 Hi, I'm Zor. Welcome to a new Zor education. Let's talk a little bit more about graphs. We used to draw graphs of function of one variable which looks like this. On the coordinate plane where this is x-axis, this is y-axis, and for every x from the domain where this function is defined, we have the point which has coordinates x and y equals to f of x. Now, with every x on this particular line, well, whatever belongs to the domain of the function, we can draw this particular point. And then all these points we can basically connect together, and that's what basically forms the graph. All right. Well, this is not the only form which is used when certain graphs are presented. I would like to have a little bit more general case where this is actually a particular case of that more general case. And here is the more general case. What if instead of expressing our function as some kind of algebraic expression with one variable x, and the result of the algebraic expression is y, which is coordinate on the y-axis, what if I will have an equation like this, where f is the function of two variables, whatever the algebraic expression is. Well, let's just remember the definition of the graph. The graph is, in this particular case, it's a set of all points with coordinates x, y where if you substitute the x into this formula, you will get the coordinate y. Well, for this particular equation, the definition is exactly the same. So a set of all points on the coordinate plane with coordinates x, y, if you substitute x and y into this formula, you will get 0. This is basically the same thing, and any function expressed in this particular way can be expressed as this, where this is a function of two variables, and this is a particular case of this more general case. So that's all there is. It's just a small generality which I would like to introduce into this picture. Now, if you can resolve this particular equation for, let's say, y, and get y as an expression with one variable x, then you will basically reduce this case into this case. Like, for instance, if I say x plus y equals to 0, this is my function of two arguments. Yes, you can always resolve it using this, and this is a familiar function which we dealt with before. But sometimes it's not possible, and you would still like to present the dependency between x and y expressed as this particular equation in the graphical form. And that's what I will be talking about today. Okay, so we understand what this particular kind of graphs are all about. These graphs represent a dependency between coordinates expressed as this particular equation, where f is a function of two arguments. We are talking about real values, obviously, x and y are real names. So let's just have a couple of examples. For instance, the first one is this. Well, this is a typical example of an equation of two variables, where it's really very easy to resolve it for one of them. So we will probably resolve it for y is equal to minus 2 third x minus 4 third, right? And the graph of this function is very easy to draw. We know that y equals to minus 2 third x is a straight line, which goes through the beginning of the coordinates. And it goes from the second to the fourth quadrant with certain coefficient. This minus means it's this direction rather than this direction. And minus 4 third, which means we go down by 4 third. So altogether, it will look like this. This is y equals to minus 2 third x minus 4 third. So this is easy. And it's absolutely straightforward. There are no underwater currents or any kind of peculiarities of this. This is simple. Can we do it differently, this particular graph? Well, we probably can. Here is one of the ways you can just consider drawing this particular graph. Well, first of all, you understand that this graph is supposed to be a straight line. Just because this is x to the first degree, this is y to the first degree. So everything is just linear. Linear dependency between x and y is expressed as a straight line. So for the straight line, all we need is just two points. And then we will be able to find out exactly how the line goes. Well, let's do it this way. Coordinance x, y, 0. Let's find where is the intersection of our line, which we don't know yet, with, let's say, an x coordinate. Well, if it's intersection with x coordinate, that means y should be equal to 0. Now, if y is equal to 0, then x must be 2x plus 4 equals to 0. So x is equal to minus 4. That might minus 2, sorry. So this is minus 2. So our straight line always goes through the point where x is equal to minus 2 and y is equal to 0. Okay, now let's find intersection of this line with the y axis. Now, in this particular case, if our point is on the y axis, it means x is equal to 0. So if x is equal to 0, what follows is 3y plus 4 is equal to 0. So y is equal to minus 4 third, which is here. So x is equal to 0, let me just put it differently. And y is equal to minus 4 third. So this is another point. And obviously, our line goes through these two points. So no matter how we draw this using this technique or that technique or anything else, we will come up with something like this. So this is an easy case. Now, let's consider a little bit less trivial, because this is trivial because it's resolvable for y. So it's