 I am Zor. Welcome to Unizor Education. Today's topic will be Graphs, Function Graphs. Well, first of all, let's very briefly remind what the function actually is. If you remember, function contains the definition of function includes the set which is called domain, another set which is called qualemain, and certain rule which puts into correspondence every element of the domain to a corresponding element, a value of qualemain. When we are talking about graphs, we are talking about one particular kind of functions. Functions when both domain and qualemain are real numbers. And here is why. Graph is something which is supposed to represent on the surface on the plane the behavior of the function. Usually graphs are used for studying qualitative behavior on the function, much rarer quantitative. Now we have computers, they can calculate everything much more precisely which we can graph. However, it's always very interesting to research how the graph behaves because it represents the function of the behavior. So why planes and why real numbers? Well, for a very simple reason. On the plane we can very easily construct a graphical representation of the function which has domain and co-domain real numbers, and here is how. Imagine you have two perpendicular straight lines on the plane. In this case, this is the board. Now, I will associate this horizontal line with real numbers, and you know that points on the line correspond to real numbers quite easily. So I will associate every point on this line with domain of real numbers for the function which has this type of a domain. And the crossing of these two lines I will associate with the number zero. And then let's assume we have certain unit segment which represents the number one. So using this unit segment I can basically mark points which correspond to every integer number on this line, and obviously everything in between this also can be defined. Now, as I said, the horizontal line represents the main real numbers, arguments of our function. Now, the vertical line will represent the values, the co-domain of the function, and I can use the same unit segment to basically associate every point with first integer numbers and then fractions in between the duration over there. So basically, right now I have a representation of domain and co-domain on the plane using my two coordinate lines. This line is called x-axis, and this line, the vertical, is called y-axis. A different name is obsessive-ordinate. Alright, now, so I have represented domain and co-domain on the function. Now, how do I represent function? Because the function is basically not only domain and co-domain, but most importantly it's a correspondence between the elements of the domain and the corresponding elements of the coordinate values. If I know that every point in the domain on the x-axis represents an element of the domain and it means that it should have a corresponding representation of the co-domain, well, let's do this. For instance, I'm looking for a representation, an image, a value of the function if my argument is equal to this particular real number. Well, very easy. Let's say the function tells that the corresponding number is whatever, this for instance. So, I can always put two perpendicular lines and on the crossing I have a point. So, this point represents that this particular number corresponds to this particular number in the function, in the value of the function. So, the value of this argument is this. So, every point on the plane represents a pair of argument and function. Okay, fine. If I have a function which has, let's say, all the real numbers as its domain, let's say the function can be algebraically expressed as y is equal to 3x plus 1. Well, obviously if I will put for every point on the plane, if I will put a corresponding point on the plane which really represents the function of dependency, let's say x is equal to 0, y is equal to 1, which means the argument is 0 and the function value is 1. So, it's this point. This coordinate is 0, 1. It belongs to our function, right? This function represents the value of the function. This point represents the value of the function if argument is equal to 0. Now, if argument is equal to 1, let's say, then it will be 4, right? So, this is 1, 4 is somewhere over there. So, this is another function. If argument is equal to minus 1, that will be what? Minus 3 plus 1 minus 2. Okay, so minus 1 minus 2. This is another point. Argument is equal to minus 1, value is equal to minus 2. So, we have minus 1 minus 2, etc. I can put many different points and in this particular case, this graph will look something like this. Now, we can definitely prove that something like a polynomial, the first degree is a straight line on the graph or a polynomial of the second degree, something like that would be a parabola, looks like this. So, there are many different things which you can do with graphs but I would like to concentrate more on qualitative characteristics of the graph and here is what's very important when you're dealing with graphs. First of all, we have to really very clearly identify the function domain. If you will take a function which looks like this, what's the domain of this function? Well, obviously, 0 is not among the arguments of this function because you cannot divide it by 0. So, in this case, arguments can be any number on this line except this point 0. So, that should be clearly identified. There are many ways to basically show on the graph that this particular point or this particular segment is not really included in the domain. In this particular case, you can either use a couple of arrows in this case or make a little circle or whatever. It doesn't really matter, there are many ways but it should be somehow identified. These two ways using little errors and circle are more traditional but it's not the matter. The matter is that you should identify, at least for yourself, that it's not every real number which can be a domain, can be included in the domain of this function but some of them really are not there at all. Function is not