 Here's the plan. I think I'm going to guess some of you have questions about graphing by hand. I'm going to do some examples of graphing by hand first. Then what I'm going to do is I'm going to talk about reciprocal trig functions and weird stuff. Then I'll do questions from the reviews. I have a feeling that's kind of the order of confusion or stress that there is out there. Okay. I will be posting this online and if you stay till the end I can also print up a hard copy of this tutorial for you so you can decide when's the end going to be when we finish. Although North Carolina is playing Duke tonight at 6 p.m. and I got company coming over to watch the games. I'm hoping to be out of here by about 5 30-ish but I'm PBR-ing it and I told them my students come first so if they got to wait till I get there they can't get in my place anyway. Okay. Now on your test I'm going to give you two options to graph. On each page I put both the lines and a grid. You can choose which method that you use. Just make sure you clearly indicate to me which one is your final answer. In other words if you try it both ways I'll just mark the first one I come to but if you clearly circle arrows this one mark this one whatever it takes for me to figure it out then I'm good. Okay. So let's look at some graphing. There's always a risk when I make these up but these might turn out to be yucky. I'm going to graph with grid paper in the background because I think it'll make it easier for me to have my scale accurate and help you figure stuff out. You can decide whether you want to use a grid paper or just the lines. So for example the first graph that you're going to have on your test is going to be reasonably simple. It's not going to be one that has a vertical displacement or a phase shift. It'll be something like this. Here's hoping I don't make up exactly the same question. There's always that risk as well but something like this probably negative just to make it interesting. For cos or sine I'll go with cos of x over 2. I would still make a list and on the on this question I would also then I think the question Channa will say something like this. Graph the following and fill in the blanks and you'll see it'll say right here amplitude equals question mark and then it'll also say period equals question mark and it'll say phase shift it won't be abbreviated I'll write out the words equals question mark and then it'll say vertical displacement again that won't be abbreviated that'll be written out equals question mark and I would no matter what would always start out by filling this in anyways. What's the amplitude of this graph? Careful four good. What's the period? Careful it's not one half the period was defined as 2 pi over b which is going to be 2 pi over one half which is going to be I think how do I divide by a fraction flip it multiply 4 pi right phase shift there isn't one when I say phase shift what are you looking for a number in brackets next to the x being minus or plus and alarm bell make sure it's factored although I told you I will give it to you factored if I ask you to graph it but I will give you one on the test in the multiple choice section that's not factored and say tell me the phase shift and it's probably going to involve fractions because you guys suck at fractions vertical displacement what's the vertical displacement that would be something plus or minus out the brackets then excuse me while I eat I gave you two more things to find I said you're going to find it really really handy to take the period and divide it by four I won't ask you to fill in that blank because that's not actually a math term that's mr duix own little invention and the reason that is Julie we want to when we graph remember when we graphed our original sine and cosine we had the four dots that match those four corners of the circle we want to find where those are on my new period so I take my new period divide it by four that tells me how far apart each dot's going to be along the x-axis tells me how far to go along before I put my next dot how far to go along before I put my next dot how far along to put my next dot and remember we said the pattern was middle top middle bottom middle top middle bottom middle top this that's a pretty easy pattern to remember if I take this period and divide it by four I get it looks like each dot is going to be pie apart and then the fifth thing and again I won't ask you for this Regan because one I made up I said I like to find what kind of a scale I'm going to go with now I'm running out of room normally I'd write scale down there do you mind Regan I'm going to write scale up here if that's okay and by scale what I really mean is my horizontal scale my vertical scale was almost always going up by ones or something like that because the amplitudes we kept and vertical displacements we kept those pretty straightforward what's my scale it's the common denominator between all of my horizontals and I don't know if you remember I said an easy way to remember which one of your horizontals they're the ones that have letter p in them horizontal horizontal horizontal what's my common denominator sorry one now I should be careful I said common denominator what I should really say is least common multiple because I think what I really want to do is I want to have a scale going up by pie I think because if I want each dot to be pie apart wouldn't it make sense to have the scale go up by heights this this is an almost one because there's no vertical displacement it actually almost made it tougher Normally the phase shift, what about a fraction like pi over 3 and you would have said oh, it's gonna be pi over 3 is my common denominator I've said common denominator Do I want to say common least common multiple denominator thing? Anyways scale up by pi, so you'll see why in a second Then what I would do is I would sketch some kind of a grid Again on your test you'll have the option to have graph paper or the blank lines I'll use graph paper here And I spend more time setting up the scale and setting up the graph than I actually do graphing. I think it's worth it First thing I do is the vertical stuff What's my vertical displacement here the trick question zero? What's the amplitude? Four so how high is this graph gonna go? How low is this graph gonna go? Well, not four negative four. Okay, so I would probably go one two three four one two three Negative four if you're free handing this what I would probably do is go four and four and then I would just divide that in half If I so if I would have this I would also probably then go that's two that's two that's one That's one. That's one. That's one. But since I have my graph paper in the background. I'll just do that Okay, and I would add dotted lines. I'll change colors. I add a dotted line along the top I Add a dotted line because that's the middle of my graph My vertical displacement wherever it is and I add a dotted line along the bottom because Regan What this tells me is my graph is gonna bounce like that. That's a huge advantage. At least I know I'm gonna get this part, right? It takes all of one second two seconds to do it But it's really the next thing I would do is I would label my x-axis scale What's my period how long is one graph according to this question? So I would need to make sure that I at least fit in four pie and Then I would convert my scale to some kind of common denominator since I'm going up by pies Since each dots gonna be pie apart. I guess to keep it simple I could go one pie to pie three pie four pie Or I could go up by twos if I really wanted to with the graph paper to make my life easy But to keep it simple and straightforward I'll go like this That's four pie one two three That's eight pie So that would be two pie that would be six pie and in between each of those would be one pie I'm putting the hash marks in because I think I think when I hit print Julie I think it'll print like that. So if I it won't print the graph paper in the background So those you that want copies so I'll put the hash marks in there And I've done the hard work now What's the phase shift? Zero that means my first dots gonna be on this line somewhere because you always start Horizontally wherever your phase shift is what's my vertical displacement? That means my first dots gonna be on this line somewhere as well. Oh, no