Let's check it out. Nice. Hi everyone, this is Chichou. Welcome to another live stream. And today is June 7th, 2019. And this stream is a drop in math tutoring session, number 31. Basically, let's do some math and open discussion. And that's basically the intro. We've done a few of these, as you can tell, with the numbering system. We've probably done more than 31, but I wasn't calling them drop in math through sessions before. It's just, let's do some math, open discussion. But officially, number 31, maybe up to 40 or 45, I don't know. Okay, it's the end of the year right now. So there's a lot of, I'm personally busy with a lot of students. And I expect there's people that do need some help. Some help. So I'm here basically putting aside a couple hours for anyone that does need help with mathematics. Usually the conversation, we try to focus it on math, but it is an open discussion. We can go roam around, talk about different things. We've done a lot of physics on these streams. Haven't done too much chemistry. We've talked a little bit about astronomy, biology, and a few different things. Obviously some current events and politics comes up, but we are holding a live stream tomorrow at 7 30 a.m. PDT, Pacific Coast, West Coast Canada, my time at 7 30 a.m. So if you wanna talk politics, current events, news and stuff, tomorrow's stream is probably a better place to be than here because math takes center stage in these streams. And if we're talking about anything that is not math related, or even if it's math related, and if you have a question regarding mathematics, just ask the question. Hopefully I'll catch it in chat. And what we've been doing is creating a list here on this side and taking things off as we deal with them. Okay. Aside from that, welcome to a stream. I hope you're having a fantastic Friday. And what we're gonna do is just, you know, wait until people start rolling in. So usually it takes a few minutes. Lord, how are you doing? Good evening to you. Good evening to you. For me, it's afternoon and the sun is coming up now. It was all cloudy overcast and we might get sunned out. The sun is coming through the skylight, I think about an hour until the sun hits here and it gets really bright. So hopefully it's not going to fear with us riding on the board. I should have timed this better. Hey, Chicho Hanna says we have an outage at work. So we are all bored, nice. Well, I don't know if it's nice or not, but I'm glad to be doing a stream when you're bored at work. So you can just chill. Belgium had a little storm too. Still warm though, nice. Yeah, here, the weather is crazy here. Hanna is probably getting the same thing. It's like windy, cold, rainy, super hot and sunny, like August temperatures and then it goes cold again. So it is what it is, right? The chaos of weather, the chaos of weather. It's, I'm glad summer's coming though. I'm glad spring is here. I'm glad summer's coming. And just like all of my students, I'm glad the school year is ending because everyone's prepping for the finals and stuff like that. So everyone's putting a lot of time in. We were sending rain showers your way to Georgia, are you? Yeah, we had it earlier today and last night as well. I guess it's coming again. It's supposed to be coming. It wasn't even supposed to be sunny today. Man, I love spring. Probably favorite time, yeah. I like fall too. I like all seasons really. For me, spring is summer. I love summer. I like warm temperatures personally. I really like warm temperatures. Randall, how are you doing? Good morning, afternoon and evening. Yes, yes, yes. Very heavy rains. Okay gang, if we get very heavy rains coming up from Hanna's part of the world, power goes out and I'll see you guys tomorrow morning. Great lasagna, how was life? I'm doing good man, doing good. Been crazy busy. I shot a video yesterday, a comic book haul that I'm gonna try to edit it later on today and tomorrow. If I can't get it edited in the next couple of days and processed and loaded up, then it's gonna be a week delay before I'm able to get it up online.  Winter can have fun things. Winter is amazing too. For me, I get a lot of work done in the winter, just back end and stuff and I do a lot of marathon watches on different things that I've been meaning to watch. So I don't mind winter, but in our part of the world, me and Hanna, Hanna is Seattle, I'm in Vancouver, we get a lot of rain. Like it's crazy rainy. It's like what do you call it? Temperate rainforest, where we are. So a lot of people move here in the summertime that come see it in the summer and they go, oh, this is beautiful, beautiful, my God, so nice. And they decide to move here and the winter kills them. They don't realize that more than 50% of the days in Vancouver anyway, it rains. Like it rains, I forget what it is, like two thirds of the year or something, overcast a lot. Gicho is a random question, but did you ever consider having NA and come to Europe? I don't know what NA is. Consider leaving North America, having NA, leaving North America and coming to Europe. Index, how you doing? Welcome, welcome. You know what, Lord? I went to Europe in 1998, right? And I tracked around. It was beautiful, loved the diversity, loved the people, but I found Europe to be a little claustrophobic. I found rampant racism in Europe. Really, there was tremendous amount of racism in Europe. At the time, did I have a beard? I don't remember if I had a beard, but I did have facial hair and I was dark, you know, I think it was after the, I went there in the fall, end of summer, fall. I love Hennessy, I love Hennessy too. And I had really dark skin, so I was tracking around in different countries. I was getting nailed by police, just random passport checks, right? And then they would look at me and they'd be amazed that I was Canadian. Then they go Canada, and then they hold me, question me for a bit and then let me go. I didn't mind Europe. There's certain parts of Europe I really liked. I loved Greece, I loved Ireland. I love Budapest. Budapest was amazing. I love Southern Spain, Alacante and that area. And I spent two and a half months, three months tracking around Europe. It was fun, it was fun. I liked Berlin, Berlin was fun, just little raw, it was good. I love all seasons, I'm from Sweden, yeah, Sweden. Swedish winters must be hard, hard, hard. Yeah, I can't deny the fact that racism still is very active in Europe. For example, the N-word is still very acceptable here, really, for example, at least in my area of Europe, wow, wow, sipping Captain Jack this week, I'm sure, that's a lot of sugar, I think, I don't know, that's a lot of sugar. One thing I found in Europe, too, that I really, it was the alcohol, the drinking was over the top in Europe for me. I've gone through periods where I drink a lot and I don't drink a lot and right now I'm not drinking very much. Back then I was in 98 or so, right? But I found it crazy that beer, wine, spirits not so much, but beer and wine were daily activities of a lot of people. So I found a huge chunk of Europe to be functioning alcoholics and that, to me, was just exhausting. Even though I had amazing times doing, you know, partying and drinking with friends and stuff, so alcohol was insane. What's on tap this weekend for you, Chico? Any hikes? I'm gonna be, I'm busy with my students, man. I'm gonna be this weekend, like last couple of weeks, three weeks or so. It's been mainly four weeks or so. It's been mainly focused on doing, dealing with students, doing these live streams. I haven't done too many videos where I'm shooting and editing because I've been crazy busy, but this weekend is just work for me. For the next week, couple of weeks, it's just work, which is basically I'm gonna do editing. I'm gonna work with my students and do some backend stuff. Yeah, we love our alcohol, even I do. I guess it's part of our culture. Beer for sure, living in Belgium. Yeah, so much beer. Racist stuff in EU at the moment. Seems very bad, you're right. Even back then, for me, it was insane. Like, there was a racism that you would have encountered in North America or Canada, my part of the world. Like 20 years before, I was encountering again like in 98 in Europe. I was like, wow. We even have something called aperitifage, aperitifage, which pretty much means drinking alcohol before, aperitif. Aperitif, I think they call it in French, no? And most traditional households in Belgium do this. Yeah, I don't mind a little bit and stuff. It's the excessive beer drinking and I took part in that. Like, I went to Ireland and I loved Ireland, but Irish food at the time was so heavy. Same with Czech, like I went to Prague and the food was so heavy that I found myself drinking so I don't have to eat. Mask of Raven, how are you doing? Aperitif, yeah. And we do have, Armenians have it as well, like pre-food, little drinks and after-food for sure, right? People don't even notice regarding the racist stuff. Yeah, you notice when you're coming outside of, from outside of Europe, especially from where I am, not that there isn't racism in Canada. There's tremendous amount of racism in Canada, right? I've encountered it, huge. I've been in fights because of that. Me and my younger days, right? But in Europe, it was cultural. It wasn't, it was within the society.  It was a core identity of some of the areas I was in. It was very weird. People hate others because they have different colors, skin, sap, very sap, or different religion, or different eye color, or different anything, right? And I think that's one of the reasons. I mean, if you think about it, Europe has started two world wars, right? Like crazy. And in Europe, there are countries that are very small, so they have to fight off invasions. There's a lot of history in Europe and a lot of violent history in Europe. Also, every Friday, all students and working class people go to cafes and bars, clubs to drink, yeah? Here is the same. I don't know if it's everyone. One of the things in Canada, alcohol is pretty expensive here. So students, there's a lot of binge drinking in Canada. Like tremendous amount of binge drinking in Canada, as I found there was in Europe as well. And I participated in that. But after a certain point, you just get exhausted and it's very bad for your health, right? Lucky for me, I'm in part of a world where cannabis was accepted. So cannabis consumption is a huge chunk of people if they want stress relief and stuff like this instead of drinking cannabis they partake in. I think it's more of a cultural dissonance or however you wanna call it. Yeah, yeah, European, average European people don't really like outsiders. It's a harsh truth. For me, it was weird. It was weird encountering it. Because I grew up here and I encountered racism in Canada a lot. Well, I've gone through phases. It goes through phases depending what the corporate propaganda mainstream machine is doing, right? Getting people all stirred up and war rhetoric and stuff like this. But, all