Fifty-nine. Oh, sorry. It's not fifty-nine, it's sixty-five. And somehow I'm missing a page, apparently, which worries me. Sorry. They jump numbers like crazy. Okay. Maybe the person who typed them did that. Oh well. No, no, no, no, no. Eighty-two. I didn't create this one. Someone else did. Maybe ninety-nine. Yeah, sure. Hang on, let me double check. Yeah. Scary algebra. One hundred. Ah, one hundred and one. That one is fair game because you don't need a calculator for it. One hundred and two. One hundred and three. One hundred and nine. Two hundred. Sorry. One hundred and ten. I was looking at the two hundred degrees, apparently, that was in the question. Good gosh, Mr. Duke. One hundred and eleven. Your brain works weird, yes I know. But it works. One hundred and nineteen. One hundred and twenty-six. One hundred and twenty-seven, which is worded as confusedly as I could imagine, but such is life. One hundred and twenty-eight. That looks like a job for the arc length equation. One hundred and twenty-nine. Sadly, you can't do any of the written yet. That's the first one. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty-one, two, three, four, five, two, six. So, thirty questions there. Then if you can look at trig review number two, which is this one here. I have my little note. Did I attach a little note to teachers on this one? Yeah, I forgot. I got to remember to nuke that. Anyhow, questions you can do. Three, five, thirteen, fourteen, fifteen, twenty, twenty-four. That's the decimal answer. Thirty-one, thirty-two, thirty-six, thirty-eight, thirty-nine, forty, fifty-eight, sixty-one, sixty-two, sixty-three. So that's some good practice. For some reason, I have a decimal there instead of a comma. Now, for both of these, if you go to the website, pitmath.com, but click here where you access the notes. I didn't put it actually where the video lessons are. I put it where block A is on my sky drive. I think this works. I've double checked. It works on my iPod and most small mobile devices if you want to. Block A, trig part one, and you will see trig review one and answers, trig review two and answers. At least that's the theory. Okay, good, good, good. All right. I also got to do a lesson today. That's okay. You're not going to get a lesson Monday. It's pressing pause. We've been looking at how to read trig equations and figure out trig equations. What we're going to look at today is graphing trig functions by hand. This is not in the book. I've created my own little handy-dandy guide. So what we're going to look at is graphing trig functions by hand on your test. I'm going to give you one or two questions on the written section where I actually ask you to graph a trig function by hand. It's tricky. It's a process. But we're going to try and make use of, we're going to try and be clever here. I'm going to add a few things that I've never seen in a textbook that I think I kind of, when I was in high school, figured out, made my life easier. One of the tricks we're going to use is this. Don't write this down. Just watch. What's that a graph of? Ryan. How many chunks is it nice to divide sine up into four? That tells me middle top, middle bottom, middle top, middle bottom. And that also works for cosine. We're going to use that trick. One of the things we're going to do is whenever we figure out the period, we're going to very quickly divide the period by four because that tells us on the graph when the next point is going to be one quarter period you're going to be at the top. One other quarter period you're going to be in the middle. One other quarter period you're going to be in the bottom. One quarter period you're going to be in the middle. I've never seen a textbook mention that. I'm going to add that little detail. And we're also going to try and be clever and pick a good scale with a common denominator. I've lectured you guys about making your life easy, finding a nice common denominator. Show you how it works. It says, graph this. I always start out by making a list of the four things, amplitude, period, phase shift, vertical displacement, and then I'm going to add two more inventions of my own. So, over here what's the amplitude? Eric. Two. What's the period? Two pi over b, which is pi. What's the phase shift? Pi by three. What's the vertical displacement? And then here's Mr. Duke's two inventions. What's the period, Alex? We said that it's really easy to take the period and just divide into four chunks because that tells you how far apart each middle top, middle bottom, middle top, middle bottom, middle top, middle bottom. So I actually write that. I always write down period over