wait wait wait wait wait wait wait What am I graphing? Where does cosine start? I'm not graphing cosine We're okay. I already know then where my first dots gonna be it's got to be where The phase shift and I can't say the middle the coast or the sign wherever it starts appropriately intersect first thoughts gonna be Right there Where's the next one gonna be middle then top then middle then bottom then middle in other words It's gonna go boom boom boom boom boom boom. I just need to find how far apart the dots are That's why I divided the period by four shout I got a question I Feel free to ask because I don't think I've done a great job of explaining how to graph by hand So if you have the question probably you're not the only one and I'm not quite sure Because I taught this lesson so disjointed this where we had some short classes and stuff away and things top What if it was signed? Here what if it was negative us here and the next one would have been top What if it was negative sign here and the next one would have been bottom Right sign starts in the middle and if it's positive goes up If it's negative goes down which kind of actually is easy to remember positive goes up negative goes down that almost sounds like physics, okay Let's keep going. So the reason I divided the period by four shattered. I answer your question, okay The reason I divided the period by four was to tell me once I find that first dot When is the next one and again? We're always relating it back to the four corners of our circle Normally we would go to pi divided by four every dot was pi by two But here my period is four pi divide that by four every dots going to be how far apart Pi which also happens to be my scale here in other words bottom middle top middle bottom Middle middle I Can keep going bottom middle top middle bottom I want I think on the test ask you to graph one complete period and if you can't quite fit your graph where you want Don't forget you could also go backwards if you just if you run out of room going forward because you're like I didn't make my line long enough Regan or I'm getting crowded Go backwards because Regan if I'm the bottom here, what am I gonna be here? middle What about here top? It was gonna look something like this something like that How would I mark this? I would first of all give you a half mark each for these four things in other words if you can list this You'll pass the question Try to I did that on purpose Then the remaining two marks would be for the graph probably I would do something like this I would give you a half mark if you've got the up and down railings in the right place And I could see that you've gone between negative four and positive four I Would probably give you a half mark if I could clearly see that you'd started at the bottom It's somewhere I could see your starting point and you'd recognize it's negative cost good For a more complicated graph I'd give you a half mark for the vertical displacement and a half mark for the phase shift here I probably just give you one mark for the proper shape or something like that If I'm marking this to be honest what I do is Scan down these I can do that really fast and then what I do is I find two points And if your two points match my two points, I assume you didn't look into it I assume the rest of the graph is right because I'll notice if one of the points is wrong your coast graph won't look symmetrical Let's do a more complicated one this one had no phase shift and no vertical displacement that will be what your first question is like oh There is a simpler way to do this one only because of the fact that it has no phase shift and no vertical displacement This was the approach that I also gave you near the beginning of the unit I said for these ones because we're not sliding it left right or up down There was a simple cheat and that was to just change the scale so Method number two and this one you can see I've nuked to the graph paper. I Would do this and this This graph here is based on negative coast You see it hidden in there What is negative coasts look like where does negative coast start? Negative coast start okay negative coasts would look like this This would be negative one and positive but ignore the four nor the over two ignore all that It would start down here, and it would finish down here. How far? how far How far pi by two how far negative coasts would look like this And what I told you you could do as a cheat if there's no phase shift Vertical displacement remember we say sometimes it's easier to change the graph paper than it is to change the graph Because I'll show you what I can do here to get one full wave Better late than ever but better never late What really should be what number hello what number should really be here for this graph not a negative one but a what what number should really be here for this graph not a one but a what or and What's the period you'd still have to list everything you still have to find the period But what do we say the period of this graph was I've scrolled down a little bit what four pi in other words from here To here should be how long oh? So if you put a four pi there What's that please do the simple math to pi You know Now that trick only works if you have no phase shift or vertical displacement And please be careful to make sure you notice whether it's negative sign or positive sign or negative coasts or positive coasts Will that work if there's a phase shift or a vertical displacement nope you have to do a lot more work but If you're looking for a kind of a nice easy bailout or a way to check your answer after you've done it a long way That'll get you there. I suppose you just got here late What I said we're gonna spend some time talking about how to graph by hand because I think that's where most of the confusion is Then I'll spend some time talking about the reciprocal trig functions and what I'm gonna ask the weird stuff That's until it's domain all that junk and then I'll take questions from the review Let's do one with a phase. Let's do one with the whole smear Why equals? Let's go negative again. Go three Let's go sign Let's go Try and do some math in my head shirt three x plus five pi by six What is tough as I'll throw at you first of all I would hope you could still at least not get zero You can list the stuff and I would still ask you to do the same thing So if I gave you a question like this, I hope this works out nice I was trying to do a bunch of math in my head to not get a terribly terribly yucky scale. We'll see What's the amplitude? What's the period? It's not three Okay, you need this part need this part You just need to memorize the period doesn't actually appear in the equation. We always have to calculate it What was the period defined as? Two pi over b what number sitting where the b is that's my period two-thirds of pi phase shift is what? Whoa Five five by six isn't quite right. Oh Did someone say a word after that sorry? I say it Left or you know what can I just write negative? It's moved to the left remember back brackets backwards last unit stuff, right? Don't be scared to ask I Don't think I Don't feel I nailed this lesson in fact Whoever I taught it to first that morning. I totally chucked it and we did the whole lesson So whether it was my block ease or my block X I can't remember which ones I had first that day, but I suck that was the worst lesson I've done in years terrible really oh One more vertical displacement One and then there's the two magic ones that I've added you know what I'll write them both over here So that they're not in the list of the four. I Always took the period and I divided it by four Here's my original period if I divide that by four really what I'm saying is put an extra four in the denominator It'll be two pi over 12 although in lowest terms pi over six, right? And then the last one I said I always find the scale What I mean is the X scale how I want my X axis to go to make it easy to count What's my scale going to be here the common? I'm gonna say common denominator should have a pie in it I'm gonna look here here and here I'm gonna look here here and here. I Think a scale that would be useful would be Pi by six and To make my life as easy as possible Jen I'm even gonna convert my period to six. This one's already six. This one's already six. This one's already six I'm