right, 200 like you. But the racism in large part is in the subculture now. It's not prevalent everywhere. Not even about skin color. I would say it seems more nationalism. Like my country is the best about everything else. Yeah, that's what I encountered in Europe mainly. But Chico is not even against just different looking people. Also against fellow white people, for example. Belgian people are really harsh against French people. Yeah, it was crazy. Like for me, let me tell you a story about Ireland. Right, I was in Belfast. I made an intention to go about it. There's a video I plan on making as an intro. By the way, if anybody has any math questions, please post it and we'll do math. But because I've been teaching a lot of mathematics, just reviewing, cramming, I don't have anything planned out to do. I'm just making myself available. So just so you know, if you have a math question, we'll deal with it right away. We'll stop all conversation, right? But just to give you examples as a conversation is here. But both speak French for me, they are all the same. Same with me. I treated everyone the same, but I didn't really realize that I couldn't even tell the difference between the different beliefs. I want to use medicinal marijuana, however, I am afraid because I don't know what I am putting in my body and I don't know how I will react. Research, Irwit is amazing for research. And there's a lot more news and information available. And Hannah, you're in Seattle, Washington. So there should be a lot of research available to you. Not true. The North of Belgium speaks Dutch, the South speaks French. Define medical marijuana. And is medical marijuana only? It's a good question, Blueberry. Hello, Blueberry. How are you doing? It's a great question. Define medical marijuana. Like a lot of people self-medicate. You don't want to be in a centralized system. The database, right? Because that database is shared within different countries, it gets hacked and stuff like this. So a lot of people self-medicate without a doctor's note, which is a legit thing to do as far as I'm concerned. It's okay, most people don't know. Most people don't know. Do, do, do, do, do, do, do. Most people don't. So the story about Belfast. So I went to Ireland and I really wanted to visit Ireland in 98, okay? And during that time when I was there, there was still a little bit of flare-ups in Northern Ireland, okay? But I wanted to go to Belfast, so I went to Belfast. I want to use it for my own medicinal purposes to relax and chill. Liquor is also medicinal, I self-medicate with it. Yeah, I agree. Alcohol can be medicinal as well. It's just a binge drinking aspect of it. And everyday drinking and how are you doing? How's life? And everyday drinking is too much. Like if you're drinking two to three beers a day, your liver is receiving it as if you're an alcoholic.  I know this because I know people in the medical industry. I've done a fair bit of research into this, okay? So you definitely don't want to be drinking, you say, oh, I don't drink too much. I'm a moderate drinker. I drink two to three beers a day. You're doing the same damage as an alcoholic is doing.  Here's my suggestion. Clear your mind for negativity, get a joint and maximize smoke, have a joint. And you don't need blueberry, good recommendation and Hannah. Like people assume you have to smoke a joint or you have to consume a lot or hit a bong bowl or something like this. You don't, right? You could use CBD oil. You could use stuff that's not, not, not THC. You could just take one bath, pull, right? Or one puff, okay? I only drink one to two a week, only two to three drinks max, good. I binge once a month. Some months I take off, good. Well, the binge, you know, North America, the teenage binge drinking is insane Youth 20s, good for me. I don't drink, yeah, very good. You're saving your internals. But in the long run, it's better not to, unless you, I can't say it because you can use it for relaxing, right? Stress is number one killer in the world, number one cause of cancer and destructive behavior and stuff. So it's okay to relieve your stress with certain foods and drinks, right? So you have to balance. Personally, I am in person, personally a very stressed anxious person. Also a very bad sleeper. Medical marijuana really helped me with that. More dead people because alcohol as with other drugs together. Yeah, alcohol is crazy, very destructive. I binge drink beer all the time, mainly because beer is 3.2% alcohol here. Where is that, Utah? I'm assuming U.S. In Canada, beer is between rounding up between five to six and a half, seven percent. Okay, four and a half. Let's say four and a half to seven and a half percent.  You could, like Canadian drinkers American drinkers under the table. Really, I've done it multiple times, right? Because Canadian beer is stronger percentage, right?  For the first time ever, I'm experiencing being flat broke.  Yeah, however, blueberry, if you're flat broke, adjust your spending habits. I've gone through multiple phases of being flat broke. Right now, I'm on the boundary, really, right now, right? I owe someone a little bit of money that I need to pay back. Okay, so that's what I'm working on, paying them back, not much, right? I don't like to be in debt, but I'm pretty much flat broke right now. But since I've gone through multiple phases of being flat broke and having very, being very, very flush, like really, I go through waves of it, right? Over the years, I've adjusted my spending habit and I've learned how to do things that bring me a tremendous amount of joy when I don't have money. Okay, walks, reading some of my comic book collection, writing, creating content, getting in touch with some old friends, catching up on news, binge-watching TV shows or movies I've been meaning to do, just going fruit-picking and going to swimming in the summer, lake-jumping and stuff like this. How many more years until it's federally legal? Good question, I wanna buy at the grocery store. I know you're in Washington, you should be able to buy at designated places, but federal government, United States, I don't know. Canada federally is legal now. So in Canada, we're good to go.  In most states it's like five or 6%, okay, okay. Beer in Germany. I better don't say anything about it, yeah. German beers. There's amazing beers in Europe. Yo, Dante, how's life? How you doing? Welcome, welcome. What are we talking about? Drinking Donna Matt's drink. The beach is free, the beach is free. I spent, Anna and Blueberry, I have spent many, many, many, hours on the beach, reading, writing, note-taking, organizing. I do a lot of work at the beach. I tell people that's one of my office locations. Index, Chicho, I'm living in Washington, DC, working for the government for the summer. I know, I know, but my internet at my apartment isn't great, so I'm not sure how much I'll be able to watch your stream for the next few months. No worries, index. Thanks for your heads up, okay? I just want to say hi and let you know I'm still around. Okay, thank you very much for the heads up index. I hope you enjoy Washington, DC, and I hope you don't get in trouble working for the government, right? There's some, I'm guessing there's some really, really nasty people in that area, right?  And no worries about being able to pop in or not. I know you've made it pretty clear you're gonna be around for a while, man. Thank you, I appreciate it very much, very much. The support is phenomenal, brother, phenomenal, brother. Thank you for that, thank you for that. Rep pop up, you can't smoke if you work for the government in the federal, for the federal government, no? In Canada, you can because it would go against if it's legal, it would go against your rights if you can't smoke. So I don't know what it is. Yeah, in US, yeah, in US federal government, no. I don't think so, and it's weird because the federal government likes to pass laws where corporations can test you for drugs and stuff like this, right? But last I checked the US government, federal government, people in Congress and stuff, they never get tested for anything, right? Not that they should, no one should. At the same time, it's nice to be able to sip beer all day on the weekend without getting super drunk. It is, it is. I don't really drink many hours at the beach, high as kites. Oh yeah, just in the US. Well, you can't smoke at work, that's for sure. Yeah, it's just like you can't drink at work. If you're, definitely if you're operating anything, you know, you gotta be mindful, but in private, I don't think anyone can touch you if usually, if it's legal. Yeah, if you're clever, you can get away with it. CBD is usually okay. Like I've known people in Canada, one thing that happens in the factories and the tar sands and different industries and stuff like this, there's a lot of, what do you call it, places where people work, camps and stuff like this, or even if they live offsite and they go, they do drug testing, right? But there's things you can buy to filter out your piss, basically.  It's just all ridiculous, really. Canada for Germans are like friendly Americans. Like not the crazy ones. Thanks for the trust, I guess. There's a lot of crazy Canadians around. I can tell you that. Technically, the feds could rate legal dispensaries at any time, yeah. And they've gone, you tell you're correct. And they're going after people who are doing things that are legal in the state that they're in, but federally they're illegal. And one of the things they're doing, they're going after people that are politically active, right? Weird.  My tolerance is very high, though. Yeah, one thing you can do, rare people, is just stop consuming for a while, go on a sabbatical, right? Which is something really good to do. If you ever find yourself that you're addicted to a drink or food or some kind of medication or anything like this, like really, even coffee or tea, go on a break, take a break, right? Don't become addicted to anything. Addiction is one of the most destructive forces in the Western world. Maybe food, alcohol, drugs, gambling, sex, anything. People here get addicted to so many things. It's insane in gaming, whatever it might be.  Take a break from it, get your life in order, and then you can do it in moderation if you can handle it. If you can't, stop it permanently. Let the laborers out. A lot of the biggest white and nationalist names are from Canada, are they? I didn't know that. Cheecho, can one smoke too much medicinal marijuana? I like calling it cannabis, Hannah, because marijuana, I found this out a while ago, but marijuana was a derogatory term that the US federal government, FBI, I believe, or I believe it was the FBI that came up with, or at the time, whatever it was called, I think it was the FBI that they came up with to make people think that marijuana was something different than cannabis, because cannabis, everyone knew what it was, hemp and cannabis and stuff like that, so they needed to name it something to make people be okay with criminalizing it. It's just crazy, word play, right? The war of words. In terms of, can you consume too much cannabis?  