 All right. Hey, Martin, what does it say at the very, very top? Memorizing. Yes and no. I don't like memorizing. I'm teaching that, folks. I don't like memorizing stuff, unless I absolutely have to. What I'm going to do along the way of this lesson is to show you how, if you can do a bit of arithmetic in your head, you don't have to memorize these. You can derive them in about a second. If it took 10 seconds to derive them, then maybe I'd say it's worth memorizing them. Just 10 seconds on a test over and over that adds up. But you can derive them in about a second. So the first one, it's what we call the quadratic equation. The basic one is this, y equals x squared. What is that a graph of you learned this one last year? This is a parabola. Parabola is a quadratic. It goes through zero, zero, one, one, and then two, four, and then three. I think nine is, I think my graph goes eight high. So nine is one off the graph. Did your teacher last year teach you one, one, two, four, three, nine? Okay. I don't know if every teacher does it. When I teach, in fact, I think Mr. Gerard speaks it a bit. His is one, one, one, three, one, six or something like that. I memorize one, one, two, four, three, nine. Those are the key points. In fact, when I teach it in math 11 blame, we call it our mantra. A mantra is a chant that you repeat over and over to relax yourself. But I stole that word for a parabola because I said anytime you do a parabola, I'm thinking one, one, two, four, three, nine. One squared is one, two squared is four, three squared is nine. What's the standard form of a parabola? Remember, Y equals A bracket X minus P all squared plus Q. Remember that bad boy from last year where P and Q Leslie told you the vertex. Why was that minus? Because then the brackets was backwards. A told you if it was positive or negative, A told you which way it opened. And it also told you the vertical stretch, what you had to do with that one, one, two, four, three, nine. There is also general form for a quadratic equation. That's the one that you're used to seeing when you want to solve a quadratic equation. It was this one, Y equals AX squared plus BX plus C. That one is handier to do math with. Standard form is handier to graph with. And remember doing that thing called completing the square last year where you could rewrite parabolas as standard form. We won't be doing it again this year, but in calculus, you probably reviewed it when you did conics. Those of you that are in calculus. Just because I'm a nerd and I like it so much. Hey, what is the quadratic formula? X equals what? Negative B plus or minus? What is it? But I'm going to give you a candy for that. Can we say it backwards? A2 overall? Are you serious, Mr. Good? Sure. C A4 minus square B root square minus plus B negative equals X. I'm enough of a nerd. We love the quadratic formula, Blaine. To us in math, it's almost, I tell my students when I show it to them in grade 11, it's almost like iambic pentameter in English. It's almost poetry. It's one of the only times in math that you guys don't know this because you don't know all that much math yet. I got a math. It's one of the only times in math where you can get a formula that works for anything, any quadratic. There is no cubic formula. They've looked long and hard. In fact, they spent about 200 years looking. The turn of the century in that position actually proved that there isn't one. You can't find one. There is no to the fourth power quartic formula. There is no to the fifth power formula. In fact, in all sorts of other equations, you'll notice, well, have I given you an equation solving formula this year? I could give you steps. So for exponential equations, I said, well, get the exponent by itself and then take the log of both sides. But there was no formula yet to tweak it a little bit. For log equations, I couldn't give you a standard formula. So we like it. x equals negative b plus or minus the square root of b squared minus 4ac all over 2a. Make that a little larger. Clara, you do it differently in Germany. I believe you have an equation that has the letter p and the letter q in it or something like that to solve a quadratic. If you can solve it that way using the p and q method, good, I don't care. In North America, we use this thing here. And you don't need to freak out. I'll show you how it works another day. So just copy it out if you want to. Although you know what? This unit, you won't need it. And the unit that you do need it, you'll have a formula sheet by then, which actually has it on here. In other words, staff, you don't need to technically memorize this in math 12. But it's so important. This is one of the few times that my nerdishness overcomes my unwillingness to memorize stuff. OK. Domain. What's the domain of this particular parabola? In fact, what's the domain of any parabola? All reals. It goes from negative infinity to positive infinity. Now, we're going to define that a little more fussy. What we're really saying is this. Madeleine, in this generic standard form for