gonna convert my period. I'm gonna say four pi by six or since each square is gonna be pi by six four squares long If I'm not back where I started from after four squares in terms of my height I've moved let me show you just for a second why I'm so Paranoid about finding appropriate period Take a look at suppose your grid paper looked like that. Don't write this down. Just watch So I'm gonna go clip I'm gonna go clip. Come on. Okay, right here. This is why it's worth finding a period That's useful Covering up the writing mr. Do it how long is one wave? Two pi by three How long is two-thirds pi here? Can you see it doesn't fit into the squares at all because here each square is worth one-quarter pi How can you tell how far one period is or how far apart is each dot? Each dot is going to be pi by six apart. Well pi by six is a About that sort of kind of sort of maybe kind of sort of So that's why it's really worth planning your attack I want that confused you or help you. Let me nuke the graph out of the way I'm gonna bring back my grid paper again. You can freehand this if you want to or not So if you're freehanding you'd have a line and you'd have a line. I always do verticals first gen What's my vertical displacement? What's my amplitude? So how high is this graph going to go? How low negative two so I would make sure if I was free-handing That I had hash marks that went for high and Negative two You know I notice on this sketch that I've done. I have eight squares right here So I'll go up by twos. I'm gonna go one Two Three one two one two four And I'll go negative If I hang on mystery look you didn't label the negative one negative two negative three Negative four Jen. How high did my graph go? Got it wrong. How low did my graph go dotted line? What's my vertical displacement? Sorry Middle just taking the time to do that to put your vertical displacement and then amplitude up and amplitude down from there That gets rid of a lot of mistakes. I mean clearly I can see what this graph is gonna look like at least vertically What's gonna bounce between horizontal scale now? What did I want each square to be worth? Pi by six so if I was free-handing this I Would say well, there's pi There's three pi by six and then I would break it up into thirds since I have graph paper I'm actually gonna count six squares one two three four five six One two three four five six there's pi one two three four five six There's two pi and let's go backwards one two three four five six There's negative pi by doing that Sarah by getting the scale, right? By getting the vertical displacement and amplitude, right? I've cut down on most of the room for mistakes. I hope what's my phase shift Sarah Negative five pi by six. What's each square worth one pi by six? I think what you're really saying is my phase shift is five squares left Right in other words my first point is gonna be somewhere on That line right there. I'm gonna get rid of that though cuz that clutter is things up too much But my first point is gonna be somewhere here Where well am I graphing sine or cosine? I'm gonna be fussy. I'm not really graphing sine either Okay, so it's gonna start in the middle. That means it's gonna start right there But my next point is gonna be top or bottom bottom go down. You ready? Let's graph Phase shift of five squares left Middle cuz it's sign when I divided the period by four By a fluke it ended up being the same scale. That means each dot is going to be One square apart what it ended up being for example five pi by six Well, then each square will be five each dot will be five squares apart or it ended up being two pi by six Pi by three each square would end up being two each dot would end up being two squares apart But here hi by six matches my scale my next dot is gonna be one square over where bottom One square over where middle One square over where top One square over middle one square over bottom one square over middle Top it's gonna look kind of like a compressed graph. It's kind of squashed. That's okay Something like that and that's more than one period that would be enough to what we do one more or is that okay? I'll do one more And then we'll do some reciprocal trig functions, and then we'll do Q&A Let's try and make one up that doesn't work out quite so nice so that we can do a challenge here and find the test ones easier Sure, let's try that this could be ugly, but that's okay. Why equals to two coast Bracket bracket X minus sure Pi by two close bracket plus three Hoping this one will be a bit yuckier than the two that I made up. I'm still gonna list everything amplitude do Period don't say five Two pi over five which is probably gonna be a fairly yucky one, but that's okay phase shift Vertical displacement and then the two that I add Take the period and divide it by four to find out how far apart the dots are going to be Dividing it by four means putting an extra four in the bottom. You'll have Two pi over 20 or pi over 10 and this one also was working out a little bit too nice So give me one second. We're gonna change one number without changing a bunch of stuff But let me do some math in my head if that's okay Let's uh instead of making this pi over two. Can you all look up, please? Can you all look up, please? Hello Instead of making this pi over two take that two take that two making both fours Sorry about that little interruption now, uh That's period divided by four that means each square each dot is going to be pi over 10 apart And the other thing I said is let's pick a scale and our scale is going to be a good common denominator between anything with a p And any horizontal what do I have on the bottom here? five here four here You know what I think pi over 20 That's what I was trying to get something a little yuckier pi over 20 And I'm going to find a common denominator then I'm going to write this as Two pi over 20 Jen every dot is going to be two squares apart this time. That's what I'm trying to get two squares apart not one square apart Oh, and uh the phase shift is going to be over 20 over five pi Over 20 I'm going to move five squares right before I graph And the period is going to be times by four times by four eight pi. That's an eight mister do it Over 20, uh one whole wave should be eight squares long if I'm not at the same height as I started from every eight squares I'm doing this wrong We'll do do you want me to use graph paper or do you want me to freehand? We'll do democracy who would prefer graph paper left on Who would prefer freehand graph paper turned off? Some of you weren't paying attention and don't care, but the freehander is one so we'll go with that okay Is my phase shift to the left or to the right? It's my phase shift to the left or to the right So if I was freehanding then I would say you know what let's put my y-axis way over here And give myself lots of room going that way. How do I change a five into a 20 multiply by what? multiply by Right good old math eight common denominator stuff In fact I would I'm going to do one more thing now if you've already written this you're fine if your pencil you can fix this because um What's my vertical displacement? Three amplitude How high is this going to go? How low You know if I was really clever what I would actually do is probably put that line Down here because everything is going to be up there now if you didn't do that It's fine, but if you're going to erase it great because how high did you say I was going to go? We're doing verticals first now right how high did you say I was going to go? There's five This is a bit of a tough one to eyeball for me So I want to divide that into five roughly equal well as not roughly equal chunks This is where if you don't have graph paper You have a bit of advantage I hope you have a ruler you can measure it and divide by five and put hash marks I'm going to eyeball it. I think it's going to be one two three four. Ooh, that was pretty good For me How low Three up and from there two down how low What I'll label that and vertical displacement It's kind of weird because I don't think you've ever labeled to graph one three and five before Usually you labeled it too, but it's going to be way easier for me to do that And I'm not going to take marks off Graphing is you know scales are meant to make it easier to be convenient. Oh, I would add my dotted lines I would add my dotted lines I would have my dotted lines scale What do I want on the x-axis each square to be worth? Pie over 20. Yeah Put your pencils down and watch I could do this. This would be yucky. So don't do this I could go. Well, let's make that high What would that be? Pie over two or you know what? Let's make this 20 pie over 20. What would that be in terms of 20s? 10 that means I'd have to try and put 10 hash marks in here. Uh, yeah Not good I still want to use as much as the graph paper as possible But I think uh, how long is one wave? How many squares? Eight You know instead of calling this pie 20 pie over 20. Why don't I make this pie by two? That's going to be 10 pie over 20. That's 10 squares. Uh, how many do I want to fit in to get one whole wave? Uh, that should I'm going to go 10 pie Over 20 What would that be? 5 pie over 2 pi over 4 5 pi over 20 And then I'll try and divide these into 5 chunks again. 