 320. Any questions that you would like me to go over? This is your chance to ask. One E, okay? One E is tricky. Okay? What's the exponent in one E? A four, okay? It's not a quadratic. It's going to show up as a quadratic. Now, I do say what's the first thing that you always, always, always, always, always, okay. Check for a GCF. There is not a greatest common factor. Count the number of terms. How many? Two. Is there a minus sign? Is it perfect square, perfect square? So this is going to start out as difference of squares. It's going to be the square root of secant to the fourth is secant squared. Is that okay, Amy, so far? And then it would be plus one, minus one, or minus one, plus one. But when you look in the back, they haven't stopped there. And the reason is they've looked at each factor. Look at this one, first of all. GCF? No. How many terms? Minus sign? Can't factor that one. GCF? No. How many terms? Two. Minus sign? Oh, perfect square, perfect square? This one goes further. So I'm going to drop the unfactorable term down. That's a factor. And I'm going to get, come on, then. Oh, there's a sneeze for the internet. He already has, thanks. Secant theta plus one, secant theta, and minus one. Sir, do we put one like that on the test? Probably not. Probably not. You know what? That would show up on a provincial as, ooh, we've got to put a couple of tricky multiple choice questions. What can we do? This would be in that vein or that level, because I would hope it would be multiple choice. I would hope you would glance at your answers and you would say, oh, some of these answers have three factors, but I only got two. Oh, I bet you one of these breaks down first. Like, I hope you would make that chain of reasoning. As a written, this would not get on the provincial, and I would not put this on the test. Is that okay? Oh, and if this was an equation, you have equals zero, you would get roots here and here. Here you would get secant squared, that's supposed to be a plus sign. You would get secant squared equals negative one, and when you try taking the square root of both sides, you can't take the square root of negative, this one actually has no roots when you try solving the actual equation. These two would. Any others? Ah, great question. Six B. Six B is sneaky. Okay? How many terms are there, grand total, even before we start? Four, I won't put this on a written. I feel okay as a multiple choice. I graph left side, graph right side and find where they cross. Six B, this is going to be called factoring by grouping, not grouping. Okay? What I'm going to do is I'm going to, first of all, leave this here temporarily. I'll come back to it. GCF, and the fact that this and this are the same tells me why this is going to be a factoring by grouping. I'm going to move this guy over now. Factoring by grouping says this, when you have four terms, you cover up the last two. In my mind, I would cover up these two, and you factor a GCF out of the first two. You already have. Then, you repeat that same procedure. You cover up the first two, factor a GCF out of the last two. Now, there is one, negative one. You got this? I'm impressed. You're smarter than you look. Not well. Make that a compliment. My pen is working like a slug. I probably have to reboot my machine a bit later. And why does this work? Well, and again, this is an advanced one. This will not be on your test. Tyler, what do you see here? What do you see here? That's also a GCF. There is a cos x plus one in both terms. I can factor out a cos x plus one, assuming my pen decides to write. And I can factor out a 2 sin x minus one left behind. That's what that factors into. And now, you can say, what are the roots? We'll talk later, sir. I'm not even going to let you roll a dice of eight, because I had a promise from you. I thought, I thought, I thought. You would get this. Cos x equals negative one. There's one root. Sin x equals one half. And both of those are very solvable. We do have a triangle with a one and a two in it. Negative one. Oh, alarm bell. Unit circle or sketch the graph. I'm not going to give you a factoring by grouping on your test. I always assign it, because it's good discussion to let you know, hey, think outside of the box. And I'm kind of curious to see who could actually come up with that. Because I could argue, all I did was factor out a GCF and then factor out a binomial GCF twice. But that's a little weird. You got far? I'm impressed. Any others? Then you want to get out your graphing calculator. We have two short lessons today. I'm going to do two, because they're both short. Lesson three, if you'd be so kind as to turn to page 325. I think it's, yeah, 325. And it says, as it's heading, equations which require a graphical solution. This is why you need your graphing calculators and it's got to be a graphing calculator. You need your graphing calculators there. And you want to make sure you're in radians, if you're not already. You know what? I better boot up my graphing calculator and I better make sure that I'm in radians mode. Ah, I am. And it says this. Some equations cannot be solved by an algebraic approach. In these cases, we use a graphical approach to estimate the solution. Now, here's how you're