Yes, hello everyone. You know, yesterday applications related to functions We had tried. That's where it's particularly important. We had examined function graphs. One points where the function intersects the axes We had determined it. Increasing or decreasing function the ranges are positive and negative determining the ranges in which it takes values We had seen it. Minimum and maximum We talked about values. Again today I'm quite familiar with function graphs. We will publish. But as of today The topic we will be discussing next is the parabola. Well actually, when the term parabola is used The first thing that comes to mind is this It happens, second degree They are functions. Yes. Actually more To give an accurate definition, I can say this. Second-degree functions We call their graphs parabolas. We are giving. So actually, this That's what I mean. What we call a parabola Time usually starts like this, with the letter 'u'. or curves resembling the letter 'n' Actually, we're just saying... uh... But in reality, these are their side-by-side versions Here's the letter C, or vice versa. These curves in the image also have names. It is a parabola. But what we will be dealing with These will be parabolas specifying functions. You know, this is our summer term In this camp we held, right from the book... I'm not moving on to the questions. First of all, regarding the subject... all such fine details related to I'm explaining. Then from the book We are moving forward, and in the book, I will again tell you... I will explain the features in more detail a little later. We will examine them one by one. But immediately Before moving on to the book, based on this parabola... I have all the fine details, whatever they are. Let me explain. And before that I told you at the beginning of the camp, I mentioned it in the presentation, and I even said it yesterday. Make sure you have a blank piece of paper, a... your notebook, that is, your paper, your pen Let it be there. A little later, with that source, nothing at all. parabola-based that you won't encounter such subtle features I'll talk about it. Make sure you write these down. Now, when the term parabola is mentioned, actually a second-order function Let me tell you. Here, of course, is the function. When this is said, it has a domain. from the range it is defined It should be mentioned. The opposite of the domain set of real numbers unless I say so Let's accept that. a and c are real entities let a be a number and a be different from 0. Therefore, f(x) = ax² + bx + c is of this type. with second-degree polynomial functions We'll try. Now, a second one like this. polynomial functions of degree The graphs are roughly like this: It looks like the letter 'n', or like that... We'll have all of these later, at your disposal. as you are used to I will name it. I'll say the arms It can be upwards or downwards. That could be true. Here is that U always arms up at the thing in your appearance being correct, the image in view of N with the arms pointing downwards We will say that is the situation. Of course, now about the subject... related, perhaps general information You have. Here's another part of the equation. As soon as you look at it, if something is positive... Arms up like this, then down like this x As soon as it cuts its axis, all of this happens. as if we knew nothing, starting from zero Let's have a talk. You know, that one you know by heart You know, there's information. If A is positive, then the arms up. No, if delta is like this, then it's like that. It is possible. Let's talk about the reasons for this. This summer period, even the summer... I've also put that period aside. Mathematics If you want to learn what and why You need to know what you're doing. Well like this, just memorizing dry information. Impossible. So where does memorization fit into all this? Now Let's be realistic. without memorizing It isn't happening. So naturally, you too Even I do this with everything we do. 100% because of this, and that because of that. therefore, or mastering the whole of it to be able to. Especially an exam like this. While the truth exists, like this, one by one, all of them To examine it so meticulously is really... difficult, but at least very easily most of the things we can understand We'll talk together. In this way more learning in a meaningful way We will have made it happen. Like this Let's begin. Now I will give you various Let me write the function equations. Well Let me write quadratic functions. Then we have some graphs of these. Let's get started. As you know, even yesterday... We were talking about graphic illustrations, weren't we? Remember this. Basically, basically any function The way to draw the graph is actually always They are the same. You write y instead of f(x). The function equation is here: yille x It represents a relationship between them. again, it shows 0 and the graph intersects the x-axis. or the point or points where it cuts You will find it. Will it cut off somewhere now, 10 Will it cut on the ground or not? Look at this Isn't that a separate question? So it doesn't necessarily have to be one the graph of the function intersects the x-axis They don't have to. Then we set x to 0 and y You find the points where it intersects the axis. If only those points are your business If it doesn't see it, the graph will show the passage of the function. find a few more points and these By combining them, the graph was created. your lesson. But of course, here too... as I said from the beginning As I said, now it's secondary the general graph of a function u or We need to know that it is in the shape of the letter 'n'. It is necessary. Even linear functions You know, the first-degree relative I mentioned yesterday. polynomial functions, theirs too that the graphs indicate a line It is necessary to know at least. Now, during this period... We'll be talking about all of this with you. √x or 2 to the power of x, or something like that, or 1/x things like this will come up frequently These are the general functions, you know. knowing the general structure of graphs There are huge benefits. Let's do this In short, this is a second-degree polynomial. the graph that we call a function I will give examples of parabola-related functions. Let me give it. The examples we gave drawing the graphs ourselves Let's work. Let me give an example like this: Let's begin. Our function f(x) is a little I said it before, I'll say it again. Let me tell you. Let's face it. Erde define that is, the domain of real Imagine they are numbers. Or else to your head Without specifying anything, just take this. I couldn't tell them to draw it. Not writing every time I'm saying this for that reason. set of real numbers Let's assume it's defined on it. Let this be our function. The graph of this Let's try drawing it right away. y instead of f(x) You will start by writing. Look, y = x² - It's 6x + 8. If the equation I gave if it can be factored Factoring and writing it down is used in most problems. It will make things much easier for you. Of course. which is expressed as x - 2 and x - 4. I can also separate the factors. Later What were we doing? Look, it's the same every time. I'll do that. Either write 0 instead of x where the graph intersects the y-axis I will find it. Substitute 0 for x in the given equation. If I write it, 0 squared is -6k = 0 + 8, that's it. You'll see that he cut it at 8. So at 8 What did cutting mean? Yesterday was always I had emphasized it. There is no such thing as an 8. in mathematics. It intersects at point 0a 8. That is, the point where it intersects the y-axis Its ordinate is 8. Okay, we found it. Like this Let's even write it next to it. 0a 8 The graph will pass through that point. More Then, in the equation above, substitute 0 for y. Imagine I gave it to you. And this time, equality. It comes down to this. 0 = x - 2 x x - 4. x - 2 If multiplying x by x - 4 results in 0, then what about x - 2? It is 0, or x - 4 is 0. x - 0 of 2 For that to be possible, it might have to be x2. Or, or Let's show it like this again, or x - 4 If it's equal to 0, it could be x4. So this What does it mean? If x is 2, then y is 0. or if x is 4 y becomes 0. Of course, look here. We were lucky. So when I assign 0 to y, uh... that real numbers x that satisfy the equality We were able to find it, but not always We might not be able to find it. Now let's use all this information. Like this a Cartesian coordinate plane on the edge Let's create and animate the graph. Let's work. This is the origin. Here As always, the x-axis. This Our axis is the y-axis. Now it cut x The scores were 2 to 0 and 4 to 0. Roughly speaking... So, x is also at the point with x-coordinate of 2. He's going to cut it at the point with an abscissa of 4. E Y It will intersect at the point where the ordinate of 8 is 8. Let's make this one an 8. Of course, now... very suitable for units and so on I'm not doing anything... uh... anything. There, it's a guess. Let's say there's a 2, then a 4, and that one's an 8. I said. Now let's think about this. You know, us We know that this function actually The graph is either in the shape of the letter U or N. In the form of a letter. And from these three points Imagine it as if it will pass. One Visualize it in your mind. Like the letter N Is it possible? So, going and doing something like this... We won't imagine it, will we? Well So, we took a shortcut and this became 4. And here we are at number 2. Well, sir, it's right here. And let's make sure this place corresponds to 8. No, that's it. We need to have a grasp of that overall graph. Already This animation I've made, this drawing I've made... That thing cannot even represent a function graph. You know, specifying a function graph. What were we doing for it? A vertical shape like this We were conducting the correct test. The one we drew The vertical line is always a single line for the graph we will draw. We were saying it has to stop at that point. TO The above data are what we found. from points, you know, when you set off on the road, actually The graph looks roughly like this: We can say that he/she possesses it. The letter U It looked like that. It's impossible otherwise. Even more so. Make sure you note this. forehead. Both engrave it in your mind and... Make sure to take note. Polynomial functions their graphs are similar to each other It possesses certain characteristics. Or a polynomial function What did I say? Here are the first-degree, second-degree. polynomials of degree 3, 4, 5, and n. the graphs of functions They share similar characteristics. So what? I mean similar characteristics? You know In polynomials, there is something we call the leading coefficient. There is that thing, the highest degree term coefficient. For example, here we are What we are dealing with is a secondary issue. Since it is a function, it has the highest degree. The term becomes the x-squared term. O x squared If the coefficient of the term is positive, then x squared if the coefficient of the term is positive the far right arm, that is, the right arm of the graph It goes upwards. So plus y's towards infinity plus infinity I'm talking about the situation if he leaves. And this The thing I'm talking about has an even degree, even degree or odd degree for all polynomial functions It is valid. So what does this mean? This Let's understand this right here. For example, 1st from a first degree, from something of the first degree Think of it as what I'm talking about. Not at all near him or anything It doesn't matter what I write. Look, we're trying to imagine the graph of this. Let's say we're working. First degree head the sign of the coefficient is positive I saw. What if it's positive, the right side of the graph? the arm goes upwards like this have to. Ok. Of course, this is the first one. because it is of degree, it is linear It will have an image. Here is our f(x) Let it be this. It starts with 4x² and continues from there. Let the outgoing function be a second-degree function. So, a graph is a parabola. let it create. Look, the leading coefficient is positive. Then its right arm goes up like this. It has to go the right way. And from that comes the following: We can actually understand. If the head If the coefficient is positive, then necessarily in a parabola If the right arm is going to go upwards The image had to be like this. What If the time leading coefficient is positive, then f(x) = ax² + bx + c animation starting coefficient x Isn't 'a' the coefficient of the square? E o In that case, but if it's positive, then it appears as u. To this too We say arms pointing upwards. Reverse In some cases, the opposite is also true: Let me give a few examples. For example, f(x) Let's say our function is this. -x³ continues Let it be a function that keeps going. 3. a polynomial function of degree and head We observed that the coefficient was negative. Most The right arm will go downwards. expedition. So this time the image looks like this: Something will happen. Of course, the things before that and all that. We don't know. We with the part on the far right We are trying. And similarly, if -x If we are dealing with something that is grid-like - x-squared And this time, the part on the right is like this again. It will go downwards. In this case Our parabola has its arms pointing downwards. The letter 'n' which we expressed as "will be" It will look like this. When? Vaccine If it's negative. As I said, today is a parabola. This thing we're talking about is actually the whole thing This applies to polynomial functions. Now as we progress in these matters, and ÖSYM (the Turkish Higher Education Entrance Examination Board) also has the final especially in years like this, third degree quite a lot with polynylon functions and so on... He started to try. Quite a bit about these. They also ask questions in exams. This is it We will become familiar with the general characteristics. Well We will not fall into this or that mistake. Yes. If the leading coefficient in the parabola is positive, then: It is possible. If it's negative, that's fine, but 3 degrees 4 5 n There are some common characteristics. Yes. Now Let me give another example. Look For example, this is the first drawing I've ever given. In the example, our function moves the x-axis to x It intersected its axis at two different points. He even cut off the place where he cut the y. let's talk. I emphasized this a lot yesterday. Today Let me emphasize this once more. E y-axis The number that is the ordinate of the point where it intersects It became equal to the last constant term. So what about this? Is this a big surprise for us? No. Because the y-axis of a function What to do to find the point where it intersects? were we doing? We were writing 0 instead of x. Well In a different way, we were calculating f(0). Look, it's already here: f(x) = ax² + bx + Calculate the value of f(0) in expression c when I want, that is, the point where it intersects y When I want to find it, I enter 0 for x, then a x 0 squared, whatever is next to it It was lost because of the x-squared term. It's the same thing. b x, that is, b x 0, went there too. It automatically gives us C. Hmm. HE Look, at this time we also have something like this... We discovered it! A second degree as soon as you look at the function's equation The constant number at the end is actually this y It represents the point where it intersects the axis. Yes. Let's give some other examples now. Let me go down a bit. Other Let's use this as our example. Let's say The equation of our function f(x) is -x² Let it be + 2x + 15. Now, let's look at its graph. Let's try to draw it. Immediately say y e= I started. Now, of course, this in a way that can be factored I gave this example. Put that aside too. Let's separate them like this, if you'd like. Even this x When the coefficient of the square is negative, I usually the expression in minus parentheses I'm getting it, but come on, it's like that here. Let's not do that. Let's call -x x. when I hit 15. The sum of the cross products also gives me + Look, it's become 5x, bringing it down to 2. 3 It became -3x. I gave him the one in the middle. Of course When writing, these will be placed side by side. That's how we were writing it, right? This is it These two and these two. This is x + 3. Alright We managed to factorize the expression. Then we do the usual things. We will do it. In fact, even never before Before drawing the graph exactly, this a quick glance and a daydream Let's begin. Now, the coefficient of x squared. negative. It starts with "Hmm, minus...". -x² e Therefore, it appears as the letter 'n'. The arms should be pointing downwards. E at the end The constant term is 15. Okay then. The ordinate of the point where it intersects the axis is 15. will be. There's also the part where it cuts x. If we can find the places quickly, then the expression... When I factored it, it cut off x. Can't I find the places easily either? I'll find it. Or the one that resets this It is value, or it is the value that nullifies it. we will say. And then suddenly the graph It will be resolved quickly. Now the thing I won't write. I assigned a value of 0 to X. He cut Y I found the place. I gave Y a 0. He cut X I found the place. Look, F0 is already at the end. It was a fixed term. The point where it cuts Y is 15. This I understand. The places where it cuts X are either like this Or values ​​that reset it to zero. TO Of course, what is the value of x that makes -x + 5 equal to 0? will it be? It will be 5. x + 3'ü 0'dırlatan What will the value be? It will be -3. Now Let's quickly imagine a rough graph. Let's say, come on over. He cut Y The ordinate of the point where it intersects the ground is 15. He's going to cut X at -3. Let's say it's -3 Let's say so. And he's going to cut it off at 5. Let's make this a 5. And already We knew it, or even if we didn't know it, this a parabola that passes through the three When I want to draw it, it takes the shape of the letter U. It will be in the shape of the letter N. This It will pass through all three points. So this is the graph. It will be on top of all three. Of course. It would be in the shape of the letter N. Here it is: Let's come, let's get it, let's draw it. Here it is this time such a thing with its arms pointing downwards It has a graph. And one more thing It must have caught your attention. Look, whether you want U Whether it's in the form of a letter or the letter N in appearance. For example, the graph is U If it's like the letter, then downwards like this. It's coming. So it's decreasing, decreasing, It is decreasing. It starts to increase after a certain point. It's starting. In the form of the letter N, it is... It's increasing, increasing, increasing. From somewhere Then it starts to decrease. Throughout the topic That's exactly the extreme point, and we're a bit... We'll name it later. peak we will say. And specifically with that point We will have tried. As you can see in our second animation with a graph where the arms point downwards We met. And in this example I gave... again, the function moves the x-axis to two different sides. He cut it short at that point. Now another one Let's do a dramatization. Another example I'll give it to you. Let's say this time Let's give the following example. f(x) Let the function be equal. What should it be equal to? Let it be equal to this. Come on. x² - 10x + 25. Now let's try to draw a graph of this. I started by saying E equals y. The expression I factored it. Pay attention! If you ask me, I gave you something special. So it's actually a perfect square. If you separate that it is separated as x - 5 x - 5 You'll see. In other words, this x - We can also say it's equal to the square of 5. We do the usual things one by one Let's do it. Let's say I gave x a value of 0. So I calculated f(0). And I know that It was already equal to the constant term at the end. F0 25. I found where to cut the Y. Y It will intersect at the point with an ordinate of 25. To E Y I'll give you 0 and then find the places where it intersects X. And that's why we're resetting this place to zero. I will find the values. In fact, either this or that. I'll find the people who reset this, right here. It must have caught your attention. Or two 5s It's coming. E x - 5 0 isa x 5. x - 5 0a x is still 5. So this means... if the number y is 0 It seems it can be x 5 and 5. Or now x 5 and What does number 5 mean? You know, we have this when imagining the graph, it cuts x We managed to find the locations, right? two in the example we gave and in two different places He had cut it. But this time, it seems like there's something... There are 10 points. And that's 5. So, what then? He cut off the Y, come on, you're 25 too. You I wrote it too. Okay, but now I'm going to go and set the x-axis to 5. it cuts and from here a linear one I can't draw graphs or anything like that. Why? Because this x squared second-degree, like the letter 'u'. I'm going to draw something, or the letter 'n'. I'm going to draw something that looks like this. This is it and having equations like this in the graphs of functions The graph you will draw is tangent to the x-axis. It has to be. In fact, something happened yesterday. I mentioned it. I said that any If the graph of the function is tangent to the x-axis It develops a double-layered root system there. Double-decker What do I mean by "root"? Here is what you have The equation remains the same when factored. It has factors that give the roots. Or specifically a second-degree integer It has a square equation. So it's a whole It is the square expansion. X - 5 squared. And this for this reason, in a full square view these second-degree functions The graphs are tangent to the x-axis and It has two equal real roots. Well x is two equal real roots It becomes 5. As I said, it's reflected in the graph. tangent to the x-axis, for example It is called. In the previous examples, x its axis, but always in two different places. He was cutting. This is what we understood from this. Means so that the x-axis is always two different He didn't have to cut it on the ground. Now Let me give you another example. HE From the example we will give, we can reach the following conclusion: Let's say we've arrived. Let's take this as an example. Okay, this time. function f(x) Let x² be + 4x be + 10 be. Now this in the examples I have given so far You were lucky because it immediately multiplied. They were leaving. roots easily We could find it. Let's see this time. Can I factorize the expression? The expression x² + 4x + 10. What now? I'm thinking if I split 10, it would be 1 to 10, 2 to 5. -1 I can't reach the middle 4 in any way. Okay, now listen, I've asked you a question. Let me be. E cannot be factored There is a second-order expression and we We are looking for the roots of this. So this