1 2 3 4 5 Didn't work quite so well. Mr. Duke a little larger Nope too much 1 2 3 4 that's better 1 2 3 4 Oh, well This one graph paper would have been way more helpful, but you guys voted I've done the hard stuff Can you see first of all what the graph's going to bounce between? Right uh What are we graphing sign coast negative sign or negative coast? Coast I'm going to start up high You're going to start somewhere there Oh phase shift What's my phase shift? It was pi by 4, but I found a common denominator that matched my scale 5 pi over 20 What's one square worth 1 pi over 20? How many squares right? 5 1 2 3 oh right here conveniently enough Where am I going to start right here? Up high When's my next stop going to be when I divided the period by 4 I got pi over 10 found a common denominator Each square is pi over 20. How many squares apart is each dot going to be? See it so 1 2 uh middle I'll drop back down to the x-axis to help me count again 1 2 bottom 1 2 I ran out of room. Oh, no, no, no, no. What's the other direction I can go? Let's go backwards. So starting here 1 2 squares backwards, where will I be? middle 1 2 squares backwards, where will I be? bottom One, well, you know what I've run out of room But I'm pretty sure I got a whole wave because don't I have two points that are at the same height along the bottom That's one whole wave. It's going to look like this Now some of you may say mr. Do it that looks like negative cosine. Well, no, no We started there to get positive cosine. I guess a negative cosine graph I could have phase shift one square right and it'd be the same graph from what it's worth, but I'm not fussy on how much of a section you give me Like I said, I usually look for two or three points and if you got those right I can tell if you aren't symmetrical That's probably a little bit of a yuckier scale than you'd ever have to deal with on your test 20s It's usually going to be six through halves or quarters or thirds or the most I've ever done at twelfths But twelfth actually works nice because three goes into it and four goes into it six goes into it Two goes into it. It's actually kind of a nice scale Pi by 12 15 degrees for those you first work, you know Okay, yeah The phase shift tells you where to start your graph. What it's really saying is Where did this line get moved to because normally whenever I just sketch sine or I always just started right there It's really where that got moved to just like your vertical displacement Is actually where your x axis got moved to How far apart Again, it comes from this One period was once around the circle before we did anything with it We divided it into four chunks because that was the four corners of our circle We want to continue to use that because it is a handy pattern middle top middle bottom middle top middle bottom How can I use that? Well, whatever new period they give me if I divide it by four That's got to tell me where those four corners are how far apart each of those four corners is on the graph Does that make sense? I hope I made that clear. I hope I hope I hope If not watch this tutorial again when you get home. I hope But that's enough on graphing. Let's talk about the reciprocals of this Okay So then what we said is I did a bit on graphing. I figured that was where the most confusion was Then I'm going to talk about reciprocal trig functions. Then I'll do q and a questions from the reviews Okay reciprocals And when I say reciprocals because the reciprocals have like asymptotes and weird stuff I'm going to include tangent here in my explanation because tangent is the ugly cousin even though it's technically not a reciprocal Although it is you just don't know that yet. Um, what's it a reciprocal of well, it's actually sign over cosine But that's a whole other issue. Um I'm going to kind of lump it in here. So start out like this Give the domain and I'm going to start out I'll use the word easy. I don't think any of these are all that easy. I'll start out less complicated And I'll go beyond I think what I feel comfortable asking you on the test so that hopefully you'll find the test So domain y equals tan x. What's the domain? No idea Well, that's a lie. I've since I'm teaching this fresh. I've taught it to three classes in a row I have actually memorized it finally. I normally never did I do a sketch Up tangent What have I told you to memorize about tangent? What does tangent go through? Zero zero. That's a fairly easy one to remember. I hope So I would go like this Tangent tangent tangent. Oh, let's use the funky graph paper that I have zip zip I would say it goes through zero zero remember it's got asymptotes right there And right there Where is right there and right there? What are the x values of those asymptotes? Now you got to be careful because every time I've been doing this last few days people have started to say Pi It's 90 and negative 90 if you're looking for an easy waiter because 90 degree angle That's one of the ones we you know, it's a right it's a right angle and a right angle. You can't take the tangent of 90 What's 90 in radians not pi Because yeah 180 is pi it's pi by 2 Oh hang on mr. Duke the tangent graph looks like that. Okay Its domain is all reels except Because it repeats itself and then goes like this Like this it continues going like that Its domain is everything but its domain is everything but its domain is All x is except that's why its domain is an x not equal to not equal to what? Where's my first hole? Where's my first asymptote? Where's the first location where tangent doesn't exist? And then they occur every so often how often well how far apart are these two? Exactly pi. That's why I memorized that picture. It tells me they're pi apart it's going to be pi over 2 plus multiples of pi Where n is an integer What if instead of give the domain? What if I said give the asymptotes? It's almost identical the asymptotes are the equations of these lines sitting in this gap They're vertical lines. What's the equation of any vertical line x equals y equals is horizontal x equals is vertical Where would they be? Well If I asked you instead for the equations of the asymptotes It would be x equals pi over 2 plus multiples of pi Where n is an integer and if you're thinking that looks an awful like domain. Yeah, you're right except no no Domain is where the holes are where the where tangent does not exist So it's a not equal to Equations of the asymptotes is hey, where do those asymptotes exist? It's an equal to And I kind of remember that because I will always ask you for listen closely Equations of the asymptotes Domain don't learn equal sign not equal I don't know whatever works for you, but gotta remember it somehow. What about um b y equals cotangent x No idea Oh I can figure this out. What does cotangent go with? Tangent that's why I desperately tried to leave that tan graph up on the screen there. I was going to scroll down. I don't want to do that because Cotangent will have asymptotes as well Where it will have asymptotes where tangent is how high? What remember our reciprocal what gave us asymptotes? zero high Where is tangent zero high? What point did I tell you to memorize for tangent? What does tangent go through zero zero where will cotangent have its first asymptote then x equals zero And then how far apart will they be? What was the period of tangent? Guess what the period of cotangent is Okay, I'm gonna write this but this is my brain process