four, which in this case is pi by four. Now, amplitude, horizontal or vertical? Vertical. Period. Horizontal or vertical? Horizontal, right? Phase shift, horizontal or vertical? Vertical. Vertical displacement, horizontal or vertical? Period divided by four horizontal. Here's this second thing that I came up with. I said, you know what? To help me count easier I always added the horizontal or x scale. This is what I'm going to have each square be worth on the graph. You've heard me say, how many squares are there? What's each square worth? I'm reversing that procedure and I'm saying, what do I want each square to be worth? And it's going to be the common denominator of the horizontals. What's the denominator here? It's invisible. Here. Here. The best scale you can pick is pi by 12. And you'll see why that's so nice in a second. I'm going to get clever. What's my scale? Justin, nice and loud? Why would I leave that over 4 and that over 3 and that over 1 and make myself think more? You're not going to find a common denominator right? I've got a 12 times by 4 times by 4 times by 12 times by 12 times by 3 times by 3. Now what I'm going to start to do, I'm going to start to put numbers in circles next to these. This is the order that I attack my graph. The very thing I do is I label my scale. What's one square worth? I don't want to label each square because that's going to just mess up my diagram. How about like this? 1, 2, 3, 4, 5. I think I would label 1, 2, 3, 4, 5. I would call this 6 pi by 12. Yes, I know it's pi by 2. Don't bother reducing the fraction. Why make yourself have to think more? 1, 2, 3, 4, 5, 6, 7. Hang on, Mr. Duke. 1, 2, 3, 4, 5, 6. Here's 12 pi by 12. Oh heck, let's go in the negative direction. 1, 2, 3, 4, 5, 6. Negative 6 pi by 12. If I need more squares because it's hard to count, I'll label a few more. But Vlad, keep it simple. Label, you know, every third, every fourth, even every sixth square. I'm assuming you can count. So the first thing I do is I label. The second thing that I do is I take the vertical displacement. What is the vertical displacement? Sabina, I put that on my graph. 1, 2, 3, I put a big negative 4 there, and I draw a little dotted line. Right there. Because that's the middle of my graph. The third thing I do is the amplitude. What is my amplitude? Justin, what's my amplitude? Which means I'm going to go 2 up. How about to negative 2? And 2 down. How about to negative 6? And I put a little dotted lines on those as well. These are really helpful. Why are these so helpful? Show you. Once you've got those dotted lines in place, look up for a second. Can you see? I think the graph's going to be bouncing like this. I've already, I know I'm not graphing anywhere up here. The fourth thing I do is I identify the graph having positive sign, positive coast, negative sign, or negative coast. Positive coast? Remember that. The fifth thing I do is identify the phase shift. What's the phase shift? 1, 2, 3, 4. I'm going to hover my pencil 4 squares to the right. What am I graphing? Positive coast, negative coast, positive sign, or negative sign? My first point is going to start there up high. Because coastline starts up high. Do you know when my next point is going to be? We divided the period by 4 for this very reason. Because it's going to be every quarter period. Every pi by 4. No, wait a minute. I was really clever. I even found a common denominator. Every 3 squares! See it? So the next thing I do, now what am I on? Is this. I go 1, 2, 3 squares middle. 1, 2, 3 squares bottom. 1, 2, 3 squares middle. 1, 2, 3 squares top. 1, 2, 3, middle. 1, 2, 3, bottom. Oh, and I can go backwards if you want me to. Middle. 1, 2, 3, bottom. 1, 2, 3, middle. 1, 2, 3, top. 1, 2, 3, bottom. Looks like that. There's a fair bit of setup at the beginning. But if you take your time and you do that, honestly I think it's just connecting the dots like grade 3 when you're done. And kind of cool connecting the dots. It's like a little wavy. Let's check our answer. Let's get at our graphing calculators. Make sure we're in radians. Clear any equations you have in your y equals. So press y equals. And I'm going to very carefully type this in. 2, cos 2 bracket x minus pi over 3 close bracket close bracket minus 4. But I'm not going to hit graph yet. I always like to graph two more things. The top railing and the bottom railing. The top railing was this one right here. How high is that one, Maria? The bottom railing was because this way I know if my graph is bouncing off of those, I know I've nailed this. What's my bottom railing, Maria? And before I hit graph, I go to my window because I want this to