 Hi, I'm Zor. Welcome to Unisor Education. I would like to continue today talking about linear functions, and especially about graphs of linear functions. So we are talking about graphical representation of a function. Well, let me start with something which everybody knows, that the graph of the linear function is a straight line. I'm sure every student of high school or whatever knows that this is what it is. Well, but let me ask you this question. Can you prove that the graph of the linear function is a straight line? Well, that's not such a simple question because we really have to think about how to do it. All right. Here's what I suggest. This is the coordinate plane. This is x, this is y, and our function is ax plus b, where a not equal to 0. It has some graph, how straight is it? All right. Here is how I suggest to approach this particular proof. Again, by the way, there might be many different proofs. I'm just suggesting one of those. We all know from geometry that the shortest distance between two points on a plane is along the straight line between them, right? So if you have two points, then the shortest distance is along the straight line. Well, by the way, this is something which I have covered in geometry part, and you definitely can recall or go into this part and check how it works. Okay, this is one property which I would like to use to prove that the graph of this particular function is a straight line. Now, graph of this function goes through certain points, right? Point x is an argument which is here, and y as a result of this application of this function would be here. This is x, this is y. That's how we basically draw the graph. What I suggest is the following. Let's have three points on the graph. Let's call it u, these are arguments, and w. These are values. This is a u plus b, this is a v plus b, and this is a w plus b, right? I would like you to recall another property. Again, it's geometrical, basically. It's the Pythagorean theorem. If you have two different points, let me just put another drawing. If you have two different points, one with coordinates a and b, another with coordinates c and g. So this is a, this is b, this is c, and this is b. What's the distance between these two points? Well, obviously you can draw the parallel line here and here. It would be the right triangle. And this is hypotenuse, and these are two calculatives. So this calculative is c minus a, right? c minus a gives you points of these calculatives. And this calculative is g minus b. So using the Pythagorean theorem, the distance between these two points would be a square root of the square of these calculatives, which is c minus a square, plus square of this calculative, which is g minus b square. So square of hypotenuse is equal to the sum of squares of topazity. That's why this is the formula for the distance between these two points with coordinates a, b, and c, d. So I will use this to calculate the distances. So basically that's enough of the preliminary information which I need to prove that this is a straight line. And here is how. I will calculate the distance, let's call this u, v, and w points. These are lower cases, and this is upper cases. Now I will check the distance between u and w, and then I will compare it with sum of distances from u to v and from v to w. Now I know that from u to w the distance is the shortest distance, it goes along the straight line. Now if I will be able to prove that some of these two distances is exactly the same as the distance from u to v, it actually means that we cannot lie outside of the straight line, because as you know again from geometry, it's the straight line which is the shortest distance. So if sum of these two distances is exactly the same as this, it's supposed to lie on the same line, on the same straight line. So that's why this is not the case. So all I have to do to prove that the graph is a straight line, I have to choose randomly completely three points, I call it u, v, and w at the argument, draw or construct whatever three points on the graph, and check the distance between two extreme points and sum of two distances between the left extreme to the middle and from middle to the right extreme. Now when I'm saying middle doesn't mean it's really in the middle, it's somewhere in between basically, right? So three completely no restrictions, let's just assume that u is less than v and v is less than w for simplicity, and then after I choose these three numbers, I substitute these three numbers to linear functions expression to get these three values. Now I have coordinates of all these three points, now having coordinates I can find the distance using this formula which I was just talking about, and then I compare the sum of these two distances with the sum of the distances between u and w. So all I have to do is to prove that the distance from u, let me use the upper case, the distance from u to w is equal to the distance from u to v plus the distance from v to w. I use the distance between two points as an absolute value and these are points actually on the plane, u, v, w, these are points. So that's what I have to prove using the formula for the distance. If I do that it means v lies on this line between u and w. So if I have any three points and connect the two extreme points with a straight line, the any point in between should also lie on it, which means it's one and only one straight line. So how can I prove this? This is just a technical issue nowadays, right? So let's calculate what's u w, distance from u to w? Coordinates of the u are lower case u and a u plus b, coordinates of the w is lower case w, a w plus b. So the distance between them would be square root of square of the distance between w and this plus square root, plus square of the distance between two values of the y. So it's a w plus b minus a u plus b square. This is u w. Well let's simplify this. Obviously if you will open these parentheses you will have a w plus b minus a u minus b. So b will be reduced. So you have a, you can bracket it out, factor it out with the w minus u, right? So that's exactly what I will do. So u w distance is equal to square root of w minus u plus square plus