If they give me cotangent I go sketch tangent somewhere and I really remember my reciprocal stuff Oh by the way I don't think I'm asking you this but I've seen them ask once in a while in the provincial exam Where would cotangent have zeros have x intercepts have roots Wherever tangent has asymptotes because we said this for reciprocals Zero high gave us asymptotes asymptotes gave us zero high when we did the reciprocal thing Anyways, let's do the domain then I'll make we'll make a little note where tan theta equals zero Cotangent theta is undefined We'll have asymptotes so Where does tangent equals zero conveniently at zero not too bad and then Pi further pi further pi further pi oh So domain that's what x can't be It's everything except Zero plus multiples of pi where n is an integer although I think I've told you if this is multiple choice Sometimes they don't bother writing the zero plus because what's zero plus pi n I think just pi n but in our notes. Let's leave the zero plus to remind us. That's how we found it We said let's go find the first one And then their pie apart What if I asked you for the equation of the asymptotes of cotangent? And if you're thinking looks an awful lot of domain Lorentz, you're right because The asymptotes is hey, what are the equations of those vertical lines where tangent where cotangent has its gaps? So far so good Yep, we're gonna we're gonna get there next yep I'm gonna do she can't and cosy can't really and then we'll do let's muck around and stick stuff in there and see what happens okay Still okay I'm hoping to be done this section by about quarter after four and then we'll go q and a so I'm trying to budget your time I think we can get there But yes in answer to your question. Yes. Uh, oh see cosy oh Mr. Duke y equals cosy can't x They want the domain or they want the equation of the asymptotes. I'm no wait a minute. I'm smart. I can figure this out What does cosy can't go with? Sign that one. I know I would do a little sketch of sign right now What does sign look like sesame street is brought to you by the sign graph the letter s sort of And this is our basic sign graph before we've changed anything So good old amplitude of one and negative one one how high negative positive one how low negative one period two five Here's my question Where will the reciprocal Have asymptotes wherever sign is how high? Oh, this sounds very similar to tangent and cotangent Wherever sign is zero high now. Where is sign zero high here here and here Oh, I can tell you those values zero What's this value right here? pie What's this value right here? Can you spot the pattern when would the next one be? three pi then Four pi then but in fact, I think the domain Is all x's except starting at zero multiples of pi where n Is an integer Now you may be noticing a lot of these answers are looking the same Hey, that's this also happens to be the domain of cotangent It's because it just happens to have asymptotes in the same location You can't just memorize say oh, they're always going to be the same They won't always be the same but for what it's worth these two are Oh, what if I asked you for the equation of the asymptotes instead of saying x can't be zero plus multiples of pi You would say equation of the equation. Oh, yeah equal x equals zero plus multiples of pi That's the equations of the vertical lines Okay I'm going to add one more though for cosy cant range you see Andre cosy cant we already agreed with me. We have asymptotes there There in there What point would be invariant? What was the invariant height when we did reciprocals? What what heights never changed? See that would stay that would stay and then closer to zero shoots off to infinity closer to zero shoots off to infinity this black graph Is what cosy cant would look like what's its range? Well, it's actually two graphs see it It's going to be two ranges There's no easy convenient way to write this as a one-line statement. Don't even try It would be Everything below and touching negative one see it right that's this graph Comma or and whatever you want to write Everything above and touching positive one. That's this graph So if they change this and they ask me a range question if they do vertical stuff here This is what I would fall back on still always saying, you know what I know signs so well I can sketch whatever I need to see can't And then we'll do weird ones Y equals secant x Domain, I don't know Wait, but no, I'm not going to take a zero on this. I can figure this out Ah secant goes with which trig function I'd be silly if I didn't at least sketch cosine Uh cosine. Oh, that was the one That started up high and finished up high And that meant it was down low over here and it was middle middle Because the pattern was top middle bottom middle top middle bottom middle top in fact it looked like this Where will secant have asymptotes wherever cosine is how high Yeah, it's going to have them here in here And continuing but let's see if I can't spot the pattern What is sorry pi by two And Three pi by two by the way, what number is in front of the pi here? It's invisible One I think it's one pi by two three pi by two. I think the next one's five pi by two Seven pi by two. It's odd seems to be odd numbers of pi by twos I've spotted the pattern though. It's this x Can't be Where's the first hole occur? pi by two Plus how far apart are the holes? Conveniently once again exactly pi What if they wanted the equation of the asymptotes? x equals pi over two plus multiples of pi where n is an integer What would this graph look like? well Remember it would have asymptote there asymptote there Andre what were the heights that remained invariant? Do you remember? That means this height this height and this height would stay the same Let's do this one here getting closer to zero shoots off to infinity getting closer to zero shoots off to infinity Let's stand here getting closer to zero shoots off to infinity getting closer to zero shoots off to infinity in fact It would it's off the page, but those little u-ish kind of shapes Look something like that So I can tell you the range. Oh conveniently same range as Cozy cans Why less than or equal to negative one comma? Why greater than or equal to one? So hopefully you've already noticed lots of symmetry lots of similarity between these Now let's spend 15 minutes or so looking at weird ones And let's try and eventually get a little tougher than I would ask on your test Okay e Find the domain of y equals three cotangent of 2x Bracket So they're asking me to find the domain What what would the range be remember cotangent and tangent have a range of all reels? So, you know what the range is still All reels what's that three in front do vertical or horizontal? It's a vertical stretch. So if you're already from negative infinity to positive infinity You make it three times as far here still from negative infinity to positive infinity. So range isn't going to change I mean that probably won't ask you a range question about tangent or cotangent Boring just like I won't ask you a domain question about sine or cosine boring the sine cosine have all reels with no gas But i'm asking domain here Is there a horizontal transformation occurring in this question? what Ignore it temporarily First I would find just pretend there's no two there I would find the domain of just good old cotangent. How would I do that? I don't know the domain of cotangent What's cotangent go with? Okay, I would go like this zip zip tan looks like that. Oh boy. That's terrible Tan looks like that. Oh, yeah, mr. Dewick says that cotangent is going to have an asymptote where tangent is zero Normally the domain would be like this x Can't be zero And then since the period of tangent multiples of pi now, that's normally what has this done Horizontal what? Compression by a half and what that's going to do is that's going to put an extra two In the bottom of everything So that instead of your asymptote being here and then your next one over here It would divide this one by two, but what's zero left right divided by two Still zero left right, but it would also move this guy over To there half as far away In other words, they're not going to be as far apart The domain if I tidy it up would be zero plus multiples Of pi over two Or n as an integer what if instead of a two, what if they put a five