parabola, you can put in any x value. No extraneous roots, no rejections, no. Martin, you always get a date on this graph. A smile on his face. The relief on his face. So we don't write all reals. Though we write x belongs to the set of all real numbers. That's the shortest. Well, that's the second shortest way to write all reals. Those of you that are in calculus, you know the shortest way is that. But that's interval notation. And so you're not going to use that. What's the range? Well, first of all, what's the range of this particular parabola? Greater than or touching zero. Now, this is a specific example. Can you generalize what's your range here? It depends on whether it opens up or down. Good point. So I'm going to say this. Y is greater than or equal to zero for our specific one. For the generic one, it would be Y is greater than or equal to, or it could be less than or equal to, but the letter P, sorry, not the letter P, mister, do I mind your P's and Q's? The letter Q, the Y coordinate of the vertex told you what your bottom or your top was. I'm not going to give you a generic one. Like I'll give you specifics, but we're just trying to refresh your memory from last year. And most of you, since you did the parabola at the beginning of the year last year, we're brushing some serious cobwebs off of your brain. Vertex. What's the vertex of this particular parabola? What, comma, what? What's the vertex of this parabola? What, comma, what? Okay. Let's do that one generic. The vertex is P, comma, Q. And if there are no numbers in those locations, it's zero, comma, zero. Do you remember the equation of the axis of symmetry? What was the axis of symmetry? That was the line that ran right down the middle of your parabola and exactly bisected it. Pardon me? That's the slope of it. I heard it correctly over here. Angelo, you just impressed me. What? Isn't it Y equals zero because it's vertical? No, it's not. And this is one thing we've got to recognize. Okay? You were right the first time. Vertical lines are X equals. Horizontal lines are Y equals. So the equation of the axis of symmetry, except Angelo, I'm going to tweak ours just a little bit. I'm going to say it's actually X equals whatever slide your vertex has done, X equals P, except what's my X coordinate of my vertex here? Zero. So for this specific one, Angelo, X equals zero if I want to generalize it, X equals P. And then there was key points. For the parabola, the key points were the vertex, and then they were one, one, two, four, three, nine. Blaine, what I'm really saying is, remember we said that the parabola was based on Y equals X squared? What's one squared? That's Y one comma one. What's two squared? What's two squared? That's Y two comma four. What was the third one? I said three, nine. Can you see why I sort of said you can memorize the one, one, two, four, three, nine, or it's sort of here so you can derive it. What we did last year was when you were asked to graph the parabola, you did not make a table of values. What you did is you found the vertex and then you moved those key points. That's the strategy we're going to bring to this unit for generic graphs. For a generic graph, Chelsea, we're going to learn its key points, either just read it from the graph if they give us a shape, or if it's an equation that we know will memorize, derive it. And then we'll say, okay, we can figure out how this graph has been moved left to right, how it's been moved up or down, and once I figure out where this graph starts, sort of like the vertex from the parabola, I'm going to do its key points for that graph. It's going to cut way down on a lot of your arithmetic. You're going to find by the end of this you'll be going, oh, this is nice. Example two, the square root function. Did I get everything on example one? Key points, yeah, okay. Example two, the square root function. Strangely enough, y equals the square root of x. Oh, I guess I have a y equals down there, Mr. Good. y equals the square root of x. What does this look like? Well, what can't I take the square root of negatives? I've probably already figured out my domain, by the way. So let's start with positives. What's the square root of zero? So this graph is going to go through zero, zero. What's the square root of one? One. What's the square root of two? Yaki decimal. In fact, what's the next nice number to take the square root of four over? What is the square root of four? Two. What's the next nice number to take the square root of nine, which is one off my graph? What is the square root of nine, Alex? It looks like this. Here's how I remember it. Unfortunately, this one would not be green piece happy, but that's okay. Remember when you were little kids and you were drawing little sunsets? So you would draw a little sun like that, and then if it was like a little beach, what was that supposed to be? A seagull? This is the wounded seagull. If you chop a wing off, okay? So there's a seagull. Green piece is not happy, but that's a square root graph. It's a wounded seagull. It