a w minus u square. So I put square here and square here, right? Square of this entire expression is square of this times square of this. Now I can factor out w minus u square, which means what? Which means square root of w minus u square and in the parentheses it will be 1 plus a square, not equal. Now I will use the w is greater than u so I can just put it outside of the square root and that's what I will get. That's it. That's all I need. So let's put it here. The distance from u to w is equal to w minus u square root of 1 plus a square. Okay. Now we will do very similar operation with this and this and check what happens. If I know how to calculate the distance from u to w and this is this formula and it's important that w is greater than u so this is positive, then very analogously the distance from u to v would be equal to v minus u square root of 1 plus a square, correct? I mean there is no difference. I mean I'll just use letter v instead of letter w. What's important is that the first letter is greater than this one and v is greater than u, right? Now similarly for v and w from v to w, I will also use the same thing instead of u I will put v so it's w minus v square root of 1 plus a square. Now we basically see that this sum of these is equal to what? v minus u plus w minus v square root of 1 plus a square, right? Minus v and v and we have w minus which is exactly the same as this. This is a very easy proof that the line which really represents the graph of the linear function a is equal to ax plus v is a straight line. I think it's important to not only know that this is a straight line but also being able to logically prove that this is a straight line. The only purpose of whatever we are doing here is to basically to stimulate you to think about certain things logically. Don't take it just from the face value. If somebody told you this is a straight line, prove it. It's very, very important. That's what words do. They prove the points, right? That's what doctors do. They prove that their diagnosis actually is correct diagnosis based on whatever symptoms they observe to the patient, etc. So there are many examples. What's important is never take anything just as a belief, as a statement without any kind of a baking. The proof is always very, very important. Okay, so we have proven that the line which represents the graph is a straight line. Now, what about the graphical meaning of coefficients a and b? That's actually very important as well, and let me go into this, but this is something which I'm sure you've heard about, again, question is why. So everybody knows that a is called a slope and b is called an intersect, right? Intersect. Why? Okay. Why a is a slope? Let's take two different values of argument x is equal to a and x is equal to b. And y is equal to, let's call it b, it's equal a a plus b. And another y, let's say it's q, is equal to a, sorry, this is capital b, b plus b. Now, so we have two values of argument and two values of function. So let's just do this, we don't really do this quite, it's confusing. Let's do the following. Let's compare the increment of the function relative to the increment of the value. So let's just for simplicity assume that a is less than b, these two points. So increment of the function is the difference between p and q, right? So which is a b plus b minus a a plus b. That's an increment of the function. Now if I will divide it to increment of the argument which is b minus a, what happens? Obviously, if you will open these parentheses, it will be minus a lower case a and minus b. b would be, this is plus b, this is minus b, so you can just get rid of it. And you will have a b minus a a, a outside of the parentheses. So it would be a a minus, sorry, a b. So it will be a b minus a a divided by b minus a. Now a, you can factor out, it would be b minus a divided by b minus a. And as a result, you can reduce this, you will have a, it's not right. So that's why the coefficient a is called a slope. Because if you will compare it graphically, let's use the same graph. We don't use the third point. So if you will take two different points on the graph. This is a, this is b, this is p, and this is q. P being a a plus b and q being a b plus b. Now, what we see is the difference between p and q is this, this is q minus b. This is increment of the function. From this value, it grew up to this value. That's the increment of the function. Argument is grew by this. So this is b minus a. So if you divide this into this, you get basically a slope, right? That's why it's called a slope. The steepness of the graph obviously is in this particular coefficient. The greater the coefficient a is, let's say it's positive. And the greater it is, the function would be monotonically increasing. But the greater this ratio is, which is a, the steeper would be the graph. Because the function would grew by a greater degree relative to the increment of the argument. Now if a is negative, then the absolute value actually of the a decides the same thing. Then the function would be decreasing, but the slope, the steepness of the decrease would be the greater if the absolute value of a being negative, the absolute value of a would be greater. Because again, for the same increment of the argument, the function would either grow if a is positive or decrease in value if a is negative by a greater ratio. So that's why a is called a slope. Now, what is b? Well, substituting to this, x equals to 0. Now graphically, if x is equal to 0, it's this distance. Which means that's where the graph intersects the y-axis. And that's why the b is called y-intercept. So this is the value of b. For a positive b, the graph intersects the y-axis above the x-axis. For a negative b, obviously the graph would intercept the y-axis below the x-axis. But in any case, the value of b is actually this particular value, positive or negative. That's why b is called intercept. So y equals ax plus b. A is a slope and b is called a y-intercept. That's it for today. That's all about graphs for linear functions, which I wanted to talk about. There will be some other properties of the linear functions. But that would be the other lecture. Thank you.