not really very interesting to represent it in this particular form, because it can be easily transformed into another more familiar to us. How about this? Well, this is kind of strange. What is the graph of this particular equation? I'm not saying function, because you will see that this is not really a graph of any function in a classical meaning of that word. But what is the graphical representation of this dependency between x and y? Well, I'm not going to do something very naive. Let's divide both parts by x. And I will get y equals to 0 divided by x is 0. So I will have this. And what is y is equal to 0 on the coordinate plane? It's basically the line which coincides with our x-axis, because only all the points in this x-axis have the y-coordinate equal to 0. Well, that's obviously not exactly the right way of doing this. Because you can not just divide by x and not considering what if x is equal to 0. So what is that if x is equal to 0? Well, if x is equal to 0, y can be basically anything, which means any point where x is equal to 0 and y is anything, which means any point on this particular y-axis also belongs to our graph. So our graph is the set of all points which lie on both coordinate axes. Well, obviously, this is not a graph of any function, because our function in our classical definition function is some kind of correspondence between a single value of one variable, which we call the argument. And the result will be one concrete value of the function. Now, in this case, if argument is somewhere here, then yes, the function is always equal to 0. But if the argument is here, there is no concrete value of the function. Anything can be a value of the function. So this is not a function, so to speak. This equation does not represent a function. It represents the pendency between x and y, which is graphically represented as the set of all points on both axes. But it's not a true function in the classical sense. All right, but that's fine. So equation is not necessarily a function. And in this particular equation, that does not necessarily define a function. But it's nevertheless a certain dependency which you can find from some practical experiments or anything like that. Some equation between x and y can be derived in whatever way you want. And then it can be expressed graphically again in some way, which does not have to be a graph of a function, so to speak. OK, next example. x squared plus y squared equals r squared. It probably can be attempted to resolve it for y. Well, yeah, you will get something like y squared equals to r squared minus x squared, right? But then when you think about the next step, you can just have plain this. This is not right because this is arithmetic square root, which means it's always positive. But y can be negative. I mean, if y is equal to, let's say, if point with coordinates x and y belongs to this graph, then the point with coordinates x and minus y also belongs to the same graph. So it's basically like minus y also would be equals to this. So it's not, again, exactly a function now. Since we have more than one value of the function, which can be the value, it's not really a classical function. So let's forget about reducing this particular dependency to a concept of a function. Let's just try to deal with this as it is, as it is given. And here is what's given. Let's consider this particular point with coordinates x, y. And let's think about what is x squared plus y squared. Well, if you project it down, you will have this as x, this as y. And if you connect it to the beginning of the coordinates, you can consider this right triangle. And this would be the square of the hypotenuse, right? So no matter where x and y are, maybe it's here. It's exactly the same thing. Now in this case, x is negative, y is positive. But we're dealing with x squared and y squared. So it's always the length squared, length of this segment squared and length of this segment squared. That's why we always have the length of this segment squared. So where is this graph? I mean, how are all these points, which belong to the graph of this equation, positioned where are they? Well, obviously they are on the same distance from the center. So any point on the coordinate plane, which is distance, which is distant from the beginning of the coordinate by the length r, belongs to our graph. So it's a circle. That's what it is. So the graph of this particular equation, which represents certain dependency on x and y, is a set of points on the coordinate plane, which are on the distance r from the beginning, which is a definition of a circle, basically, right? So what is a circle with a point o as a center? All the points which are on this particular distance from the center. So this is an equation of a circle. Is it a function? No, it's not a function. Again, same thing. Take any value of the argument x, and you will have two different values of y. If you remember, y is equal to square root of r squared minus x squared and minus, same thing. So two results mean that this is another function. However, it's a legitimate graph, which represents this particular dependence. Now let's do a little bit more technical stuff. The more technical stuff is this. By the way, I hope you are not really discouraged by the