is. Sorry. Pardon me? No. Square root is defined as the positive answer when you take the square root of a point. In other words, this is defined as the positive. Now, you are also noticing, I'll get a little math speak for you, this is the top half of the inverse of a parabola. Is square rooting the inverse of squaring? Yeah. Oh, except the square root, it's defined as a well-known positive answer. Okay? By the way, key points. What did you tell me the square root of zero was? What was the square root of one? One. What was the next one we did? Comma. Two, and then nine comma. In fact, if you, in math 11, I made my kids memorize one, one, two, four, three, nine, and the reason was by doing that, they'd also memorize the square root, which is one, one, four, two, nine, three. Which, oh, how do you find an inverse? Switch the X and Y around. Really? Are they inverses of each other, the key points? Yes, they are. Oh, but instead of a vertex, we have an origin. In fact, you know what the origin is? Zero, zero. In other words, Kellen, if they asked me to graph the square root graph, but they've moved it around, if I can figure out where the origin has moved to, from there, I'll just count one, one, four, two, nine, three. I won't do any arithmetic. I'll just one, one, four, two, nine, three. There's my dots. What do we say the domain was? Everything bigger than or touching zero. Right? Pardon me? I can't hear you. Oh, greater than or equal to, which one? Q is rationals. You could say X is greater than or equal to the whole number system. Natural numbers from math 10 are one, two, three, four. Whole number, you add a zero. I can remember because zero looks like a whole, a whole, serious. You saw me thinking for a second there. My brain was going, okay, natural and whole. Which one has the zero? Zero goes with whole because it looks like a whole, the whole numbers. You guys not figure that one in grade 10 when you have to memorize those stupid number sets, which I always thought were kind of a waste of time. But anyways, hey, what's my range? Also the same thing. Everything above or touching zero. Key points, one, one, two, four, four. One, one, four, two, nine, three. Oh, and zero, zero, your origin. What I mean by at the top when I said memorize these, here's what I'm going to say. Chels, if you run across a graph that looks like this, hopefully you'll say wounded seagull, that's probably a square root graph, and I'll see if I can figure out how it's been moved. If you run across a graph that looks like this, hopefully you'll say, that's parabola. Oh, and we've broadened our math horizons. If you run into a graph that looks like this, hopefully you'll say, I bet you that's an inverse parabola because it looks like they switched the x and y's. In other words, we're going to broaden our what's out there. Memorize, recognize when you see it. It's really what I should have put at the top of the page, but that was too much writing. Graph three. Here's your first new one, the semicircle. I'm going to give you a specific example of a semicircle. It's going to look like this. Do a great big square root, and then inside the square root, you're going to have 25 minus x squared. Okay, now just take the square root of 25 and the square root of x squared. Isn't that the same as just 5 minus x? No, you can't square root one term at a time inside a square root. That's as simplified as it gets. But we're going to try and generate some numbers here. The first number, if I don't know what a graph looks like, the first number is I always try zero. What's zero squared, Justin? What's 25 take away zero? What's the square root of 25? When x is zero, y is 1, 2, 3, 4, 5. That's a point on our ground. How good are you guys at arithmetic? What other numbers could you put in there so that this would work out evenly? Five? Good suggestion. What's five squared? What's 25 take away 25? What's the square root of zero? So you're saying when x is five, y is zero? Okay, I like that. Oh, you know what? Not only positive five. You know what else would work? Not only positive five, because we're squaring negative five squared is also going to give me 25, 25 take away 25 is zero. You know what? Negative five is going to go through zero high. Justin, what? Three. Nice catch. What's three squared? What's 25 take away nine in your head? 16? What's the square root of 16? So when x is three, y is? So let's go three over four up. You know what? Not only positive three. Negative three gives you the same thing. It gives you 25 take away nine, square root of 16. Negative three is also four high. Three gave me a four as an answer. Why don't I try plugging in a four? That would be logical to try. What's four squared? 