there? Divide by five divide by five be zero plus pi over five times n Ooh, what if they got nasty? What if they gave me a fraction we better practice that one y equals secant of Two x over three Can you pretend that's written as a fraction secant of two x over three? That no relax. Where's haley? Oh not here. Okay First of all Is that a horizontal? So it's going to change the domain But i'm going to ignore it first I'm going to find the domain of just plain old secant And then i'll make whatever adjustments that I need Oh, I don't know secant Secant goes with which trig function Coast Coast looks like this You'll notice i'm getting a bit sloppier in my sketches because i'm hoping you're getting a bit better at this You know what secant's going to have asymptotes there and there What is there and there? I heard it over here already Pi okay, normally x can't be Pi over two And then it's every pi apart because that distance there is pi where n Is an integer. What's this two doing to the x? divide divide What's this three doing to the x? dividing times times By the way for what it's worth Two x over three. That's an expansion factor of three over two But if you want to you can just treat them each as separate little factors. Oh times by two That's going to divide the graph into divide by three. Oh that we've multiplied by three now I tidy this up clearly Mike. I'd write this as x can't be three pi by four plus multiples of three pi by two Where n is an integer whatever period thing they throw at me I can adjust I can tweak I go back to my original always Now I keep saying that some of you may have memorized because I already heard right when I drew the Or right when I wrote down c can't someone already said the domain You may have memorized the domain already and great, but I'm you know, I'll always fall back on a test on my safety I know sign cosine and tangent inside and out I can figure out anything about the reciprocals from those that I need to you. Let's do a phase shift phase shift Little trickier And then we'll do a period change and a phase shift and then I think that's all I can throw at you here I think that this tough as I can get equals Let's see. I've done cotangent. I've done secant. Oh, uh Cosy can't thank you. You can remember the function cosy can't uh What did I say a phase shift phase shift, right? x minus I don't know where the pi button is Stupid mister do it Pi over oh pu that's very good pi over four Can you put a pi symbol there, please This one is a tiny bit trickier Can you all look up for a second? Okay, you did watch a little bit of visual illustration here Cosy can't let's suppose the asymptotes are this far apart And they're right here When I phase shift it. I'm sliding it left right In this case. I'm sliding it to the right because it's minus Will that change how far apart the asymptotes are? In other words phase shift. I'm only going to change my starting point, but not my plus multiples of whatever That's the only slightly tricky part, right? Although in some ways it's easier because it's less work, but Don't get carried away with always changing. So, um Cosy can't I don't know uh, uh cosy can't goes with uh sign sign looks like that Okay, normally it was here here and oh normally it was x Can't be zero Plus multiples of pi where n was an integer But this whole thing this whole graph has been moved to the right. How far Pi by four it's been moved pi by four to the right So, you know what I think instead of my first asymptote starting at zero Where will it start at? Pi by four Will the distance between them change for a phase shift though? No Instead of starting at zero You'd move pi by four to the right. So that's where your first asymptote is going to occur But since you haven't stretched or shrunk it at all They're still going to be pi apart. Do I want to do a combined? That's harder than what's on on the test. I'm giving you one or the other but not both. Okay I've done cotangent secant and cosecant Tangent behaves similarly Right Now I am going to ask you for range of something to do with Reciprocals and since the range of cotangent is all reals I'm going to ask you a range question for secant or for cosecant where there's a vertical stretch And maybe a vertical displacement. We did one like that last lesson. I think in the example Somewhere I'm going to give you ask you for something to do with the absolute value of a trig function Probably what it'll be is I'll give you the trig function with an absolute value and say not tell me the range Anything that was below flips up anything that was above stays You have to visualize it a little bit first because there's a big chunk above and a little chunk below That was what I'll let you think about it. You'll sketch if you're not sure Otherwise most of the rest of the multiple choice while it's going to be either Special triangles exact values or I think there's three or four questions that involve x y and r and one with numbers Oh the point three comma seven is on your graph. Tell me the Cotangent as an exact value. Oh, I better find r and then cotangent y. That's x over y I would never ask you that because I gave you x and y so ask me for cotangent. We do boring I'd ask you for secant or cosecant. At least x y and r stump And I've already told you the written section is going to have two graphs and three Equations to solve The first two equations are going to be between zero and two pi One of them is a reciprocal equation And then the last equation is going to be sine coser tan and it's going to be a domain shift I can't remember what I picked for my domain shift I don't think I picked negative pi to positive pi because I did that with you as examples I think I picked something a bit different just because I was bored Okay Be prepared when you're solving the trig equations Remember the weird ones that are on here when you have a one or a zero or an undefined Usually that means you're right on one of these and your task will no longer works Then we fell back on the whole unit circle idea. We spent a day looking at that Otherwise, I don't think you'll find Too many surprises. There is look at your quizzes You're going to see one question that you'll say hey, mr. Duet didn't do a big song and dance above Oh, wait a minute. He gave us one like this on the quiz. Ah That's why he gave us one like this on the quiz. Okay I've given you three quizzes right this quiz one quiz two and I called it quiz 2.5 I think the answer keys are online in terms of video And I think I put the answer keys online as pdfs as well. If not, I'll do that tonight So having rambled for a while and finished four minutes ahead of my 415 thinking Question and answers for the remainder of this. I'm totally willing to go now You know what? Let me ask who's planning on staying longer for question and answers or who's gonna Ask it the other way. Who's planning on leaving right now then? None of you. Okay, I'll keep going then I just want to know if you were leaving if I had like 15 of you I was going to print up a bunch of these Tutorials that you're sticking around I'll print it up when most people are ready to leave You can get one tomorrow or if you email me tonight, I'll email you one, but you remember to email me I'm gonna be watching North Carolina play duke, but I'll have my laptop in front of me And yes, that's how much I Life I have I do schoolwork while I have fun depressing So from the reviews, which ones would you like me to go over? From which there's the first one or the second one Love to And I think I'm going well, I'm going to say I at least partly like number 18 I at least partly like number 18 number 18 has some good stuff Whenever they give me something like this um Read the question Algebraic You know what this is going to be an x y and r question Okay They want me to find tangent tangent is what over what in terms of x y and r Lawrence Yeah Yes So I'm going to say this tan theta is going to be y over x now that almost always means that they give me another trig function and that trig function Will give me two of x and y and r. Here's what they gave me except I see this That's what I see I see it as a fraction because sign is what over what You know what I think y is m and r is one. I'm sure of it Now don't make me have to give you that hint on the test if it's not a fraction It is a fraction. Everything