 I'm Zor. Welcome to Unisor Education. Today's topic will be graphs and problems actually for graphs. We'll start with problem number one. This is actually a repetition of whatever we were talking about during the main lecture, about the transformations which graph can take when you change the argument or the function. So let's assume that you have a function and its graph. So we have four questions. What happens with function's graph if I add something to an argument or add something to the entire function or multiply something by an argument or the function? These four questions we will consider right now. So let's do it one by one. First, we will add something to argument and let's assume that this constant is greater than zero. Okay, that's simple. Consider you have a point with coordinates, let's say capital A and capital B which belongs to original graph. So this is x coordinate and this is y coordinate. x is equal to A if it is equal to B. And this point belongs to the graph. Now, obviously if you would take the point A minus small lower case A and B, this point belongs to this graph because if you add x plus A, you will get A and f function of capital A is equal to B as we know. Because the fact that this graph contains point A and B, it actually means that if we apply function f to the value of argument A, we will get B. So same thing actually will happen here. Function f of A plus A, that's the x plus A, sorry, minus A here, plus A comma B equals B, function of A is equal to B. So if point A B belongs to the original graph, then point A minus lower case A B belongs to the new graph. So we have to shift to the left because do you remember we started with A greater than zero. So let's say this particular segment has a length A. So now we get the point which has x coordinate A minus lower case A and the same B. So if this point belongs to original graph, then this point belongs to the new graph. Same thing here, shift it to the left, same thing here, shift it to the left and you will get an entire graph which resembles, which resembles our graph, but shift it to the left, shift it to the left by A units. Okay, so that's how it happens when we shift to the left. Now obviously if we subtract positive number, everything will be shifted to the right. So you can always say that adding positive number shifts the graph to the left, adding positive number, so subtracting positive number shifts the graph to the right. Whatever the graph is, the shape of the curve is preserved. So if originally we had something like parabola, which is y is equal to x square, then if I want a graph of y equals to x minus 3 square, it will be shifted to the right by 3 units. So if this is 3, this is 0, then the whole parabola will be started from here. The same curve just shifted to the right because this is subtraction of the positive number by this number. Let's say that covers adding or subtraction of the constant to the argument of the function. So the graph shifts left or right based on the sign of addition or subtraction. Okay, that's number one. Number two, number two, we add a constant to an entire function, to an entire algebraic expression which defines our function. What happens here? Well, again, we'll do exactly the same thing. If point AB belongs to the graph, which means B is equal to function of A, then obviously point AB plus lower case A belongs to this graph because if I will substitute A for x, effort x we know according to this is equal to B. So on the right I will have B plus A and on the left I will have B plus A. So basically this point belongs to the new graph. So how does it look if this is my original graph and let's consider that this is point AB. So what I'm saying is that the new graph, the graph of this function, will have a point same A but B shifted, let's say A is positive, so I shift it by A units up. So this point belongs to the new graph. Similarly, every point can be shifted by the same segment of length A and the new graph will be just above the original one, just shifted A units up. So if we are adding positive number to an entire algebraic expression on the right side of this equation, of this expression to the function, then the whole graph shifts up. Obviously if A is negative, it will be shifted then. Okay, now we will do multiplication. The problem number 3 will be, we will multiply argument by a constant and again we consider that the point AB belongs to the original graph which means B is equal to f of x. Now if we can say, if we will substitute A divided by lowercase AB into the new graph, what happens? It will be f of A A divided by lowercase A, that will be on the right, right? But we know that this is function of A and function of A is equal to, as I should have actually written here, A. So function of A is equal to B, so this is B. What it means? It means that this particular point with these coordinates satisfies this new equation, which means it belongs to the graph. And now what we can say is the following. If AB belongs to a graph, then A divided by lowercase AB also belongs to the graph. Now where is it graphically? Okay, here is where it is. This is my original function, let's say, let's consider