Okay, the level that kills you is basically your lack of oxygen, like you're consuming too much burnt matter, so you die because you're not breathing, right?  So everyone to their own means, right? To their own tolerance, right? Some people are allergic to strawberries.  Some people can sit down and eat a bowl of strawberries, right? If you find out a certain food does not agree with you, don't consume it, it's not good for you. Your body's saying no, period. So Canada definitely has crazies. Canada definitely has crazies. 100%, I've come across a few of them myself. True, I stop a few times a year when I travel with family, good, good. I take tolerance sprays from alcohol and coffee from time to time. If you use it every day, it begins to lose its effect unless you keep taking more and more. Yeah, which is part of the problem with many things. Addiction can be very complicated, but you're right. Yeah, addiction is complicated, very complicated. There's multiple causes for addiction. Trauma is one of the main reasons why people get addicted to different things. And stress is one reason and whatnot. But in general, for me anyway, for my life anyway, if I find out that I get addicted to anything, I cut back on that thing. Even if it becomes difficult for me to do so, right? Even if it becomes difficult for me to do so. Okay, I'm gonna stand, stretch a little, instead of sitting on the stool, right? I guess people must be busy studying. It's weird, because my students are in panic mode, some of them, some new students have gotten where we're trying to get them to a reasonable mark or a passing grade where we're doing a lot of work, trying to get them to that level, right? So that's what I'm doing right now with mathematics. So I'm assuming right now we've got no one on the stream, which is desperate to do some cram sessions for their math, right? Which is funny, which is funny, which is like for my students, next week the final exams start in two weeks and three weeks. So in the next three weeks, they have a lot of exams, right? That's crazy. I'm ready for math. Me too. I could teach some stuff, but depends what you guys, if there's any questions, if there isn't, we could do something. What should we do? What have I been doing with my students? Find the minimum value of this. And that guy, we could mask over them. And sure, let's do a quadratic. Quadratic is one of the main ones, right? So we've got this. X squared plus two X, plus two X minus four, right? Plus two X minus four and F of X. And if we're talking about minimum, this is a quadratic, it's a parabola. So this thing opens up because the number in front here is positive, is positive one, right? So the parabola looks like this. This is gonna have X intercepts for sure. So it'll look like this, something like that, right? So minimum means we wanna find the lowest point, basically the vertex of the parabola. So you gotta complete the square through this. You gotta make sure the X squared is by X squared is a one in front only, and it is. So all you gotta do is just take this guy, divide it by two, you get one, square this thing, you get one. And then you sub that back in side the brackets, right?  Grab this two and keep bringing it out, becomes minus one. So this factor is that guy. Two numbers that multiply to give you one, I have to give you two, one and one. So this becomes X plus one squared, minus five. So the minimum is negative five, right? So the vertex of this thing would be one, negative one and negative five. So we're actually on this side, right? The Y intercept is negative four, and that was one, negative one and negative five. This was zero and negative four, and the axis of symmetry is this two, so you go one more over. So this would be negative two and negative four, and you can graph your problem and it goes like this. So the minimum is negative five, right? Like this is speedy Gonzales style, this is grade 11 mathematics in my part of the world. Which equation mirrors the female menstruation cycle? I think you made a video, but it's been a few years since I have watched this. I don't know what the equation is, but I've been drawing it.  I'm a math tutor, right? I guess I teach, you can call it teaching, but I don't, I think the word teacher means you're certified to be teaching in a centralized institution. Is it educated problem? I don't know. I teach mathematics, I teach mathematics. The female menstrual cycle, let's do chicho female menstrual cycle. Let me find you the video that I was referring to. It's a great video, I love it. Chicho menstrual cycle video. Here's the link to the female menstrual cycle. All right, menstruation, ovulation. Let's put this on here. Fun, here is chicho, the chijo, the chijo, maybe I'll start using that. Chicho, the chijo, so that's the ovulation. So basically the reason I did that is because really as males, I'm assuming there's a lot of us that are male here, right? Not female, I guess, I don't know. We don't menstruate, right? One thing I highly recommend, have an appreciation for what women, females go through, okay? For me, I really didn't have an appreciation for this, and I mentioned this in this video that I linked, right? For me, I really didn't have an appreciation for this until like graphically it was like, wow, right? Or thinking about it on the mat, lens of mathematics where you see a cycle going through, right? So basically the graph is this, right? You have very general research for this for that video. I'm going by memory. This is the graph that I have in mind, right? I just looked up the problem you saw. I don't know there was a formula for quadratic functions. Yeah, literally never learned this in school. You never learned that in school. The x is equal to negative e plus or minus the square or the b square minus four is equal to a. That's for the x intercepts, but completing the square that had to be, that had to be part of it, no? But the female menstrual cycle is this, right? There's menstruation and then ovulation, right? So it goes something like this and this is a 20, I mean, my graph's a little bit too thick. I never understood this stuff until I started dating how complicated the female organs are. Yeah, it is crazy. They are so complicated, so much they have to do to take care of themselves. Yeah, no, I learned some calculus, but not this, wow, Dante, that's crazy they didn't teach you that stuff. That's like core essential. Dante, you're Canadian, no? You're Canada, Ontario, I believe. They didn't teach this to you guys in Ontario? So here's menstruation, let's do this. And then ovulation, right? Again, this is a brutal graph. I have to look at the graph to make sure I know what it is. And this cycle, that's called a whole cycle, 28 days. Would it go, hold on, let me draw this properly even better. Hold on a second. So then you would have this, right? And then you would have menstruation again and so on, right? So from here to here would be the one period, period. So if this is menstruation, so if this is menstruation, this is ovulation, ov-vue-lation, I'm not a speller. And then this one would be menstruation again, right? So the period from here to here is 28 days. And it goes through a moon cycle, right? So that's why there's a lot of people that say the year should be broken up into 13 months instead of 12 months with the days changing between the months, right? You could make it 13, 28 days. And then every four years or something, you adjust the time a little bit and it works itself out. So it's the lunar calendar, right? So female menstrual cycle goes through the lunar calendar, right? Menstruation is when they're shedding the lining and the ovaries, the aches, right? What does the y-axis? This one would be intensity, right? Intensity. We call this intensity in the graph that we made, right? Intensity meaning when your menstruation is in peak. So this period, the period is 28 days, right? Where you go through it again. The cycle begins again, right? This thing, this menstruation, this thing varies from female to female, okay? On average, it's about five days long, right? So when the bleeding starts, a little bit of bleeding starts, you can count that as day one. And in day five, you're getting your last bleeding. Usually in day two and day three is the heaviest bleeding, right? And again, I'm a guy, I just know this because for the graph and girlfriends, female friends that I've had and I've made it, I sort of spent a little bit of time to really appreciate what it is that they are going through because as males, we don't go through this. And there's chemicals being released in their bodies, right? Hardcore and ovulation. This is when the eggs are forming and it's going through the whole thing, going through tubes and coming out in the linings. Like really, I don't know the terminology. I just know this is where they're making it. This is when they're shedding it, right? If you want to think about it. Some people actually calculate this. Now, what is that? Semi-scogger. Oh, I'm not going to cling on that, don't they, just because it's a PDF. I don't want it to lock up anything. Sometimes PDF lock things up, okay? So this is intensity. So in day two and three, you would have the most bleeding, right? And there's cramps and like literally, there's so many chemicals going through their bodies. And there's a lot of women that have iron deficiency because they're bleeding, right? So there's a lot of women that take iron pills to get their irons back up again and stuff like that. So there's huge, huge things going on. And if you, so if this is 28 days and if you extend this to a whole year, right? You basically get 13 of those cycles, 13 of these guys, right? Do, do, do, back to back, right? So you basically see this and this and this and one is menstruation, one is ovulation, right? So this would be one and then do, do, do. So you get 28, sorry, 13 periods in a year, right? So 13 times in a year, there's serious release chemicals being released. They have to deal with thing, heavy, heavy. They made a model for the menstrual cycle. There's a model. There's actually a function for the menstrual cycle. I gotta click on that piece of, I'm gonna click on it. If we lose the string gang, I'm gonna come back right away, okay? Let's see what we get. A mathematical model for the human menstrual cycle. Ah, sweet. Let's check this out. When did this come out, 2013? When did I put out my video? I don't know when I put out my video. I forget when I put out my video. Ah, where's my, when did we put out my video? I put this out in 2012. I didn't, I haven't seen this one. Abstract. Do we read this? The abstract says this. A simple mathematical model framework is developed to describe the hormonal interaction of the human menstrual cycle along the hypothalamus