going to know this. First of all, in real math life and then for the test. In real math life, you'll have an equation that has some trig and some non-trig Xs. Or several different trig functions and you can't do anything with them. That's how you know in real life. On the test, here's how you'll know this. It's going to be, what does MC stand for on a test? Okay, it's going to be multiple choice and you'll glance at your answers and you'll notice that all of your answers, instead of pi by somethings or three pi by somethings, all of your answers are decimals. And in fact, what I'll start teaching you for your next test, not your Valentine's test, but your next one is on the multiple choice, always glance at the answers first. That's your trigger watch. Yeah, I know it's already running twice. Maybe that's why you're running so quick. Look up. So here's a, where's my provincial exam? New course provincial exams. Here is from April 2001, I think that is, or January 2004. Let me make this a bit bigger. But you'll be able to see or recognize a trig question that you have to do. Oh, right there. How did I spot it so fast, Jesse? Look at my answers. See them? That easy. What that means is, don't try and do this algebraically. Don't bother. Graph left side. Graph right side. Find where they cross. And in fact, that's the first part of today's lesson. So it says, consider the equation three sine x equals x. Use the following method to graph between zero and two pi. Here's what I'm going to do. I'm going to get up my graphing calculator. I'm going to let y1 be my left side. So my left side of my equation is three sine x. I'm going to let y2 be the right side x. Now I'm noticing in the notes he has y1 be x and y2 be three sine x. Doesn't matter. Just graph one side, graph the other side. Before I hit graph though, I'm going to change my view window a little bit. What domain do they want me to look for a solution in? So I'm going to go x min, zero, x max, two pi. Good scale. If I'm not sure, I almost always pick pi by six. That's 30 degrees. That's kind of my fallback scale. What about my y's? Well, oh, what's the amplitude of this? I'm going to go from negative four to positive four then because that graph goes three up and three down. And I would like very much to center it. Y scale, yeah, one x res, never touch graph. There's three sine x. Oh, first of all, how many answers, how many solutions are there? Can you see? Two, you'll often find on a provincial, you can cross off half your wrong answers just by looking at the number of solutions. And you'll often find, so let's support what we'll do this one in just a second, but let's find the roots of this. How do I find the solutions? It's going to be second function. Where was the intersection find where they cross feature? Calculate intersection. Then it asks first curve, that is the first graph I want it to use, so I just hit enter. Then it asks second curve, that is the second graph that I want to use, so I just hit enter. Then it wants to make a guess. I'll move my cursor close to that root there. You know why? I'm pretty sure I can tell you what the other intersection point is. What is the other intersection point look like it is to you? Zero, zero. So I'm not going to waste my time. And I hit enter. 2.278. We have x1 equals zero, x2 equals, oh they only said to the nearest tenth even easier, 2.3. Now the only tricky part here is we've talked about the general solution that was where the graph repeated over and over. Take a look at this equation here. Are they going to cross as I move to the right ever again? Nope. I think they probably cross back down here. If they did ask for a different domain, I would just change my view window. So I'm not going to worry about the general solution. Let's try this one together. So you guys, is that big enough for you to read? You know I can make it larger. Here's the equation. Solve sin x equals 3 cos x. And right away Alex, I know that it's graphing calculator because I glanced at my answers with their decimals. So I'm going to graph left side to sin x. I'm going to graph right side cos 3x. What's my domain? 0 to 2 pi. So I'm pretty good I think with my view window. What's my amplitude here? You know what? Negative 4 to 4 will work. I'm going to rush because I'm writing a test. I'm going to hit graph. There's 2 sin x. There's cos 3x. Maybe not getting it? Sorry I should ask. Are we okay? You don't get the left side. Right side. Right side. Y1 is this. Y2 is that. What you're saying is where these are equal is where they'll cross on a graph. By the way how many answers? So I would hope on the test you would cross off C and D. Now look at my remaining two answers. My first answer for A is 0.3. My second answer for A is 2.83. Do you think that x value right there is 0.3 or 2.83? If this is 0 to 2 pi that's really 0 to 6.2. I think 2.8 is over here somewhere. You actually don't even need for this particular example. You really don't need to even bother finding where they cross. You can just clue in. It's A but let's find where they cross just for the practice. Second function. Calculate. Intersection. First curve. Second curve. Let's go find the first answer. My guess is oh right about there I'll hit enter and that should get me right at the intersection point and lo and behold it does. And I would quit here because it's multiple choice and hopefully you're good enough test writers now that you've learned where you can cut corners on your test. Oh if I finished the test early and I got time to kill yeah I'd probably come back and find the second route. Just for practice though because this is not a test we're doing as an example. Let's practice finding the second route because there's some is it working by the way we're good. Second function. Calculate. Intersection. First curve. Second curve. Now I'll move my guess to where they cross over here. Looks like they cross right about there. Enter. 