16? What's 25 take away 16? Square root of nine? Oh, you know what? Not only in this graph does three give me a four. Four gives me a three. That's a fluke. I made this one up for you guys nicely so that does work. That does not work with any other semicircle, unfortunately. So you're saying four over three up and also negative four over negative three up. If I connect the dots, do you see why I called this the semicircle graph? Looks like that. Oh, huge, huge. Semi-circle is a subgraph of a group of graphs called conics. They include the parabola, which you already looked at. The circle, the ellipse and the hyperbola, they show up all over the place in physics. In fact, almost any type of architecture that involves curves, like bridges or the church arches or any of those are almost all conic sections that they distribute forces very, very nicely. They have wonderful force distributing properties. Our planets move in ellipses. Orbits are circular or semicircle. Well, not quite semicircle, but circular. Comets move in ellipses or hyperbola. They show up all over the place. Huge. You just don't know that yet, but huge. Hey, Martin, what's the radius of this semicircle count? That's the diameter. What's the radius of this semicircle? Louder, please. Five. Now, look at the equation that I gave you. Can you see a five hidden in that equation anywhere at all? Where, Chels? What do you think if I wanted a semicircle of radius seven? What would my equation be? Big square root of 49 minus x squared. What if I wanted a radius 10 big square root of 100 minus x squared? In fact, if you want the standard form of a semicircle, it's this y equals great big square root of whatever you want the radius to be squared minus x squared. A semicircle doesn't have a vertex like the parabola. You know what it does have? It has the center. What's the center of our semicircle here? What, comma, what? Zero, zero, which is a really nice one. In other words, if I knew a graph was a semicircle and I knew the radius, if I could figure out where the new center was, if they moved it around, say the center was seven, comma, nine, and the radius was four, I would go seven over, nine up, and then I would go four right, four left, four up, draw a semicircle. That's how you do a semicircle. You don't actually know this table of values things that Justin and I did. You don't do that. It was nice to find our first one. Well, I'll try a three. I'll try a four. I'll try a five. You don't do that. You find the center, find the radius, and good enough. Oh, what's the radius? The radius of hours is five. The radius of this one is R. Martin, the other reason I like this graph is it's got a more interesting domain. Our specific graph, how far left does it go? Negative five. Touching? Yep. How far right? Touching? Okay. How do I write everything between and including negative five and positive five? This is the one that we did it this way last year. You did the smallest number first. Domain, is that X or Y? X. Leave a space. Put an X there because it's domain. You did the bigger number next. Leave a space again. By the way, can you see that the X is between negative five and positive five, which is kind of what I'm trying to show with my diagram. And then if you always did the smaller number first and you always did the bigger number next, it was always less than, less than. Oh, but Alex, did you say we're touching? Or equal to or equal to. Now what's the statement really saying? Cover up the right half and it's saying X is bigger than negative five, which is true. Everything to the right of negative five. And at the same time, everything to the left of positive five. That's the mathematical way to say it's in between those two numbers. I don't really memorize that. I just know if I do the smaller number first and then the X and the bigger number, it's less than, less than. And that part of it kind of makes sense. So starting between the two numbers, I want all the points between those two numbers. So now we figured that out so we can do the range. How low does my graph go? Zero. How high? Touching and touching? Smaller number. Range is Y. Bigger number. Less than, less than touching, touching. By the way, if I wanted to generalize this, negative radius less than or equal to X, less than or equal to positive radius, zero less than or equal to Y, less than or equal to whatever the positive radius is. This one for key points does not have a little mantra. Sorry. This one. Find the center. Find the radius. For me? Can the range? What do you think would cause a semi circle to open upside down? Do you think? Come on, bring the parabola back home here. Putting a negative in front of the whole thing? Really? We're going to generalize all those parabola rules. We're going to generalize them to any function. And yes, you can flip it. Put a negative in front of it. What you didn't do last year, Alex, was you didn't do very much of flipping stuff horizontally. And the reason is a parabola is symmetrical. If I spin a parabola, I still get a parabola, which isn't very much fun. You flip them vertically. And this one, flipping it horizontally, won't make a difference because it's so symmetrical. Ah, but we will eventually have non-symmetrical graphs. All the stuff you learned how to do up and down and vertically, we're going to add horizontally as well. And the nice thing is it all