x values ​​that reset the expression We are looking. There is no other way to find this. Is there no way in mathematics? So, necessarily Are we going to perform the calculation by factoring? Of course not. What I mean You should have understood by now. You know, number 2. In equations of degree, there is a word called delta. There was something. We used to say B² - 4ac. long long with formulas these types of equations We were finding their roots. But of course, now. Let's talk about this too. You know, today we... what we call a parabola, that is, a second-degree parabola. We're dealing with functions, you know, inside it. inevitably secondary He also has to deal with equations. We are staying. However, after one or two headings... all quadratic equations We will discuss it in detail, but in small steps. They appear before us in this small way. It will come out. Mathematics is a bit like this for AYT (Turkish University Entrance Exam). Mathematical topics are somewhat intertwined. It's nestled together. That's exactly what we are. Without disrupting the integrity, for example, today, uh, we When talking about parabolas, we said little things like this... Especially those of you going from 11th to 12th grade I'm saying this for that reason. Like derivatives, integrals concepts such as threats, etc. I'll mention it intentionally, but never... Don't worry. You know, a very detailed one Not a derivative, but how the topics are related in harmony or nothing in mathematics That's the point, just like that, on its own. as if it were an independent issue We cannot evaluate it. Oh, that's a lot of extras. Would there be any examples? For example, possible Even in what we call probability, actually What is the general title? Probability function like. Now, how do I draw a graph of this? Really? I assigned a value of 0 to Ex. So the value of f(0) I found. Its Y value became 10. Okay then. Y We found the spot where he cut it. Let's go from 0 to 10. E to Y. I'll give it a 0. When I assign 0 to y, x² + 4x + 10 = 0. And with an equation like this... I encountered it. It cannot be factored either. If they're not leaving, let's remind them immediately. ax² + In equations of the form bx + c = 0 When it cannot be factored, first the delta We had something called... That's b² - 4acydi. Why was it like this? Where did this come from? As I said, one or two headings later, the second one. We will be studying multiple-degree equations. There I will explain it in great detail. For now, you can have this information as readily available. Consider it as such. Delta = b² - 4ac We used to start by saying "it's about to happen". Later We were saying -b + root of delta / 2a this It is the first root of the equation. -b - inside the root What we call delta/2a is the equation. It is the second root. Of course, the one I gave here... Have you noticed anything in the formulas? It should be. Or there are expressions involving square roots. TO now written inside the square root expressions things if they are 0 or positive If it's a real number, then those expressions are also real numbers. indicates. Let's say our delta is negative. The result was equal to the number. E is negative inside the root. I can't say there's a number that's absolutely impossible. It is possible. The root when it contains negative numbers The values ​​you find will not be real numbers. In mathematics, there are also imaginary numbers. We had number systems, which we call numerical systems. E o Over time, this leads to the following. If in these function equations I have given If the delta of the equation is less than 0, then the expression... It has no real roots. If there is no real root You know, that thing we call the x-axis... that which we draw on what we call the y-axis These are lines of real numbers. So it's virtual on the x-axis. numbers, complex numbers, that is, those i things indicated by the letter We can't show it. This is why... We say, "This doesn't intersect the x-axis." X It will not intersect its axis. Because if it cut x the real roots of the equation we wrote It would be. Now look, this is really it. In the example I gave, we call b² - 4ac We observed that the delta was negative. Because its delta is negative, this equation... It has no real roots. It has no real roots. Now, some habits are the way to go. Let's eliminate it while it's still early. Don't do that Don't construct those sentences. Professor, this equation... It has no roots. No, every equation has a root there are. That's what we primarily emphasized there. What is it? It has no real roots. Its roots Virtual numbers will be generated. So what about this? What will this be reflected on the graph? Now Let's try to bring it to life. Tangible What information is available? I gave it as X². Good morning my baby. We know the sign of the leading coefficient. Then their arms will be pointing upwards. TO I know exactly where he cut the Y. He's going to cut Y off at 10 too. Let's say this is 10. let it be. with the arms pointing upwards, that is I'm going to draw a parabola that looks like the letter U. Or Two things come to my mind now. Should I draw it like this? Should I draw it like this? So, both of them It has the shape of the letter U. Look at the X-axis. I didn't cut it short. As you can see, it's not a tangent. It didn't happen, nor did it touch the x-axis in the drawing I made. chart. But for now there are two possibilities. Which one will it be, I wonder? will be? And that's where the determining factor comes in. Do you know what's going to happen? A little later As I said, I will explain it in great detail. This extreme point, or if I am at this extreme point... What is the x-coordinate of the point? Finding it I will draw the graph if I can find a way. So even if it's symbolic, it's in accordance with the procedure. It would become a graph. Of course, you know. How is it for now? I won't explain, but still, what I've given... let there be a suitable graphic representation The graph of the example I gave is roughly... We can say that this is the case. X It does not intersect the axis. The point where it intersects Y ordinate 10. with arms pointing upwards It's a graph. Actually. Look, up until now... There are three different situations in what I've described. We realized that we might encounter each other. Now, those things we found... uh... Looking at the overall latest charts Let's try to make some comments. Is it possible? In the first of the animations I provided, the graphic It intersected the x-axis at two different points. In one of them, it was tangent to the x-axis. This is the last one In the animation, however, it did not intersect the x-axis. So there are 3 possible scenarios. We were meeting. 2nd degree In the graphs of functions. Look again. Keep that in mind. This What I'm about to say is also very, very important. What is the order of an equation of the polynomial type? If it's a certain degree, then it will have 100% that many roots. I'll repeat the sentence now. A polynomial to the degree of the equation It becomes the root. 100%. For example, second degree Is that something? It has two roots. This definite. The really important thing here is either that or... Are the roots real numbers or imaginary numbers? Really? But what if those roots are virtual roots? If it comes, then the x-axis on the graph We say it's not enough. So for the parabola I'm talking. For example, something like this: It is possible. For example, a third-degree polynomial It becomes an equation. Or one root of this real root, the other two It could be a virtual root. And I am the third one. graph of a degree polynomial function If I wanted to draw a real root... I mentioned it then, that x-axis... It only seems to cut off at one point. So the x-axis is only in one place. The fact that it looks like it's cutting... It has real roots. So this is the first one. It does not mean it is of a certain degree. Ours These are the virtual roots that we can't see in the graph. There are also. Here too, this is the latest e In the animation we made, actually... We've dealt with this. So now this... some of the natural situations we found In conclusion, let's write something. Now then We talked about this. So we said that y = The second equation is given in the form ax² + bx + c. graph of a function of degree x It intersects its axis at two different points. It's either tangential or it doesn't intersect. So what? Time cuts off at two different points? TO When I set the given equation equal to 0, two If I find a different value, it's only because of what? When will that happen? Look back at that animation. Let's get there. And we have this equation If we can find its roots in this way, then... The value we call delta is a positive value. If it's a number, we find two different numbers. One is found here, the other one is found there. But If delta equals 0, then delta equals 0. If it becomes +√0, then the one below will be -√0, and the other one will be The parties are already the same. E √0 is 0. It will have no effect. Then x1 and x2 Those values ​​that I call interconnected They will earn equal profit. So if delta is equal to 0 There are two equal roots. Each other How does this look on the graph when there are two equal roots? was it reflecting? It was tangent to the x-axis. TO And finally, if delta is less than 0 He said it doesn't automatically intersect the x-axis. We became. So if its delta is greater than 0, then like this: I'll write briefly. I said a little later These are good in our book. It was given as information notes, but we... as if nothing is written there ourselves From the beginning, with animations like these... Let's see if this information is correct. that it is. If Delta is greater than 0, the graph... Let me write it briefly. different x-axis It cuts at two points. At two different points It cuts. So if its delta is equal to 0, what is its delta? If it is equal to 0, the graph will be tangent to the x-axis. Because they are two equal real roots. He was coming. If Delta is less than 0 because the equation has no real roots The graph does not intersect the x-axis. It remains suspended in the air. First, look at the places where it intersects the x. how we should comment on it We understand. What are we controlling? The Delta. That given function the delta of the equation. Then Y Let's talk about the place he cut. Where it cuts the Y It's always found in the same way anyway. X to 0 You are giving. When you give 0 to EX, naturally found the value of f(0) you become. And this is always the constant at the end. The term tells us this. ax² + bx + The value of c in the expression represents the y-axis of the graph. It is the place where it was cut. Let me write it like this. Y-axis cut We do this to find the point. The ordinate of the point where it intersects the y-axis is each Time is the constant term in that equation. Well, we understood this too. In fact, this And I said it's not just about parabolas. That is, according to the sign of the leading coefficient. The right end of the graph I will draw will point upwards. Will it go straight or downwards? We had determined this as well. But the parabola Specifically, the value we call 'a' So the coefficient of x squared is if a is less than 0. If it's large, the arms tend to point upwards. If the value of 'a' is less than 0, then... Their arms were pointing downwards. Of course, now, you know, around here... We will see the things we wrote in many places. things. Now we have these in the questions How does it appear to us? How else? can it be expressed? One of these let's talk. And then there's something very, very important. Let me mention one feature. This is just the beginning. See section 1. In fact, primarily that Let me give you one of the features. X-axis one for those who cut in two different places let's talk. Now, move the x-axis to two different sides. Even a colorful one cuts across the ground here. Let me use a pen, or let's say... This An important feature. Imagine that the x-axis is different. There is a function e that intersects in two places. In our possession. And this is what I want. HE The graph of the function on the coordinate plane It should be located in all 4 regions. Look at the period He has height, you know, like the height I wrote above. Delta is less than 0. Okay, these You can find them everywhere. everywhere You can hear it. They are frequently exposed in questions. You stay. But sometimes it's 1:35 like this I will be giving a lecture on the topic of clocks, but in between... Subtle sentences like the ones I'm using right now. I will construct each of these sentences as a question on the exam It might come out. Now I'll ask the question again. This is what I want. There is a parabola. Well There is a second-order function. HE whichever condition it satisfies, the graph will have coordinates It passes through all four quadrants of the plane. What then? What do I mean? from all four regions By "valid," I mean, for example, like this: If only there was a graph, or a graph like this... if Here's an example related to tangent: If it were a graph, for example, the first one I gave... In which regions is the example located? This 1st region. This part of the 2nd region 3 is located in 1, 2, and 3. Uh, Not at 3, but here of course at 4, 1, 2. and it's located in 4th place, but it's in the 3rd region. He doesn't accept it. I want it to be in all 4 regions. I want. For example, the second example, the third. He passed through the region, he passed through region 4. This It passed through zone 1, then zone 2. past. Let me tell you the condition. Absolutely! Take note. I'll write about it too, but if... that if the two functions have opposite signs If it has a root, if it has two roots with opposite signs opposite signs, meaning one is positive and the other is negative. If it has two negative real roots, then... time graph coordinates in this thing It passes through four regions in the plane. Opposite If there are two signed real roots, the graph It passes through all four regions. Now, I will discuss these during the period. They always appear as questions I will present it. I even say it from time to time. to that WhatsApp group. Now, this... I emphasized it, didn't I? That's exactly the question. regarding its usage within I will give examples. Later period inside, you know, beyond the ordinary questions, turbo repeat series these while doing so, use these and similar locks. to include more information I'll start. This is extremely important information. Really. A simulation like this: Let's do it. Let's see. Opposite sign Let it have roots. One is positive, one is negative. the one which. Let the negative root be this. Positive Let its root be this. Let's draw. Arms Let it be downwards. You know, your arms that the direction doesn't matter Let's understand. Then let's say the negative root Let this be it. Let this be the positive root. Also a graph with arms pointing upwards Let's do a dramatization. Look what I drew. Pay attention to the graphs. In the first region as well It has a part. It's also in the second region. 3 of It exists too. It's also in number 4. The same thing I drew The same applies to the other one. Of course, now... This question comes to your mind... uh, this question It should have occurred to you. TO Okay, it has two real roots with opposite signs. TO So how do I use this? Well two real roots of an equation with opposite signs It existed. Two numbers with opposite signs comment about the result of the multiplication You can. If they have opposite signs, these are Their product is negative. Oh look at the comment! The comment was born. Then we can also claim the following: I can. I have a second-degree one in my possession. There is a function. The question told me: Someone said, "The graph of this is coordinates. It passes through all four regions of the plane." So we will say, "That means the opposite of this." It has two real roots. Then that The product of its real roots is negative. Look multiplication of roots. And of course, you know, that too. It's not random, is it? You know There is also a way to find the product of the roots. Remember how we used to call it C/A for short? E C/ A They said it would be negative, and that's where it came from. We will be able to start solving the problem. You later made a note of it. Note this You got it. Now from another feature Let's talk about it. Let's make sure that feature is this one. You will encounter them so often throughout the semester. so you'll get tired of it now and in topics you would never expect Then the following feature will appear future. Those that don't intersect the x-axis I'm speaking for that. You know, the x-axis for those that do not intersect, that is, those that do not have a real root So, for those with a delta less than 0: We need to investigate the situation. Or delta Of course, delta is less than 0. Let's say a, e, and a are both positive. Even Let me lower it a little. A is positive an animation with a delta less than 0 Let's do it. And also, its 'a' has a negative delta. a portrayal that is again negative Let's do it. Now, the one where 'a' is positive, that is... delta with arms pointing upwards The x-axis, which is less than 0. not cutting. No, it won't intersect the x-axis. one with its arms pointing upwards Let's visualize a parabola. Here, take this. Let's say it looks roughly like this. Let's say. Here are the x-axis and the y-axis. The other one Let's create a visualization for that as well. Vaccine negative, meaning arms pointing downwards. Its delta is also negative, meaning the x-axis is negative again. It won't cut. Let's do a simulation. Let's say it also has an image like this. Let's say so. Okay, so this is how I downloaded it. He arrived. He arrived. Alright. So now, from here on out, something like this... We can say that. For example, the first one. The first one Which one on the graph do I need? Let me examine the point carefully. Look No matter which point I examine, what Do you know what I mean? For every real number x for. Now, what happens for every real number x? What do you mean? For example, consider this point. Think about this. Think about it. Think about it. No matter which point you consider, it makes no difference. It doesn't. You know, these dots, these points You know how abscissa values ​​are? Well, this one's is 0. I won't show it. Its abscissa, this its abscissa, this abscissa, this abscissa. Well Instead of x, you can use this, or that, or... Whatever number I give, give me this number, give me that number, give me that number. The result is, Let's make the results colorful too. Let me show you. Even though I don't know what it's equal to, take a look. I see it as a positive value. I see it as a positive value. Or Now, whatever you write in place of x, that's what you'll get. conclusion. And what was x? What was Y? You know, ours Isn't that what we generally agree on? y = ax² It was the expression + bx + c. Whatever you write for X, that is, every x is real. for each real number x, this the results corresponding to the x's, that is y's, and what I mean by y is this ax² + bx + c'di. So the result of the expression ax² + bx + c is always It would be greater than 0. When? If α is positive and delta negative for each x, that is, whatever you write for x, The result of the statement always came out positive. These are questions posed to us in reverse. The question is being asked. So, it's a secondary type. It gives. He says this is always greater than 0. Since that's the case, we'll have to think about what to do. So, no matter what you write in x, it always works the same way. The outcome will always be positive, right? At work its condition is that 'a' is positive delta It turned out to be negative. Let's examine the other one as well. In the other one too for which x value would you consider Think about it. Look, on the graph... Imagine any point. Which Whatever you take, x is the reason this happens. In that case, this will happen. In this case, the results that came out this time were all the same. It corresponds to negative numbers. Look, always The result came back negative. So then this We will tell you. y = ax² + bx + c in the expression for every real number x, for every real number x This time, the result I found was from 0. It turns out it's too small. Whichever condition is met If A is negative and delta is negative It's so important, look at the parabola today. these features we talked about We will use it to address inequalities. Tomorrow one We will use day in the derivative. Always to explain This is what I'm working on. From time to time I emphasize this. Mathematics as a whole We will see it as such. Yes. The main focus Our point is this, which we call a parabola. graphs, but these are the investments. We need to do it now. Yes. Now Let's take a look at this. You know, this ax k In an expression of the form + bx + c, complete a They will be real numbers. Absolutely It will be different from 0. Acceptance. What about B and C? Is there a possibility that these are 0? Or B What does the 0SA graph look like? Is it happening? What's happening at C0sa? Now only We will only examine these situations. Of course There is no possibility of A being 0. Yes, that's right. If that were the case, it wouldn't be a second-degree relative. Let's say and in this statement, something like this In the expression, if B is 0, then a star like this... Let me guess again, in case B is 0, then b Let me give you an example. With more examples We understand each other easily. For example, let's say f(x) = this let it be. Let x² be -4. And its graph Let's try to draw it. E graph We can easily draw it, that is to say: I can factorize y = xk -4. A difference of 2 squares occurs, right? You know, a - b a + b x - 2 x + 2. E roots themselves He made it obvious right away. Here it is The value that resets is 2. The other one The value that resets to zero is -2. Automatically So I found the places where it will intersect x. E y He already looks at the equation for the place he cut. I could find it without looking. So how is that? were we finding? Let me remind you again. Come on. F0 we were saying. I entered 0 instead of X. Y I also found the place where he's going to cut it. Now these Let's combine them. The title is already there. I know the coefficient is positive. It will have a boot pointing upwards I know that too. Now it's -4 here, and 2 there. let it be. Let's make this place -2. E from the three It will pass. Here's a rough outline of it. We can say it has a graph. Okay then. Now, why is it done this way specifically? Did I emphasize it? In fact, here's another example below. I'll give you more. Let's talk after that. Let's say the graph of the function f(x) Let's say something like this happens. Let's say it's 3x². + + Let it