3.448. There it is. 0.31, 3.45. Would they ask for a y value you mean? Nope. No because you're solving what variable was in the equation x. You're solving for x. Okay. It says example one turn the page. Page 326. It says use a graphical approach and the zero feature. I'm not a big fan of the zero feature. I'm actually going to cross this out. You can make these equal to zero and remember we did find roots but you had to do left bound, right bound, sandwiched. The intersection one is so much faster. I will always graph left side, right side and find where they cross. So let's solve this. How do I know that I have to use a graphing calculator? Well they said it. Yeah okay. What if they didn't? Do I have some trig and some non-trig? Graphing calculator or calculus. Have you guys learned Newton's method for solving equations yet? Okay. One of the great many great things about derivatives is you can use derivatives to solve any equation. You use it to make a guess and then use the guess to make a better guess and use the better guess to make an even better guess and within about four trials you're accurate to about eight decimal places. It's called Newton's approximation method because it never gets you the exact answer. It approximates it but if you're accurate to it in eight decimal places that's pretty close. Ready? Left graph. X to the power of three minus x squared. X, x, come on Mr. Dewey, squared. Right side, two, sine, x. Did they give me a domain here? No. So I'm going to be a little bit paranoid. I'm actually going to go from negative two pi to positive two pi just in case it crosses in the negative area because I have no idea what this looks like. So I'm going to go from negative two pi to positive two pi just in case there is a negative solution. And how high? Well, amplitude of two, you know what? Negative four to four works pretty good. Let's see what this looks like. Ah, look at that. There is a negative solution. I might have missed that. Oh, in fact, how many solutions are their grand total? Can you see? Three. Oh, by the way, every once in a while they'll simply ask you how many solutions are there as the multiple choice question? Okay. Three. Let's find them. First one. Second function, calculate. First curve, second curve. And let's find the first one, which is over here. Negative point nine. What does it say? Give answers to the nearest tenth. So I guess this is going to be negative point nine. X one equals negative point nine. X two equals, I think, zero zero, but I'm not quite positive. I'll double check. Since this isn't multiple choice, I'll be a bit more paranoid. First curve, second curve. Yeah, I thought so. Zero. And find X three on your own. I'll freeze the screen. Let's see if we get to the same spot. I got 1.7. Okay. Oh, by the way, remember the quadratics we did yesterday? You can use this to check your answers. The only problem is we did some with exact values, and this gives you decimal answers, but could you convert pi by six to a decimal to see if it was the same decimal as yours? You can on this next test, check your answers. However, when I mark the trig equations, I only give you one mark out of four for the answers. So if you use this to find the answers and you show no other work, the best you're getting is 25%. So that was lesson number one, short lesson. What's your homework? Try number one, number four, number five. What was that, Miguel? You want to do more? Is that what I heard from over there? I heard you just, Mr. Duit, please give me more math work. I'm going to, because I'm going to do a whole new lesson. Can you turn to lesson four, please? I thank you. No, you're the best. I don't know if you remember, but last day, when we did the quadratic equations, I said, when we were finding the general solution, what's the period? Is there any B? No. And I always said, we're going to put a B in a couple of days. Here it is. We're going to change the period. So this is also nice because it's good review of solving trig equations, which is on your Valentine's Day test. It's a little bit tricky, and I don't think the book does enough practice. So I'm actually going to, once we're done this, find a blank page and we're going to do a couple more. It says this, solving equations involving multiple angles. And the first thing it says is we're going to introduce a graphical approach. I've already told you, I'm not big a fond of the graphical approach. We will use cast reference angle, and then I'm going to add one more little trick, substitution. The author