piturary ovaries axis. The framework is designed so that it can be readily extended to model processes that disrupt the normal functioning cycle. The model in its most basic formation exhibits multiple periodic solutions. One of which shows the key characteristics of a menstrual cycle, whilst the other indicate possible abnormalities, sometimes observed in women of reproductive age. The basic model is extended to encompass receptor down regulation. Regulation as a mechanism to describe the desensitization of the piturary to continuous, simultaneous, I don't know. So I'm gonna keep on, I'm gonna look for the graphs. I wanna see the graphs. Is there graphs? Oh, there's this. There's little charts. Oh, look at that. Look at the formulas. Cool. Cool. Oh, look at that. There's the graphs. I love the graphs. Look at that. So if you go to page eight, you see the graphs, right? That's cool. 128 days. There's two cycles. So they grafted LHE2. So we'd have to find out what the axes are they're talking about. So these must be, these are the chemical releases. Yeah, I looked at some of the stuff when I was doing the video. There's some crazy graphs as well, but I didn't find any mathematical equations for it. It was mainly describing what they were and the different chemicals being released. This is cool. It's like a heartbeat, right? The heartbeat is a crazy trick function. That's cool. Look at that. It looks like a heartbeat. Wow, look at that. They got tables and references. That's cool. I'm gonna bookmark this. Let's do this text. Draw. Cycle. Math. Thanks for the, sorry, gang. When I find stuff like this, I like to um, what do you call it? Graph, there you go. I like to bookmark these things. Oh, it's not bookmarking, it was a PDF. Son of a, I gotta grab it another time. My bookmark thing, Dingo doesn't diggy, whatever it is, the Jingo doesn't do it. I don't know what that word means, brother. I never understood this until I started dating. How complicated the female was. Oh, I've read that again. I should calculate this, yeah. That's cool. That's cool. So it's, it is complicated. And the chemistry, the biology of this, like I've seen graphs of this when I was doing the stuff and I embedded some of those graphs in that video where there's different color tones for the different chemicals being released and stuff. And there's pain associated with this as well, right? So every 28 days, some, some, some women don't go through it, some do, right? Where the pain, the bloating, the exhaustion, it's just the headaches, it's pretty intense, man. I don't think as, as males, we really fully appreciate what women go through, which is our loss, really, because we don't have as deep a connection with our bodies as women do to a certain degree, right? But once you learn it, it is brilliant. Once you learn it, you cannot unlearn it. That's the beauty of it, right? Once you learn, you cannot unlearn, which is fantastic in my opinion. Really learn it, right? You can make it a part of your life and incorporate it into your life for good, right? For good. So that's one math. What else am I doing? I've been doing what with my students. We're all over the place with my students. Whatever I'm doing, that's everywhere. We're everywhere with it. There's definitely a lot of quadratic stuff, logs. There is, right now, I just have one grade 12 students. So the grade 12 stuff is, I'm not as actively, there are years where I have a lot of grade 12 students and some years where this is one of the few years where I only have one, usually I have two or three or four sometimes. I mean, there's square root stuff going on, like just preliminary stuff, right? It's one place where a lot of people lack how to do things. For example, one thing I ask my students, in general, you need to know how to do is something like this, right? 27 to the power of negative two over three, right? I ask people how to do this. The majority of people have no idea how to do this, right? Even though they teach this in grade nine, grade 10 heavy, right? And the solution to this is basically you have to appreciate that a radical, right? Whatever the number is on the radical, that's equivalent to writing whatever this is and this goes in a denominator in the power, right? So radicals are really powers, but the denominator of the power. So all you can do is bring this to the radical and go cube root of 27 to the power of negative two. Cube root of 27 is just three, right? You're looking for triplets.  So three threes can come out of that thing as one three because the number here basically means as many, many clones as you have over this, you can bring them out as one thing. So this thing basically comes out as three and is power negative two. Negative power, it does things. Anything to the power does things to this base number. It doesn't directly interact with it. So negative power, what it implies you flip the base number. So this becomes one over three squared which is equal to one over nine, right? So there's some of the stuff I do with students just to give them refreshers, remind them how to do these things. Pretty important, really important, really. And these are just exponential radical. And one way, one of the things you get from radicals the function anyway for people, if you got f of x is equal to x squared, right? That's a parabola, right? That looks like this. And the radical functions basically we get by taking the inverse of the switch, x with y. So if we take a function like this and say, hey, what's this function that'll look like if we flip it over to the line y equals x? That's the line y equals x, right? You're using that as a mirror to go to, right? So this thing goes, right? Goes like that. So the graph is gonna look like this, right? The line y is equal to x at the point one, one, that stays where it is, right? So here's a point one, one, so the graph looks like this, right? Now if you're gonna call this a function, you can't have both the top and the bottom of this thing because that's a relation. That's what it looks like visually, you're flipping it. But what you're really concerned about is just either the top part of it or the bottom part so you can kill this, right? So radical function looks like this. Algebraically, it's this. You switch these guys around, so this becomes x is equal to y squared, right? And what that's done is switch the x and the y, which means you're flipping along the line y is equal to x. And then you gotta get isolate y because that's your function, right? So you take the square root of both sides. So y is equal to square root of x. So y is equal to x squared looks like that, y is equal to square root of x looks like this, which is half the problem. That's actually this guy here. That guy graphed there, right? So that's what a radical is, is related to the exponent to a certain degree, right? To the power, okay? But the first thing they do, they don't teach you this in grade eight, nine, 10, right? They teach it to you in grade 11, but you've already done dealt a lot with the radicals, right? So anything to the power, here, let's do a large one. Let's say you're gonna take the fifth power or fifth root of 64 x to the power of six, y to the power of seven, z to the power of 10 over the fifth root. Okay, let's make this nine w five q seven. So you can break down 64. Here, let's do this up here. So if you're gonna break down 64, 64 is eight times eight, two, two, two, two, two, two. And we're looking for the fifth root of 64, right? Means you're looking for five of a kind. So you can grab these five, five twos. You can bring them outside the radical symbol. So five, not five, two comes out, right? Five twos come out as a single two, right? And then what's left inside the radical is two. So this is the fifth root of two. So if you're solving for this or simplifying this, on the outside you got two, okay? In here, I'm gonna leave enough room to fill in whatever we need, whatever's left over, and we had a two left over, right? And then what you do, you take care of the rest, okay? If you're looking for five of a kind, x to the power of six means you got six x's, right? So let's just do it for this one so you see. The fifth root of x to the power of six, this is the fifth root of one, two, three, four, five, six, there's six x's multiplied together. So take five of them and you can bring them out as a single x, right? So this becomes x comes out and you got an x left over and still inside.  You ask yourself, y to the power of seven, how many y's is that? That's seven y's. You're looking for five of a kind. You can bring five of a kind out as a single y, right? So five of these y's come out, become one y and there's two y's left, so this is y squared, right? Okay, let's add a m to the power of 23 so you see how this works as well. This is z to the power of 10, right? So there's 10 z's, right? How many groups of five are there in 10 z's? There's two groups of five, so z to the power of two. So basically if you wanna think about it, this is like saying the fifth root of z to the power of 10 and like we said, this goes in the denominator in the power, right? So this becomes z to the power of 10 over five. 10 divided by five is two, so the whole thing comes out. So z squared comes out. You got two groups of z's, five z's together, okay? So we took care of the top. The top's done, no more z's left inside, right? Then you can do the bottom. Nine is three times three. There's no groups of five to be had, so nine stays in the bottom. W to the power of five. We got five W's. Well, we can bring five of a kind out as one. So it's just W in the bottom. Q to the seven, that's like y to the seven. Five Q's come out as a single Q. And you got Q squared left. And you got M to the 23. How many times does five go into 23? That's what you're really doing, right? How many groups of five are there in 23? There's four, right? So 23 divided by five is four. M to the power of four, and you got three left over. M to the power of three. This guy, simplified as this guy. Looks more complicated, but it's not, okay? It's easier to deal with, to a certain degree, depending on what you're doing, right? But that's what you can do. So this is something that you have to know how to do in grade. In my part of the world, by the end of grade 10, going into grade 11, you have to know how to do this, okay? Ideally, in grade nine, you should know how to do this. This is something that I teach my students in grade nine. If I'm working with them, I teach them grade 10 stuff, if I can, if they're receptive to it, right? What else, what else? Let's do, let's do, let's do, let's do, what else should we do? What have I been doing with my students? What have I been doing with my students? Let me take a little bit of tahini, spoonful of tahini. This one is with honey, tahini and honey mixed together. Very good, power food, okay? So, let's see, what other math matters should we do? Let's do translations. Here's something I'm gonna show you. If you're dealing with translating functions, okay? It's something that they really don't teach the way I do it in school here. So, whenever I get a student and I show them this math and they're like, oh my God, that's so much easier. I know, I don't know why they don't teach it that way. Right? So, let's assume we have the following function. All right? F of x is equal to negative two, square root of x minus one plus four, okay? This is, again, a radical function. So, what you do is you start off with your base function, okay? Your base function for this is this. F of x is equal to square root of x. That's your base function. It's like saying f of x is equal to x square is your most basic quadratic function. We're saying f of x is equal to x is one of your most basic linear functions, right? And these are certain types of functions that you have to recognize, okay? It's good to know, right? This line is this, okay? This line or this function is this, right? This function is this. It's the base function, right? So, you can create a table for this, for this value. And that's what you do. And once you're doing this, once you're embedded in this, it's the easiest way to do this. So, you're gonna go here, let's call this, let's call the top one h of x, right? So, we don't wanna get confused by things, by the terminologies. So, let's say this is h of x, oops, h of x, right? So, let's get some points for f of x, right?  