 If you assume that this piece is A, and let's go up by B. So for each point AB on the old graph, you can find a point A plus lowercase AB on the new one. So if old graph was something like this, then the new graph will be something like this. So for each point on the old graph, you can shift it to the right by A, and you will get the point on the new graph. So my point is that if you know how this graph looks, then all you have to do to find this graph is just to shift the whole picture to the right. Now, what happens in our example Y equals 1 over X minus 1? In this case, lowercase A is 1. Well, let's just draw it. First let's draw it, and then we'll think about what happens. Let's draw it without any... Okay, we used to have this as a function 1 over X, right? Now I'm saying that this graph should be shifted to the right by 1. So this is 1. Now if function was going to infinity around point 0, the graph, then the new graph, since it's shifted to the right, then this axis, actually this line, vertical line, will be shifted to the right, and new graph will look like this. So we just shifted the whole thing to the right. Okay, that's basically it. So we know how to deal with this type of thing. Well, as an example, if you have Y is equal to 1 over X plus 1, well, obviously in this case A is equal to minus 1, so it should be shifted to the left. That's kind of obvious, right? So we know how the function behaves if you shift an argument to the left to the right. Great. Now, next exercise. What will be with the function if you multiply argument? What's the difference between original function and the function when you multiply argument by something? Okay, very easily. Let's start from original function B equals function of A. Now, for this particular case, let it take as an argument the value of A divided by K. If I multiply it by K and apply my function, it will be F of A with no its B, right? So basically, what I can say that A over K times B belongs to this graph. If A belongs to, A B belongs to this graph, then A over K B belongs to this graph. What does it mean graphically? Well, very simple. This is A, this is B. So if I will squeeze this particular segment from 0 to A by the ratio of K, so it will do this, then the value of the function will do this. So for each point on the original graph, if I squeeze this towards the vertical axis, then I will get the point on the new graph. So if my original graph, let's say, was something like this, and I squeeze every point K times, I will probably have something like this, right? This point is squeezed to this one, this is smaller, but so this one will also be smaller, et cetera, et cetera, down to 0. It's just an example, but whatever the graph original is, you can always imagine, again, it's not quantitatively, it's only studying, it's only qualitative behavior of the graph. You can say that the graph is squeezed towards the y-axis, okay? Obviously, if you have y is equal to f of x divided by K, it's quite obvious that it will be stretching by K times, right? It's obvious because then, in this particular case, if A B belongs to the original graph, then A K comma B belongs to the graph of this function. If x is substituted with A times K divided by K is A, so it will be B. Now this function, this point, relative to this one, is just stretched to the stretched K times versus this graph. So if original graph was this, then the new graph will be something like this, stretching from the y-axis to the right. Now obviously, if you change the sign, let's say minus K or a negative, et cetera, it will change the direction of that thing, that's obvious. All right, so we know what to do with the graph if our argument is changing. Now, the most complex modification of the argument in this case would be y is equal to f of, let's say, x times A plus B. This is the most general kind of linear transformation of the argument. How to draw this graph? Well, first you draw y is equal to f of x, fine. Next what you do, you do y is equal to f of x times A. That means you squeeze it towards y-axis by A times. When you draw that, all you have to do is to add argument B, which means if you add argument B, these positive shifts to the left, basically, by B units. So that's how you transform from this graph to this and from this to this purely qualitatively. No real studying of the behavior of certain points, et cetera. All you have to do at the beginning. What does it actually mean for you? Well, it means that something, if you know that, let's say, function y is equal to x squared is a regular parabola, which basically goes like this, then you can very easily draw something like y is equal x squared plus 7. You just shift it to the left by 7. You can very easily draw something like y is equal to 3x squared, because you can always say, well, I won't use that example. I would use it for the next line. But in any case, if you have something like y is equal to 1 over x plus 1, what does it mean? You draw 1 over x, which is this, and plus 1, it means I have to shift the whole graph to the left. So my vertical axis will be here, and it will be this type of graph. All right. So we know what to do with argument. How about the function? Well, that's even easier. If you add something to the function, obviously it shifts