defined. So, these are kind of special points or segments, usually it's points when you're dealing with graphs. And these special points which are not included in the domain usually have their peculiarities. Now, what kind of peculiarities we might expect from these special points? Well, in this particular case, for instance, you know that if x is, let's say x is some positive number, but if x is diminishing, let's say first x is equal to 1, then 1 half, then 1 quarter, then 1 eighths, et cetera, et cetera, then its reverse will be increasing as x goes down to 0 to this special point, the value of the function will be bigger and bigger and as you probably understand, it will be infinitely big. So, whenever we approach 0 from the right, from the positive side of the argument, the value will be greater and greater so the graph will go to infinity. Similarly, if x is negative and also approaches 0 from this side, from left to right, same thing would happen, but in this case infinity would be a negative, so it will be something like this. So, this qualitative property of the graph that when you're approaching a special point where graph is not, what function is not really defined and graph neither, but then the graph behaves this particular way. That's what's very important. Now, special points, that's number one. Secondly, what's very important is how graph behaves when the argument goes to infinity. Now, in this particular case, we probably can make exactly the same type of logical conclusion. If x is increasing, obviously the reverse is decreasing down to 0, basically. So, function will probably be something like this. It will be smaller and smaller and smaller and as our arguments are getting bigger and bigger. Same thing would be in a negative direction if I go to minus infinity on the x, on the argument side, then the function will also diminish in its absolute value, but staying negative. So, it will be under this x-axis and approaching it. Alright, so that's how we have to really pay attention to where exactly our arguments are. Special points and infinity. Now, how are the function behaves? Well, as we have just seen, around special points, we usually investigate it very thoroughly about how it behaves. And same thing on the infinity. There is also another very important set of points. Points where the function is equal to 0. Usually, these points very clearly identify the behavior of the function and you will basically see that the function changes the sign, for instance, around these points, etc. Here's an example. If you, for instance, have... Let me draw a new one because this is already... If you, for instance, have a function which looks something like this. It's a function, actually, it's a parabola. Because it's a polynomial of the second degree. x squared minus 3x plus 2, right? Now, on the graph, so these are x's, these are y's. This is 0. This is 1, 2, 3, 4, minus 1, minus 2. 1, 2, minus 1, minus 2. Now, what's very important and it really strikes the I, that the function takes the value of 0 when x is equal to 1 or x is equal to 2. So the function is definitely, function graph is definitely crossing these two points. x is equal to 1 and y is equal to 0. Same thing, x is equal to 2 and y is equal to 0. That's obvious, right? Now, secondly, let's go to infinity. When x is very large, obviously y grows as well and it's always positive. So function is always positive when the x is large. So here, somewhere, it looks like this and it's increasing, obviously. On the function, when the argument is negative, same thing happens here because it will be negative and this will be negative and negative times negative will be positive. So function is positive again and its absolute value will grow as well if absolute value of the x grows. So somewhere here is also growing. If x goes to minus infinity, y goes to plus infinity. Now, what we have here is the following. Function is positive here and positive here. These are the only two points where the function is equal to 0. So obviously, your conclusion is that the function goes something like this and changes the sign when it crosses its root, actually, when the function is equal to 0. So qualitatively, the function should look like that. Well, yes, we know that this is a parabola and, basically, it's probably more like this, more smooth, etc. But that's the detail and the general behavior of the function is basically like this. It goes to infinity on both sides. One and two are two roots, so it changes the sign when it crosses this, crosses the point f equals one, it changes the value from positive to negative, same thing here. So that's just one of the very important characteristics of the function and, exactly, it takes the value of 0. Another very important characteristic is functions can be odd or even. Now, odd function is the function which changes the sign when argument changes the sign. Example, y equals 3x. If x is positive, that's a positive value. If x is changing to a negative sign, let's say from 5 to minus 5, then the function will change from 15 to minus 15. What does it mean graphically? Very simple. If this is one of the points on the graph, let's say 1 and 3. So x is equal to 1 and y is equal to 3. Obviously, it belongs to the graph. So I know that if x equals 1, y equals to 3, belongs to the graph, then if I change the sign, both x and y, and it will be the same, but a negative sign, so minus 1, minus 3, minus 1, minus 3, will always be on the graph as well. So together with 1 point, 3, we have minus 1, minus 3. What does it mean? It means it's centrally symmetric. Let's take another example. y equals to x cubed. If x changes the sign, y changes the sign as well. And the absolute value will be exactly the same because 2 to the third degree to the power of 3 will be 8, minus 2 to the third degree will be 3 times multiplied by itself. It will be minus 8. So again, on my graph, if my point to 8 belongs to the graph, my point of minus 2 of minus 8 belongs to the graph. So again, it's centrally symmetric. So all the points, which means that the whole