is going to tie together. In fact, I almost argue that it makes the parabola easier when you figure out why the parabola behaves weird in certain situations. No, it doesn't actually. It's behaving weird because you have to be specific instead of a general rule. The cubic, graph number four. The cubic is y equals x cubed. Once again, I'm going to stick in some basic numbers. And once again, I always try zero. What's zero to the third power? Yeah, this graph is going to go through zero, zero. What's one to the third power? One. It's going to go through one, one. What's two to the third power? Eight. This is going to go through two, comma, eight. Sally, what's three to the third? Twenty-seven. Off my graph. Who cares? Oh, you know what? What's negative going right? Let's go left. What's negative one to the third power? What's negative one times negative one times negative one? Which one? Negative or positive, folks? Okay. Negative one goes through negative one. Oh, and what's negative two to the third power? What's negative two times negative two times negative two? Not positive eight. Negative eight. And Sally, what's negative three to the third power? Negative 20. Is that off my graph? Okay. The cubic looks like this. Okay. What else am I supposed to... I'm limited. It's not a common shape. So, John Travolta graphs. Okay. This one doesn't have a vertex because a vertex is where the graph turns around on itself. This graph does have a center, though. Where does this graph look like it's centered symmetrically? Zero, zero. What's the domain? People on the video. Okay. Zero, zero. Zero, zero. Zero, zero. Zero, zero. Zero, zero. What's the domain? People on the video, anybody who's wait any day is watching this on video. They're like, What the heck is everyone laugh about? Come to class, if you're away. So, we do like this in that it has, really, the simplest possible domain in a range. All reels, all reels. Mr. Dubic, Y. What that saying is, I can put in any x value between negative infinity and positive infinity, and any answer you want to get a y value. I can find an x value that'll give you that answer, those Those types of graphs those types of functions have a special name, but I can't remember what they're called They're not one-to-one. They're not on to the homogeneous. There's something else. Oh key points Zero zero one one Two eight and then I'll trust that you can remember also negative one negative one and negative two negative eight But I'm not gonna write those down. I'll figure that out because it's a John Travolta graph Do I have to memorize those key points or can you just derive them by going? What is zero cubed? What is one cubed? What is two cubed? Yes, you can that's why you know If you're someone who has to memorize things find I honestly Connor don't I derive them every time because I can do a Bit of arithmetic in my head next one graph number five the absolute value function Looks like this That's the symbol for absolute value. What does this graph look like? The absolute value looks like a V shape You're saying the absolute value looks like a V shape It'll be great if there was an easy way that I could remember that the absolute value looks like a V shape But I'm a terrible math teacher. I can't come up with anything. What's the absolute value of zero? Zero what's the absolute value of one? One what's the absolute value of two? To what's the absolute value of three? I think I spotted the pattern Let's go in the other direction. What's the absolute value of negative one one? negative two two negative three three In fact, it looks like this Sesame Street is brought to you today by letter V Martin may be wondering What's the point of this function as well? Those who there in physics know how often we chuck a negative and just take the absolute value of our answer We don't call it taking the absolute value of our answer. We call it a scalar right We actually do use this all the time Vertex zero zero In fact, doesn't this sort of kind of looks similar to the parabola. It's got the same domain of range domain All reels range why greater than or touching zero in the old math 12 course and in the conics unit Pardon me Just any question. No, okay We used to be interested in the slope of the arms I stuck this on just to review with you what slope was because technically you haven't seen it since math 10 Although you might have done it a little bit in math 11 or in chem This right hand arm, what's the slope? What's the rise over the run of the right hand arm one? What's the slope here? Negative ones. I'm gonna write down What letter do we use for slope and why? Because not all math is done by English speaking people And I believe is from the French Who's my immersion students anybody know the word for sneakness? I guess I'm gonna do with mountain or Mont Blanc or something like that. Anyways Not all English all not all math was done by English people. I think it's from the French or it could be from the Latin Okay, anyways, what do you say the slopes were plus or minus one depending