be 10. Now let's try to draw it. Okay, so the arms will be pointing upwards. Since the leading coefficient is positive, x is 0. If I give y, it will cut it off at 10. He will cut Y I know the location too. That's usually the case. What's the problem? Does it cut X or not? Really? If I set it to 0, then it's still 0. We were finding the places where it intersected with x by typing. Yeah. Now I'm thinking about it. 0 = 3x² + 10. For example, here, delta meltaya There's no need to even go in. Step by step as follows: I can leave the x-squared term alone. TO x² became -10/3. The square of a real number Will it turn out negative? Impossible. Hmm. So, It turns out this has no real roots. So what is the real root? no. I know that E F0 has a 10. What does this mean? It will intersect the Y-axis The ordinate of the earth is 10. E is the real root of this. no. So it won't intersect the x-axis. Arms It will be upwards. That's when It has a graph roughly like this: I can say that. This is actually what happens when B is 0. What is it that I want to give you? He too I'll write it down right away. Now Let me remind you. Dual function There was a type of function we called. Couple. Even polynomials in functions if in the given equation The exponents of terms containing x are always even numbers. If that happens, then it means he's a couple It provided. Now, let me animate this... Think about it. B 0a B 0sa so we have If there is an equation of the form ax² + c, then b 0sa e x'i above the terms I'm looking. And there's already one in the picture. It seems like there's an x, but the constant term at the end... and right next to it, x to the power of 0 We can put it there. Especially this one, you know, fixed The term is either x to the power of 0. the reflex in what we call 1000 ohms that's something we do quite often anyway. It was something. Now take a look at them. E 2 and 0 and these are even. So then this couple We call it a function. Okay, let's call it... What does it mean to say it's a binary function? Will it be beneficial? If b is 0, then it is even. It seems to be a function. Okay, but let me ask again. E name Dual function. So what are we going to do? Its name is different. What good would it have done us if something had happened? It's not the name that we're interested in. Its characteristic is that it has dual functions. graphs always on the y-axis It will be symmetrical accordingly. I'll say it again. Its graph is symmetrical with respect to the y-axis. It is possible. Let's summarize the graph as follows. Y symmetrical with respect to its axis. So the Y-axis It was acting as a mirror. Look, first... Let's look at the example I gave. So this... It was a mirror. In other words, different Look at where the x is intersected in the expression. One becomes -2 and the other becomes +2. Here's what's compared to 0a. So, we have a zero here, right? Based on that... There are two symmetrical roots and This graph we drew is based on the y-axis. It becomes symmetrical. The Y-axis graph is complete. It's like dividing it in half. So right here Let's show one extreme example. Full It cuts it in half. The same applies to the one below. The same applies to... Here is our y-axis. It was dividing our shape right down the middle. Ok. Let's also mention this extreme point. Let's show it. Divided into two equal regions, equal It seems two regions are being formed. Especially these and similar questions. I'll get to that later. This steep We will take it on the coordinate plane. We will do things like this. Here's what we'll say Or let's place a rectangle here. Oh dear, what is the area of ​​this rectangle? Let's find it. Let's find its surroundings. This is it In most questions like these, the dual function It is used. b is given as 0. Because It makes the process easier for us. So, If B is 0, then our function is even. It seems to be a function. E graph Y It appears to be symmetrical with respect to its axis. So, y The axis of our parabola is exactly in the middle like this. He's dividing it by 2. Now let's talk about this. If C becomes 0 So in the expression y = ax² + bx + c, c = 0. if e, c, if 0, automatically The thing we'll be working with is e in the format ax² + bx It will be an equation. And this is the most typical What's an example? Where does C.C. work the most? Does it work? The place where C intersects Y. E now C If it is 0, then the ordinate of the point where it intersects Y is... We say it will be 0. Now, where it cuts Y The ordinate is 0. Okay. In fact, even generally Since the equation will be in the form ax² + bx It can be written by factoring out x. When x is factored out, it becomes ax + b. Even and In fact, the main comment we're going to make is this... you know, the roots are either this or that. Those were the values ​​that reset everything to zero. And from here What will the next root always be? It becomes 0. So when you set x to 0, you also set y to 0. It is possible. So the graph starts from point 0a 0 It passes. What exactly is this thing we call the point E 0a 0? Origin. So if it's CO0, then quickly... reflexively, yes, this is the graph. We would definitely say it will pass through the origin. Or, if it passes through the origin, C in the equation We say it will be 0. If C0 becomes the graph It passes through the origin. Let's give an example right away. I will give it. Let's write it down. I already said it. Look, this is important. The point is not to memorize this information. Look, at the very bottom, two tiny lines. We are showing. In several examples as well I'm explaining. Here's a meaningful way to do it. That's how you'll keep it in your mind. But Of course, by solving questions like this... We will train together. Subject We will conduct screening tests. Here's the solution. By solving and solving and solving, it's become so fast now. We will respond reflexively. Well, there you go Look, this graph passes through the origin. Ok you will say. That's when it becomes fixed. The term is 0. Here's an example... Let's give it. Let's say f(x) is the following. -x² + Let it be 4x. So y = -x² + 4x. So y = Let me write it out by putting eee - x in parentheses. Come on. X - 4. E, the root I get from here is 0. My ancestor who came from here is number 4. So, It will intersect the x-axis at 0 and 4. Here Let it be 0. So let's make this a number 4. Head I gave the coefficient as negative -x². Well Therefore, their arms are also pointing downwards. what is happening is a graph passing through the origin. He will have it. Now, let's examine this as well. We examined the following in order. B 0sa What's happening? If both C and 0a are 0, then both B and C are 0. Also, if C is 0, then B is 0, and if C is also 0, then e is 0. The equation we have is in the form ax² It takes on an appearance. So, it has a B next to it. There's nothing there, not even in C. B and C are 0. It has neither an X term nor a constant term. For example, let me give the basis f(x) = n. Look, x² is mostly related to parabolas anyway. Sometimes, when the story is told, it goes something like this... First it gives x squared, then the function. I shifted it using the translations. The graph is as follows: I drew it. That's not how I told it. Difference You must have done so. Because what if we haven't already? We didn't discuss postponement with you either. You know, to the right Let's put it off; we haven't discussed these things upstairs. But I am giving you the fundamental laws. Basis What I mean by law is this. Today on the parabola all the applications I made when drawing the graph of a linear function I will do it too. Tomorrow, the third one. when plotting the degrees I will do it. I'll assign 0 to X, and it will intersect Y. I'll find the place. I'll assign 0 to Y, and x to... I'll find the place where he cut it. The leading coefficient I'll look at the sign. The far right arm Will it go up or down? Will he leave? Uh, I'll decide that. while finding the roots of the function from my knowledge of solving equations I will benefit from it. If I'm really, really stuck, I'll use this sentence. I remember. Or the graph of the function Every point on it is its equation provides. So what is the function for x = 1? I'll find it. What happens for -3? I'll find it. I'll find out what's happening with number 4. HE I'll identify the points anyway. Hop, that graph He would be giving himself away. Of course, now... Let's factorize x². E separated Let's make it clearer, though. If you wish. This x then comes from here. The square root of 0. And the square root that comes from here is also 0. Aa 0 too. It developed a double-layered root system. So, of course, we... When you set x to 0, then y is also 0. It's coming out, right? Our graph starts from the origin. It's passing. And it looks like this. See what happens when it has a double root system. was happening? It was tangent, right? Tget. And so, this is our graph. It had that appearance. Okay, let's talk about this now. Or this x²are What would happen if it were 3x², sir? E 3x² What would have happened if that had been the case? Think about this. I added three times the amount. I put Xar 3. What happened That number causes a change in the roots. It's not working. Roughly speaking, the graph looks like this again. It would be. So, again, exactly at point 0 to 0. A tangential relationship would occur. Arms up It would have an appearance that was accurate. TO Let's also consider the x-squared version. Let's also consider the -x squared version. And this The journey will again be tangent to the origin, but... Since the leading coefficient is negative, the arms It will be downwards. Or is it already in your mind? When y = x² is mentioned, this graph immediately comes to mind. future. Let's call it y = x squared. -x² This is what should immediately come to your mind. These visualizations are right before your eyes. It needs to have formed immediately. In fact, this That's also an option. For example, y = x² + 4 Let's say we're going to draw its graph. What I said like we haven't made such detailed postponements yet. We didn't talk about it or anything, but just a tiny bit about her. Let me mention this as well. If this is x squared then x² + 4 Take this graph and move it up 4 units. It is to postpone. And that place was zero. 0da teetlik It was forming. When you shift it up 4 units That means it will now occur in 4. And as I said just a moment ago, for people like this... What did we say? I mentioned dual function. Look, B is 0. X is the coefficient of the term 0. E Therefore, it is symmetrical with respect to the y-axis. will be. Yes. The Y-axis is such a mirror. It will split the task right down the middle. It will serve as an axis of symmetry. we can say. Here you go, in order. we are roughly the second of a parabola. a polynomial function of degree In what situations do we encounter the graph? These things might come up; we discussed them. A's, B, C, that is, B and C being 0. The delta's position is such that as soon as it intersects X, its arms... Upwards and downwards, for example. Now a little more details, you know... Let's begin. In this thing we call a parabola will constantly come up with the concept of a peak point Let's start paying attention. Now the hill What is this thing we call the "point"? Firstly Let me talk about that too. Actually, in yesterday's lesson, there were one or two questions... It just happened that way. Minimum of the function point or maximum point. Oh yes, of course. when we say the vertex of a parabola And that's exactly the point we're talking about, right here at the edge. point. Of course, there's also one with the arms pointing downwards. Let's give an example. Here's the point... We call it the vertex of the parabola. Now this What do I mean by "point"? For example, their arms for an upward parabola I can express it like this. Function It was decreasing, decreasing, decreasing. Then it increased, and increased, and increased. H function to the point where it transitioned from decreasing to increasing If the arms are pointing downwards, it increased, it increased. increased or decreased, decreased, decreased or to the point where it transitioned from increasing to decreasing I can even say this: Look, here's an example right now. Let me give it. Keep your arms raised. I use this function on it For example, this point is one of these point. From this point, there is a function Let's imagine I've drawn a tangent line. Tangent actually. So what does tangent mean? From here I touched it at one point and passed by. And another Let me mark a point on it. Take this I determined it. Let me draw a tangent from there as well. Now, recall yesterday's lesson. Yesterday's Remember the lesson. This green thing I drew straight to the right, i.e., tilted to the right. slopes of horizontal lines We were saying it was positive. For example, this truth The slope is positive. Our red line is to the left. straight and tilted. Tilted to the left The slope of the lines is also negative. I told you. Now, the main question is this. Alright What color is it from right this point? Shall I show you? Let's go with this light blue one. Let me show you. From the very extreme point, to this, that is... If I draw a dot from a point, boom, I've got it, I've drawn it. Look, I'll draw that line on the x-axis. They became parallel. Such horizontal lines What will the slopes be like? It becomes 0. The slope will be 0. So, in fact, which extreme point are we talking about? It possesses certain characteristics; what are they, and what are they not? We're talking about them, aren't we? Look now Where am I connecting this graduates issue? He must have understood. You know, this... this right now What I'm talking about is actually, yes, the hill. point but another matter I enter its borders little by little like this. If you notice, uh, from 11 to 12 And let me say this for those who have passed away. At work This is the concept of tangent. Tangent teet teet If there is tangentiality, then there is a derivative involved. So, what I'm actually doing is this: I don't call it that. The name of the topic I'm not saying anything, but those topics too... We are building the infrastructure. So tomorrow One day we will be preoccupied with this. Here's a function graph they'll provide. to us. They will say that this is on it Draw a tangent line from the point. Here I am like I did. Oh my goodness, that tangent Please comment on the slope. At work Is it positive, negative, or zero? like, what is it equal to, that derivative Within the topic we mentioned, this is also included. We will look into it. So then the peak of the parabola While I want to express this point, I want to say this: I can say that too. If the arms are pointing upwards, the arms If it's pointing upwards, then this is the function. not the point where it reaches its minimum value Is it? The point at which it reaches its minimum value It expresses. If the arms are pointing downwards the greatest value the function can take It indicates the point it has reached. Look like this. I can also express the peak point. Or And that point is such a point that from there the drawn test is parallel to the x-axis The slope for it is 0. So the parabola causing the slope to be 0 That's the point. So it's tangent at that point. When I draw it, we say the slope of the tangent is 0. TO Of course, all this is well and good, but what's our real problem? What now? give us the parabola equation They will give. Find the peak of this. They will say. The peak How do we find the coordinates? Now him Let's talk. And there, right away... Let me talk about it Actually. Actually, look over there... Before we move on to the animations below Something else came to my mind first. Let me tell you about that too. For example, the first one from that very peak point, vertically like this: If I draw a straight line, or if I draw it here... If I were to draw it here, what I would actually draw is... The shape of the lines is such that they are symmetrical. It divides it into parts. So what I drew is correct. It becomes the axis of symmetry. So the shape is perfect. It cuts it in half. From the peak point such a vertical line that it will pass when you draw It divides the graph into two equal parts. We call it the axis of symmetry. And from this point onwards... In conclusion, we can say the following. Look, nothing at all I didn't give any formulas or anything. The formula is not yet available. I didn't give it. Logically, we ourselves are a hill. that point which we call the point How can we access their coordinates? Him Let's try to understand. For this, we need to do it right away. Let me give an example. A function Let's liven it up. So, let's say what? let it be? Let X² be the value. Let the roots be 3/5. X - Let 3 be the value, which is X² - ​​8x + 15. Now this Let's draw its graph. And then this to reach the coordinates of the peak Let's work. Now I started with y e=. I factored the expression. X - 3 at X - It's 5 now. I bought it right away. There I'm starting to create my animation. Its roots are 3 and 5. multipliers I could see it clearly once I separated them. He cut Y The constant term e is 15 in location F0. I saw that too. And then your arms are pointing upwards I know that will happen too. Roughly speaking... Let's do a dramatization. This is number 3. let it be. The other root is 5. He cut Y I know the location too, it's 15. Now mine What is my main problem with this question? Right here Could this be what we call the peak point? What will the resulting value be? And this I know. And I conclude from this that it is true. When I drew it, the shape was divided into two perfectly equal parts. It was dividing them, wasn't it? Look, right here... Focus your mind. Give this piece a try Concentrate. Is it ok? This part through which we actually do the job We can finalize it. And like this... We can do it. This is the point, and this is the point. According to this, it will have a symmetrical appearance. Let me put it more clearly. These lengths will be equal. TO This also has a meaning. So then this If you want to find the point, you'll come across it here. To find the resulting number, use this and that. You'll find the perfect middle ground. E two numbers How do you find the exact middle ground? Collecting You find it by dividing by 2. 3 + 5/2 So the number that will appear here It was 4. So this point is the peak point. I'll refer to it as TN from now on. Top The abscissa of the point is 4. Well, he is one point. Point. What I call a point also has another meaning. Will there be no ordinate? So, to this point How to determine the corresponding y value Will I find it? Simple. This key word It was kicking in, wasn't it? Any the graph of a function from a point if it passes or a point if it is on the graph of the function That point satisfies the equation of that function. So this point is actually where x is the x-coordinate of 4. Since it is a point, finding the value of y In the function, wherever I see x, I enter 4. I write. So I calculate f(')ü. There too When I write 4 everywhere I see x Let's do a quick calculation. 16 - 32 I added -16 15. It is equivalent to -1 I found. So the function's vertex The coordinates of the point are 4e -1. This The red line I drew is the axis of symmetry. we say. Our axis of symmetry is x = 4. It turns out it's true. So right at that peak point Our line passing through the abscissa is x = 4 It turns out it's true. Now in all these explanations Teacher, we always do it this way... Draw the graph. Find the roots from there. No We won't do that, but we'll do this. How Have we reached this point? We have received this information as well. Because the places where it intersects the x-axis, that is... right in the middle of the roots. Gathering the roots I divided it in two. Indeed, 5 out of 3 here those were the roots of the function. So then, generally As for me, couldn't I do something like this? Let's say we have a parabola. Here's how I drew it symbolically. Let me call the value here x1. The value here is also x2. Let me name it. Of course, this is a x² + bx + c In that statement, when I reset it... Not when I reset it to 0, but to 0. When I equate them, that is, when I give y 0 and x... I was able to find the places where he had cut. This that has formed the roots of the equation formed by this equation Let x1 and x2 be the roots. Now this I also included a diagram of it above. Look The values ​​that reset the equation to zero are... They were the roots. So the places where it intersects x are x1 And let it be x2. I automatically climb the hill I was after that point. And just a moment ago I told you. The value of this point To find it, add x1 and x2 and get 2. I'll divide it. Let's show it like this. x1 + x2/ It is 2. So now, second degree... This information I know from the equations I use it. How do I find X1 + X2? Now, like I said, this one or two times... I will present the title and the reasons later. but also being able to say it quickly and move on There is a benefit. You know. X1 + X2 - B/ A It was found in this form. So, to the equation when we look The coefficient of the x term, x squared We divide by the coefficient of the term. Also There's a minus sign in front of it. He gave us the sum of the roots was giving. So then, here's x1 + I replaced x2 with -b/a. And also divide There are two. I edited it. The most regular naturally -b/ a x 1/2 also includes the 2 in the denominator I included it. Here's one formula that has emerged. It happened. What have we discovered so far? Any the vertex of a second-degree function the abscissa of the point, that is, it will come here as soon as you look at the number in the equation -b/2a We were able to find it with their help. And of course, how do we find the ordinate of that point? Will I find it? So, corresponding to that point How can I find YD? A moment ago How did I find it in the animation? I said this Find number 4, and the rest will be easy. Because this If the abscissa of the point is 4, then F4. He takes it right away I said I'll write it instead. I would express it as follows: Let me do it. F -b/ 2A here's what I found -b/2a I'll write its value in its place. peak I will have determined the coordinates. Means the vertex of a parabola If you want to find the coordinates, do this. Its abscissa is -b/2a and its ordinate is also o take the value you found and write it in its place You can find it. Now, let's clean this up a little more... Let me explain. In fact, let's take a look at this place. Let's enlarge it. Heh. Let's say f(x) = ax² + bx + c expression, that is, the vertex of this function the point which is usually the peak When lettering, it is lettered as follows. R becomes k. Or maybe we should just call this K. I don't want to, but you know, that mouth... What does the 'K' in habit mean? But here I am