graph will be centrally symmetric, which means if we all turn it by 100 and 8 a degree, it will basically coincide with itself. And by the way, the graphic of this thing would be something like this. Very centrally symmetrical. If you turn it at 100 and 8 a degree, it will coincide with itself. The consequence of the function being odd is that obviously point 0, 0 also belongs to the graph. Every odd function crosses the 0, 0. So official definition of the odd function is the following. f of minus x is equal to minus f of x. So function of this function. I will take x cube minus x cube. This is function of minus x. Obviously, this is minus x minus x minus x equals to minus 1 to the third degree and x to the third degree, which is minus 1 times x minus x cube. I know that's kind of trivial. We probably should write it in one line rather than three. But anyway, this is kind of an obvious quality of one of odd functions, the x cube. But this is the general characteristic. If you substitute minus x for x into the formula, then basically you can have the same value, the same absolute value as if x would be used, but then it's supposed to be negative sign should be applied. Now, with the even function, let me start from the definition and then we'll go to examples. That's the function which does not change the sign if you change the sign of the argument. Example. An obvious example is classical parabola. If you change the sign from x to minus x, minus x squared will be exactly the same thing as x squared. Now, what's interesting about even functions from the graphical standpoint? Again, odd functions are centrally symmetrical, so you can turn the whole plane and the edge of the grid will coincide with itself. Even functions are symmetrical relative to the vertical y-axis. Why is that? Because if A, B, a point, A, B, let me just put a new coordinate point. This is point A, B, which means B is equal to F of A, right? So this point belongs to the graph. It means this. Now, since the function is even, if I change the sign of the argument, it will get exactly the same function. So what actually follows from it that the point minus A, B belongs to the graph. Now, if this is A, B, this is minus A. So for each point, there is another one, which is just a mirror image relative to the y-axis, which also belongs to the graph. So for even functions, we can always say that they are centrally symmetrical, and you see parabola is centrally symmetrical. Another example can be x squared plus 1. It's also parabola, and it doesn't really cross zero. Crossing zero is only for odd functions because they are centrally symmetrical. Even functions don't have to cross zero, obviously, but they are supposed to be centrally symmetrical. So in the case of x squared plus 1, it will be almost the same thing, but shifted up by the unit, virtually. Okay, so these are odd and even functions. Now, why is it important to differentiate all the functions in certain categories? Well, because we know certain properties, and sometimes it's easier to do graphs for something more elementary and then construct it in a more complete fashion. For instance, if I know how a function behaves for positive argument, x, and I know that the function is, let's say, even, I don't really have to think about how it behaves on a negative side. I can just symmetrically draw another piece. Similarly, if you remember, in the beginning I was talking about function y equals to 1 over x, and in the beginning, again, we said that it approaches infinity on the positive side when x is approaching zero and it's approaching zero when x is approaching infinity on the positive side. I don't even have to think about what's happening on the negative side, because obviously this function is odd, because if I will take a function of minus x, which is 1 over minus x, it's equal to 1 over x with a minus in front of it, let me just look this way. Minus 1 over x, which is x with a minus sign. So, f of minus x is equal to minus f of x, which is the definition of the odd function. So the function is centrally symmetrical. I mean, the graph is centrally symmetrical. So I can just turn the whole thing on a 180 degree and I get this piece. But I don't even have to think about this, because I know that it's a centrally symmetrical graph. So it helps. Now, if you have a certain graph, you can actually manipulate with this graph using some very elementary techniques. Example, let's say you have a function f of x, and now you would like to know what happens if instead of x you would put x minus a. What happens with the graph? Because sometimes function looks like this. But you know the function and graphical representation of this function. Example, y is equal to x minus 1. Well, you know when the f of x is 1 over x. You know the graph of this function and you have to draw this function. Well, I mean, immediately it's not really easy and you can obviously start by thinking, aha, this function is not defined when x is equal to 1. So 1 is a special point in exactly the same way as for this function. 0 was a special point. But we have already made all our studying research of this particular graph. Can we use it to grab this function without thinking much? Yes, we can. And here is how. Well, let's assume that you have a point on the graph which is one of the pairs which satisfies this particular equation. Let's say you have something like capital B is equal to f of capital A which means function has a graph with A B on it. Now, here is an interesting point. If you will substitute for this function, if you will substitute instead of x, you will put A plus lowercase A. What would be the value of y in this particular case? Well, it will be f of A plus A minus A which means f of A. But we know this is B, right? We have started from this graph. So what's interesting is that if argument is equal to this, then the function is equal to this. Which means the point A plus lowercase A, B belongs to the new graph. So if A B belongs to the old graph, then A plus A B belongs to the new graph. Now, where is A plus A? It's this.