walks through lovely graphical examples. Forget it, forget it, forget it, forget it, forget it, forget it, forget it, forget it, forget it. Page 330 it says solve a multiple angle equation using a graphical approach. Forget it, forget it, forget it, forget it, forget it, forget it, forget it, forget it. Page 230, 331 it says solve it using an algebraic approach. This is what we're going to do. Take a look at example two. Is there something in front of the X? Okay. That you have to spot. That I can't help you spot Amy. That you've got to pick out. Okay. Soon as I see that the first thing that I do in big writing in the margins so that I don't forget is I say the period is 2 pi over b which in this case is pi because I'm worried that when I do my general solution I'm so used to going plus 2 pi n I'll just go plus 2 pi no period here it's gonna be plus pi n so I started doing that a few years ago. Big letters. What don't you like about this equation though? The answer I think for most of you is that 2x. If that was just an X hopefully for all of us this would be a yonor. I got a triangle with a 1 and a 2 in it. Cosine is positive. Castrel is the 1, 2, root 3 triangle. Which angle has a cosine of 1 half the top one? Pi by 3 and away you go. So what don't we like? The 2x. My method not using a graphing calculator is well I don't like the 2x. I'm gonna temporarily get rid of it. I'm gonna say this. Temporarily I'm gonna let the letter a equal 2x. What I'm gonna solve here Matt is this equation. Cos a equals one half except let's bring the negative in and let's make that two work sorry. Cos a equals negative one half. Okay this I like. In fact Jesse which trig function? Negative or positive? Let's go castrel. C A S T. I am positive we're here or here. Do I have a triangle? Oh it did say exact values so I know I must have a triangle with a 1 and a 2 in it. Why yes it's the 1, 2, root 3 triangle. Which angle has a cosine of 1 half the bottom one or the top one? How big? So if I hear you correctly this is pi by 3 and this is pi by 3. Now I'm not solving for x. What letter am I solving for right now to make life easier? So can you tell me what a1 is and can you tell me what a2 is? By the way this is totally fair game for Valentine's Day yes? Okay so what is a1 what is a2? 2 pi by 3 or pi by 3. In other words my first step area I write the period and then I temporarily get rid of the thing that bugs me except we weren't solving for a what were we solving for really? Alex well x how are a and x related can you read that out to me please? So I'm gonna argue that a1 is the same as 2x1. Is that okay Dylan yeah? How would I get the x by itself? Ah you see how I pulled that off? Rather than deal with that stupid period change ignore it temporarily put a different variable there just so you don't get confused solve for that different variable and then do the period change after you've done all the tough trig. How would I get the x by itself? You were right say it again. I'm gonna have an extra 2 in the bottom x1 is gonna be 2 pi dividing by 2 would put a 6 down there because it's already a 3 down there yes? Oh in lowest terms what is x1? There's my first root area how are a and x related can you read that out to me please? That was our little substitution that we just made up right? So if I want to find x2 I would argue that 2x2 equals 4 pi by 3. I'm replacing a with 2x but since it was a2 I'll call it x2 just to be organized. In area how would I get x2 by itself? Divide by 2 can you see you'll get 4 pi over what? 6 can we reduce that in one step are we okay with that? I'm gonna write down here x2 equals 2 pi by 3 here's where this gets a tiny bit more challenging there could be more in our domain. I have found 2 now what I'm going to do is I'm gonna add the period to each of these I'm gonna find a coterminal angle and if that's less than 2 pi it's also a solution let's find out add pi to this am I really gonna add pi what am I really gonna add in my head 3 pi by 3 so if you add 3 pi by 3 is there a third x value that satisfies this equation that's still less than 2 pi x3 is going to be pi by 3 plus pi 4 pi by 3 is also a root it's also a solution between 0 and 2 pi and I'll prove that to you on my calculator in a little bit I'm also gonna do this with x2 if I add the period to this am I still less than 2 pi that's also a solution then there is an x4 which is gonna be 2 pi by 3 plus pi plus 3 pi by 3 it's gonna be 5 pi by 3 that there may be another one I'm gonna add the period to these two and if I'm still less than 2 pi those are also solutions in our given domain if I add pi to this one am I bigger than 2 pi okay I'm out of my domain put your pencils down and look up see here's really what's happening my domain is 0 to 2 pi my first equation is cosine of 2x my second equation is negative a half now if I didn't have this 2 here if I just had that we would have this graph there's cosine there's negative a half how many answers to but if I put a 2 there that's a horizontal compression by a factor of 2 there's going to be two cosine waves occurring in the same distance that's why I got four answers I found the first two cleverly and then I said well I'll add my period to get that one I'd my period