And plug in values for x. When x is zero, square root of zero is zero. So, when x is zero, f of x is zero. When x is one, square root of one is one, right? One and one.  So, I'm looking for, I'm gonna plug in values for x, where I can take the perfect square root of it, right? It's a perfect square. So, if you put in four here, when x is four here, square root of four is two. It's plus or minus two, but because we're talking about it as a function, we're not gonna, we're not looking for the bottom guys, right? So, basically what we're doing is graphing this thing, right? We're going, plug in x is equal to zero, we got y is equal to zero. Plug in in x is equal to one, one. We get y is equal to positive or negative one. Square root of one is plus or minus one, right? So, we also get this, where this is one and that's negative one. But because we gave this notation f of x, it means it's a function. So, vertical line test has to apply. So, we arbitrarily decide to keep the positive aspect of it. If you wanted to keep the negative, we're gonna make this negative, okay? And then I plug in four and you get two, right? So, that's all we're doing. We're finding points on this graph, okay? We'll do one more. Nine, square root of nine is three. That's good enough for now, okay? Now, if they ask you, so the question is this, here let me erase these guys, we'll get confused. So, the question is this, if they ask you to graph this thing, start off with this, okay? And then what you're gonna do is, you're gonna do what this equation, this function is telling you to do to this, because this is derived from this, okay? This number here means multiply the f of x by that number. So, you take these guys and multiply them by negative two. All of these. So, you get zero, negative two, negative four, negative six. This number here means it's the opposite. It's x minus one. So, it's not subtracting one from the x's. It's the, whatever you do to the x's in the function, you do the opposite in the table. So, you're actually adding one to this. So, plus one to all of these. One, two, five, 10, okay? This one is affecting the y and it means add four to these guys. So, this becomes plus four, so that becomes four, two, zero, negative two. The points that we graph in our original function, which were zero, zero, one, one, one, four and two, and nine and three. Those these guys, right? Have become these guys. So, all we're gonna do is just plot these points and graph it, okay? So, let's do it. One and four. So, one, one, two, three, four, two and two, two and two, five and zero, two, three, four, five and zero, 10 and negative two. One, two, three, four, five, six, seven, eight, nine, 10 and negative two. So, we took a function that looked like this and now it looks like this. And you can do this with all functions given to you. You can have a core function and then just find the points for the core function, the basic function, and then do whatever you're being told to do based on the sentence really in mathematics, which is an equation, a function, right? So, these are the rules that you have to apply to this and you get your function. And you can do this for any type of function. That's the way I teach my students out of graph functions. May they be trig, trig, law, exponential, law, whatever they may be, okay? Which is fun to do. And once you learn this method, this system, you're gonna apply it everywhere, okay?  It's pretty cool. What else should we do? What else should we do? We're doing power math. Fun. What else should we do? What else should we do? What else should we do? What else should we do? Binomial theorem? Should we do binomial theorem? Let's do binomial theorem. Should we? Oh, binomial theorem. Should we do binomial theorem? No, we need a little bit more space to do binomial theorem. Let's see, unless we have a specific question. What else should we do? More tahini. We should eat more tahini. Let's eat more tahini. Look at that. Deliciousness. This is amazing. If you cut up apples, I use apples as dip for this. I eat it with apples. So good. And of course with bread. Dried like flat bread, like pita bread and stuff. So good. And kick it back with water. So good. Let's take a kick it back with tea and water for sure too. But with tea is amazing. What else should we do? What else have I been covering a lot of? Graphing functions, we just did one. Solving. We can do synthetic division. Should we do synthetic division? Sure. Let's find the x-intercepts for polynomial, polynomial function or should we graph a radical? Let's graph a radical function. Okay. Let's assume you have this. f of x is equal to x plus 1 over x squared plus 4x plus 3. Okay. There's a certain process you have to go through to be able to graph these things. Okay. First thing you do, you find your restrictions. So you're going to go restrictions. And I always do it on the side. Restrictions in mathematics are only true real restriction we have. We can't divide by zero. So you take the denominator and say the denominator can't equal zero. So you're going to go x squared plus 4x plus 3 cannot equal zero. You factor this then. Cannot equal zero. xx plus 3 plus 1. So x cannot equal negative 3 or negative 1. Okay. Hopefully you can see that. We're just factoring this is a simple triangle. Write it a little clear. xx plus 3 plus 1 cannot equal zero. Two numbers, the multiplier to give you 3, add to give you 4. Multiplier to give you 3, add to give you 4. So x cannot equal negative 3 and negative 1. And then when you've done this, you found your restrictions, like literally these are your restrictions. Whatever happens to this function, x cannot equal negative 3 or negative 1. Because in the original statement, that would make the denominator zero. And we can't divide by zero. And then what you do. So step number one is find your restrictions. Step number two is factor the original polynomial. So this becomes x plus 1 over x plus 3, x plus 1. The next step is reduce. So if you're reducing this, x plus 1 kills x plus 1. So this becomes 1 over x plus 3. Okay. If you end up killing anything, if it reduces, then what you ended up reducing becomes a whole in the function, in the graph. Okay. So this guy here, this negative 1 is really a whole at x is equal to negative 1. Now, if you're graphing this on a Cartesian corner system, let's put it here. And as soon as you find your whole, what you need to do is start putting your restrictions on the graph. So you know what things are going to start looking like, right? So if anything cancels, it's a whole. And what it means is there is a point on the graph where you can't have an x or a, speaking too fast. You can't have an x or a y value there, right? So we found the x, but we need to find the y. And this is our reduced function now, right? So you need to find the y associated with this x. So all you're going to do is find f of negative 1. So you plug in negative 1 for x. So this becomes 1 over negative 1 plus 3. So the y, f of negative 1, f of negative 1 is 1 over 4. So x is equal to negative 1, y is equal to 4. This is a coordinate system, negative 1 and a quarter. You have a whole. So here's negative 1. Let's say that's negative 1. And here's negative 1 here. So here we have a whole. The function comes up to that point and leaves that point, but cannot be that point. Now, once we found the whole, now we look back at our asymptote restrictions, we've got to find our vertical asymptotes. So vertical asymptotes.  And we've got x cannot equal negative 3. So we have an asymptote, a vertical asymptote that x is equal to negative 3. So 2, 3. I should probably graph this a little bit over. So I'm going to erase this. I'm going to move it over a little bit so we see. We get a better representation. So let's bring this over.  Negative 1 and a quarter. Here's negative 2. Here's negative 3 and we have an asymptote here. Vertical asymptote. That means the function cannot touch this point or cross this point. Go across it or touch it. So we found our vertical asymptote. It's just whatever the x couldn't be that didn't cancel in the simplifying process, reducing process. Then we have to find our horizontal asymptotes. Now there's a rule to find your horizontal asymptotes. I'm just going to write it here. Horizontal asymptotes is this. If you have a function of a x where you have a x to the power of n plus whatever it is divided by b x to the power of n plus whatever it is. There's three things that can happen. If the power up top n is greater than m, if the power up top is greater than the power in the bottom for the x's, then there are no horizontal asymptotes. No horizontal asymptotes. If n, the power up top for a function, is equal to the power in the bottom of the function, then the horizontal asymptote is y is equal to a over b. So it'd be a line wherever this is. And if the power up top is less than the power in the bottom, then the horizontal asymptote is y is equal to zero. So for our case here, the power in the bottom on the x is more than the power up top because then the power up top is x to the power of zero. Really that's how we get over one. So the power in the bottom is bigger than the power up top, which is this situation. So we have a horizontal asymptote as y is equal to zero. So there's a horizontal asymptote here. I hope that makes sense. I'm going through the speedy ones out. This is more like a review if you got a final coming up. And when I do this with my students, if there's any questions they have, we go over it. So horizontal asymptote