which arm you look at? Hey, what are the key points of the absolute value graph? one one two two Three three Justin, can you figure out the next key point? How about four four? Four four what will the next one be? Five five now Justin actually said how about negative one comma one. I don't memorize that. I know that the absolute Value graph is shaped like a V. So you know what if I write that I can get the rest just flip it over, right? In fact, I'll be honest. I don't even memorize these I can absolute value is an easy enough function to do in my head that I would just try zero one two and three Somebody yawning the linear function y equals x This is a line that goes through One one two two well goes through one one two two three three four four five five six six seven seven eight eight Zero zero negative one negative one negative two negative two it looks like that That's the simplest possible line y equals x and then in math 10. We generalized it You remember what slope intercept form what the equation template was equal to y equals what x plus what? Okay, hugely overrated who's in calculus You've learned a better form. Have you not? Point slope form have you learned that one yet? If you haven't yet, you will be way more flexible In fact, they're finally in the new math 10 curriculum, which is this year. They're doing point slope form Good for them. Hey, what's the y intercept? zero comma B in My specific graph here is zero comma zero. What's the slope? The number in front of the x what's the domain of our particular line? All reels does that mean every single line has a domain of all reels There's one specific line that does not Negative I think that'll have a domain of all reels still good gas the good thought Which line would not have a domain of all reels? Who said that what? vertical Because it only has one x value doesn't move left right or off So let's pretend we're going to ignore the vertical and we're going to say the domain of any other line Is all reels and the range of any other line is also all reels unless it's a horizontal line Let's pretend we're ignoring the two extremes. What's the x intercept? How do I find an x intercept? Mathematically remember how? Angelo Okay, I'm just going to write down here Let y equals zero and solve Ooh wait a minute my physics 12 students Put a zero right there in your mind and get the x by itself negative b over m comma zero. I Don't memorize that Hannah I can derive it from this because I've done so much physics and rearranging it remember how hard that was at the beginning of physics 11 And now you're like, please I could do this in my sleep just about yeah, it's a useful skill But for what it's worth if they ever said what is the generic x intercept of a line? There it is Equation of a vertical line x equals or y equals for a vertical line Angelo remember Yeah x equals and I'll say a some number, you know the equation of a horizontal line is Y equals B. I did use B because it is the y intercept. That's how high that line is going to be so By the way, a horizontal line has a fancy schmancy name. It's also called the Constant function and those either done physics 11 or arguing physics 12. So oh, yeah when the V was steady We call that a constant velocity or when the distance versus time graph was steady That was a constant velocity. That is the constant function. Okay. Oh Constant function function didn't see that word there. I'll do that point slope form Really handy Well, it used to be really handy in math 12. It's not quite as handy as it used to be But really handy in calculus. There's a better form than y equals mx plus B You see for this to work and If you want to graph it B can't be a fraction because how the how high is 17 over 3 along your y-axis. You're kind of guessing an eyeballing So point slope form says this If your graph goes through a point, we're going to call it x1 x1. Mr. Do it comma y1 if you know any point on your graph and You know the slope the equation is this Calculus of mr. Kamosy showed you that one yet He probably will have you guys talked about tangent lines yet This is by far the best equation to find a tangent line with because you already know the point on the tangent line They gave that to you in the question because they said find the point at x equals 7 common And all you're doing then the spotting is really plugging in the slope, but it's really plug each up It's the best way to find a tangent. So you're gonna tangent. Uh-huh. Uh-huh Now then I wrote down in the blue Alberta book try Number page 5 numbers 1 to 3 and number 10 am I good with that? Let me see page 5 1 to 1 2 3 and Number 10. Oh sure That's only about 10 minutes worth of homework And I think I did that on purpose when I set this up because I also think that some of you were probably thinking Hey, mr. Do it could like to maybe do work on my log review or do a bit of studying or something like that Remember on Wednesday your log review is also due That's This big thing here, right? Okay And I'm gonna turn you loose, but just give me one second