too. I can't break this habit. I sometimes say K. From time to time K I say. Usually the letters R and K It is being used. The abscissa of the vertex R, The ordinate of the vertex K. R is this That's how we find them. -b/ It's 2A. That's how R was discovered. Finding K and the value I found here for that We find it by substituting it into the function. Ha, if I really want to, I can find a formula for that too. I'll give it to you, the book will probably be there later. has given. It's not very important. Well Wherever you see x above, add -b/2a. in summer. You've already reached the conclusion. Of course, we should also mention this here. Look, let's say, let's just say like this... Let's have a parabola. The peak Let's say the coordinates are these 3. Here Let's say it's minus ten. Give us the hill Find the point. That's the peak point. Find this, find that. He usually doesn't say that. They use the concept of the axis of symmetry very often. in the questions. So what was the axis of symmetry? The axis of symmetry is exactly at that vertex. The line passing through the apse. So, symmetry We say our axis is the line x = 3. And this They framed the question in reverse for us. They'll get watered. It will provide a function. He will say that its axis of symmetry is x = 3 actually. So we will say, "That means..." the abscissa of the parabola's vertex, that is Our r, that is, our -b/2a, is equal to 3 we will say. So when you want to ask about R It uses the concept of the axis of symmetry. Using its ordinate, that is, the k value and something even better when you want it He mentions it. Look, right here, small... Let's do some role-playing. Imagine the arms There's a parabola that points upwards, right? with arms pointing downwards There is a parabola. That's what we're looking for. this point ordinate. We are looking into this. This is not what we were looking for. We are interested. My first drawing According to the animation, the arms are raised. The correct parabola corresponds to this point. The number is actually the image of that function. Isn't it the smallest element of the set? A different variable function the smallest value it can take. Look, this is it. Let it have a function. This is called a function. It is the smallest value it can take. Smallest value. And when the arms are pointing downwards... The value that corresponds to this point is the function It is the greatest value it can obtain. So nobody told us the peak point It won't ask you to find the ordinate. Second It will give a function of degree. He will say find its smallest value or the smallest Find its great value. We always encounter this... They will ask questions and derive answers in various ways. Now, here's an example of a function. Let's give it. That function on it here is the axis of symmetry, or what it can take. to find the greatest and smallest value Let's work. Let's say our function is this. Let's say we want to get x² - 6x - 6. What do we use to get there? Let's say +8. Immediately the multipliers Let me separate them. I gave it in a detachable type. already. Let x be x - 2 and x be x - 4. in the graph Let's just roughly imagine it here. The roots are already making themselves known. X - It's 2 out of 2. X minus 4 equals 4. He's going to cut it off at 2 and 4. Your arms I know it will go upwards. Because our leading coefficient is positive. Also, I actually did. Right here, at this hilltop that its abscissa is exactly halfway between 2 and 4 I know how to apply the formula, but... Let's do it. Let's get used to it. -b/2a I was finding it. I was finding R as -b/2a. - Our B is -6 / 2 a's. Therefore r Its value is 3. We already have it This is what we expected. Its 'R' is 3. It will immediately correspond to these 3, the equivalent. the upcoming value, namely F3 Let me calculate it. 3 squared 6 x 3 + 8 9 - 18 It's down to -9. Did 8 become -1? This process of mine Let me check. 9 - 18 -9 + 8. Yes. That's when we can say this. For example, is this the axis of symmetry? was asked? The axis of symmetry is the line x = 3. We can say that. This function What is the smallest value it can take? TO The smallest value it can take is -1. I can say that. Or they already have arms If it's going upwards, the maximum it can get Does its great value ask us? He doesn't ask. TO Because it's going upwards. So that Plus, it's moving in the right direction. For this reason, if the arms are pointing upwards, then... He asks about the younger one. If their arms are pointing downwards He asks which one is the biggest. Of course, here. Look, there's something very important here. Let's talk about it. The peak is more like this. These are the kinds of questions we encounter. any everyday life problem It will come up when we solve it. Already You will see it happening more and more as we go on. It would never even cross your mind. They will model a problem. We We're going to spell it out, man. We'll see. Something of secondary importance. The question also gives us He'll keep asking that question. Here it is, here it is What is the greatest value it can obtain, the most What is a small value? The biggest or the largest if small value is emphasized If the function you have is also of order 2... The ordinate of its vertex is k. The question being asked is what we call it. This Don't forget. Let's talk about this too. Look, we have something to say about the peak. In fact, one of the comments we will make is It is this. Let's say such a function exists. There is. What is the peak point here? Should he come? Let's make it a 5. Our function is defined from r to r. Already What did I say at the beginning of the day? On the contrary unless I say so, the domain is real Assume that it is a set of numbers. I said. So, what about this function in its current form? Is it an exact replica? Exact, one-to-one function. So I'm saying... so that different values ​​are used instead of x. Will you always get different answers when you ask? Will you get it? Well, let's test this too. easy. No, it's not an exact match. Because Look, there's an x ​​value here. This When you substitute the result, for example, the result is 10. But let me proceed a little further. Look, there's an x ​​value here too. When I substituted that, it came out as 10. But each different x value to be one-to-one I found a different result for it It was necessary. But it doesn't work that way here. TO So, that means it was normal under normal circumstances. By conditions, I mean this. Domain As long as it is the set of real numbers, then 2. Functions of degree are not one-to-one. I'll repeat the sentence. domain is real numbers, that is, everything in place of x If I can write, if there are no restrictions When you think like this on the graph This function is not a one-to-one replica. Because The same answers are obtained for different x values. However, for it to be exactly the same, each different x finding a different result for its value We had to. Now, let me get to my main question. And me What if I played around with its domain? Well What if I say, "Okay, this is the function, but you..." Don't think of it that way, all at once. For example, 5 and Take the part after number 5. And like this... Let me show it in green. Just this green Let's focus on that part. Let me underline that as well. Such a if it had a graph, that is, a domain This time, if it were like this, let me just say TK. From 5 to infinity, from something to something. If it had gone like that. So our domain is... If 5 were infinite. Look, those 5 are random. I didn't choose. Right there, that peak point I used it. That's when the graph is just from the piece I showed below in green would form. That's when it would be exactly the same. Or consider the part up until number 5. If I had said that, I mean, in a light blue color... Let me show you. Focus only on this part. Look. On that piece No matter which x you think about, it's always You will find a different answer. At that time too We would say the graph would look like this. So, the definition the domain of the set is minus infinity If it is in the range of 5 The function would be one-to-one. So one The domain of a parabola is the set of real numbers. As long as it exists, yes, it's not exactly the same, but... by manipulating its domain, its one-to-one We can make it happen. This is one. Secondly, you know, the peak point that we're talking about... with the ordinate of the relevant vertex the biggest thing when they ask related questions We said its value is its smallest value, didn't we? I wonder if that peak ordinate in every situation, under every circumstance, that second one the smallest or most basic of something Is there a point where it is too big? Look at every What I mean by chart is this. Understand this from that. Don't let the wrong idea form in your mind. As before, if I use the function If I play within the defined range, the peak That point might not be useful at all. This is what I'm talking about. Let's say... Let's say we have a function like this: There is. This Let its abscissa be 4. And what if I were to say to you, "What about the graph of this?" Actually, that's normally the case, but I... I'm going to narrow down its domain. Definition set the domain is closed between 2 and 5 Imagine the situation where it's within that range. And this time we'll do something like this... automatically, you know, starting from this point... I call it F, but actually it's more like this: Feyelim Let 'ha' be the name of the person next to it. Normally when there is something whose graph is like the one on the left I'm saying, what if it's within its domain? Let's make it 2 to 5. So it's the same with which parts of the graph then Will I try? Here's a 2 of x. the situation. Let's make this one a 5 as well. Or just take this piece and put it aside Imagine I drew it, look at just that part. I took it and drew it on the side. Here's the definition. the endpoint of this set is the endpoint of the range The point was 2. The other end was 5. This place too It was 4. Now, in this case, the maximum of the function comments about these of little value Let's do it. For example, in this animation of mine... That's why I'm speaking. Peak He is taking an active role again. Look The ordinate of the vertex of the function It's still the lowest value it received. E the biggest Who plays a role in determining value? Here it is. The numbers corresponding to these. Of course I mean, I'm saying this symbolically now. I drew it and showed it to us as a function. They will give us the equation, so we'll take that end. write the points to the function value and the resulting output We can find the results. But whatever happens Okay, look at this animation I've made. again, the vertex is the smallest of the function I made sure it received its due value. But if If I were to say that f is f with 6 Think of it in an infinite range. Okay then. HE Let me think about the part about time 6 and infinity. of the graph. Let's make this one number 6. 6 to infinity. Just place that green piece right here. Let me bring it to life. Of course, the continuation of this is actually like this... It comes and goes like this. Our hill Our point and everything else is also staying here, but... This part doesn't concern us. Why? Because we are only with the green part. We are trying. Look, in a function How important is the domain of definition? You know, to number 2 I defined something from 5. This one to 6 as well I defined something from infinity. Because What comments will you make, or which ones are for the x's? Is the condition met? First, this... You think. According to him, you look at the graph. You interpret it accordingly. So then this under these conditions the function takes the maximum The smaller value appears to be equal to f6. There is such a thing as the greatest value Can't he talk about it? It's going towards infinity. He's leaving. So what happened? This is the peak The attempt was completely unsuccessful. Then if the domain of the function is like this If it's being compressed and delivered, pay a little attention. We will. Like in that first example. If The abscissa of the apex is one of these intervals If it's an employee, there's no problem. Peak It still works. Look, 4 is one of these intervals. He was a member. But like in the second example The abscissa of the vertex is one of these intervals. If it's not an element, then the apex is useless. It's not working. These and similar applications More on the book a little later We will do it. Now, the peak of the questions. the most frequently used Let me mention one feature. Look at the hill Imagine the point as follows: You can. Let's illustrate with an example. Again, with the arms pointing downwards like this Let me give you a parabola. This place too Let's say it happened to be 7. You know, exactly the way it will pass through this number 7. If I draw it correctly, that will be our axis of symmetry. It was happening. In fact, we can now even do these things. We knew. 100% exactly with this and that The middle was equivalent to a 7. This means For example, let's say this part is equal to 3. 3 7 that is 7 will be exactly in the middle. one in three things The middle is 7, adding something in three parts and giving it to 2. When I divided, I divided 7 by 2. So... When you add them up, it will be 14. So this is 11 I can say that it will be. So actually, this I did. Did you notice? I gave R. I gave a 7. From the places where it cut X I gave one away too. And the other one, we'll take it. We said we could find it. Because this, that is, this more precisely, equidistant from this point the apses located at equal distances from this the points found in the function The images are identical. What does this sentence mean? Is he coming? Look at this point and this point. Aren't these two points equidistant from each other? Two points equidistant from each other. I mean, really This place and that place are equal. In that case I'm saying that between this point and this point Their ordinates are equal. Or is this already the case? It's clear they're equal in the animation. X the place where he cut. The ordinate of X is also 0. But this only related to where it intersects this x It wasn't a feature. What I said The conclusion that can be drawn from the sentence is actually It was this. Well, the same animation... Let me do it. Let's make this a 3. This is number 7. This is the 11th. Now 7's The value two units to the left is 5, right? The value that is 9 is 2 units to the right of 7. Look, over there, our 5 and 7 are equivalent to 5 and 9. Our values ​​are once again at their peak. Equidistant from its abscissa. between 5 and 7 2 units. The difference between 7 and 9 is also 2 units. Then this situation occurs. So then both 5 and 9 The values ​​in the function are equal to each other. It's coming out. So f5 and f9 are equal. It's coming out. That is, equidistant from r The images of the points found are equal. Their ordinates are equal. E and in these questions This provides such incredible convenience. For example, another animation Let's do it. We need to get your hand used to this. It is necessary. A function like this: Let's assume it's a graph. Come on, let's go over there. Let it be 10. Look at the apex of the vertex. 10. Two points equidistant from 10. Let's choose. Let's move the 7, which is 3 units to the left. Imagine I bought it. And also 3 units Imagine I took the 13 on the right. Both The value that 13 and 7 will take in the function The values ​​are equal. So f(7 to F13) equals. The reason is because of these lengths They were equal to each other. This is the hill. This is the most common use of the point. Both these simple graphs appear in the questions. we should make this kind of comment It's expected, and look what... I'm going to say in the question now: 2. There is a function of degree. In fact, first of all, an infrastructure at the top. Let me create one, for example, here with F7. The F13s turned out to be equally matched. And we are from here Don't we understand this? 7 and 13 in full the middle, that is, half of their total, is 10a equal. So R is the abscissa of the vertex They were equal. These are the questions we ask. They present it to us like this. He says f a function of degree. F11 + √3. Now What is F11 + √3? Instead of X, go to 11 + Should we write √3? How do those processes work? What will we do? E = F 7 - revenge. F is naturally of the second degree. He says it's a function. So already We're talking about parabolas again today. And so... He gave some information. What is this information? Well that's exactly what's expected of us. Substitute 11 + for x in the function equation. "Let me write √3" is not the same as "Let me write 7 -√3". Isn't it? Here's what we'll do right away. We will understand. How was that possible? F some number, some number in F It has become equal to the number. So, that means... these, that is, the sum of these two Half of it gives r. So these two are R. According to him, it was in a symmetrical position. We do this We understood that. So here we have information that is being conveyed subtly It's actually the R value. R, who is there too. The established figures and others ended up killing each other. 18/2 a Its 'r' is 9. How did I find E r? I was finding it with -b/2a and then I'll proceed. expected. Is it ok? So much like that Ugly, very strange things are being said. They even do things like this. Let f be m - 2e = - m, I mean, I've passed √3, and now there's another one like m. an unknown, okay, this is second-degree. If f something equals f something then this means Half of the sum of the two is m - 2 and 18 - the sum of m divided by 2 We'll say it takes us to R. Then r - b/ We'll say it's 2a. Oh, and of course, this emphasis too. Let me do it, you see. If he says fm if it is equal to ft So what can we actually understand from this? Look at this This time we can understand two things. Ya m t and t can also be equal. Well What does MT mean? F3 equals F3. F5 to F5 It is naturally equal. So this time two I need to examine that thing. Or that M and T The thing called is equal to each other or either again, half of the sum of M and T It is equal to R. Oh, sir, the two above... in the example they should be equal You haven't examined it. I haven't checked it because anyway... I also see this. 11 + √3 and 7 - √3 that they are not equal to each other. And right there Let's take a look at the second one right away. And like this... I need to write it as the second one below. Because m - 2 is equal to 18 - m The situation is 2m = 20. And here m10 is... It seems it's possible. Look, if it's m10, write it instead. f8 - 10 is equal to F8. Like this one. We should examine fiction in both ways. So they gave unknowns within the fin. but when you give something numerical The things written inside are already the same. I see that it's not there. So the man told me He said, "So F3 = F, what should it be?" 27. E How is F3 equivalent to F27? 2nd degree If R = 3 + 27/2, then it becomes. From here find the value of R and from there arrive at the result. We'll go. Now, the final point regarding the peak. Let me also mention this. Look the vertex of any parabola it is located right here on the parabola it can take or intersect the x-axis Let there be space. Look, that's the peak point then. Where is that? Think about it. Or if the apex is on the y-axis Let's consider the situation. So here's the thing Let's talk privately. Or first It could be like the example I gave. Ok. So it just happened to come across somewhere. it could be. Peak point 1st region, 2 3 4 but it also lies on the x-axis can get it. It is also located on the y-axis. can get it. Here we are at this peak x if it is on the axis or the apex If it is on the y-axis, their status is special. Let's examine it from that perspective. Now, right away Let's even note it down in red. Now the vertex of a parabola, peak if it is on the x-axis now the vertex of the x-axis on it Here's a rough estimate of what the graph looks like. it could be. Maybe that's true, maybe. Their arms will be pointing downwards. Him We don't know the ropes. Symbolic drawings I'm doing this so that we can visualize it in our minds. TO One thing that catches our attention in all these drawings There is something. Or the vertex of the x-axis The graph is tangent to the x-axis. It is possible. Tangent. The graph is tangent to the x-axis. It is possible. So under what conditions are we Our parabola was tangent to the x-axis, right? If the delta is equal to 0, look at the interpretation. She gave birth. So someone tells us, uh... How can I say it, giving something away? parabola give the equation and state that its delta is equal to 0 If he wants to say that, then actually it is this: He also constructs the sentence. Some parabola If the vertex is on the x-axis, then... Therefore, it must be tangent to the x-axis. The condition was that the delta should be 0. Also Let's talk about this. Off the screen Let's not overdo it. Let's also talk about this. If the vertex is on the y-axis if it is on the y-axis And look, there's something different here too. Let me explain this to you using this approach. Any point on the y-axis So what happens then? Now consider the y-axis a few examples of points on it I'll give it to you. Point 0a 5 is on the y-axis It is on it. 0a is above -3y. 0a He/She is over 40 years old. Look at the y-axis the abscissa of a point located on it It is 0. And now our peak point is... If it is above the axis, it automatically becomes a peak. The abscissa of the point is 0. Something to 0, to 0 For example, 7 to 0, 3 to 0, -2 to 0. So its r is 0. We also don't find E r as -b/ 2a. were we? Therefore, -b/2a = 0. For E -b/ 2a to be 0, b must be 0. It should be. B becomes 0. In fact, even, what if this 'b' is 0? We had already examined that situation in detail. In this case, our equation is ax² + c We said it would be in that form. So our function It will have a dual function. Look, every comment we make is actually... They support each other. So the vertex of a parabola is x. If it is on the axis, the graph is on the x-axis. It will be tangent. If the vertex of the parabola If the point is on the y-axis, then B is 0. It is possible. So its R value becomes 0. B being 0 It is related to B being 0. Yes. Now, look at the parabola in particular. on this Cartesian coordinate plane... This is the graph they give us. Parabola inside, outside, square, triangle, placing shapes like rectangles They do ask questions. Now too much I have something important to say. Like him how we should approach the questions Let's have a talk. Now from this I'm talking about it. Look, symbolically again... Let's try to visualize a parabola. What if I said, "Let's have a parabola like this..."