we got y is equal to zero. That's this guy here. Now we can graph this thing. We have one zone, two zones, three zones, four zones. Now for us, if you have your function is up here, it can't be here because it wouldn't be a function. The vertical line test doesn't hold. And if the function is up here, it can't be here. And if it's here, it can't be there. Now we know the function is down here. How do we know that? Because there's a hole there. So the function has to touch the hole and come out of the hole. So the graph of this most likely looks like this. But what we can do is plug in x is equal to zero to see what y is going to be if you want another point. So if you plug in, this is negative two. And this is a hole. It's not zero. This is negative one. If you plug in x is equal to zero, which puts you on the y-intercept, then it's one-third. So we know the point is there. And you usually want to get close to the vertical asymptotes and the horizontal asymptotes or vertical asymptotes anyway. So plug in negative two here, one divided by negative two. So we're trying to find f of negative two. That's going to be one over negative two plus three, which is going to be one over one, which is equal to one. So when x is negative two, y is one. Wait a second. Negative two plus. Did I make a mistake here? Oh, I made a mistake here. Check this out. This one is negative one and a quarter. The hole is up here. And this guy's not down there. It's going to be up top, right? So the hole is up here. That's negative one. That's negative two. If you plug in zero, it's a third. So it's here. And if you plug in negative two, you're at one, which is here. So we know the graph does this. And this is where my students usually would correct me, by the way. So this guy goes like this, goes like this, goes like this. So we've graphed what this top function looks like on this side of the asymptote. And then what you need to do is graph this side. Now this is at negative three, right? So you can plug in negative four to see if the function's up here or up here. Below that, x-axis, or above the x-axis. So let's plug in negative four. f of negative four is going to be one over negative four plus three, which is going to be one over negative one, which is negative one. So when x is negative four, y is negative one. It's here. So we know the graph does this. And this guy is the graph of that guy. And this would be more grade 11, grade 12 in my part of the world. They're fun to do these graphs. I actually really appreciate doing these things. They use a lot of the things that you learned over the years to apply to a function to be able to graph it. Fun to do, fun to do. What else should we do? What else should we do? Let's graph inverse of a polynomial function where we don't know the equation of the function, but we know what it looks like. Right? Let's do this. Let's assume you have the following function, graphically. We have this straight up polynomial function, easy. Here, let's make it, yeah, let's take it up a notch. Let's go, let's kick it up a little bit. Let's assume it goes up like this. Okay. Let's put this at negative one. Let's put this at one. Okay. And these points here are your accent to suffer this function. Now, we don't know what the graph of this function looks like, right? Or what the equation of this function is right now, because I haven't given you any points on this. Let's call this f of x. What if I said I want you to graph the following function? I want you to graph h of x, h of x, where h of x is one over f of x. That's what we want to graph. Now, what that means is you're taking a reciprocal function, flipping it. And what you have to appreciate is f of x is really your y. So this is your graphing f of x versus x. So your x positions are the same, but your y positions change. y values change. So if we graph this thing, let's draw another thing. Let's draw a graph which mirrors this thing. So if this is your y value, the function, this is where y is equal to zero, right? And if we take h of x is equal to one over f of x, wherever f of x is zero, because this is f of x is zero, f of x is one, f of x is two, f of x is negative one, right? Wherever f of x is zero, if we go one over zero, we get an asymptote, a vertical asymptote. So at this point, we have a vertical asymptote. At this point, we have a vertical asymptote. At this point, we have a vertical asymptote. At this point, we have a vertical asymptote. And at that point, we have a vertical asymptote. As for the horizontal asymptote, the same rule applies as before. The power up top is less than the power in the bottom, right? The power up top is one. The bottom is, this is at least, minimum, at least x to the power of one, two, three, four, five, right? So the power in the bottom is bigger than the power up top, so you've got a horizontal asymptote as y is equal to zero. Now there's one other point, one other type of point you have to put on here. Wherever y is one, wherever y is one, here, and here, and here, and here, and here, and here. So wherever y is one or negative one, the point stays exactly where it is in the flip version. Because if y is one, you flip, you put one over one as one. If y is negative one, then one over negative one is negative one. So these are called invariant points. They don't change when you change the function around, when you generate the new function. So here, this point, let's mark these as one and negative one. So this point, this point, stay where they are. This point stays where it is. We've got that point stays where it is. We've got that point stays where it is. Now we can graph this thing. Asymptotes act like magnets. They push a function away. So if you have a function coming in, if you have an asymptote and you've got a function approaching it, it can't hit this function. Now you could do this and go down on the function, where you can go asymptotic with a function like this. It can't go through it. And it can't touch it and bounce back because it wouldn't be a function. And it can't touch it, period. So when you have an asymptote, when you have a function approaching it, it can do this or do this, or sometimes it comes across and does a little dip and goes back up or goes up and comes back down. Our function is pretty generic, pretty simple. And we have a horizontal function. So for example, if you have this vertical asymptote and you've got a function approaching it and you've got another horizontal asymptote which is perpendicular to this, then you know the function is not going to go down because it can't cross this horizontal asymptote. So for example, we know that this point and this point are the same. So we know the function goes through that point. Now the function is not going to go up because there's an asymptote here. It's not just going to go straight vertically out because there's an asymptote there. So these asymptotes act like magnets.  So the graph of this guy is going to look like this. This part of it. Okay. Over here, these guys, anything on this side goes up and anything above the one value is below the one value here. So for example, if you had this 2, 3, 4, if this point was the y part of it was 4, we don't know what the x part is because I haven't put any scales on it. If the y part was 4, then 1 over 4 would be a quarter because we're flipping the y's really, what we're doing. Whatever the point here is, it's x and f of x. f of x is your y. So if the original function, the y was 4, then h of x is 1 over 4. So that point, the vertex becomes a quarter. So it's over here. That's a half. That's a quarter. So it's right there. So the graph does this. It goes up, goes up. And so on and so forth, right? Over here, the graph would do this. Wherever the x intercept that. Over here, we didn't do anything over here. I'll check this out. Now the one value is here, but those are less than one. Anything less than one, you flip it, it becomes greater than one. So the graph of that one is going to look like. So for example, let me give you a point here. Let's assume the y value is here. We don't know what the x is. I haven't given you a scale. I have given you a scale on the y, right? So let's assume that was a half. Well, 1 over a half is 2. So this becomes 2, or negative 2. So it becomes negative 2 and you know the function does this. And this guy here does this. And that guy there would do this. So we just graphed the reciprocal function for this, which is pretty cool. And this applies in trig as well, right? When you got the sine versus the cosecant or the cos versus the secant or the tangent versus the cotangent. They can give you the reciprocal function and ask you to drive the original function. They can give you a function like this and give you a whole bunch of points and ask you to come up with the equation of the function. So there's a whole bunch of things that can be done. Whole bunch of things that can be done. What else should we do? What else should we do? I've been talking like mad. Going crazy. I like it. We got about half an hour left. What other mathematics should we do? Let me think about this for a second. What have we got going on? What have we got going on? What's some of the stuff that we're doing? There's some basic stuff that we're doing. Great 10. I've got some great 10 students that I'm working with right now. So great 10 is a lot of quadratics, a lot of factoring. Complex trinomial, simple trinomial, quadratic formula. You find areas of things. You can do your special triangles, trigonometry. So for example, one of the tricks that we're doing is this. Your special triangles. I went through a little bit of sessions with a couple of students where I was trying to explain to them what a special triangle is coming. And special triangles, if you're doing trig, I explained to people that if you're studying trigonometry, the main reason or one of the main reasons you're studying trig is because you want to understand a circle. So in general, when you try to understand something, you try to break it down into smaller and smaller pieces. So let's draw a circle. And let's assume we want to analyze the circle. And the reason we want to analyze circles is because they represent the perfect cyclic function. We've talked about this. There's a whole trigonometry playlist that we have and random trig videos that we have. So if you want to analyze this, first thing you do you put on a coordinate system because you need points. So you do this, you do this, you do this. You got your x and y. So every point on this circle has an x and a y associated with it. And once you put it on a coordinate system, you've simplified this thing by a power of four by four.  