fact that I put x squared plus y squared equals r squared. It's not exactly this form, but if you really insist, I can always do this. r is a constant. r squared is a constant, so it's the same thing. So no matter how I write it in this way, or equal r squared, it's exactly the same thing. Same thing stands with this one. I can put it minus r squared is equal to 0, or equal to r squared, same thing. But now, question is, now, what's the graph of this particular function? Well, here, let's just think about. If you remember, when you were dealing with functions, we were talking about that if you have a function like this and a graph of this function, then if you would like to have a graph of this function, you have to shift this one to the right by a. Well, if a is positive, it means to the right. If a is negative, it means to the left. But anyway, it's to the right by a. So in which case, the whole graph would look like this. It comes up like this. Well, it's exactly the same story with this particular case when you have an equation where x and y are mixed together. If you add something to x coordinate, it means the original graph of this function, let's forget this. I put equals to r squared. It's better. So the original graph, and compare it with this one, let's say this is y squared, not y minus v squared, it will be a shift to the right by a. In case a is positive, if a is negative, it would be to the left. But anyway, it's always to the right by a. Why? Well, for obvious reason. Because if the point, let's say, p and q belongs to this particular graph, then point p plus a q obviously belongs to the graph where this is y squared. Why? Because first we subtract a. So p plus a becomes p. And then we have p squared plus q squared. And we know it's already r squared. So if point p q belongs to this graph, then point p plus a q belongs to this graph with this y squared. Now, if this is y minus v squared, I will just make it this. So again, if p q belongs to this, then p plus a q plus b belongs to the graph of this dependence, which means that the whole circle, which is all these points p and q, which satisfy this particular equation, is shifted by a to the right and b to the left. So the center basically will be shifted, but the radius will remain the same. So this is a new center, and the radius will be the same. So from this, we just shifted the whole picture here. So this is also a circle. And this circle represents all points on the distance r from a center at a d. So now we know how to draw a circle algebraically on the coordinate plane. We could not do it, by the way, using function representation because this is not a function. Circle is not a graph of any function. Circle is a graph of a dependency between x and y, which is expressed by this particular equation. OK, what's next? Next is this. What is this? Again, let's go back to the graphs of the functions. And if you remember, if you have a graph of function y is equal to f of x, and then I was asking, so what would be the graph of this function? Let's say p. And we were talking about the whole graph actually being scaled by p along the x-axis. So if I had something like, let's say, this graph of the original function, and let's say p is equal to 2, then it would be this. It's squeezed. y, again, to the same reason. If point p, q belongs to this graph, let's not use the letter p. Let's use the letter k, let's say. Then point p, k, q belongs to this graph. And now I see I made a slight mistake. Because if k is integer, it's not squeezing. It's actually stretching. I should have drawn, instead of this, I should have drawn stretching by k, if k is equal to 2. So when x is divided by k, you're stretching by k times. But obviously, if k is a fraction, let's say 1 half, it would be squeezing. So it all depends on what the value of k is. But for the k equals to 2, it's stretching. For k is equal to 1 half, it would be squeezing. So anyway, obviously, this is a proof of this particular quality of the graphs, f at x and f at x over k. Absolutely the same situation would be here. So what I'm saying is that if point belongs to this particular graph, then point p, x, q, y. Sorry, it's lowercase. p, k, x, q. This point belongs to the graph x square plus y square is equal to 1. So if you have this particular graph, and this is a unit circle, a circle with a radius 1, then this should be actually stretching by p along the x axis. So if this is my unit circle, and let's say p is equal to 2, then x over p square plus y square equals to 1. This would be a stretching by p. Let's say p is equal to 2, so it would be like this. Not very beautiful, but you understand the point. So we are stretching. Now, if this is y divided by q, it will also be a stretching by q upwards. Let's say q is equal to 1.5. So the points will be here. So the result will be here. Sorry, we don't stretch x any longer. So I'm stretching by p horizontally, and I'm stretching by q vertically. And what's the result? Well, it's called ellipse, basically, or ellipse sometimes. OK, so this is an equation of ellipse or ellipse with these factors being the factors where stretching the unit circle horizontally or correspondingly vertically. That's it. So you know how the ellipse can be expressed algebraically on the