the whole graph upwards by a unit. Why? Because if AB was a point on this graph, then if you substitute A here, it will be A comma B plus A. It will be on this graph, right? So for each point, AB of the old graph, the new graph will have the same A and B upwards. This is the segment of A on the y-axis. So if you add something to argument, let's go back, then the graph shifts left or right horizontally, depending on the sign you're shifting. If you add something to the function, the graph shifts up or down, depending on the sign of this constant. Similarly, if you multiply the graph by something, then for obvious reasons, it's stretching. In this case, B times K will be a point on this graph. So for the same argument, A, you used to have AB. Now you will have AB times K. So every point goes up or down, depending on the sign of the K. Okay, so that's even easier. So we can multiply graph by constant or we can add a constant to the graph. That's why my example which I started to write is easier to consider an original parabola, y equals x square, and then multiply by 3 means stretch 3 times. At the same time, you can consider this to be... Can I use the number 4 instead of 3? You can use 2 times x square. And that actually corresponds to the rules which we were discussing before, when we were multiplying an argument. Now if you remember, if you multiply an argument by certain constant K, the graph is squeezing. Here's a very interesting property of parabola in this case. This is the original parabola. If you multiply it by 4, then it will be stretched upwards. So it will be something like this. So each number would be stretched 4 times. From here, you go up to here. Used to be on this point, now it's in this point. Hope it's visible in the video. So that's how you stretch it by 4 times. But at the same time, I was telling that you can squeeze an argument by 2 times, basically the same formula. So a parabola is such an animal which if you squeeze it from these sides by 2 times, you will get exactly the same thing as you will stretch it vertically by 4 times. Horizontal stretch by the ratio of 2 is equivalent to vertical stretch. Horizontal squeeze, sorry, by the ratio of 2 is equivalent to vertical stretch by the ratio of 4. And you get exactly the same graph. So both ways you get exactly the same picture. This is just a little easier, I believe. Okay. So you can multiply arguments and functions and you can add constants. How about adding functions? Not just constant, adding new functions. Let's say you have y is equal to f at x plus g at x. Two different functions. And each of them is a simple function so you know the graph. Let's say y is equal to x plus 1 over x. Well, it's not that easy to draw it by what? By using some key values and calculating, that's not easy. What simpler thing is, let's just add these two functions together. y is equal to x, f at x is one function and this is a straight line which goes like this. Now, y is equal to 1 over x. This is this function. Hyperbola, by the way, it's called. Now, obviously this function is odd because if I will change the sign of the x then the sign of the y will be changed as well. Which means I don't really have to think about how the function will behave in a negative side, on a negative argument. So let me just forget about this for a while. I'll just think about how it will behave on a positive sign, south side and then I will just symmetrically reflect it. Alright, so we have two functions. This is one which is a straight line and this is hyperbola. Obviously zero is a very special point because it's not defined when x is equal to zero. How can we add these two functions together? Well, let's just think about it. They're obviously crossing at one one. Now what happens before that? Before that, one component which is one over x is really very, very big. This component of another function is very small. So if you will add a small component to a big component you will obviously get a little bigger but not really by much. And this extra piece which we are adding to a hyperbola and this is the original hyperbola and this is the piece which we are adding so we are adding that much. And then we are adding smaller and smaller and smaller so basically the new graph will be very much like the old one. It will be approaching the same infinity but maybe a little bit to the right of the original one. So this is the behavior of the function on this particular place. And this should go to one plus one is two actually from this point. So this is the function. Now let's move to the right of the one. Here the one over x is smaller and smaller which means this component y equal x is most important and for the result of the addition of the bigger component to a smaller we get just a little bigger and that little becomes even smaller and smaller as we approach infinity. So basically this extra piece which we are adding to a straight line becomes smaller and smaller so the graph is actually approaching our straight line from above always being above it but smaller and smaller distance. By the way these two lines which are basically they represent some angle where our graph is approaching both sides and approaching basically in such a way that the difference becomes smaller and smaller. They are