But once you break it down to four quadrants, then if you know what's happening here, that means you also know what's happening here, what's happening here, and what's happening there. Because it's just a reflection of that mirror. So the problem is a quarter of what it was if it was a whole circle. And then what you do, you break things down even further. So what you can do with this quarter, because this is 90 degrees, cut it into 45 degrees. And if this has a coordinate system, x and a y, you draw your right angle triangle. The x value is the distance here. And the y value is the distance there. So all of a sudden you've taken a quadrant. We're doing some unicircle stuff. Basically, yeah, Jacob, hey guys, how are you doing? So basically what you're doing is you're breaking this quadrant down to semicircles, 45 degrees special triangles. So you have one special triangle, which is 45, 45, this, and the ratio of the sides, right, is one, one, two. This holds for any triangle that has a sides that's on a Euclidean geometry flat surface, right? 45, 90, 45, right? So if I give you a triangle, and I say this is 45, and you know this is going to be 45 because of some of the angles and triangles equal to 180, right? And if I said this is 13, I don't want you to find x and y, all you got to do is say, oh, these are similar triangles. That means their proportions are the same, right? It's like taking you, if you happen to be six feet tall, and your arm is three feet long, and making a bigger version of you, that is 12 feet tall, and saying, hey, how long is your arm, right? If you're 12 feet tall, right? We've multiplied you by a factor of two. So all you got to do now is go, oh, okay, this is proportionality. You would say six over six is related to 12. So six over 12 has to be equal to three over x, and you cross multiply. Six x is equal to 36 and divide by six, so you get x is equal to six. So your arm is going to be six feet tall if you're going to be 12 feet tall, if you want to keep everything proportional, right? That's exactly what you're doing with the triangles. They have the same ratio, they have the same angles. That means the ratio of the size has to be consistent, has to be the same. So all you do for this is go, oh, if you want to find x and y, you're going to go one over 13. One over 13 has to be the same as one over y, has to be the same as root two over x. And then you can solve for this. You combine these two guys, cross multiply, so y is equal to 13. So this is y is equal to 13, and you combine this guy and that guy. So you got one over 13 is equal to root two over x cross multiply, so x is equal to x is equal to 13 root two. That's all you're doing, proportional. Right? So that's one special triangle you have. The other special triangle you have, where they teach you, there's more special triangles, of course, right? You can take a quadrant and break it into three parts, right? Because 90 divides by three, 30 degrees, right? So this would be 30 degrees, and here's 30, here's 30. So you have a 30 degree degree. That's 90. That becomes 60. And you also have a 60 degree and a 30 degree. So all you really need to learn is 30, 60, 90. And the ratio of the sides is one square root of three and two. That's where you get your special triangles from. You're breaking a circle, quadrant circle to smaller, smaller pieces. Now there's nothing stopping you from creating a circle or creating a special triangle, which is 10 degrees, 10 by 80 by 90. All you would do is say, hey, I'm going to break this into a certain number of pieces where this is, into nine pieces. So I'm going to make these 10 degrees each, right? So this becomes 10, 10, 10, 10, 10, 10. So you're going to have these triangles. So if you create another special triangle, 10, 90, and 80. So your special triangle would be 10, 90, or 10, 80, 90, whatever you want to call it, right? I don't know what the ratio is in size with you, all right? But we could figure it out. Okay. We could definitely figure it out. This is rare. You don't do, you don't do special triangles at 10, 80, 90. Wolf, Alexander Wolf, Ichicho, longtime YouTube follower here. First time catching you live on Twitch. Love your stuff. Well, dude, brother, well, dude. Thanks. Or sister. Or sister. Thanks, man. I love the little symbols you got. Twitch unity, bleep purple, bleep purple, and a heart, and a heart purple, purple heart. And of course, the happy face. I love the happy face. So that's where your special triangles are really originating from, why we have them. And they come in handy. And you can take this. We've taken this far in Trinometry, a lot more to come. Thank you for being here by the way on this stream. What fun to do. By the way, I'll let you guys know. I'm not sure what the deal is, but I had Twitch contact me, right? Let's see how many people are here. I got an email from Twitch from one of the reps saying they've noticed that I've been doing political streams.  We've been doing political streams, something that we're going to do tomorrow at 7.30 a.m., right? We've been doing political streams and they mentioned that Twitch is trying to expand their library or the services they provide beyond gaming. So they wanted to talk to some people who've been doing different types of streams, one of them being political. So they wanted to talk to me. So I said, sure, I'll talk to you guys. So I think next week, I'm going to be either Skyping or Hangout, Google Hangouts, or I don't know if Twitch has, or Discord might have a, I think Discord does have a video thing. I haven't done it yet on Discord. Hell, yes, political streams. Yeah, tomorrow morning, 10 a.m., at 7.30 a.m., right? So Twitch was interested in the political streams that we've been doing. Glad you agreed. Yeah, for sure. I'll talk to them. I think maybe they're legit. They do want to expand their library, what they're promoting, because that's the direction it's going. It's not just going to be gaming. It's going to be live everything. What I'm doing right now with the math live streams in coming years is going to become more and more prevalent. Hopefully, on more platforms, there'll be a lot of people providing a lot of help for mathematics to many, many people who need the help, right? Because it's pretty important for us to teach people mathematics. It's already lots of different stuff. Yeah, for sure, Dante. So I think they're interested in that. And I mentioned this, sent their email to a friend of mine. And my friend is a little cynical. It's not cynical, it's not the right word. Very funny and more cautious. And they said, oh, chicho, they just want to find out if you're radicalizing people on Twitch and stuff. I just started laughing. Hopefully, they're not concerned about the politics, the content that we've been doing. I hope not.  But it is what it is, right? Absolutely. We've been branching out to try to support other categories. Video games are great, but we want to include all interesting topics. 100% agreed. 100% agreed. That's what we need. We need the dialogue, the discourse to be opened up for people to participate and create content. There's going to be live streams of stuff coming up that we didn't even imagine people would be live streaming that stuff. Right now, I've been trying to convince a friend of mine, and he is in the process of trying to get his stuff together and do it, but I've been trying to convince a friend of mine to come on Twitch. Diversity is the key. 100%. Trying to convince a friend of mine to come on Twitch and start teaching Tai Chi. I learned a little bit of Tai Chi from him to do Tai Chi meditation and teach the Tai Chi techniques and Bagua and Shigong and stuff like this. And I've told him I'll help him out. And once he's ready, we'll probably start off by streaming some of that stuff on our channel here and set something up for him as well, and try to get people subbing to him and start doing things, right? But for sure, it'd be amazing to have... I'm pretty sure there is already live yogas and live everything, but we need more of that. More of that. And definitely we need more mathematics, much, much more mathematics and sciences and stuff, right? So I'll let you guys know how it goes. The conversation goes with Twitch. If it ends up happening, most likely next week, I'm going to be talking with them. Possibly. I told them that I'm really busy right now with students, so my window of opportunity is very limited. Hello, Barbarian. Yeah, there definitely is on site already. Need more, need more, much, much more, much, much more. Gaming is amazing, but I think it needs to expand beyond that. Really. There's many things in life to appreciate, not just gaming. This time, I didn't get a little notice from Twitch saying drink water. Oh, well. Hello, Barbarian. Is this your first time here? Hope you're enjoying the math stream. You're keeping it chill. Definitely. So we've been going for about an hour and 45 minutes or so. It doesn't seem like there was too many math people interested in mathematics today. Sometimes you're getting a lot of mathematics. Blue waves. I just, you've taught jazz. I just know it's your blue wave thing. Fun. Gicho. What is the next stream? Next stream had a tomorrow morning, 7.30 a.m. Everyone acing their exams apparently don't. My students aren't. Well, I'm trying to get them to ace it, but usually I'm actually, I do have students that do very, very, very well, right? I've had students there, you know, they're getting 100% of the stuff, right? But in general, once I start working with students, they stop paying attention in class because it takes class two weeks to teach them something that I can teach them in half an hour, right? I'll be there. You'll be here on it. Awesome. Awesome. I wish the streams were longer than two. We'll get into it at some point. We will for sure. I've been doing, taking care of the plants as well, taking cuttings and we fixed up our patio. So in the summer, we'll be doing streams in the patio a little bit. Okay. Taking care of plants and just chilling in the patio and catching some sun and talking. We got grapes growing on our patio and a couple of days ago I set up strings so they're going to come towards the house and maybe create a canopy for us outdoor streams. Love it. Yeah. I want to do some outdoor streams. There are many gifted students whose problem is that they can't pay attention 100%, 100%. Are you married? No, no. I don't believe in the institution of marriage and I don't, I believe in, I'm monogamous. I have partner and I don't believe, it's not my thing. You know, I'm okay with it. People want to do, maybe at some point I will. I don't think so, but maybe. But I do have a partner, I believe in relationships, growing the relationships and stuff, but I don't believe in signing pieces of paper to allow governments, churches, centralized institutions to be the gatekeepers of your relationships. I think that's just mainly for tax purposes. The pace of the class is too slow. Too slow. They get bored out of their minds or the presentation is too boring as well. But in general, in my opinion, too slow. We're dumbing down the kids. We're dumbing them down to a level which is detrimental, detrimental to the well-being of all of us globally, really. We