to get that one oh hey hey hey what if there had been a you know how many cosine waves are gonna fit in three there's gonna be six answers I would put a 3x there I would get a 1 and a 2 and I would figure out what x1 x2 I end up having to keep adding until I'm out of the domain keep adding the period until I'm out of the domain so four answers b says state the general solution the general solution is gonna be my first two primary ones which were pi by 3 and 2 pi by 3 plus multiples of the period not 2 pi n what did I write in great big letters right away what's my period plus pi m where m is an integer if they give me a period change what's my strategy first of all make a big note that there's a period change and what's the new period then I ignore it I'll temporarily substitute a nicer variable in solve it do all the trig and then once I finish that I'll do whatever arithmetic I need to do to take the period into account then I'll keep adding the period until I'm out of my domain oh it says complete the following statement the general solution consists of answers which differ by pi radians because the cosine of 2x as a period of pi radians it wants us to jump into the homework we're going to go back one page please to page 330 but instead of solving this one with a graphical approach we're going to solve it with an algebraic approach given tan 2x equals root 3 between 0 and 2 pi find the exact values of x and I'm going to cross out using a graphical approach you have to notice that there is in this case a B what is the B what's the period ugly cousin right period here is not 2 pi over B it's pi over B the period here is pi over 2 temporarily to make things less confusing I'm going to solve this I'm going to let a equal 2x why do you use letter a because they hardly ever use it as a variable so I know it's a I'm going to solve tan a equals root 3 root 3 over what by the way do I have a triangle with the 1 and the root 3 okay I can do this Irwin which tree oh tan I can't ask you Matt which trig function C A S T what quadrant are we in a positive right math so here and here sorry we don't have much room to write because they're expecting us to do this graphically you have to kind of smoosh your writing a little bit we said we have a triangle we said it's the 1 2 1 2 root 3 triangle which angle has a tangent of root 3 the bottom one or the top one top one once again this is going to be pi by 3 oh the answers are always pi by three no it's just the two examples happen to have that in there so Matt can you tell me what a 1 is pi by 3 now let's see how good we are take a look at this how would you get the x by itself divide by 2 so can you tell me Matt what x 1 is going to be it's going to be a divided by 2 there's going to be an extra 2 down there it's going to be what pi by 6 is that okay are you asking us to do some math I had your mediocre at fractions now yes we have entered the land of mediocrity we can expect you to do some basics dividing is the same as putting that number in the bottom of a fraction oh what's a 2 Matt 4 pi by 3 what's x 2 let's see it would be dividing by 2 an extra 2 down there except I don't want you to say to me 4 pi by 6 please reduce a fraction in your head we're mediocre now what are you giving me I think we're going to pi by 3 okay I better write it out it's 4 pi by 6 which is 2 pi by 3 okay ah there may be more now I'll be honest because you're going to be allowed to use your graphing calculator on the next test you could graph this between 0 and 2 pi and you can find out how many answers there are I'm just going to say what if I don't let's see if I can anticipate I'm going to add the period oh what's my denominator here I'm going to add 3 pi by 6 because I'm going to find a common denominator right away ready add 3 pi by 6 to 1 pi by 6 what do we get 4 pi by 6 you know what we've already mentioned that right so let's add the period to this one 4 pi by 6 plus 3 pi by 6 equals 7 pi by 6 is that less than 2 pi yeah okay so that's a solution x 4 would be let's see if we add the period to this one are we still less than 2 pi and hopefully you're recognizing you can kind of even do some of this in your head 7 pi plus 3 pi is what you 10 pi by 6 is that less than 2 pi what would 2 pi be in terms of 6 Troy 12 really I'm saying as long as I'm less than 12 pi by 6 I'm good so x 4 is going to be 10 pi by 6 although I'd probably reduce that to 5 pi by 3 is there a fifth solution well if you add 3 pi by 6 to 10 pi by 6 13 pi by 6 oh we're just out state the general solution I think the general solution is going to be your first one pi by 6 plus multiples of the period I think that one actually generates every single following answer yeah you okay with the pi by 2 2 times what is 6 same to the top right 3 over 6 is the same as 1 half okay yep you want me to make up one more or do you guys feel okay we do that's why I'm asking you want me to make up one more do you guys feel okay one more okay do you have room where your answers are at the end of this lesson is there room at the bottom of the page no which one oh previous answers 320 there's room okay I'll do mine right here apparently page 320 you have a lovely room page 328 that's what I said can't you boy your