coordinate plane. What else do we have here? OK, the next one is parabola. OK, well, we know that this is the function graph of which is the regular parabola, like this. Now, this can be expressed in this language, right? So basically, it corresponds to the function of two arguments. And this is the dependency between these two arguments. And the dependency is expressed as a parabola. Now, what's interesting is that let me just consider a similar function. But instead of y, I will use x. And instead of x, I will use y. What's the graph of this? Now, before answering this question, I would like to address the following point. If you have a graph of the function this, and now I have the same function, but I will exchange the arguments. How these two graphs would correspond to each other? Well, let's assume first that this is the equation which can be resolved for y. So it would be y equals to some function of x. Now, that means that this can be resolved for x, right? So it would be exactly the same function, f. But now x is expressed as a function of y. Now, so let's reduce our problem of correspondence between the graphs to these cases. So what if you have an equation which looks like this, and you would exchange x and y places? What actually that would be? Just as an example, y is equal to x square. And this x is equal to y square. This is exactly the case which we need, right? Now, this might not be the same. Now, this is the function. This might not actually be the function, because the function is the y as a formula of x. But that's not always possible, right? But anyway, what kind of a graph we will have if we will try to draw this one? Well, if point x, y belongs to this graph, then point y, x belongs to this graph, right? It's the same thing, formula is the same. But how are these two dots, one represented by coordinates x, y, and another represented by coordinates y, x? How do they position relative to each other? Well, let me just give you an example. If you have point 1, 2, this is 1, 2. Where is 2, 1? 2, 1 is here. So obviously, these two points are symmetrical relatively to bisector of the main coordinate angle. Now, I hope it's obvious because we're changing whatever was x here becomes y. Whatever was y becomes x. So we can very easily, purely geometrically prove, look at this. Now, this is a distance, which is a square of this distance is sum of two squares, x-coordinate and y-coordinate. Now, this is y-coordinate and x-coordinate, but this is exactly the same thing. So these two are exactly the same. And so are angles. And so are these. So the congruence of these two triangles is obvious. And from the congruence of these two triangles, follows the congruence of these triangles. So that's why these two points are symmetrical relative to the angle bisector, which means that graphs of this and this are supposed to be exactly symmetrical, as well as graphs of these two, for the same reason. Because if point x, y belongs to this, then the point y-ax belongs to this. So these two graphs are symmetrical, which means these two graphs in our particular example are supposed to be symmetrical. Now, this is a parabola. Now, what's the symmetry of the parabola? What if we will just reflect it relative to this particular angle bisector? Well, it would be this, right? These parabola and this parabola, they are symmetrical to each other. Now, granted, this does not represent a function, because again, for a single value of x, we can have two different values of y. But that's not what we care about. We care about equation, not about the function. So equation, this one, would be satisfied for all points which are flying on this parabola. And again, let's just remember the main principle. If you consider these two equations with the same function f, just exchanging the arguments, then the graph of this and graph of that would be symmetrical relatively to the angle bisector, all right? Now, let me just exemplify it with some other example, which probably would be a little bit easier to understand, because it is about function. So if you have a function this, or y is equal to x cubed. Now, we all know this is a curve which looks like this, right? This is minus 1, 1. This is plus 1, 1. Now, if I will change places, x is equal to x minus y cubed equals to 0, or y is equal to, or x to the power of 1 third. Now, the graph would be, let me get another color, would be this. And as you see, the black and red are symmetrical relatively to this. Now, you remember that the power function, x to the power of something, power of a, it depends on the a. If a is greater than 1, it goes this way. If it's less than 1, like in this particular case, it goes this way. But it's always around the point 0, 0. All these graphs, the greater the a, the steeper goes this one, the smaller the a, the positive obviously, but it goes closer to 0. The closer this particular branch of this curve will be towards x-axis. But in any case, these two graphs, this one and this one, they are symmetrical relatively to the sector, to angle by sector. OK. And finally, just for fun, mass sometimes can be fun, but you have to really struggle to get to this fun. Here is something which I have found on the internet as an interesting curve. x to the force minus x square