parabola. Now this is A and B. So where is, in this case, point A divided by lowercase A? Well let's consider this constant is greater than 1, like 2, 3, whatever, any positive number greater than 1. It means we are actually squeezing A divided by lowercase A that's squeezing the point to the left. So instead of this segment 0A, I will have a segment 0A divided by lowercase A. Let's say lowercase A is 2, then this will be exactly half. So every point AB, which belongs to the original graph, you can consider that the new graph will be by squeezing this particular segment A times, so it will be like here. This point will be here. This point will be here. So the whole graph will go like here and similarly on the left side. So we are squeezing the graph by this constant A along the X towards the axis of Y axis. Now obviously if this constant A is less than 1, let's say it's 1 half, then squeezing by 1 half is actually multiplying by 2, obviously. So we are stretching the graph. But in any case, whatever English word we are using, stretching or squeezing, whatever, what it actually means in this particular case is we multiply the lengths of every X segment, we will divide it by this number A. So if number A is greater than 1, it's squeezing. It's less than 1. By absolute way, it's stretching. By the way, here is an interesting point. What happens if it's negative? Let's say it's negative 1. Let's start with this. Well, that's actually very simple because again, if you're original point, let's say you have a graph like this. If you have original point A, B, let's consider that this constant A is equal to minus 1. So in this particular case, A divided by minus 1, which is minus A, B. This point belongs to the graph. Now where is minus A, B? Well, obviously it's a reflection relative to the y-axis. So with this point, we will have this point which belongs to the new graph. With this point, we will have this which belongs to the new graph. This point, which is already on the y-axis, actually stays where it is because x is equal to 0 in this case. So if we multiply it by minus 1, 0 times minus 1 will be 0. So basically the whole graph will reflect relative to the y-axis. So whenever you multiply by a positive number, depending on whether it's greater or less than 1, you're basically squeezing or stretching the graph. But if it's negative, not only you're squeezing or stretching, you're also reflecting the graph relative to the y-axis. And the last, which is obviously similar to all these guys, will be what happens with function if you multiply it. The whole function will be multiplied by A. Again, we start with original. B is equal to function of A. That means that point A, B belongs to the original graph. Obviously, if you take new function, then A, comma, A, B belongs to the new graph. Why? Because if we will substitute A for x, we will have f of A, which is B, times A. So we will have A times B. Now, graphically, how it belongs graphically, if you have a function like this, again, and you have original point A, B, then the same value of argument A will result in stretching. Let's again consider that A is greater than 1, positive number greater than 1, like 2, 3. That means that the function will actually stretch the y-axis, the ordinate by corresponding with like 2, 3, whatever the A is. Instead of this segment, we will have segment which is A times greater, and this point belongs to the new graph. Now, instead of this segment, we will have A times greater, which is this one. So the whole graph will go higher. So every segment is stretched by, let's say, ratio of 2 or 3 or whatever. Now, obviously, if A is positive but less than 1, you are squeezing the graph towards the x-axis. So it will be something like this. And finally, if A is negative, then not only you are, let's start with a minus 1. If it's minus 1, then A comma minus B will be a point on the graph. So if this is a point AB, then A minus B will be here. So basically, the whole graph will reflect this way. Together with this, on the original graph, you will have this on the new graph. Instead of this, you will have this point. So this point is transformed into this, for example. So we are reflecting the graph relative to the x-axis. Now, if it's not just minus 1 but minus, let's say, 2, then it's both a reflection and stretching by 2. And if it's, let's say, minus 1 half, it's reflection and squeezing by the ratio of 2. Okay, that basically concludes all these four cases when we are adding something to an argument or adding something to a function, multiplying something by an argument or multiplying the whole function. If we are adding or subtracting number to an argument or from an argument, that shifts the graph left or right. And if we are edging to the whole function, it moves the graph up or down. Similarly, with multiplication, with multiplication we are squeezing or maybe reflecting relative to the y-axis, the graph along the x-axis. And finally, if we are multiplying by a number of the entire function, we are stretching or squeezing vertically and maybe reflecting as well. So these are four cases I wanted to present in this particular problem number one. Thank you very much.