listening skills are terrible sign I'll make it equal root 3 over 2 so it's going to be an exact value lovely lovely I'm gonna give you a zero less than or equal to theta less than 2 pi let's do something kind of yucky here let's see 5x over 2 I guess technically that means there should be an x there you don't need to change yours but I will because I'm just fussy okay what you know what the most common mistake that I see and I should mention this because it drives me crazy when student a lot of stuff drives you crazy yeah when students get a question like this I'll see them go oh divide by 2 and they'll make that a 4 they'll somehow divide something from inside a trig function you can't do that you can't do that so we've got to just deal with it first of all what's the period well it's 2 pi over b how do I divide by a fraction it's gonna be gonna be what 4 pi over 5 right temporarily I'm gonna let a equal 5x over 2 and I'm gonna solve sine a equals root 3 over 2 gonna use the cast rule sine is positive here and here do I have a triangle with the root 3 and the 2 in it yep our famous 1 2 root 3 triangle which angle has a sign of root 3 over 2 oh for pizza it's pi by 3 again I told you miss you do it shut up they're not bye bye 3 bye by 3 so here's what I end up with then my first root before I bring the period back in is pi by 3 my second root before I bring the period back in it is what 2 pi by 3 okay how would I get the x by itself well what's the five doing to the x mathematically times in so I'll divide what's the two doing to the x mathematically dividing so I'll in other words I'm gonna argue that x1 is gonna be the two on top and the five on the bottom along with the three yes by the way this is probably just slightly tougher than I've ever seen those is this well they could probably do this on multiple choice I guess because when there's multiple choice answers it's a little easier written I a little bit of top yucky fractional period um x2 is gonna be this one here times by 2 divide by 5 times by 2 divide by 5 Amy what's my denominator in both of these by the way I'm gonna write the period out of 15 as well to make my life easy how would I change a 5 into a 15 times by top and bottom 12 pi by 15 Alex what's my denominator what's 2 pi gonna be then 30 out of 15 once I get bigger than 30 and equip is there an x3 well x3 is gonna be the first value plus the period what you get 14 pi by 15 is that less than 2 pi yep is there an x4 well that's gonna be the second value plus the period I get 16 pi by 15 is that less than 2 pi yes is there an x5 let's go back to x3 and let's see if we can in our heads add these two together what is 14 pi by 15 plus the period 26 pi by 5 you know what there is an x5 26 pi by 15 x3 plus the period third value plus the period is there an x6 I'll add the fourth value in the period let's see the fourth value was this one 16 pi by 15 and 12 pi by 15 is 28 oh hey hey hey there is an x6 28 pi by 15 is there an x7 oh I hope not let's find out if I add the period to this why I'm over this one had six answers I didn't know that when I made this one up ahead of time I figured it might have six I figured with that fractional period might only have five I wasn't quite sure where it was gonna end up where the cutoff was gonna be this one had six just barely had six what would the general solution be write the first two plus the period times and that's how I deal with period changes I would rather Michelle turn it into a question I already know how to solve I don't like the way the textbook really emphasizes using the graphing calculator problem is your provincial has a non calc section and this could be because you'll notice I haven't used a graphing calculator this could be considered fair game on the provincial for the non calc section will this be on your Valentine's Day test no oh well solving that will be and getting those but not period change and the rest of it not general solution okay what was that Miguel more homework you want more I would love to give you more homework number one but instead of using a graphical approach use an algebraic approach number five use an algebraic approach ooh see can't yes he can't now I'm gonna assign number six but instead of two root three over three this is really two over root three that's it's the one two root three triangle they rationalize the denominator okay nine you know what I'm gonna go back and give you what I've given you three I'm gonna give you one more to practice let's see I did say nine I did how about let's do two but again don't do a graphical approach do an algebraic approach so I've assigned one two five six and nine or was that a German student saying no sorry I assigned number I think I did didn't one two five six nine eleven twelve thirteen four okay is that okay folks you got 20 minutes left here's what you cannot do I'm gonna get a visit and die it's a Friday or no because you do have a test next class you can either work on the review or you can start working because a lot of this homework these three lessons have very much overlapped with the skills that I'm testing on your Valentine's Day test we've added stuff but you we've been using the stuff from that okay use this time wisely please