called asymptote. Now so these are two asymptote and they basically characterize the behavior of the function when argument goes to zero which is one special point or infinity which is another thing which we have to research. So basically that's the line. It's this type of thing and it's basically inscribed into this angle. Now using the oddness of this function because it's symmetrical relative to the zero central is symmetrical I can basically draw the same thing here. So that's the graph. One curve and then another graph and I did it just by adding two graphs together. Well let me make one more interesting operation. Adding is easy. How about dividing graphs? Okay we can do it too. It's not big deal. Let's say you have a function y is equal to x squared plus one divided by x. Okay. I think I have to change. It's probably not very visible. I'm not sure this is good. Alright so let's divide. We have graph of x squared plus one divided by x. Now what is x squared by one? x squared plus one. Plus one is a parabola x squared shifted upwards by one. So it's like this. Now x y equals x is a straight line. Like this. Well let's divide them one by another. Obviously if we approach zero which is a special point and let me just make a note that this is also odd function. If I change the sign of x this will be the same because it's a square and this changes the sign so the function changes the sign. So it's odd function. So forget about the negative side. Positive x. As x approaches zero my denominator goes to zero. My denominator is relatively small. It's always around one. So if x is very small so if I divide something quite small which almost equals to one by infinitely smaller and smaller number I will get infinitely bigger and bigger result. So when I approach zero my resulting graph will basically go to infinity. So as I approach zero my graph goes to infinity. Now if I approach infinity if I approach infinity this x square is gross significantly faster than x. So the result of their division will be bigger and bigger and bigger. So obviously my function will go to infinity when the x goes to infinity. So somewhere these two things are connected to each other. So I have something which is very much resembling my graph which I draw before that the function of x plus one over x well it's not a coincidence because this is the same thing x square plus one this is the same thing. So by dividing graphs one over another I kind of came to the same general shape of the graph. The only thing which I did not really get from here is that the graph is closing and closing its distance towards the line y equals to x. So y equals to x the straight line is obviously a signature for this particular graph and it's not so obvious for this one just the general became where you can see. But in any case you divide it as normal things and everything seems to be working fine. One more division example and that would conclude our graph's representation. One over x minus one times x minus two. Well first of all let's picture the denominator x minus one times x minus two. Well obviously this is something which has one and two as zero points if x is equal to one is equal to zero and x is equal to two is zero. Other than that this is the parabola this is the polynomial of the second degree and it's positive on both bigger and negative and positive side and it's negative in between so it's something like this. This is our parabola. This is our denominator. Now I have to divide one into this by this actually. One by this. Well if my denominator goes to infinity obviously one over this would go to zero. So as parabola goes to infinity here and here my function will definitely go to zero on both sides. Now what happens around the special points one and two? My denominator goes to zero that means that our result should go to infinity. So we obviously have two asymptotes here and the function will go to infinity here and infinity here as we approach roots. Well obviously the same thing should go if we approach these roots from inside of this segment from one to two but in this case my denominator is negative so the whole function is negative and infinity will be negative here and here and it will go something like this so it will stay negative all the time but again it will approach the same infinity as the x is approaching one and two. So these pieces represent the graph of this function. So that's from basically purely qualitative standpoint. This was my original just to make it more visible and this is the resultant graph. So that's how people draw the graphs and they help them to study the functions their behavior especially again very very important special points and infinities and asymptotes. These are major characteristics of the graph. You can again manipulate your graph you can add constants to arguments the function which shifts to the left or to the right or to the up or down you can multiply arguments or functions by certain numbers which will stretch or squeeze in vertical or horizontal direction and that's how you manipulate the graph and you basically can draw anything which is expressible in this formula kind of thing. That's it for graphs. Thank you very much. I'll probably do some exercises, problems, etc. but that will be an extra. Thank you very much.