 Hi, I'm Zor. Welcome to a new Zor education. I would like to expand the previous lecture where I was drawing certain graphs of equations like this. So I would like to expand to basically talk about different set theory manipulations of the graph. What I mean is like union and intersection, these set theory things. So let's consider that your graph should consist of two separate pieces. How can you represent algebraically in this form, basically the equation which represents this combined graph? Or alternatively, if you know that the points of your graph are actually intersection of points which belong to one graph and points which belong to another graph, how to algebraically express this intersection of two graphs. Actually, I did touch this particular point in the previous lecture when I was explaining how to draw the graph of this equation. So I was first talking about like naive dividing by x. So you have y is equal to zero, which is this basically line which coincides with an x-axis. But then since we divide by x, we have to really separately consider the case of x is equal to zero and x is equal to zero is a vertical line. So the combination of these would be this. I would like actually to use exactly the same technique and explain how to do this. Well, basically that was union of two graphs, right? So I will use exactly the same technique to unionize two different graphs. So let's say you have two different functions. Let's start with the functions first, one function and another function. Now this has certain graph and this has certain graph. Well, let's consider that the domain where the functions are defined is the same. How to represent algebraically the union of these two graphs? Well, let's just go back to this example, which I was just mentioning. X, Y is equal to zero. Recall that if you have a product of two different things, which is equal to zero, then either a is equal to zero or b is equal to zero. Both actually result in the whole equation to be true. Now, that means that if I will use y minus f of x and multiply it by y minus g of x and put it equal to zero, consider this particular equation. Now, when is this true? Under what circumstances, what points with coordinates x and y belongs to the graph of this equation? Well, either those points which nullify this part or the points which nullify this part, which means points which satisfy this equation or that equation. So, again, the fact that this is or, these or there. These points or these points. This actually means that we are unionizing all the points which belong to this and unionize with all the points which belong to that. This means that our two graphs, separate graphs, will be unionized when we are represented by this particular equation. In more general case, if instead of this type of functional dependency between x and y, you have an equation dependency. Basically, it's exactly the same thing. So, if you have one particular dependency and you have another dependency, then this is some graph. This is some graph. The combination of these two graphs, which is a union from the set theory point. E is represented by multiplied by f2 xy equals to 0. So, this is how you unionize the graph. Well, apparently everything can be done in an opposite direction. What if your equation actually looks like this in the very beginning? And you are asked to draw the graph of this particular dependency between x and y. Well, you can obviously say, hey, this equation is equal to 0 only if f1 of x and y is equal to 0 or this is your unionization. So, all the points which satisfy f1 is equal to 0. Unionized will all the points of f2 is equal to 0. So, these two equations must actually be represented as graphs and unionized. And this is an easier task to grab these and these separately and unionize than to graph the big thing. Now, from this position, let's go back to our original example, which I was just talking about, x, y equals to 0. Well, this is exactly this case, right? So, x is equal to 0 should be unionized with y is equal to 0. So, x is equal to 0 is vertical line and y is equal to 0 is horizontal line. So, the combination of these gives you the same graph we were talking about. In this case, we came up with the same result using a little bit more general approach to manipulation with graphs. We basically unionized them, all right? Now, okay, union we have covered. Now, let's talk about intersection. Basically, we are using the same trick. I mean, we will find an algebraic equation which is equal to 0 if both parts are equal to 0. So, whenever we are multiplying an equation to 0, then either this or that should be equal to 0. And that's the source to unionization. The formula which will give you one of the formulas, actually there are many others, when we are requiring that both A and B are equal to 0, can be something like this. Now, when is this is equal to 0? Only if both A and B are equal to 0. So, if this represents one particular set of points and this represents another set of points, then this represents an intersection between these two sets of points. Both A and B must be equal to 0. So, if you have a function which looks like, let's say, y minus f of x square plus y minus g of x square equals to 0. What does it mean? What's the graph of this particular equation? Well, obviously, this is a combination. This is supposed to be equal to 0. So all the points which are represented by the function y is equal to f of x must be there first. Then this also must be equal to 0. Which means the second piece is all the points which represent the function y is equal to g of x. But since both of them must be equal to 0, it's only intersection between these two graphs, which would be a graph of this particular equation. So, how to draw something like this? Well, first you draw the left part, then you draw the right part, and then intersect them. And obviously the same thing is true for more general representation f of 1 of xy square plus f2 of xy square equals to 0. So obviously this also is an intersection between this graph, f1 of xy is equal to 0, and this graph, f2 of xy is equal to 0. So that's what it is. Next, what's next? So we have basically, by the way, instead of a square plus b square, I can actually use something else. For instance, let's say absolute value of a plus absolute value of b is equal to 0. Now, again obviously since this is non-negative and this is non-negative. So it's only when it's 0 and this is 0, the total will be 0. But absolute value is sometimes a little bit more difficult to deal with than the square. But that's kind of a personal preference and probably depends on the problem in question. All right, so that's the theory of this. And now let's just do a couple of examples, which will basically illustrate this particular approach. How to draw certain things in certain cases when it's not really obvious, but using this approach we can do it. So I have three examples here. Example number one is x square is equal to y square. What is the graph of this particular equation? Now, if you wish, you can rewrite it as this. Okay, that actually kind of brings up to a very interesting