is a fraction. It's always over one or if it'd give me a fraction like m over n I would say oh m is y and n is r um They also told me that we end in quadrant four That's here Will tangent be negative or positive Shannon is that okay right terminal arm is in quadrant four the quadrants go one two three four right So tangent Will they want me to find that will tangent be negative or positive? This would be a multiple choice question by the way I'm just saying tangent is positive here. There's got to be negative here, right? Oh, let's find r. How are r and x and y related y we have that handy Pythagoras little x squared plus y squared equals r squared in fact I think that x equals the square root of r squared minus y squared I think it's going to be this let's see y m Over which makes sense by the way would not y be negative down there. I like that uh x oh square root of Should be what it says in the back. I hope I think open it go Negative m all over square root of one minus m squared Answer key has a positive there. Let me think for a second Okay, hang on. Let me sorry. What were you going to say john? Okay, this is what I would oh Good catch. I consider this a cheap question. I didn't notice I didn't read this one and catch that they've done this once in a while on the provincial I will never do this to you In this quadrant is sign negative or positive negative Is there a negative sign in front of the m? Then where is the negative sign john it must be hit the m itself must be Negative it must be a negative number So since the m itself is actually a negative number and that's what the negative number is coming from here for tangent as well I guess they didn't put a negative in front because they're arguing that m itself is negative I hate that question. I love everything else about this question, but what I would do is I would put a negative in front To me that's so cheap You know what though? I know the guy who wrote this and I bet you he wrote number 18 after a question Just like this came on the provincial exam question like this came out. I think in 0405 I hated it because almost every kid was so proud of himself for catching this negative and saying uh-huh And that was one answer to pick from and they were mean they even made that the first answer to pick from And the correct answer was the positive one because Sign which was supposed to be negative didn't have a negative in front of it So it must have a negative built in that's where that negative was supposed to come from. Oh, please Back, you know what I got a positive video It was a proud moment Is that Kevin? Is that what you're wondering about why it was positive everything else about that question? I love Okay, I just think that's a really cheap loophole. I will not do that to you on your test And I don't think they should ever do that to you on the provincial. I think that's That's too obscure If they didn't have the negative answer to pick from I would live with it like if they didn't have that on the multiple choice to choose from then I would say Okay, then the kids will clue in oh, my answer is not there. There's got to be something. We oh because John saw it in a minute. John so you didn't oh well next Jim 8a from which review Second one of the first half for second First one Yep, I said those are decimals Since you can't do them without a calculator. I won't give those to you on test next test when you can't use a calculator sure so nine arc length Okay And 15 that sounds I think 15 is an x y and r 1 if I recall Number nine. What is the length of an arc right away? I would say oh, this is arc length And I have a handy way to remember the arc length formula. It's a equals r theta. It's the word arc sort of Uh they oh and they want me to find the length of the arc. So they want me to find a Circle has a radius of 15 And the angle is pi over five Hey divide by five divide by five the answer is three pi That should be what it says in the back or to have decimals You got them wrong That's the disadvantage of the esl In english, we know that's the word arc and you're going to memorize in a different language Sorry, my trick doesn't work great for esl students, but I think it works great for the english kids Sorry So it doesn't say three pi in the back because we have it as a decimal like nine points something three five I that would be an exact value kind of a question Okay, and again, there's going to be one arc length question And I will ask you either to find a or r or theta But I'll give you the other two be really really careful once in a while instead of giving you a radius They'll give you a diameter What do you have to do to find the radius then? Bye bye two Uh and once in a while they'll give you an angle and degrees No, no arc length formula doesn't work for degrees change it to radians I don't think I'm doing that on your test, but it happens on the provincial And I'm pretty sure at the end of the year on one of my mock provincials. I'll do it then But by that that'll be at the end of the year That'll be right around when you guys are all thinking about grad and not serious about school. That's when I really hammer you guys And which one 15 15 yeah I like 15 Yeah, sure. I like 15 I had to quickly read it to see who has another cheap one. I don't think it is That's why I said I like I said that for the other one now. I'm going. No, that was a cheap sneaky one with the extra negative um What do they want me to find secant Um, I see they've given exact values, but this is not special triangles. I think this is x y and r question So right away. I'm going to say well secant theta Is secant goes with cosine secant is r over x now. What did they give me here? cotangent cotangent is what over what in terms of x and y and r I think x is three. Oh, by the way, I'll do that right away And y is root 13. So I need to find r r squared equals x squared plus y squared r is going to be the square root of x squared Plus y squared r is going to be root 22 Now Let's double check the positive or negative because it tells me the terminal arm is in quadrant. What? three One two, we're in this quadrant in that quadrant is secant negative or positive Negative by the way in that quadrant is cotangent negative or positive Oh, so they're not trying to sneak a double negative sneaky move on me like they did in that last question You know what there's got to be a negative there which makes sense because in this quadrant here my x coordinate would be negative As with my y-coordinate. It is negative root 22 over 3. I hope in the back. Yes I haven't proof read the answer key, but my friend who made these is a teacher from terry fox. He's More careful than I am. I doubt there's any mistakes But maybe any others second review To see or not to see that is the question This he didn't ask to be 2d as well These are both way way way way way way way way way tougher than I would ask Okay, so I'm going to walk through these with you to see I've seen them as the nasty that year on the provincial 2d But if you understand me going through 2d, then you're probably in good shape So let's take both of these then I left them in your homework though because I wanted to That was the only secant and tangent ones that I could see that weren't totally gross Both of these Look, look, look, look, look both of these not factored off Those mean people Both of these I'm going to have to rewrite Before I actually do any stuff So I'm going to have to rewrite this as y equals Wonder what that yell is going to sound like on the internet You know I should really set people up what I should do is I should talk quietly for about 30 seconds So that they turn the volume up on their computer and kind of move back and then lean in and do the yeah Oh, that'd be great. Note to self Blow the speakers X minus what uh when I factor a three out what mathematical operation am I performing? Division that's the same as adding an extra three in the bottom dividing the same as putting it in the bottom of the fraction It's going to be high over six We'll tackle the secant in a second first of all Tangent, you know what the range is It's all reals. So range. There's your answer all reels all reels and for cotangent. You know what the range is all reels even that you know what If they slide it up That's still all reels if they stretch it. It's still all reels even if they