need to start feeding information to the youth harder, faster, and connecting everything. I love baseball stats. I should start looking into, I know soccer stats to a certain degree, but soccer, the stats in soccer, nothing compared to the baseball stats. At some point, I'm pretty sure we'll tackle baseball statistics and look at the ratios because that'll directly connect up to what some of the things we've been doing in personal finance and some of the things we will be doing in personal finance and investing in Wall Street and whatnot. We will. There's so much I got laptop to do, right? Just taking my own sweet time. I don't know if my own sweet time. I just shot the video for a 700th video that we have live online, right? That I'll be editing today and tomorrow. Hopefully have it out. Hopefully, fingers crossed. Tomorrow evening, latest, hopefully by Sunday morning. If it's not Sunday morning, then it's going to be a week later. So it's going to be very quiet on YouTube and bit shoot because I won't be loading on any videos because I want it to be the 700th, right? So I want to take the 700th spot. So we'll see. Hopefully, I can get it done next couple of days. Considering how many students I got to deal with and what I got lined up. We'll see. We'll see. It's a comic book. If you've been on the live streams before in the last couple of months, you would have seen about more than half of what I'm presenting in that video. And there's some stuff I picked up that I haven't shown you guys yet in the live streams. Okay. I love the comic book calls. It's absolutely fantastic. How many hours a day do you tutor? It varies. There are times where I don't at all. But right now, it kicks up a little bit. Not much compared to what I've done in the past. In the past, I've had students knocking on my door random with parents saying, my kid needs help. I'm like, man, what are you doing here? It's like 9am on a Sunday. You can't come knocking on my door. That's like unacceptable. Right? Oh, come on, please, please. Oh, my God. So when it reaches that level, I cut back. I don't give up my number. I tell my students, don't pass on the word. I'm trimming down. Right now, I'm in the process of kicking back up because I moved here and I'm getting students here. So the word of mouth is going out. So it varies. Really, it varies. There are days I don't do it all. There are days I only do one hour, two hours. Right? But I put my heart and soul into it. Right? For me, ideally, I don't want to do more than five hours a day. That's tiring because you're dealing with, because kids, teenagers are high intensity, high energy. Right? So when they're high energy, they, and I pay attention to their behavior and stuff. So I'm very specialized. They're really, it's very specialized.  I mean, the kicker is, I don't want to price certain people out that need the help. Right? I want to be able to maintain my life, you know, do some of the things that I need to do as a human being to saving and investing and paying my bills and have a little bit of play money and stuff like this. And I've had flexible rates as well. There are people that, you know, I charge a certain rate. And then there are people that I know if they're single mothers or they need help. I charge them much, much lower. And sometimes I do pro bono stuff. Right? Just because they need it. It's good for our society if people learn mathematics. Right? It's not as good for my pocketbook or for my sanity, but I can do short stints of that. But I do have to kick things up a little bit. I don't like being, like Blueberry said, on the broke level. Right? Sports analytics is fascinating. They focus much more on pure stats now than ever. They have to. They should have. If you were managing a sports team back in the 60s and 70s and you were applying full blown stats to your team, you would have been world champions for decades. You would have blown away all the competition. If you were just looking at the stats, bring in the mathematics. I don't know if they did for baseball. Nicholas, how are you doing? How is life? Are you still in Europe, man? I think you should be back by now, no? Or did you extend for another month? I hope so, man. I hope you extend it for two more months. Enjoy your stay. Last we heard from you, your own whales. With a new love, which is fantastic. It's fantastic. I hope that's going well or went well. Right? I am just on my way home tonight. Oh, you're flying back tonight, man. Safe travels. Safe travels. Stay away from the Boeing planes. Joking, of course, but semi-serious. Right? Safe travels though, brother. That was a good trip you were on, eh? Fantastic. Fantastic. I hope it rejuvenated you, man. And we all need a little bit of rejuvenation and a little bit of separation from chaos of our daily lives, right? Billy Bean and Moneyball. Yeah, Moneyball. I watched that movie. It was fantastic. Changed the game forever. They used stats and analytics and destroyed other teams with one-tenth of the payroll. Amazing. Amazing indeed. Right? Like the entertainment you can get, like if you stop watching blockbuster movies on the same level, right? Stop watching blockbuster movies, right? Watch movies that are low budget that have a script, which is amazing. Those movies blow away the rest. By the way, is anyone going to play WoW Classic on EU? I don't know what that is. Wow. World of Warcraft craft. Is that World of Warcraft? Classic on EU. EU? What's the EU stand for? It can't be a European Union. EU must be the platform. So, World of Warcraft Classic, like the map view that you were playing with stuff. I love that. I played that a lot. Like I was playing that. It's EU region. It's EU region. Cool. EU servers. Oh, that's cool, man. Just to let you know, World of Warcraft Classic, back in the day, like I can't remember. Late 90s. There were servers that you can join. You have to pay per month, but I found cracks and I was going through Cali and I was playing World of Warcraft using Cali, so you don't have to pay with people all around the world. And I had my little cousins would come over and play that. I love that. The Classic World of Warcraft is absolutely brilliant. They are re-releasing the vanilla version this summer.  That's crazy. I didn't know this. Yeah, I needed it. I am still in contact with the girl, but we shall see what comes of it because of distance.  Thanks, Nicholas. Awesome, and I'm glad it's working out. It worked out well with the girl, man. Sometimes it's short-term relationships. They're what you need. They're phenomenal. In the NBA, the Warriors were one of the first teams to fully embrace analytics and have won three out of the last four championships because of it. Now everyone is trying to copy that. Cool. I can't believe it took them this long to embrace it. Like crazy. That shows you the lack of math literacy in our society, right? We're in 2019, and it's just recent in the last three, four years that NBA and other football leaves are embracing analytics like data. Insane. I would fire all that everyone that wasn't using it. Really? How could you not use it? The highest I got in math was English GCSE, great C level. Now I need to improve for my work. Good, good. You got all the tools or disposal, wealth to do. Games you grew up with are always the best. I saw a movie about a baseball team which won, that's the Moneyball, I believe, Dante. Forgot what it's called, but it was good. No, Moneyball was that baseball or football. I think Moneyball was football. Which one was the baseball one? I don't think I've seen the baseball one. Or was that the baseball one? Moneyball was baseball, I think. It's not even that complicated to realize that three points shots are worth more than two points shots, I agree. Which is crazy. And people were spending like gazillion dollars for these players, superstars that were temperamental and bad for morale of the team when they could buy the whole team which was shooting above averages. Right? And win everything. Basically, this concept is the same concept as what do you call it? Baseball is Moneyball. Okay, cool. That was a great movie. It's the same concept. This concept of using analytics is differential accumulation. Same thing as Jonathan Knitz and the videos we've put out. If you do chicho and differential accumulation, it's the same concept. It doesn't make a difference what you're making in the absolute. It's relative to everybody else. Chicho, differential accumulation. I'll give you the main one. I think we've put out three videos on differential accumulation. Hey, where is it? Differential accumulation. Differential. Come on, find it for me. There we go. So this is the last video we put on differential accumulation and it links up with the rest of the stuff too. And it was a message that I got from Jonathan Knitz and he was a person that introduced it. Differential accumulation. It's the same concept as that. I hope I linked this right one. Did I copy the right one? Yeah, if you look at that, it's a half an hour video that talks about some of the other videos we put out and some of the concepts of it. And personal finance talks about it. I can't believe it's taken this long for a sports team to include that stuff in their game, really. It's crazy. Fun stuff gang. So tomorrow at 7.30 a.m., we're going to do current events, okay, politics, economics, news, whatever, whatever you guys want to talk about. Okay. And stream after that, it's going to be next week. This is going to be a little bit delay. I'm going to be really busy with my students for about a week. And hopefully I'll have the video up by Sunday. If not, it's going to be another week before we get it out. Got to board now. I think, oh, you're at the gate, so got to go, folks. I hope you all had a great stream, Chico Bro. Thank you for the advice during my breakup. I needed it. Have a great stream and see you next stream. For sure, Nicholas. Safe travels, brother. Safe travels. Okay. And welcome back. Okay gang, thanks for being here. We'll call that a stream. I'll set up for the stream for tomorrow morning at 7.30 a.m. And if I end up talking to Twitch regarding political streams and stuff like this, I'll fill you guys in and let you know what the discussion was and where it went to. Okay. Aside from that, hope you have a fantastic, fantastic weekend. Oh, we got a little notice from Twitch saying, drink water. We've been streaming for two hours. Thanks, Twitch. We will, we will. Okay gang. Hope you have a fantastic weekend. And if you can make it, I'll see you guys tomorrow. Hey, Chico. What time zone are you in? I'm in Pacific PDT, Canada, West Coast. Okay, Brown. See you tomorrow morning. Sip some Hennessy tonight for me. I don't have Hennessy here, but I'll sip something for you, Hannah. So I can come tomorrow. For sure. Thanks. Okay, awesome. See you, Chico. See you, Dante. Thanks for popping by, Brown. Hope to see you tomorrow. Okay. Have fun gang. We'll talk later. Bye for now.