? The type, shape, and so on of a parabola... It doesn't matter. Inside, inside I took it and placed it like a rectangle. Let's say. Or it comes up so often which are question models. And ÖSYM too, very much so. He likes this. It's like, suddenly it's obvious... questions like these from time to time They ask. Now, in questions like these, there are two... We will pay attention to that. Rather We will first approach the question as follows. There are points that are on the same horizontal line. Really? Points on the same horizontal line. Let me write it briefly. You always do that Write it down neatly in your notebook. Same horizontal in the direction points. These points will take us to the goal. So what does that mean, in the same horizontal direction? points? Of course, I'm here now... I didn't say it, but the question will say it. This is one rectangle. The opposite sides of the rectangle The edges are parallel to each other, right? One side, as you can already see, looks like this. Let's do it. One side of the x-axis on. Then this one is on the other side. It will be parallel to this. Therefore, this x parallel to the axis. A rectangle like this Let me make it a little more prominent. Now here are two that are on the same horizontal line. There are 10 points. Look at these two points. in the same horizontal direction. Therefore, this We will use it in the question as follows. Look at these two. The ordinate of the point is the same. These two points The ordinate is the same. For example, if it were 2, this If it were 10, we would have understood this. we were. F2 and F10 are equivalent. So, where do we connect this to? Again to the peak. We will say, since it's F2... F10 and R are equal, what we call R. It is exactly halfway between 2 and 10. 12/2 It is 6. So, this is the peak point. In that case, this would be equivalent to 6. We'll say it's coming. So that's exactly the symmetry. The axis divides the shape in half. we can say. Because with this length... lengths being equal We know it's necessary. Of course they It's affecting these areas too. From there to the conclusion. We'll go. This is the first move. Look, it's the same. Are there any points that are in the horizontal direction? Control is very important. In fact, it's like this. for other types of functions as well It is valid. You noticed that their ordinates are the same. There are points that are problematic. By using those places You try to reach a conclusion. And the second one is this. Make this a reflex. You need to bring it. Let me draw it like this. Hop. For example Let me visualize a parabola. This And let's put a rectangle inside too. Let me place it. That thing I drew in blue Let it be a rectangle. Just by looking at the shape. Look, he looks He doesn't look at us as the target in solving this problem. There is a point where it will be carried. At that point with the given function geometric drawn inside with its graph It is the point where the shapes touch each other. I'll repeat the sentence. With a graph The graph of the function is also given in the question. geometric shapes touching each other point. What point is that here? point? That's the point. This point leads us that will lead to the goal in solving the problem That's the point. Now, first of all, a few similar ones. Let's do a simulation like this. Which point What will take us to the goal? First, give it a try Let's learn how to determine it. Then what? Does it carry? So, okay, that point is acceptable, but how? Does it carry? What are we going to do? He later let's talk. I drew a shape like this Think about it. Forget about the parabola. Parabola I'm saying it doesn't matter whether it happens or not. Yeah. In general, every in this format the move that will take us to the goal in the question They are the same. I included one more thing in this another corner so that it touches this Let me create a geometric shape. Here you go. Imagine I've formed a triangle. More As soon as you look at the figure, look at that function. the geometric shape I drew with its graph The point where they touch is not this point. Really? This is the point that will take us to our goal. we say. Another animation Let me do it. Let me take it this way. Here, take it. Let's go through the origin. Such a Let's say it's a parabola. Here's one Imagine I placed a rectangle. At work It can be a rectangle, it can be a square, it makes no difference. It doesn't. I looked at the picture. Inside the graph the objects I drew touching each other point. These points will take me to my goal. How do I carry it now? Let's talk. A lot There's a key thing, you know. Suddenly I'm telling you. Any point is If it is on the graph of the function the coordinates of the point are the function It satisfies the equation. For example, this second one I'm talking about the graph I drew. This Let's say the equation was x³, then x³ and this If the abscissa of the point were 2, then its ordinate would be ex. I wrote 2. 2 cubed is 8. If If its abscissa were 2. But of course, now... In the questions, we ask whether it's 2, 8, or 10, etc. We will never know. We will revive that point. We will say this: We will say this If the abscissa of the point were equal to t... I wrote t instead of x for the ordinate. T-cube It would be. This point has only one unknown. We will express it as follows. Believe me, the rest is a lot. The questions are solved easily. For example For example, the general equation of the first one I gave... Let's say this happens. X - 5 in parentheses Let it be the square of... Because someone like him It must have an equation, right? X It must be a perfect square tangent to its axis. E o Therefore, at the abscissa of this point, I say t If I were to say "let it be," the ordinate would be the equation. I wrote it down at its value. T becomes the square of 5. So our point is t, which is t - 5 squared. I'd say they'd be equal. How about the rest? Are they connecting? He tells us, "Here he goes..." Since its area is this, either it already... edge lengths of a single letter expressed as the only unknown of its kind Once you do that, the question is easily solved. Or It gives the surroundings or some other information. It gives. Here's an animation. Let's do it. Let's say this is y = 4x²e Let's say it has a graph. This is the point If I let the abscissa be t, then the ordinate is 4t² It is possible. The coordinates of the point are also 4t² It is possible. My important caveat here is this. One the period when I said bring the point to life We'll always do the height. Especially this and Look at questions like these from grades 11 to 12. The past is a completely different story between us now. We are also laying the groundwork for these topics. We will be exposed to derivatives very frequently in the future. to these. Learning moves is important. in mathematics. Here's a point. When I said let's revive it, he went and said, "Oh, that one..." I don't mean that point A or B should be the same. HE Add the second unknown. That point is this the graph of a function on it based on the letter you gave to its abscissa expressing the ordinate as I'm talking about it. By creating a point animation We will reach a conclusion. The point their coordinates are also in the same letter type We write. While doing this, we also use a dot plot. because it is on it, its equation provides. We make use of this information. Yes. Now, let's also touch upon this point. If you've noticed so far, we If the function graph is given, then most of them... We talked about this before. Or the graph If the equation is not given, then the graph of that equation will be shown. We started with how to draw. Then the graph There are various points where it passes through. I said. That's the point that makes it possible. HE We said we would make use of the information. Now Let's talk about this. Here is one The graph of the second function gives us Let them give it. Tell us to write the equation Let them ask. Let's say, for example, writing equations. Chart I'm talking about writing equations when you have the ability to do so. There's actually something very nice here too. Here. We only do the job like this second do it based on degree functions Let's not give up. Look, mathematics is a... Understand that it is a whole. It's like now... My functions here are of degree 8. The specifications I will give for it are unlike any other. Don't think of it as if it's not being used anywhere. Forget about the secondary grades. General simpler, lower grade Let's start with something. Even yesterday's In our lesson, we will be discussing linear functions. We had dealt with these as well. If you remember. Let me give you another linear function. Let's say this is number 3. Come here Let it be 15. Let this be f(x). This Let's try to write the equation. Our problem is finding the equation of this function. to write. Of course, when it's a linear function... We actually have a lot of options. It was happening. I was also starting from the slope. I was able to do it thanks to my analytical knowledge. Using the function information, f(x) = ax + b let it be. There are two points where it passes. F3 0, F0 15. Equation group A also includes B. I could find it. But now all polynomials for equations, all polynomials to be able to write the equations of functions Let's talk about the most general way to do this. Look, You know the concept of the point where it intersects the x-axis? This is a very crucial point. X-axis The point where it cuts is actually my function. It was the root, wasn't it? I claim the following: Can't I? What if one of its roots is 3? f(x) must contain x - 3. There must be a multiplier. Well If f(x) is x - 3 Yes, indeed, when I enter 3 for x, the result is 0. It is possible. Of course, there is only one thing to pay attention to here. There is something. Now, if f(x)x - 3, then x is 3. It cuts. It is also √3. Okay, let's go over there too. If there is an 8, then put an 'x' at the beginning of it. If the number 8 appears, it will cause a change in that root. That's impossible. So where did that thing come from? How will I know? Are there 8s there? -4 or 2? Is there? Are there 10? Here's what I saw at first glance We don't know. A letter there right away We will write. Here is that letter, too. How would we usually express it? A With the letter ax, that is, when I distribute it, ax - 3a It will be something that continues in the same way. So I wrote down the leading coefficient. Then that The question asks us to find 'a'. He definitely gives something extra, you know, here What additional information did I provide? Look The point where F0 intersects y is F0's 15. I knew that. So immediately f(0) I will write 0 instead of x in the function f(x). 0 - I'd say 3 = 15. From here, a's I would find its value to be -5. Therefore, the function f(x) I arrived at the equation. Here's -5 x -3. Look. When I want to write the equation like this, x The points where it intersects the axis play a key role. He's playing. The point where it intersects the x-axis is its point. It is the root. If 3 is a root, then it contains The question is: does it have a factor of x - 3? I solved it. Okay, so now what about the second degree? Let me give an example of a function. Or your inner self Relax. The technique is the same technique, look. The technique is the same technique. forehead. Like this Let's do a dramatization. Bring it immediately. Let's make this number 2. Come on. How many is this place? let it be? Let it be 8. This is equivalent to 32. Let him come. Our question is the same. The equation of FX Write it down. We write that f(x)'s We can easily write the equation. Why? Because I know its roots, you know. Look, a root Here's the other root, too. Look at that root. Isn't the concept incredibly important? X The point where it intersects the axis is the root. So much That's useful in the question. And I say: f(x) = will contain a factor of x - 2. There will also be a multiplier of x - 8. Of course and at the beginning, the famous leading coefficient. Because Maybe there's a 3 there, maybe there's a 40 I don't know. So how do I find A? Additional information Because it's necessary. Here's some additional information for you. given. In this question, we again use f(0). I gave it. I see that f(0). Every time I see x in the function equation I write 0 on the ground. a x 0 -2 -2 0 -8 e -8 So, it's 32. From this, the value of a is 2. I found it as... Therefore, f(x)'s the equation as 2 x x - 2 x x - 8 I found it. So, as long as we know its roots. general polynomial if we know its roots with the logic of writing function equations I can easily solve the question. Alright Let's do something like this right away. Under Let me give another example. Look x Let it be tangent to the axis. Okay then. Let it come Let it pass like this. Let's say it's a tangent at exactly 3 o'clock. let it be. Let me give it as 36 here as well. F in This is the graph. I'm telling you to write the equation. TO Let's write it down. I would do it using the same logic again. f(x) e= of course, the following information comes into play here enters. The graph is tangent to the x-axis. In that case What's there? It has a double root system. So x - It's not 3, the graph is tangent to the x-axis. For this, there is the square of x - 3. To his head too I'll immediately put a cross ('a'). To the thing I looked. F0 is equal to 36. I saw. I substituted 0 for x in the equation. 0 -3 -3 -3 squared, that's it. 9a 36. A 4 I found it as... So the equation of f(x) is 4 I found it to be x² - 3. So, if it intersects the x-axis at two different points Even if it's tangent to the x-axis, the logic is the same. What's the difference? If it cuts through X, then it's the only one. double root if tangent It's happening. Look, the root is very important, but look... The following should immediately come to your mind. my teacher. if there is no root, that is, intersecting the x-axis He didn't have to. Or the graph of a parabola. TO If it doesn't intersect the x-axis, there's no root in the middle. How am I going to create this animation? Let's give an example of that right away. Like this Let's give an example. Let's say our function is... Let it look like this. Come on Let him come. Look, it's not intersecting the x-axis right now. Okay, don't cut it. Of course, this time there are other additions. I will provide information. A hill like this Let me give you some information on this point. Let's say 2 let it be. Let's make this a 5 as well. At work I'm shaking it. Let's make this one a 10 as well. A lot I wrote it without thinking. Bad numbers It might come out. This is our FX. Your equation We are looking. What can I do now? I'm thinking, what if x doesn't cut it? I don't know any general techniques. And then Here's some kind of translation in the functions. Let me make use of my knowledge. I'll even change that number 10 while I'm solving it. To avoid any misunderstanding. I'll make it a 9. Let me imagine this now. Look If the graph I provided isn't like that... exactly like before, exactly x If it were tangent to the axis, to the x-axis If it had been tangent, what would it be now or then? would it be? Let's make this a little more rule-abiding. Let me try to show you. Let me clean it. Hi. Let's do this. The graph I provided looks like this. if, If it were tangent to the x-axis, it would be related to it. It was easy to write a general equation. isn't it? So actually it's x - 2 squared I would say yes. Of course, there's also the matter of adding... I would put a multiplication sign (a). But this is me in the question, the information I gave at the beginning of the question It's not the equation of the original function. Because it wasn't tangent to the x-axis. So what about me? Imagine it as if it were tangent to the x-axis. To be able to do this, how many times do we actually use this function? How did I manage to lower the unit like this? Or that I've shown in red, it's dashed. How many units higher did I raise what I drew? If I move it, what will I get above? And here's number 5. unit. It is equal to the length of that place. TO Upward translation in functions When we do that, how many units are there actually? If you postpone it, it will be a plus number at the end. because it was added, this type of when writing the equation of the function I can benefit from this. If it were tangent, then the square of a x - 2 would be It would be, but it's 5 units higher. its shifted form. Or the numbers 2 and 5 Look, the peak of this isn't 2 to 5. was it? Those 2 I gave at the beginning of the question He came here. Didn't those five come here too? Do you know where this is leading us? This This has led us to a general rule. It actually happens. In fact, let's look at this brand new one below. Let me open a page. Now in those books the general you see constantly I'll give you all the rules. Here we go They say, "Oh dear, there's only one function." There is. x-axis in two different places If I'm cutting them, it's on the condition that they are x1 x2. The general equation of the function is a x - x1 x - It is x2. Of course, x1 and x2 are numbers. You know, he recommends it. Natural single-layered roots there there are. No acceptance, that's it. Oh my goodness, this x if it aligns with the x-axis Let's call this place x1. In fact, let's make use of this ourselves. When tangent to the x-axis, the tangency The place where it formed is also the peak. Wasn't it possible? RY 0 automatically x It's on the axis. So then actually r Its value fully supports our double-layered roots. He is doing. And then I would say that a x - r's It's a square in parentheses. Yes. No, this I'll take it. I'll move it up a little bit. Or I'll push it down, it doesn't matter. Well If I know the peak point, it's here. Let's hope for the next one. You know, in the animation above... if it were exactly tangent to the x-axis x - It was becoming the square of the rin, right? Well, this is it. It is a unit shifted state. It's going upwards. It could have been postponed. Let's show it like this. Downward translational movement also It was possible. Here's what's in these books: It passes. two points where the x-axis intersects If I knew the answer, that's how I would write the equation. X I would write the TS equation like this. Peak R is K and I know the vertex. the equation of a second-degree function That's how I write. Look, these are the rules. Don't memorize it like that. This gives you nothing It won't get up. I'll explain the reasons. I am working. In fact, we generally consider the situation... Let's say. Let's invest in the future again. E is a third-degree polynomial function Let me give it. Let's say it's a third-degree polynomial. Let it be a function. Come on, you come here. Come here go. Let this be 0 -2. Whether it's 3 or 5. Let's make this one 30. Of course, the problem It states at the beginning, right below, vertically. real numbers in the coordinate plane 3rd degree f(x) defined on the set The graph of the function is given. Find the equation. And I'll find it, I mean Whatever I did in the parabola, I did in the linear plane. I'll do the same thing in this case as I did in previous cases. I start with f(x) e=. Your roots I see. Here we have this, here we have this, here we have this. Since there is an x ​​- 2 inside, there is an x ​​+ 2 It has a multiplier. For x to be 3, For the value to be 5, the factor must be x - 5. I'd immediately put a cross at the beginning. To find A, from a piece of information I will benefit from it. That is f(0). f(0) I saw. E f(0) everywhere I see 0 When written down, a x 2 x -3 x -5 = 30. At work From this, I find the value of 'a' to be 2. And I say that f(x) = 2 times x + 2, x - 3 And I can say that x is -5. See, the logic is the same. So in the future Polynomial functions, these are them. We'll talk, let's talk now. One Let me give another example. Something like this let it be. Let's say our graph looks like this. let it be. Let's have a chat here. let it happen. Just cut it off. Let this be -1. Let this be +2. Let's add +3 here as well. 4th degree polynomial function I provided the graph. Okay then. Come on, let's go over there. Let it be 6. I'm writing the general equation. Look, if f(x) = one root is -1 then x + It will be a multiplier of 1. If one root is 2 and that is a double root How did you figure that out? Double-layered root x If it is tangent to the axis, then x - 2 is inside it. It will be a square. If one root of 3 exists, and If it's single-story, then it will also be x - 3. Of course, you should immediately add a multiplication sign (a) at the beginning of the 1. I will put it there. To find A, use f(0) = 6 Use your knowledge to solve A as well. I will have found it. 4 degrees 3 1 2nd difference It doesn't. The solution is the same in all of them. You must have noticed that I arrived. But of course which is our main task today, number 2. Because there are degrees, that's what I said there. like what is constantly given to us These are the features. Yes. Now Let's talk about this as well. You know, that one until now always one parabola We've examined it. And then there are us with these... We will be dealing with them. Two parabolas will give. He will ask us this question. These Do they intersect? The graphs of these two are tangent. Is it possible? Or completely separate from each other. Are they independent? Then a parabola It will give a truth. Do they intersect here? Will it be tangent? Uh, in two different places? Will it cut or not? These They will ask us to investigate. There are many here too. Let me tell you something very important. Look Again, this is not unique to this parabola. Any two functions, any two the function, that is, these two linear It becomes a function. One of the radical functions It is possible. The other one, I don't know, might be a parabola. One is a 5th-degree polynomial function It is possible. The other one becomes a constant function. None it doesn't matter. Any two functions Do they intersect or not? This The way to understand is to share the equations. It involves solving it that way. Because, you know, us at intersection points if they intersect common point of intersection points We know that. Both of those functions common point. The graph of a function E If it passes through a point, then that point is also... that it satisfies the equation of the function We know. That's when we're looking for such points. which includes both the first function given It must satisfy the equation of the second function. So there must be a common root. Finding the common root I will also find a common equation solution for that. Again, something like this related to parabolas... I won't give an example. Look, it's always the same. Understand that I did that. Something like this Let me give it. Let's say with the line y = 2x - 3 The line y = x + 7. Do these intersect? Don't they intersect? If they intersect, then they intersect. Find the coordinates of the point. Question This. It's very easy to understand this. TO I solve equations collaboratively. Here are two There is something. That's why he puts the two one on top of the other. I'll get someone eliminated, but... Usually, this is done, or both are "y". Since this is also y and this is also y, the parts with x... common equation while the solution is being made. So from here x is 10 I found it as... Look, there's a real number. I found number 10. This means... So this The two intersect. These two They were intersecting. So, it's linear like this... Let him do it. Oh, and let the other one be this one too. One Since I found the value of x, I have one of these. I realized they intersected at the ground. Now this Pay attention to the sentence. Our joint solution what we finally found The x value is the point where they intersect It is the apse. Look, what we call a point... I always say there's also the ordinate. And this How do we find the ordinate of a point? TO This satisfied the equation for both of them. If you want, you can replace x with 10 here. in summer. You can replace x here if you wish. Write 10. The result you will find in both They are the same. Look, I wrote it down. 20 - 3 17 10 + 7 17. Here's where they intersect: 10 to 17 That was the point. Or this thing I did for two truths when a parabola is involved I'll do it, or as I said, something completely different. I also do it when there are functions involved. Let me give you an example. Look Let's understand it easily. For example, x squared with the parabola x squared parabola 2x + 3 actually. I wonder if these two will intersect? Don't they intersect? It's easy to understand. Immediately their equations Let's equalize them. A higher-order equation output. All the data to the same side Let's throw it away. Look, put all the data on the same side. After adding it, I obtained this equation: We call it a common solution equation. Common solution the equation. And do these two intersect? Don't they intersect? This is entirely within this framework. the real roots of the equation I showed This is a determining factor depending on whether or not it exists. common solution equation. Equations I equalized them. I threw them all to the same side. Now, if the real root of this equation is either... If they have roots, then they intersect. Or I'd say they don't intersect. TO Can the equation be solved? Yes, it can be solved. It seems so. It can be factored. output. From here I found 2 x values. Wow, I found two x values! In that case What does this mean? So the question is... The functions I provided are used at two different points. They were intersecting. For example, here's one One was a parabola, the other was a straight line. They intersected at two different points. The abscissa of one of the points where they intersect One has an abscissa of -1, and the other has an abscissa of 3. Look When we solve the equation we found where the values ​​we found intersect Abscissas of the points. Those points How do we find their ordinates? Any We will write it in place of one of them. So these three either take it and write it in its place, or take it Write it in place of this. The same result You'll find it. And of course, those three are in return. We'll write it here. The value we found was 3. 