flipped it It's still all reels. So, okay We're more interested in domain for this one Ignore all this junk This is really going to be yucky I would consider this a fair game, but yucky question if they didn't have the vertical displacement I'll show you why in a second Normally tangent has a domain x can't be It was where the asymptotes were do you remember it was Pi over two Plus multiples of pi where n was an integer see that Multiplying backwards see that It's going to be a phase shift pi by six to the right now. That's only going to move this It's not going to move how far apart they are We've already shrunk them now slide them sideways the pi over three n is going to stay the same But what I'm going to do is I'm going to add Pi over six to that there That makes sense. It's kind of crowded, but it's actually going to turn out to be fairly nice because I think I'm going to get this x can't be Can you see this is pi over six plus pi over six? It's two pi over six Nope running out of room. It's going to be pi over three in lowest terms. I'll reduce that in a second Plus pi over three n And so I'm going to reduce that To just plain old pi over three. Is that what it says in the back? Can you check the domain for me, please? Pi n over three But not pi over three plus pi n over three. Okay. Here's the problem Watch Here's your standard. Here's your tangent graph after the phase shift. So it's been moved over a little bit It looks something like oh good gosh. I should do it It looks something like this And your asymptote would be right here And your asymptote would be Right here. Let me think for a second how I'm going to explain this It says pi over three. Oh I know why Sorry What's that? What's that? Are you saying you can probably just combine those? And just make that a single pop that your first one is at pi over three That's when n is one your second one is at pi over three times two Your third one is at pi over three. So what the author did is he simplified this He said you know what those are like terms pi over three plus pi over three times A random number is going to be pi over three times another random number So we did do that one right and I accept that Okay This one Gotta go. It's fine I will post this online as well a pdf copy of this if I can I'll even try and email it to you guys I just don't know if the file is going to be too big Factor y equals four c cant Factor out the one half x plus. I have no idea Minus five you're dividing by a half Dividing when you factoring Dividing by a half same as multiplying by what? When you factor you're dividing by a half. That's the same as multiplying by what? In other words instead of pi over eight it's going to be or That's your patient. Okay. Sorry. I'd walk through it in more detail, but I can tell I'm losing you guys and that's fine Range let's do range first because I would consider this a fair game for range, but I wouldn't consider this fair game for domain trig function Remember c cant was everything below and touching negative one comma everything above and touching positive one What's that vertical or horizontal? It's in front. So it's vertical And it's not backwards because it's not next to the y where it belongs. It's an expansion by a factor of four in other words Set up negative one negative four positive one positive four What's that? Moves the whole graph out Yep, and since these are both y's they're going to be affected by the verticals five down Five down. That's your new range Okay, you don't want to domain if you're unlucky. Let's try domain C cant I don't remember Goes with which trig function. Okay cosine was zero at pi by two three pi by two. Okay. C cant originally was going to be x can't be pi by two Plus multiples of pi where n is an integer. That's your expansion factor everything's backwards instead of multiplying by a half You're going to times by two Times by two times by two And I know that'll simplify. I'll take care of that in a second Then I have my phase shift Now my phase shift isn't going to change how far apart the asymptotes are it's just going to Move it over so the first one's going to start somewhere different Is that to the right or to the left? So the left so I'm going to minus a pi by four from that guy. Okay The domain is going to be you know what to do this. I need a common denominator. I'm going to write this as four pi by four Minus one pi by four Can you tell me what four pi by four minus one pi by four is going to be This one's going to start off at three pi by four And then they're going to be every two pi and the part That what it has in the back I hope Two pi n to it. Yay I hope you followed that explanation. I'm not going to give you one quite that yucky range I'd feel comfy as one of the trickier ones sure Because the range you're not having to pull a bunch of fractions and junk and things Oh, these would all be multiple choice. I told you what the written is two graphs graph them One sign one coast. I'll even tell you that And then three equations solve them as exact values using special triangles or the unit circle that's an undefined or a zero or a one And one of the three the last one of the three is going to be a domain ship So solve it like it's between zero and two pi and then find the co-terminals to get whatever domain that you need to That suggests 16 multiple choice one six two to the fourth four squared square root of 256 At 1.5 each and what now again don't ask me why and I'm probably going to stop doing that next year because the provincials aren't quite as important As they used to be but I used to make sure I graded exactly the same way as the provincials since they were 40 percent of your grade They're going to be as one do it the same way. Yep four from which one 4a Oh, there's going to be one multiple choice. We're asking to find an equation. So this might be a good practice I think we'll end with this And then I'll print up over many copies that I need some of you wanted the review You want one you want one the other five that I printed they left Waste of paper they can see me tomorrow. Okay, wait a minute your test is tomorrow well They want me to write this as a sine equation I would probably go like this I would add that dotted line because I feel much better with that I would add that dotted line because I feel much better with that What I'm really interested in is where the middle is how high does this graph go? Uh, is that 10 or I think it's 12, right? Look at the scale How low negative 4 What's the total distance here? 12 to negative 4. What's the total distance? Well the total distance 16 what's the amplitude? Okay, so I know it's this eight If that was multiple choice, you might be able to cross out a bunch of wrong answers already. Oh, what did you say the amplitude was? Eight, how high does my graph go? 12 if I drop eight down 12 2 4 6 8 this should be the middle of my graph Is that the middle of my graph? It's also my vertical displacement. How high? You might be done. You might if it was multiple choice that might be enough to say I got all the rest of them Uh, I doubt it is though. Let's see Oh, they want me to use sign. Okay If I want to graph this as sign, where does sign start top middle or bottom? Right there What's my phase shift two squares right now? I need to figure out what one square is worth. Let's see One two three four five six seven eight squares make up pi. So I think each square is worth Pi by eight So my phase shift Is two pi by eights to the right? Of course if it was multiple choice, I wouldn't write it as two pi by eight What would I write it as if it was multiple choice? Okay, my phase shift is x minus pi by four Last thing I need is b not the period the period helps me find b But period doesn't appear in my equation b does well From there to there is one wave from top to top or from bottom to bottom is one wave One two three four five six seven. Oh eight squares. What was eight squares worth? ah, the period Is pi that means b is going to be two pi over the period There's going to be a two that that's the equation Multiple choice when you have four answers in front of you to pick from it's a lot easier because you can use a process of Elimination and get to the right answer often a lot sooner rarely. Do you have to find all four of them? To get the right answer usually you can get it to in two or three