try and match as well. Well, I'm going to go from zero. I'm going to graph not this whole thing, but I'm going to go from let's say zero to 12 pi by 2, 13 pi by 2, 14 pi by 2, 16 pi by 12, 16 pi by 12, and I should be at the top right there. So I'm going to go x min, zero, x max, 16 pi by 12. I could have picked different numbers. I just want to graph a nice section. That's how I'm going to check. Oh, what would I put in as my scale? Why not pi by 12? What I used on my graph paper. Now, my calculator makes it a decimal. I don't care. It's going to be accurate. Why min? I went to negative 6, but that's right at the very bottom. I'm going to go to negative 7 just so that I don't miss something on the edge of the screen. Why? Sorry? Is that okay? Why max? How about to negative 1? What's my y scale? What am I going up by on the graph paper? Each square. Oh, I'll leave that. Now hit graph. Now I said I should be at the very, very top at the very, very end. Am I? And I also notice 1, 2, 3, 4 squares in. I'm at the very, very top. Yes? 1, 2, 3, 4, I'm right. I have to be. Did you get that okay? If you didn't get that graph, I can come help you. Yes? No? Let me come help. Let's match our scale. We can use pi by the calculator handler. We can use trig functions. It's a process. But I hope you can kind of see where everything's coming from. We make a list. That's the basic. You see why I added these two? Period over 4 tells me how far part each dot's going to be. Oh, and do yourself a favor. Find a common denominator scale for all of the horizontal stuff so that you can count squares and not have to do math in your head. You'll notice once or twice we've done graphs and never graph it that way. Let's try another one. Negative 3. Okay, that thing. Let's make a list. Amplitude. Eric, what's the amplitude? Brianna, which one what? Why? Is the amplitude ever negative? Will I give you this question on your test and ask you the amplitude? Yes. Will I have a negative answer to pick from? Yes. Will the answer probably be A? Don't do that. What's the negative tell me? It's got to tell me what the graph looks like. Period. 2 pi over a half. Which is 4 pi. Phase shift. I should have asked also, Emily, is this factored properly? It is. Yay. Negative pi by 2. Critical displacement. Positive 1. And then I add two more things. I take the period. I divide it by 4. Because I want to know how far apart my dots are going to be. What's 4 pi divided by 4? Hey, that's a nice one. And then I find a nice horizontal scale. I don't care about the 3 or the Oh, you know what my common denominator is going to be here? I'm going to scale a pi over 2's. Which means I'm going to call this 8 pi over 2. And I'm going to call this 4 pi over 2. 2 pi over 2. I was noticing and going something seems wrong there. Each square is going to be that big. Each dot is going to be 2 squares apart. Back. Now, let's label our X scale. How about 1, 2 pi by 2, 4 pi by 2, 6 pi by 2, Mr. Duick. And let's go backwards. I'll add more if I need to. That's enough that I can count successfully, I think. What was the next thing I did? Okay. Looks like our Y scale is going to go up by 1's. So I'm going to put a little 1 right there. Vertical displacement. What was the next thing I did? Sorry? Railings. What's my amplitude? 3. So 3 up from here. 1, 2, 3. And that's at 4. And 3 down. 1, 2, 3. And that's at negative 2. By the way, every year there's some kid who does their whole graph but doesn't put any numbers on it. I have to give you a 0 because if you put no numbers on it, how do I know you weren't graphing in your head out at infinity somewhere? The most common one, they put the X scale but they forget to put in numbers on the Y scale and they assume do a clue in. No. Now what? Which trig function are we graphing? No, we're not graphing sine. Negative sine starts where? Middle and goes down. Where? Oh, phase shift. 1 square left. I'll go, hey, what color will I choose, Mr. Dewick? Let's go red. 1 square left. I'm in the middle. And my next one, we said negative sine starts in the middle and goes down. It's going to be bottom, then middle, then top, then middle, then bottom, then top, then middle. You okay with that? How far till my next one? 2 squares. 1, 2. Bottom. 1, 2. Top. 1, 2. Middle. Bottom. Middle. I'll go backwards a little bit. Middle. You know what? That's good enough. I've grabbed a full period and then some. You want to make sure you have at least one full wave. Two full waves if you can fit it in even better, but one full wave for clear. Clear, clear. Negative 3. What? Negative 3. Shine. 