9. The same applies to -1. Whether...or Write it here, or write it here. At work We found the points where they intersect. For example, this parabola is straight in two different directions. They intersected at that point. That's all from here Many different types of questions can actually be generated. I might ask, how can I connect this to analytics? If I wanted to. Here are the points where these intersections occur. Let's label them A and B. A and B What is the distance between the points? I would say it's a unit. You know, between two points I attribute it to distance. Points A and B Find the coordinates of the midpoint. I say. It will be right in the middle. At work Let's call that C. C's I can ask for the coordinates. And I can say this. I'd like to know the length of this beam. The beam refers to this. So there is the length of the EU. So, you could refer to it as a beam. Uh, I can ask. And then... I would say, what if A and B are different from C? symmetrical, or the midpoint, relative to C. These intersect at two symmetrical points. I can then generate a type of question from that. So the ending is open-ended, but how we're going about it is up to you. You will understand. Find a solution together. Everything It depends on this equation. Of course, from here You should also consider this immediately. Yes, sir. That's what you should say then. E is a parabola That's right, it's always two different points. They don't intersect. In this example, two numbers like 3 and -1 We found 2 values. But that's how the equation is. an equation is two equal parts It could also turn out to be a real root. Even the real root It might not even have come out, right? So that It depends on the equation at hand. If there too Let's talk about it. Here are the two of them. You equated the equations. Partner common solution of the solution equation Let's write the equation briefly. Common solution equation I would look at the discriminant. You know I'm talking about parabolas and straight lines, of course. Possible for a parabola and a straight line the situations are either at two different points They can intersect. Parabolas and lines are tangent to each other. it could be. In other words, the common solution equation Eventually, a single root appears. in the middle Honey, they don't necessarily have to intersect. These might never intersect. When can you obtain two different real roots? When are they related if Delta is greater than 0? Can there be two equal roots? What happens if Delta is equal to 0? Time, real roots, dead ends, and intersecting elements? If Delta is less than 0, whose delta is it? What do you mean? The common solution equation. Look, I've equalized the common solution. I threw it to the same side. To its delta I'll see. So, it's actually the same story, right? Wouldn't that be true for both parabolas? Two parabolas, look at two different points. They can intersect. Two parabolas are tangent to each other. It's possible, but they're not at all similar. Those who are not connected, and those who are not connected It could also be a parabola. What does it depend on? Again It depends on the delta of the common solution equation. Is it ok? So then, whether you want one a parabola requires a straight line, two parabolas We were asked about their positions relative to each other. let it be. I look at the common solution equation. definitely. Now, here I will specifically tell you... Let me ask you something important about that... Look, this is very important. Be sure to write this in your notebook. Take note. I'm telling you, it's been like this lately. You know, the things I'm saying are very solid. as selective question models You might encounter it. On two parabolas let's talk. Could two parabolas be like this? not tangent to each other, but tangentiality I don't mean that. It is as follows below: Let me show you. Could two parabolas form a single parabola? Can they intersect at that point? Just think about it. Just think about it. Single Can they intersect at that point? Yes They can intersect. Do you know how it happens? Let's say we have a function f(x). At work Let x² + 7x - 3. Let g(x) be the value. HE Let x² again. Look, both of them The leading coefficient is the same. So the highest I kept the graded terms the same. X² I said. I called this x squared. Even take Look at this. Let's say 4x². We can call this 4x². Let me start. Let 4x² - x + 11 be the equation. The question is this. Do these two intersect or not? Really? If they intersect, at how many points do they intersect? TO Let's see. I look at the common solution equation. I'll buy it right away. Their equations I would put them on the same level. 4x²e cancels each other out. I posted it here. 8x = 14. So I found one value for x. 7/4. Since I found one x, this... What does it mean? Here they are, all in one place. They were intersecting. The point where they intersect Its abscissa is 7/4. Let's not do it here anymore. I made up the ordinate of the function because... It takes away both the rule and the ordinate. Any We write it in place of one of them. So the result is... head coefficients or degrees and head two parabolas with the same coefficients Since both parabolas are second-degree, will be. Two with the same leading coefficients Parabolas can intersect at a single point. See, I spoke cautiously. You noticed Really? I said they could intersect at a single point. My teacher Why don't you say it clearly? Intersect to the pattern. I can't say. Why can't I say? Look Let me give another example. FX Let's say our function is this. x² + 3x + 7 let it be. Let our function g(x) be as follows. X² look x² x² beautiful. And what if it continues like that, 3x and so on? Let it be -4. Now, will these intersect? Don't they intersect? Think about it. Come on Let's equalize them. And this time when we equalize both x-squared and 3x terms They took each other away. E 7 = -4 remains. 7 with Are -4 equal to each other? No. Of course. not. So what does that mean? Then this The two don't intersect. So one of them is the other Like a postponed version of it. It already is If you check their C's, one of them 11 more than the other, one 11 more than the other It became the unit that was shifted upwards. HE in Let's formulate the most accurate sentence. We have two parabolas. These are the only ones head for them to intersect at a point The coefficients will be equal. This sentence Look it up in the summer. The leading coefficients are equal. The coefficients of the terms with X are also if they are different from each other These two parabolas intersect at a single point. What Isn't it just as important? So now In mathematics, questions are like this: elimination questions. It is constructed in reverse. I mean, around here... Since I explained the topic, f(x) and g(x) I give the rules and then ask. Now While solving the questions, I will do the following: you. Here are some unknowns in f(x). I will put it there. Those 4, those 7, and so on. I won't give it. Some unknowns in G(x) But I will say this sentence. I'm going to say... These only intersect at one point and I don't mean tangent either. Here you go too You will say that then they both have the same head The coefficient must be the same. Terms beginning with X The coefficients should also be different from each other. and then try to solve the question from there. You will work. Here are two parabolas, for example. Whether it's a parabola or a straight line. Here's what's possible. These are the circumstances. Who decides here? Is it happening? It becomes a common solution equation. Especially that part about the tangent... important. Look, those who are going from 11th to 12th grade. Where the concept of tangence exists, the derivative. that comes into play. But the parabola and this the tangent that was drawn, that is, saying tangent And the point we're making is this: it's true. If he is 99% of the work is done without derivatives. It's being resolved. How is it solved? Here's the partner Solved using the solution equation It's happening. Now look at these two. of a parabola or with a parabola the situations where the lines intersect or not also on the Cartesian coordinate plane We also encounter shape-related questions. They can bring them. Something like this I'm talking about it. For example, a graph like this Imagine what you saw. One parabola Let me draw it, let it be this. Another Let me draw another parabola. What is that? What let it be? To show it in this way Let me try. Let it come from here. Hop. Continue Let him go. Let's call this FX. Let its name be g(x). Another Let me do another animation. And so... let it be. a parabola with arms pointing downwards I drew it. I went and saw a parabola too. I drew it. Let this be FX. Let this be g(x). Another animation next to it. I will do it. Let this be one of them. The other one is this: let it be. Let it be FX. For example, let g(x) be the value. Now When we encounter these shapes, it's like this... These things happen to coincide perfectly without control. That's not our main problem. Look, partner. Are there any points? Focus on these. For example, the common point is this first one I drew... This is what we see. There is one of F and G. There is a point where they intersect. One common There were some points. So they have something in common It has a root. If this is 3, then it's F3. It's 0, G3 is also 0. This comment I can do it. For example, below 10 When I draw it, what condition do F and G satisfy? The point where they intersected the Y-axis was the same. So, that huge thing we have... This is what they actually want us to see. TO The place where they cut the Y. So then F0 It's equal to G0. The constant terms E, F0, and G0 are connected to each other. They were equal. Look at this shape I've drawn. This is the conclusion we will draw. Because y Give x a value of 0 at the point where it intersects. E f0 g for g think. G0 and 0 are both the same number. F0 g0'a equal. So the constant terms are equal. They expect us to say that. Look at the other one! These two have two roots in common. So they intersect at two different points. You could even say this: with f(x) Solving the equations of g(x) together and the same the common solution we obtained by throwing it to the side The equation's delta is greater than 0. From 0 It was so big that it was in two different places. It could be said that they intersect. Uh, most of them Time is like that, you know, the peak of a parabola. That's a very important point. The thing They let them use it. For example, the top of this Let this be the point. Let me draw one like this too. Let's think about it. This is the culmination of it. Oh well. Sir, they're fighting each other. two parabolas that intersect at points It existed. Okay, fine, what's the big deal? TO We are at the peaks of both F and G. We know the ways to reach it. Of course, this He can't talk about that thing this time. common root No, for example, the place where they cut off the Y. But they are not the same. In questions like these I'll just use these things. F too I can also find the peak of G. For example This is where FX's peak is. And this It's on the graph of G. Then F's The coordinates of the peak are G's. It satisfies the equation. The opposite is true. G's The coordinates of the peak are as follows: The coordinates of that point are F. Since it is on it, the equation of F I'd say it provides that. It seems very complicated. These types of questions may seem unusual at first glance, but... There must be at least one common root in those places, two common roots, the points where they intersect Y It is given as the same. Later each other at their peaks... They might be cutting. This and similar examples So, you notice things and then focus on them. They expect us to solve it. And yes, now look... Youngsters, let's delve a little deeper. Well when you pick up any book Right there, everywhere, on the right and left to subtle details you can't see Let's touch on it. Now, of course, this is subtle. When I mention these points, I always say the main thing is... Our aim is this. Yes, as of today. Our main problem is secondary. everything related to functions to learn, but in general in mathematics the topics are intertwined Something we learned today, you know... Those who went from 11th to 12th grade, it was 45 months later. see another topic while you're working on it You will use it. So we learned this today. HE This is now permanent information for us. At work Three months have passed since then. I forgot about this. Teacher, I keep forgetting. How to do this How will we prevent it? And there's WhatsApp too. My channel will be activated. Information We will keep hunting continuously. Even Let me give you a little hint right now. They will give you online screening tests from there. I will do that too. This is key information, that is. It's not about giving FX and then asking what F3 is. You know, you haven't forgotten that he was exposed to it so much. And because you stayed, here's the topic today. I also touched upon the narrative aspect. so thin There are dots. For example, look at this four of the parabola's orthogonal coordinate system For it to be located in that region, it is the opposite. The symbol would be two roots. Now I will do this. The information is still fresh in my mind as I explain it right now. you are also exposed to the heat, but not too much. You won't be stuck on these kinds of detailed questions. Where did you meet? At work It's becoming a university argument. They always say it, right? There was a question I'd never asked before. I hadn't seen it. Actually, it seems so... It is not included in most places. Here it is for you, the one you don't use regularly You're forgetting... Here is his too. We will prevent it. Look at this one Let's talk about it. Now I will talk about it here These things are very, very important. Much, much more Such profound information. You know how I say that? The concept of tangent is very important. Tangency a parabola and its tangent. Now Let's think about this first. Look at this Let the parabola be as follows. Let's choose a point. At that point Let's discuss the positions of the parabola. Look point and parabola. Our point is on the parabola. It can be found between their arms. Our point lies on the parabola. can get it. Or our point is the parabola. It can also be located outside. So, there are three possible options. There are several situations. Or in his arms either on their arms or outside of them. Now, let me ask you a question to answer. Let's say from any point... Is it always possible to draw a tet (then) line across a parabola? For example, a diagram can be drawn here. Isn't it obvious? So it's already on it Let's pass through here a little bit Let me enlarge it. One that will pass through there I draw 't's. If it's outside of E, sir. It can be drawn, that's all. Buy it even when you're outside. I draw. Let's even go from here too. There will also be a tangent somewhere on that side. Let's draw. Acceptance. What if it's inside? from this point, that is, to put it in this particular way Let's take a look. Or a point inside There is. I will pass through this point on the parabola I could never draw a tangent line. Look how this question might come up: I'll give you a point. A parabolic equation I'll give it, but the parabola equation is unknown. I put things there. I would say from this point and will pass through and be tangent to the parabola any correct form It is being drawn. You can understand this from this. Hmm. The point is located between the arms of the parabola. should take. That's the information part. APPLICATION I will show that part in the questions. So one a point inside the arms of a parabola What happens if it's in that part? For example, this setting You can. The point is on the parabola. What would have happened if that had been the case? if it were on it For example, let this be point A and this be point FX. fx e 'a' would become 'fa', right? His equation It provided. For example, if the abscissa is 3, then 3 to f3. It would be. E is the other point, and its abscissa is again a. So these are the same vertical line, but... Its ordinate is greater than this fa. That's what we used to say, right? So, right here... Let's say point t is the point. What you need to do The required interpretation is T > FA. On If that were the case, this would happen. On ours not. Somewhere above that. In more detail We will talk. Now Let's talk about this. Look, this is very, very important. I will give you information. I'll prove it. I had done it. Now this narrative part is more The fact that it takes a long time proves the same thing once again. I'd rather not, but I... these comments... Let me fix it. That's the proof I made. I'll send you the link to the video. He definitely Examine it. Is it possible? This is our problem. One parabola from a point outside, outside the perpendiculars of the tangents drawn from the point I'll say it again: the condition of intersection is the sentence. There's an extra point to this. Here it is Let me spell out the dot with a letter. Come on. This Let the coordinates of the point be x0 and y0. Or I would also call a b. I can also call M as N, though. ÖSYM also uses this language from time to time. That's why I also have a place for these unknowns. I'm giving it. Let the coordinates be x0 and y0. From here, to our parabola, and that one too... Let's reiterate that it is a parabola. ax² + bx + c. Now, this is drawn Let me draw two tangents. Someone should just say, "Okay, let's do this." Let the other one be this. Now, for these two to intersect perpendicularly... Our applicable condition is as follows. Parabola delta equal -1 - 4 x a x y0. Look, right now it's so important and so I'm describing a different feature to you. right now. Like I said, it's a bit like us. We're delving deep into this matter right now. Well For those of you who are aiming for the top, For those aiming for a degree. Now delta Whose delta? The delta of the parabola. A What I'm referring to is the leading coefficient of the parabola. What is this thing I call Y0? What I call Y0 is... That's the point outside. The question gives us The ordinate of the given point. Here's the link to the video proving this. I'll throw it away. That's the most common characteristic. At work This is how books present it to us. You will definitely be exposed to this. This You won't be exposed to its general state very often. You know You don't see this in many places, to be honest. And they say to us, "If..." tangents drawn to a parabola from the origin If they intersect perpendicularly, it forms a parabola like this. Apparently, there was. This is what it originally was. We already know that our origin is right here. That is, tangents from point 0 to 0. when we draw We got it! If these two intersect perpendicularly this as a direct rule for us They give it and say, "Here is its delta -1." equals. Of course, its delta is equal to -1. It is possible. The reason is because of what I've explained here. The thing is, what I originally gave above is natural. a result. Look, the main formula is above. The feature I gave with green. Is it ok? So what happens when it becomes the original? We We know that it is actually an external point What difference does it make whether this is the origin or not? General characteristics are listed above. E -1 - 4 x a x y 0. That is, tet, that is, the point being used