modification. You know that the difference between two squares is this, right? Y square minus x square is y minus x times y plus x. Now, we remember that if you would like to draw a graph of the product of two functions. Are equal to 0 functions of two arguments x and y. Then it's a union of this union. I will use the sign of union from the set theory and this. So what's the union of these two graphs? What's this particular graph? This particular graph is y is equal to x and this is y is equal to minus x, right? So graphically, y is equal to x is this, y is equal to minus x is this. So the combination of these two lines is a union, right? The combination means union of these two lines is the graph in the original equation. Okay? Now, is it obvious that this is exactly this particular set of two lines? This is bisecting the first and the third quadrant and this is bisecting second and the fourth quadrant. Well, looking at this, I'm not sure it's so obvious that this is really the graph. But if you transform it in this way, it's kind of easy. All right, second example. Okay, second example is y square equals 2x. Okay, I think I was talking that if you have a graph of the function f of x, y is equal to zero and you exchange the x and y places and you will put f of y, x is equal to zero. Then these two graphs will be symmetrical relatively to the main bisector of the main angle of the first and the third quadrant. Because whatever point x, y, let's say this is x and this is y. If it belongs to this graph, then the symmetrical point, so this is y and this is x, belongs to this particular graph. So I was explaining this. From this perspective, the graph of this function should be a symmetrical line to this one, right? And this is the regular parabola. So the regular parabola comes this way and if I will symmetrically reflect it, it will be something like this way. So that's what we will have to really obtain from the considerations of symmetry and this thing. Now, what if I would like to approach it differently? The way I would like to approach this particular graph is not using this symmetrical consideration, but slightly different thing. First of all, let's just consider what's the allowed value for x. Now, since x is equal to y squared, y squared is not negative. It means x is, so graph should be concentrated only for values of x greater or equal to zero. Nothing on the left from the zero is defined. Okay, we still have to define what's on the right. What I can say is the following. y squared minus x is equal to zero and x for positive or zero x I can always represent as this, right? This is x for non-negative x. Same thing, square root and then squared. So that means that I have this or again the difference between two squares is y minus square of x and y plus square of x is equal to zero. And again using our considerations about when this particular function is equal to zero and either this is equal to zero or that is equal to zero. This gives me y is equal to square of x. This gives me minus square root of x. Now you remember that this particular function or if you wish x to the power of one-half is this. We were talking about x to some power. We had the whole lecture how this particular curve looks like for different values of power. So for one-half it looks something like this. Now this one is basically the same, just symmetrical relative to the x axis, right? Whatever positive y, now you should go to a negative y. So it goes this way. So the combination of these two lines gives you the graph of the original function. So we have exactly the same curve which is original parabola, it's symmetrically reflected relative to the bisector. We just came to the same conclusion using slightly different logic. Which is good actually, I mean it's great when we can come to the same solution using different ways to solve the problem, it just confirms that we were right. It's always great when you come to this type of situation. I mean in simple cases like this you probably are sure in the first place that you were right. But in a complicated case if you have more than one approach and you're coming to the same solution that's great. All right, and the third one is, well it looks complicated but believe me freely not, y minus x square square plus absolute value of y minus square root of x to the fourth square equals to zero. Don't get scared. All right, there is a purpose in this. Now first of all this is sum of two squares, right? Now since it's a sum of two squares and it's equal to zero, it means that we're talking about intersection of two graphs. This graph and this graph. So we have to separately consider what's the graph of this? And this is obviously y is equal to x square, right? And that's a regular parabola. And how about this? Well, let's write it this way. Well, if you think about this, for positive x and y, you can just drop the absolute value, right? For y, so it's positive. And on the right, square root of x to the fourth degree is basically x square, right? So for positive or zero, actually, non-negative. You have this, which is the same as this, right? Okay, how to expand it for all other combinations of x and y? Well, first of all, since there is a square root of x, we cannot expand it to the negative x. X must be non-negative, otherwise the square root doesn't exist. Okay, so the function, this particular graph is concentrated only on the right part of the coordinate plate for non-negative x. Now, how about y? Can we expand it to a negative y? Well, obviously, yes, since this is an absolute value, it means that the graph is symmetrical. Whatever is on this side would be symmetrical relative to the x-axis. Because if point x, y belongs, then x minus y also belongs, since y is always in absolute value to the graph. Now, so for non-negative x and y, we have this graph, so it's a parabola. For symmetrical side, we will get this, obviously. So the graph of our function, this function, is a piece of a regular parabola reflected down relatively to the x-axis and we have this particular thing. So that's the graph of this particular function. Well, I shouldn't say function, this is the graph of an equation. This is an equation, so the function. Because again, for a single variable of x, we have two different values of y, so we shouldn't really call it a function. But yes, an equation. Equation of two variables, x and y. Now, let's go back to this one. What's this? Well, this is a regular parabola, which means it goes this way. Coinciding with this in the first quadrant of the coordinate plane. So what's the intersection between these two? So obviously, the intersection is this part. This piece of the parabola, half of the parabola, is actually an intersection between this and this. Well, so this relatively complicated equation can be graphically represented simply as half of the regular parabola. I told you it's simple. All right. That's it for today. This is just a little expansion on certain graphs and manipulation with graphs. There is something else which I would like actually to address, which is polar coordinates. And that will be in the next lecture. Thank you very much.