1 half bracket. X plus pi over 2 bracket. Bracket plus 1. Highest point is 4. Lowest point is negative 2. Let's put those on there as well. My guide railings. Window. Well, I'll be a bit more accurate. Let's go to this point. One, two, three, four. 5 squares back. I'm going to go negative 5 squares. Negative 5 pi by 2s. How far forward? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. I went 11 squares to the right on my graph and I was in the middle. 11 squares. Scale. 2. How low does my graph go? I'll go negative 3 just so I have it a little more centered. How high does my graph go? And what would you think I would pick? 5. Scale of 1 vertically works. One more truth. Let's find out. I notice 1 square to the right. I'm on the bottom. 1 square to the right. I'm on the bottom. That's the other reason. Those railings are really handy to spot exactly where you are at the top and bottom. I could even add one more, by the way. I could graph the middle line. Did I say I was in the middle at the very beginning? Yay! You can add that if you want to. I get that confused with the x-axis because I'm stupid. I don't do that one very often. Mr. Dirk, how will this help us on the test? You won't be able to check your answers on the test but you will be able to check your answers in your homework. I think also graphing these on your graphing calculator reinforces much of what we did to graph it by hand. Because you have to think, why did I type that? Why did I do that? Why did I type that? 3. Turn the page. Still 12 minutes left. Yeah, Kara. It's a long class but you won't have to hear me at all on Monday. Woohoo! Let's make our list. Try this one on your own, making your list. Just look up to see if we get the same thing. Yes, this is a little weird one. You get a period of 1 half. That's 2, right? Which means this X scale is not going to be in terms of pi. Ok, I'll deal with that. Phase shift of 1 to the right. Vertical displacement of 2. Period divided by 4. 1 half divided by 4 means put an extra 4 in the bottom so instead of a 2 I'll have an 8 down there. What's my common denominator going to be here for my scale? And it's not going to be pi over 8. It's just going to be 1 eighth because there's no pi terms and everything. Each square is going to be 1 eighth. In fact, if I go 1, 2, 3, 4, 5, 6, 7, 8 squares 4 squares Oh! Let's put everything into a common denominator. So this is going to be 4 over 8 This is going to be 8 over 8 1, 2, 3, 4 negative 4 over 8 By the way What's my phase shift, right? You know what? Because 8 over 8 is as far I should probably add a few more points. 1, 2, 3, 4, 12 over 8 1, 2, 3, 4, 16 over 8 Although I can always just graph to my left a bit more if I don't want to lengthen my graph paper because these waves go forever. Right? Ok, now what? We've done my X scale We've got a common denominator What was next? What did we do now? Oh, because that's my middle 2 Now what? So railings 2 up, 2 down 4 and 0 Can you already see where this graph is going to be bouncing between? I already said my phase shift is 8 over 8 So my first point is going to be somewhere on here. Where is it going to be? Are you asking me to graph? Negative cost, which starts where? Ah, first point is going to be right there. How far apart is each point going to be? How many squares apart? Oh, one square apart middle top middle bottom I'm going to move towards the X axis I'm going to head left, Jimmy middle top middle top By the way, Breanne you could have picked, for some reason a different scale So until I came up with this I had a lot of kids and they would say I'll pick a scale of 1 quarter but now you're graphing like little half squares Or you could have picked a scale of 1 over 16 which would mean every two squares If you're finding it too crowded just change your scale and just adjust everything to the common denominator and count carefully But I still find it very, very helpful to divide the period by 4 that tells you how far apart each dot is going to be and take the time to pick a good scale don't just plunge on to the graph add your railings in your middle and then I'd argue Breanne it really is clever connecting the dots I'd like to check this one but we don't have time I'd like you to try number 4 on your own I'll do it up here without saying a word let's see if we all end up in the same place do that one what the heck is that I just hit graphing three functions by hand I got that and you could have gone further in another direction but there's two whole periods I'm pretty good I double checked on my graphing calculator yay I was right this is really going to be your homework on Monday I have a practice sheet for you to try a couple of these okay here it is