plus y square equals to 0. So let's try to draw the graph of this particular equation, this particular dependency between x and y. All right. It's not easy. I mean, it's not obvious what the graph looks like. So let's just think about this step by step. Now, the first form, we can represent it as y square equals to minus x to the force plus x square. That's the same thing. Now, this is a positive value, because it's y square. Square is positive. Well, actually, non-negative. Let's put it this way. Non-negative. So this is supposed to be also non-negative, which means what? Which means that x square should be greater or equal to the x to the force, right? For this to be positive or 0, we should have this greater than this. That's obvious, right? Or if you wish, you can always put it minus x to the force plus x square greater or equal to 0. Now, we add x4 to both sides, and we get this. All right, fine. Now, how to resolve this particular inequality? Well, there will be a separate chapter in this course about inequalities. But it's kind of obvious that if one number is greater than another, then we can divide or multiply it by some positive number, and the relationship would actually remain. You can multiply by 3 would be 6 greater or equal to 3. Or you can divide by 3 would be 2 thirds greater or equal to 1 third. So manipulation of division and multiplication by a positive number doesn't really change the inequality. So what do we have from here? Well, obviously, it's either x is equal to 0. Or if x is not equal to 0, then let's divide both parts by x square, which is positive now, right? If it's not equal to 0, it's positive. So I will have 1 greater or equal than x square. Or if you wish, x square less than or equal to 1. Now, what does it mean? Well, actually, this way, it belongs to this, right? x square less than 1, it means absolute value of x greater than or equal to 1. It means that the whole domain for this particular function of two variables should be from minus 1 to plus 1 for x value. That's done. So the main for the x is this particular segment, from minus 1 to 1. x cannot be greater than 1 or less than minus 1. Because then it will force, as you see, y square to be negative, which is impossible. OK, fine. So we have determined the boundaries for our x. Now let's go to the boundaries for the y. Well, we will use the same thing. Now, what's the maximum value for y square? Well, that's what's the maximum value for this particular expression minus x4 plus x to the second. Well, let's do it this way. Let's substitute u for x square. I will have minus u square plus u on the right, correct? Now, what are the values, the maximum value for this thing? Now, this is a parabola, right? You remember that the graph of this function, it's a quadratic function. So it's a parabola with horns downwards because this is minus. And it has two roots, 0 and 1. So it goes like this, with a maximum at 1 half. And at 1 half, it's minus 1 quarter plus 1 half is equal to 1 quarter. So the maximum value for y square is 1 quarter. So the maximum value for y is 1 half, which means these are the boundaries for y. So the whole graph should be within this particular rectangle. Great, we have determined that. Also, let's just have a couple of very obvious straight points. If x is equal to 1 or minus 1 and y is equal to 0, then we have this equation satisfied, right? 1 minus 1 plus 0 is 0, right? So this point, which is 1, 0, and this point, which is minus 1, 0, satisfy the equation, as well as 0, 0, 0. So our graph is supposed to go through all these three points. Also notice that our graph is symmetrical relatively to both horizontal x-axis, because if you will substitute from y minus y, you will get exactly the same. And it's symmetrical relative to the vertical y-axis, because, again, if you change x to minus x, it will also be exactly the same, because these are all squares and fourth degrees of the power. So the graph is symmetrical this way and this way, which means let's just draw this particular graph in one particular quadrant, the first one, where both x and y are positive. And then we will just symmetrically reflect it. Now, how can I draw the graph in this particular area, where x and y are positive? And I know that graph should not really go outside of this rectangle. And I know that it starts here and ends here. So it's probably somewhere here, something like this. That's what I assume. I can probably apply a little bit more logic, but in any case, we have to smoothly connect these two points while being in this particular area. So that's the most obvious kind of a thing. And now we can reflect it down and to the left, and we get almost an infinity sign. So this graph is called infinity curve, just for fun. This is a graphical representation of mathematical symbol of infinity using this type of a formula. All right, anyway, that's it for today. Please go to unizor.com. There are notes for this particular lecture. And I do encourage you to take a look at them. And it would be great if your parents are registered as supervisors. And then you can take exams and basically take the whole course as a course with exams and get some grades, which would probably satisfy the way how you study. That's it. Thanks very much for today.