The ordinate, or the ordinate is 0. And since this is 0... For this, on the right side, the effect of 0 That area is going down, but its delta is equal to -1. It's happening. Then a parabola the perpendiculars of the tangents drawn from the origin For the parabola to intersect, its delta must be -1. They're going to be equal. This is the first one. Latter Let's present it like this as well. Let's give it like this. A parabola drawn from the points where it intersects the x-axis I'm repeating the sentence about tangents. X from the points where it intersects the axis When I draw tangents to the parabola If these tangents intersect perpendicularly, then they are perpendicular. if they intersect This time, the delta of the parabola is 1. equals. I will tell you the reason for this. I will send proof. The reason is actually Those of you who are going from 11th to 12th grade... actually the kind you can understand We'll dig around a bit to prove it, dear. graduates tangent tangent where right now How are you going to connect the dots? To the derivative. Derivative connecting two that actually intersect later slopes for perpendicular intersection of lines Saying the product is -1 and the result You can go. But what I said in the questions because you will be exposed to it very frequently It's useful to know these two features. One The tangents drawn to the parabola from the origin are perpendicular. if they intersect The delta of that parabola is -1. Parabola x drawn from the points where it intersects the axis If the tangents intersect perpendicularly, then... The delta of a parabola is equal to 1. But in general any other than a parabola from one point, from any point, that's it. If the tangents drawn intersect perpendicularly, then... The general formula, our main formula, is this. Yes. Now, let's look at this detail as well. let's talk. Think about parabolas. Let's imagine we have a parabola like this. Let's say her arms are pointing downwards. Let this be 0. Your roots I see. Let this be the function f(x). FX. Doesn't this make one wonder...? Actually? Look, by painting this area... Let me show you. There's one I painted yellow over there. There is a region. So, the graph of that parabola is x A strange region situated between its axes. How can we find the area of ​​that region? Is there a way? Look, this happened in time. He thought, you know, people from 11 to 12 Those who passed by. I'm telling you this for your sake. At work It's a strange area. Initially, it was like this: Let me give you a warning. Don't ever do that. semicircle, resembles a circle. No, this parabola. A parabola is one thing, a circle is another. So our geometry knowledge is for this region That's not enough to find its location. So, square, It's a rectangle, not a circle. What's this? One If it's just a straight line. That in between How do I find the area of ​​the region? At work areas of such strange regions branch of mathematics concerned with finding It is also integral. We will also discuss the reasons behind integrals. But you can integrate without knowing the inner integral. I painted this strange thing without knowing it. There is a way to find the area of ​​the region. Let's write down that route right away. The area of ​​this region How can I express it? Shaded area Let me tell you. shaded area equals delta. Whose delta, of course? That's what I mean, right? The parabola equation. Delta is delta/ 6a²aredir. It's a very important feature. Questions while solving it, you reach a conclusion in a very, very short time. It allows you to go. Delta ö delta/ 6a². Ok. What does it give? The one I showed in yellow the area of ​​the strange region. Even more so. In fact, look, a parabola and a Think clearly. Is it ok? In fact, two of them Think of a parabola. Among these the remaining areas formed in between It's the same when finding their areas. We take advantage of the feature. But but In what I provided above, Delta FX... It was the delta. Below We benefit from the delta of the common solution. Remember, it's a collaborative solution. Because, you know, there There are two functions. Their We find the one in between. The common solution Let me write it briefly so that we can understand it. from the language. This is its delta. Here However, there is already one that is given directly. There was a parabola. There, the delta of f(x) is this The following formula applies: Here, the common solution is to take both of them. We made them equal. What we obtained the common solution equation of the equation we were saying. That's what we called that equation. At work Starting from its delta, the delta We'll say delta / 6a squared. Of course What does the 'a' here mean? A always represents this He is doing. My main coefficient. Ax² + bx + c. ax² + bx + c. These are all key factors. Partner the leading coefficient of the solution equation We're going to use it. It's also further down the road. We'll be exposed to these a lot, but as I said... For example, we don't know integrals today. but especially that first one I gave, this area was asked. We'll find it. Then regarding the field Let's compress one more feature right away. in between. Now imagine a parabola, that's the best one. Let me give you the basics. Let's call this ax². So it passes through the origin. At point 0a 0 The vertex on the x-axis is the same. Where in time is this happening? This is exactly 0a 0 That's the point. Now on this Let's take any point. Any I got the dot. Provided that a peak point is used The top management will be involved. There was a dot on it too. Like this I formed a rectangle. Look. Even Let me spell it out. Let's say OABC rectangle. That rectangle is also one of our parabola. He split the arm like this. Two regions were formed. there. Even those regions are different. Let me show you with colors. Look, one of them is this The area I painted yellow. Resembling a triangle but the non-triangular region. And the other one... that I painted with light blue and blue area. That arm of our parabola is gone. He divided the place. Those areas always a valid relationship regarding the ratio there are. Always the bottom part If the area is S in square units, then the area of ​​the top is... It will be twice as much. Look, when? This time. X²eli something peak point provided it is used. Look, that's the hill. point. Without using the peak point We cannot make such a comment. Top point. And also any on it You got the point. You created it. Like this You'll say this place is a rectangle. If the area is S square units, then the interior part is each It is twice as much in terms of time. As graduates know, in integration, area... These questions keep popping up all the time. This is an important feature that has been introduced. Dear Those of you who are going from 11th to 12th grade, this is yours now. This is common knowledge. Look, you threw it away. to memory. Of course, the reason for this is as follows: I can't tell you right now, but during the period... This and much, much more of it progressing in this way. We will use it with you in the versions. We'll keep talking, to be more precise. Well We will say, "Look, this is why..." That's how it happens. That's why it's like this. It's happening. It's not like I said it and it happened. You But you won't accept that. Like, when you're really stuck... You'll ask, "Professor, how did that happen?" From where So, it's done, I mean, okay, Sez, we accept it. Let's implement it, there is a reason for this. The reason, and those reasons are difficult reasons. not. You know, like, right at the beginning of the day... I said it. Even in this camp promotion I said it. I'm speaking generally. We may not understand everything. I'm everything I can't explain. something that exists in mathematics and that I have I have proof of most of the things I've implemented. I can't keep up with most of them. Or rather, it's like this most of what we do within the curriculum I know some things, but what I don't know is... I don't understand, I still can't make sense of it. I have many things I'm working on. Its accuracy I know, I know it's being used, but... My advice to you is to at least take a look once. Just touch something. For example Let me give this as an example. SA 2S There was a rule like that. Yes there is. Look at the At least once yourself You may not be able to, but someone else's something that explains why it was done Watch it. Ask a teacher about a book. If you have a friend who knows, ask them. Just take a look once and see for yourself... Witness it. Later that information became clear and It becomes permanent. Look at this We also memorized this feature. S to 2s. What did we discuss above? Delta or delta/ We talked about 6A squares. On top of it We discussed the concept of proximity. Delta's 1 We discussed the situations where it becomes -1. Now let's delve a little deeper. Let's see. One of its most beautiful features I will tell you about the parabola. There too Let's do something like this. A star Let's put it there. These are the new generation of questions. It happens, you know. Strange shapes It is being drawn. Even the coordinate system It doesn't get scratched. No one is throwing the ball, the basketball You see a student throwing something. There isn't one The shape of the bridge is like this, a parabola. They give it like that. over there and stuff like that The feature I'm going to talk about is very, very useful for us. It might be useful. The characteristic there is this. Comfortable Let me explain. Look, here's the proof. Let's do that too. We'll figure it out ourselves. You know What's going on there, what's happening? Like this Let it be a parabola. This is the peak Let the vertex be r and k. And now we are the general equation of this function We could write it like this, right? peak If I know, it's in the form a x - r² + k I could write. Our concern is this: we will ask ourselves, "What if...?" I wonder Let's establish a point now. That point I'll show it in red too. Let it be this. Top I also highlighted the point in blue. With the peak point, move a little further to the right. My registration was correct. To the left as well We could have slipped. I've set a point. Let's say the name of this point is A. let it be. Even the coordinates of A. Let's try to express it this way. Like this Let's show it. Let's show you. Now, the abscissa of A and the vertex... Let's compare their abscissas. You know, mine... In my animation, the abscissa of A is the apex. It is larger than the abscissa of the point. Because the hill It came slightly to the right of that point. It happened. Let's even say we turn this place into another one. Let me spell it out. Let this be B. B let it be. How many units to the right should we have shifted? At work Imagine I referred to it as the T unit. He too Let me show it in green. This length T Let it be one unit. This length is equal to T. let it be. Of course, in this case point A What happens to the abscess? E is equal to T more than R. Okay, right? So it equals R + T. If the abscissa of the apex is 3, then this is 5 Moving one unit to the right becomes 3 + 5 = 8. That animation I did. My problem is figuring out what's going to come here. And of course, the statement that will come there It's not too difficult to find either. Why? Because a The point is on our function. Then it will satisfy the equation. So f too I will find r + t. The abscissa of A I will write x wherever I see it in the function. X I wrote R + T instead. - R squared + K It happened. So F is equivalent to R + T, which is: Inside, the cards were killing each other. AT² + It became K. Let's write it down. A x T² + K It happened. Now, what I'm really after... Now it's my turn to tell the story. My problem was this: Actually, in this question. And with yellow too. Let me show you. I wonder if the peak point is... a point located to the right of the unit with its ordinate between the ordinate of the vertex Can I establish a relationship? So actually this The length of AB is also expressed in terms of T. Can I? Because if we can do this We will have found this. From the peak point T when the unit moves away By how many units does the ordinate increase? Well The rise or fall in the ordinate... arms up and down accordingly It varies. And that's easy to say. Because of the EU that I've shown in yellow. Couldn't I actually find its length like this? From the ordinate of A to the ordinate of B I'll take it out. They are in the same vertical direction. If A's Y is 12, B's Y is 7, then 12 It is 7. We say they are five units that have a difference. TO The ordinate of A. It is equal to at² + K. The ordinate of E B is e. B and the peak are already the same. in that direction. Look, for example, at the peak. Here, point B is here. E B's its ordinate is equal to the ordinate of the vertex already. So it's actually equal to K. Like that isn't it? Therefore - K. A, look from there. The K's canceled each other out too. A x square They were equal. Who? That's the length between them. Alright What has this told us? He knows Are you? If we have a parabola, then... from that parabola, from that parabola, its abscissa is T unit to the right or left it doesn't matter. The ordinate of a point parabola Look at the leading coefficient. Because the a here What? Leading coefficient. Now with the leading coefficient how many units to the right or left are you shifting it? It is equal to the product of the square of this. The amount of rise was equal to at²are. Let's give an example right away and then look at it. Let's understand. Let's make this a little smaller. Let's say I'm sketching it roughly like this. At work Let's visualize a function f(x). This Let the function f(x) be x² - 4x, then what is the value of +? let it be? Let it be +5. This is right next to a hill. Let's find the point. Or the peak point -b/2a I substituted 2 for x in f(2) from 4/2. 4 -8 -4 5 plus 1, look, this is the peak. It was 2 to 1. Now, its abscissa is 3. Let's consider the point where there is an extra unit. Let's make this our peak point. That was it. I shifted it 3 units to the right. Think about it. If this is number 3, then it's the same as our above. What does our characteristic say? This length He says you'll find it comfortable. How do you find it? he says? You find it by using the square of A x 3. Our A was the leading coefficient. So he's saying that this place He says its length will be 9. And this place Its length is 9h and its ordinate is 1. E 9 When you go up one unit then He said that its ordinate is equal to 10. It's happening. Is testing it too difficult? No, very It's not difficult. Because the peak was 2. I Three units to the right, this point... Its abscissa would be equal to 5. So this place Something will happen to number 5. Here is that thing now Let's find one. To find something, press F5. You bring it to life. F5 went to 25, then to -20, then to +5? At work From this, we can see that the value of F5 is 10. You found it. Therefore, over there It matched the information we found. Of course you know, like this more often now while solving new questions on the book I will show it to this. They even call it quadratic increase. in the parabola. You will use the peak point. from the peak to the right or left The rise in ordinates started when he left. with the coefficient, with the main coefficient, it's here Let me repaint the inside again. Let it catch your attention. With the leading coefficient However many units you go, that's the distance you go It is equal to the product of the square of the unit. That's a good quality. Oh, sir, this... Is it impossible to solve questions without knowing the answers? It will be resolved. TO Why am I telling you this then? Shorter We can solve it in time. It seems very difficult. This is in solving some of the apparent problems. features and a couple of things like that There's more. We always use them like this when the time is right. I'll explain. Very useful to us It might be useful. Our AYT mathematics camp Our topic for the second day is parabolas. But Before we move on to the topic of parabolas, yesterday... What did we do on the first day? Like this Let me briefly remind you. Yesterday applications related to functions We had looked. Specifically, the function a function through its graphs Finding the roots, here's the graph. Identifying the roots, where they intersect the axes. finding the points, increasing or decreasing determining the ranges, if minimum and maximum points, if any concepts such as determination We had gone. Today we will be studying parabolas. So, a parabola is actually expressed like this... I can. Look, it specifies the function. There are parabolas. And also, those who don't mention it There are parabolas. But what we will deal with thing function denoting parabolas will be. In other words, the second with degree polynomial functions We will have tried. In yesterday's video, yesterday's I actually mentioned this in my previous explanation. Let me mention it again today. Look, We have two types of polynomial functions We must have very, very good control. The first of these first-degree polynomial functions, Also known as linear functions. The second one is what we will be talking about today. second-degree polynomial functions In other words, a parabola. Because these two functions other higher-order polynomials slightly different from functions It had certain characteristics. For example, linear what makes functions important There is a feature that others don't have. I told you. What did we call that feature? We mentioned the slope. The concept of slope is only for the 1st grade. A characteristic specific to certain ranks. At work In fact, we said it's so important that... linear functions. We do this in geometry under the name of analytical geometry once again as a huge unit We said we are working on it and we will continue to work on it. Now Today we will be working with parabolas. Exactly Like we did yesterday, even more than this in the explanation of most of the following topics as we did with this book Before I move on to the narrative sections, I What you need to know about this topic The fine details are now a blank page. I'll open it and explain it there. I always say Make sure you have a notebook with you. Make sure you have your pen with you. Here's what I'll talk about. features you've never heard of before Make sure to write it down in your notebook. Yes. Now, let's revisit the topic from the book. you know, its fine details to address then solve the questions Let's begin. Here is the information in the first section for each part. As usual, what is a parabola and what isn't? A definition for this has been provided. At work Let a and c be real numbers and a0 be a0. being different. Of course, from 0 They should have been different, second-degree relatives. Because it's defined from r to ry. Ax² + bx + Defined in the form of c. A second-degree the graph of a scientific function is a parabola It is called. Beautiful. What is the emphasis here? Look, it will be a secondary degree. So x It will contain a squared term. That's when he told her We call it a parabola. In the first example, here it is... A function is given. This is the graph Given that it represents a parabola, what is the value of a? that was asked. And now we're looking at x It needs to be something checkered. only one I see a term starting with 'x' here at first glance. If the exponent of this is going to be something like 1 x squared So, the exponent of that x term next to it... The written expression 4 + a equals 2. is necessary. So the value of a is -2. I can say that. In the second example, again, one The function has been given. The graph of this again It indicates a parabola. So x squared Something must happen. But here's x cubed Something caught my attention. And this When its degree is 3, we call it a parabola. We cannot say that. There is no such term It shouldn't be like that. Here's why it shouldn't be there. How do we ensure it? The coefficient at the beginning is 0 We ensure this by having it. So, more of 'a' Actually, for a - 7 to be 0, there must be different In other words, if a is 7 This place will be gone. So, this place is gone, okay? What remains indicates a parabola. Really? It doesn't specify that in its current form. Because There's something here with x to the power of 1. TO And on the side, x to the power of b + 3. Here's that again. For it to be a second-degree equation, b + 3 what we said is also equal to 2 We will say it is necessary. What b will receive from here the value should be -1 We will tell you. We found A, we found B. Now it's easy to hit them. Now side When we move on to that part, you know, that When visualizing the graph of a parabola, A's The task has been emphasized. It has been said that a is from 0. When older, arms raised; when smaller than 0... The arms are pointing downwards. We already do this just a moment ago in that short lecture part We had seen it. The applications are right below. The function has been assigned. These arms Since it is pointing upwards, e is the parabola. If the arms are pointing upwards, the head coefficient that is, the coefficient of the x-squared term is positive should be. a - 3 will be greater than 0. So A will be greater than 3. In the other, a point is placed at the vertex of the parabola. Emphasis has been placed on this. You know, the apex of the parabola. point Of course, that's a point. Now this Let's talk a little. It has an apse. One It has an ordinate. How to find the abscissa What did we say it was? -b/2a and the reason for it We spoke a little while ago. It intersects the x-axis It is exactly in the middle of the points. So actually We said it comes from x1 + x2/2. The second way is to take the derivative of the function. We said it means setting it equal to 0. Now this We found the abscissa of the point. Your ordinate How did we find it? This value we found We would immediately write it in the function. Now, actually, we can substitute it into the function. After performing the editing process, oh uh... what we call the peak Let R be the abscissa and K be the ordinate of the point. We call it. K also has a formula. There is. It wouldn't hurt if you knew, though. If you don't know, it will cause a lot of problems. I don't think so. I'll find R. in the function I'll write it instead. I'll get to the point. Now Let's see what has been given regarding this. 4. Here is the equation of the function in the example. It is given and it says the peak point Find it. Let's find the vertex of e. Now To find the peak, let's do this immediately. Let's do our dramatization. Formulas thoroughly so that it sticks in our minds every time Let me write. We were going to find R as -b/2a. Therefore, the value of r - look, our b - is... The coefficient of x is already -8. -8/a = 1. From 8/2, r = 4. I found. So right now, the peak... I found its abscissa. Its abscissa is 4. There's also this... It will be the ordinate. Here is that ordinate. To find it, we immediately need the function. We will substitute 4 for x in the equation. 4in² - 8 x 4 + 6 equals 16 - 32 - 16. 6 My additions were -10. Now the peak He found both the abscissa and the ordinate. We became. Of course, you can also tell us this: They might ask. Look at a peak when we are asked about the abscissa What is the axis of symmetry of this function? The question can also be phrased as follows: Because that's exactly it you know, from the apex of the apex drawn perpendicular to the x-axis We called that line the axis of symmetry. So x The line = 4 is our axis of symmetry. So we won't be expecting the same thing over and over. At work What is the abscissa of the apex? Please don't ask. You know, the axis of symmetry. Tell him to find it. Let us understand that from us What is desired is the r-value. So what about your ordinate? How was he asking? If the arms are up the vertex of a parabola that is correct the ordinate is the minimum value the function can take value. If the arms are pointing downwards, then it's the top. The ordinate of the point is the function's value. It represented the maximum value. And here a parabola with arms pointing upwards There is. How did you figure that out? X squared I looked at the sign. the sign of x squared Arms up because it's positive That's true. With the arms pointing upwards -10 is the ordinate of the vertex. the smallest possible value of the function It is valuable. Let's say the smallest one is d. TO That makes it -10. See, that's what mathematics is like. So there you go... The peak point is RK, which is K in the exam. It doesn't say "find, find R". This is mathematical. He expresses it as follows. The way to ask R This means what is the axis of symmetry? Here is K. If he wants to find it, the function... the greatest or smallest value it can take They ask, "What is value?" Now 5. The equation of the function is given in the question. In the equation, there is a variable like 'a' at the end. There are unknowns. And also the peak. A small piece of relevant information has also been provided. The abscissa of the apex is also not given, The ordinate is given. Okay, we'll do it quickly. First, can we find the abscissa? Let's take a look. Immediately I looked at the equation. Starting from equation E How did I find it by going out? -b/ 2a. So r The value of -b, which is the coefficient of x, is -4/ 2 a's. Our value for 'a' is 5. So, 4/10. In other words, 2/5 is actually r I found its value. So, that means the hill The abscissa of the point is 2/5. So, what if the ordinate is also 4 in the question? Hasn't it been given? Given. Peak I know. the pinnacle of his name It doesn't matter at all. To the function I know a point that belongs there. Well graph of different component functions I know a point that is located there. On it belongs to the function, that is, the function's I found a point that satisfies the equation. So, f(2/5) will be equal to 4. Immediately Wherever I see x in the function, I get 2/5. I am writing. 2/5 squared is the future. You know It continued as 5x². -4 pieces. X's I wrote 2/5 again instead. Also the 'a' at the end Adding it would make the answer 4. From here I will try to find the value of 'a'. 2/5 squared is 4/25. With the 5 above I simplified the 25 in the denominator. It came out as 4/5. -8/5 +a has a value of 4. 4/5 - 8/5 - 4/5. He too immediately to the other side + 4/5 I'll send it. From here, the value of A is 5x 4 plus 24. He also found A to be 24/5. We will be. So what was the question asking? What was the m? That was our r value. Therefore, the value of m = the value of e = r, which is also The result was equal to 2/5. The value of A is also 24/5 output. It's easy to collect the two now. Let's see what was asked in example 6. 6. The equation of the function is given in the example. X and at which points the Y-axes It was said, "Find what he cut." TO This isn't something unique to parabolas, anyway. The x-axis of any function I'll assign a value of 0 to y to find where it intersects. To find where y intersects y, I'll assign a value of 0 to x. The above data I wrote 0 instead of x in the equation If I think about it, y = 0 squared -7 x 0 - 8. So, the ordinate of where it intersects y. I'll find it. In other words, the y-axis I would say the point where it intersects is 0a -8. Now, in the same equation, I've set y to 0. the second-degree equation obtained over time x² - 7x - 8 = 0. E factors I saw that it was detachable. x - 8de x + It will be 1. From here, the value that x will take is either... I'd say it's either 8 or -1. A different In other words, at the point 8 to 0, there is also -1 to 0. The function intersects the x-axis at this point. I say. Of course, what immediately comes to mind here... This question needs to be revived. TO Does it always intersect the x-axis? He could have cut it off as well. What does it all depend on? do you know? This is the parabola that was given to me. It depends on the equation. If that equation if it has two distinct real roots As in this question, the x-axis is two It cuts in a different place. So what about this equation? There cannot be two roots that are equal to each other. was it? It would be E. Even back then, how? would it be? We'll get to that later, but in detail... It would be a square. It would have a double-layered root system. X It would be tangent to the axis. I told you yesterday. Remember. If it's tangent to the x-axis, then it's there. We were saying it would have a double root system. So what about this? Could the parabola avoid intersecting the x-axis? Yes, he doesn't have to cut it. If The delta of the parabola equation is less than 0. If so, we can use this equation: x² - 7x - 8 = 0 from that equation in the form of delta If it's less than 0, we can't find real roots. TO The root of the real root function is the root of the x-axis. There are no real roots where it intersects. In that case We would say that the function does not intersect the x-axis. Let's see what was asked in example 7. Look at number 7. The graph of the function is given in the example. The equation of the function is given below. What we are asked to do is find a, b, and c. We need to find the coordinates. Look at A, B, C They're not random points, are they? A and B X The points where it intersects the axis. C of Y The point where it intersects the axis. Again, the axes We were asked to find the points where it intersected. Let me put the equation we have here. y = 1/3 x or that x² - 2x - 3 Let me write it by factoring. My attention was drawn to the fact that it could be factored. He took it. I separated it as x - 3x + 1. Now, I substituted 0 for x in this equation. What will the y-value obtained over time be? I wrote 0. 1/3 x -3 x 1 and 3le de -3'ü I simplified it. The point where it intersects Y is -1 output. So the coordinates of C are 0a. It was -1. I found this. Then replace y with 0. When I wrote it, I edited and submitted it. For the equation to be 0, either x - 3 must be equal to... Or x + 1 must be 0. x - For 3 to be 0, x must be 3 and x + 1 must be 0. For that to be the case, x must be -1. So, of course, this means abscissa A. negative, abscissa b is positive. That is. Here are a, b, and c. I found the coordinates. As we progress on this topic, we will find the following: They will ask us to do it. Here's what they'll say. So, how about we form a triangle right here? Let's create it. Here is the area of ​​this triangle. Find it. And we can easily find the difference. If you have, these are the points where it intersects the x-axis. The distance between them is actually the base of the triangle. length. The distance between -1 and 3 4 units. The point where it intersects the y-axis is... So, starting from this -1... I'm finding the height. And this too It was -1. So here's the length, of course. It will be +1. Saying 4 x 1/2 for example, easily in the area of ​​that triangle I could find it. Of course, that's the kind of question. I'm currently trying to produce the models. For example, between points B and C Distance could also have been asked. Here are two points. I would take advantage of the distance between us. Or we'll do this triangle animation again Using the Pythagorean theorem, the length of BC is calculated as AC. We could have reached all of these lengths. New information has now been provided. Let's see what this is about. At work For a parabola, f(0) is true for x = 0. Now what does this mean? when f(0) or x is given 0 What were we finding? Here's where he cut y. How we found the point is information. given. By writing 0 to x, of course, it becomes y. We find the point where it intersects the axis. At work This is actually what they were trying to emphasize. Here. What if x is 0 in this parabolic equation? When you give e, each time ax² + b Parts will be removed. We write 0 for X. TO Who will be left behind? Natural C will remain. Oh, then look at this too. I can say. To the equation of the parabola I looked. The constant term at the end is actually that. It is the point where the parabola intersects the y-axis. for time. What does the information below say? It changed for y = 0. second degree An equation is formed. The roots of this equation where the parabola intersects the x-axis These are the points. Here's what we just had a moment ago. The statements we emphasized. In fact, we immediately Let's add these here as well. Ok Really? These are the points where it intersects the x-axis. If If it cuts, does it cut or not? What does that mean? connected? Here is the delta of this equation. connected. If its delta is greater than 0 if, then delta is greater than 0 two different real roots will be. Let this be the x-axis. Here is x It will intersect its axis at two different points. If its delta is equal to 0, then the two are equal. It will be the root. In this case, the parabola The graph will be tangent to the x-axis. If Delta is less than 0, then the x-axis... It will not cut. This is either completely on top of it. or will be located below. Now the other In the information section, it says the vertex of that parabola. how to find the abscissa of the point It was reminded. apex of the apex It was found as -b/2a. In fact, exactly If we draw a vertical line from here... that truth obtained over time We said that it is the axis of symmetry of the parabola. There's an application for it right below. HE Let's take a look at the application. He says this Find the axis of symmetry of the parabola. Look What does the axis of symmetry ask? Top It asks you to find the abscissa of the point. Top the point where we refer to R as K We call it. From the axis of symmetry What is meant is "find the letter R". TO Let's find it. How did we find it? -b/ 2a b's He is already -6. -6/ There are 2 'a's, so 2 x 3 = 1, so r is 1. I found. Therefore, the axis of symmetry is x = 1. That's true. That is, the axis of symmetry. When asked to write a line equation It is necessary. I just roughly sketched it. Its 'R' is 1. That thing that passes right here We are in pursuit of the truth. This is the truth. The equation is the line x = 1. As stated in question 9, a parabola... The equation is given. Axis of symmetry x + 2 = 0. The axis of symmetry refers to r. to you. You leave x alone from here. x The result is -2. So to me It says that what is being said is r-2. And I am this Looking at the equation, r can be found with -b/2a. Let me bring it to life. I'll be careful. One There's a minus sign in the formula. And also x's in the coefficient, in the negative parentheses, m + 1 There is. Those two negatives turn into positives. translated. Here are m + 1/2 times 'a'. a'mız da It was 2. This is equal to the r value, which is -2. I moved the 4 from the denominator to the other side. -8 It happened. I subtracted 1 from -8. The value of M I found the result to be -9. Axis of symmetry When we encounter the concept, now we know that what is asked of us Actually, it's the abscissa of the apex. The graph of a parabola is also in question 10. given. Places where it intersects the x-axis It has been said. Find the axis of symmetry. it was said. Look at the explanation of the topic. I said it was important. Remember. Each This is the peak of it for time. Let's say it's over there. The apex of the apex Anyway, this intersects the x-axis. It is equidistant from the points. Equal at a distance. So, this length and this length They were supposed to be equal. E -1 to 3 The distance between them is 4 units. E means which lengths are 2 units 2 units It seems it will happen. E3 is located 2 units to the left. The number that will come next is automatically 1. Or how can we do it faster? Could we have done it? -1 and 3 equal 2. by dividing it immediately to that value of 1 We could reach it. So, from the axis of symmetry? What did he mean? Right here This is how I drew it, how I imagined it. Vertical That's right, we were asked. So the line x = 1 The question was asked. Question 11. No. First, some information... given. It has been said that f(x) e = something or other In the parabola, m and n are distinct. If f(m) is equal to FN, then... that thing I described at the beginning How is that possible, given the emphasis? FM to FN If they are equal, there are two possibilities. Either M or N They are equal to each other, but here's what's in front of it. It's over. Or it's already in a function That's the benefit of the peak, that's its meaning. equidistant from the peak the images of the points in the function They were equal. Naturally, it's very important. A warning. If FM is equal to FN, then the value of R is M. It is half the sum of N and. Now Let's look at the example below. At work in the example the axis of symmetry of the parabola We were asked to find it. Look, this is very important. I'm going to talk about something here. In such a Cartesian coordinate plane, this is simply It's not something unique to parabolas, but you know, we... Today we are talking about parabolas. Same Are there any points that are in the horizontal direction? We'll check it right away. What So, in the same horizontal direction? Ordinates Are there any points on this parabola that are equal? on? Yes there is. Let's look at them like this Let me show you the blue one too. This point and this The ordinates of the point are the same, and those of the two are the same. Of number 2. Let's delete that 2. This point by point, like this: 0 to 2 Let's write it down. Okay, then look, we'll get here from here. This is what we understand. In F-3, F-3 and F0 They were equal to each other and both were equal to 2. They were equal. It also has a point on what it equals. It doesn't matter. Look, I wrote -3 for x. when I write 0 for x with the resulting value The resulting values ​​were equal. And this What does it mean? If f -3 is equal to f0, this parabola r, which is the abscissa of the vertex. It is exactly in the middle of them. That is, the r value -3/2 we say. So the axis of symmetry is x = -3/2 We say that's the truth. Look, it's not very nice. Really? So this time we didn't have that thing. The function has no equation. What is A and what is B? We don't know if it is, but the function ordinates on the graph two points that are equal to each other We found it. A, we said, are the ordinates of these. If they are equal, the middle of them is equal to r income. In fact, while I'm at it, let me tell you... It will seem very strange at first glance. Let me write a question. Here is f2 It's a function of degree, and I'm saying f - 3√2 = f Since it is + 3√2, accordingly this Find the axis of symmetry of the function. Or Now F is one thing and F is another. You know, if FM is equal to FN, then R I'd say that's half of the total. So R = 17 - 3√2 and 5 + 3√2 It is half of their total. 3√2s They took each other away. 17 plus 5 equals 22 plus 2. I divided it and found the value of R. That way You'll definitely find them in question banks. Things like that. It's strange to go inside like this, writing very ugly things and scaring people They work. No. Information measured there What is it? If FM equals FN then R = M + N/2. Now I'll move on to example 12. Just a little bit Let me enlarge it. In example 12, the function The equation is given. It has been said that f(m) + 4 Since f(2m - 2) is equal to m, what is m? What is the sum of its possible values? Or Of course, there are two possibilities here. First, these two numbers are directly connected to each other. They could have been equal. So, F10 to F10 equals. F3 equals F3. So m + 4 2m It equals -2. They could have been equal. TO From this, one possible value of m is 6. I found that it was. Secondly, there's what's written inside. They are different values, however. Compare the images in the function If they are equal, then actually the sum of the ricins It is half of it. So r = m + 4 + 2m - 2/ I could say 2. So, how do I put 'r' here? Looking at the equation, they can't find it as -b/2a either. Was I? Immediately with that animation, b becomes -4 I saw that it was. And there's a minus sign in front of it. It's now +4. 2 a's. Our value A is also 3. coefficient. It was 4/2 2/3. So, it equals 2/3. Up there too Let me edit it. To 3m + 2/2. From here, a cross-multiplication Let's do it. Let's try to find M. 6 If I move it to the left, it's -2 -2/9. We call it the M value. There were two possibilities. in the question. Our 13th question. Was it underneath? Ha Let's take a look. Let me see question 13. Has it gone from 12 to 14? Ah, because right here A warning has been issued. Below the warning The question is hidden. Now he says parabola every point on the parabola It satisfies the equation. Very good. We are the situation Let's generalize. Any function Every point on the graph is that It satisfies the equation of the function. This That's not something unique to parabolas, is it? Parabola, parabola, that's all in my head. We won't make a big deal out of it. ultimately second a polynomial function of degree, i.e., a parabola This sentence applies to all This applies to functions. On I emphasize this word frequently. That means the equation of that function. It means it provides. And these are the kinds of questions we ask you. uh, they should make that emphasis openly Don't wait. Let's look at an example. Some information about the graph of a function Information has been provided. The equation of the function Some information regarding this has also been provided. Or When I look at the graph now, that... on the graph of the function I see a few points that... The first of these points is point 0a4. 0a 4 the point on the graph of this function on. Ah, so what I understand is f(0) It was 4. Look, this is the first thing I saw. Secondly, this point and this point are 5 to 0. point. And this is this function on the graph. Therefore, f(5) It is 0. Look, we've found another piece of information. I became. Point E is also the function's point A. on the graph. So f(1) becomes n equals. Now, from these three pieces of information... by using it to find all the unknowns We will work. Let's shrink it down a bit. TO If f(0) = 4, I have the equation of the function. there was. I wrote 0 instead of f(0). It's 0 now. 0 It happened. It was B. And we equalize this to 4. We knew that. Look, we found B. And one more piece of advice for you. Let me be there. Look, we found B! Immediately Let's rewrite the function's equation as follows: Let's write one. f(x) = -x² + ax + 4. So I wrote it. That unknown Let it not appear as unknown. Just one There's still something we couldn't find. What else? Did we know? We knew that F5 was 0. Let me use F5 now. f(5) -25 + 5 times a + 4. This is 0 It seems it will happen. So, 5a - 21. Therefore From this, we can also deduce the value of a as 21/5. I found. Okay, I found 'a' too. In that case The equation of the function has been completely found. -x² + 21x / 5 + 4. So what happens in the end? How will I use it? F1 was equal to N. F1 was equal to N. Now To find the value of f1, use this equation: I write 1 wherever I see x. 21/5 + 4 It happened. Therefore, F1 = 3 + 21/5 36/5 It was found as such. So, the value of n too. We got the result as 36/5. I passed 14. example. Look at the function here as well. The equation is given. Accordingly which of the following are correct? It says to check the box. Let's see. First, its arms are pointing downwards. That's true. What did I look at? I've made my decision. Head to the coefficient, that is, the coefficient of x squared. X we express the coefficient of the square as 'a' Because that value is less than 0, the arms It is downwards. That's true. Latter, The point where it intersects the y-axis. Y-axis How did I find the point where it intersected? X to 0 I was giving. In the expression, I assigned a value of 0 to x. time f(0) value is indeed 0 + 0 + It is found by subtracting 7 from 7. So 0 to 7 That's the point. The point where it intersects Y is correct. We can also quickly say this: we said. I looked at the function's equation. What is the constant term? The final constant is 7. The point where it intersects Y is 7. Looking quickly We could have said it. The peak point is RK. It is R4. Let's remember. -b/2a I was finding it. -b/ We have 2 'a's, so 8/2 equals 4. Its R value is 4. That's true. Axis of symmetry x = It was stated to be -4. This is wrong. Because If r is 4, symmetry axis is x = 4 That's true. The 2 by 20 point of the parabola It is on it. Check if it's on E. If I'm going to do it, maybe to F2? I need to check if they're equal. Because the point on it is the function. It would satisfy the equation. Immediate function I wrote 2 instead of x in the equation I am thinking. -4 + 16 + 7 23 - 7 This place went from 23 - 4 to 19, but the right side It was 20. That was wrong too. Yes. Next, look, this time it's our turn. you know, the smallest 'K' at the peak value, the greatest value, we emphasize them. made. A person with their arms pointing downwards ordinate of the vertex of a parabola actually the biggest of that function It is valuable. Arms pointing upwards in a parabola, the vertex The ordinate is the smallest value of the function. So, what is its greatest value to us? Most When asked what the smaller value is What is desired? The peak It is the ordinate. Now let's look at example 15. Look. The smallest value of the function How much is it? Now, he asks, "Which is the smallest?" Is that surprising? No. Is this a surprise? No. Because the arms are pointing upwards Something. And of course, finding that K, finding K. What do I need to find first to reach it? R. I'll find R and write it in its place. R now I'm calculating. -b and our B was -8. B/ 2 There are 4 A. E r's. Immediately in the function x I write 4 wherever I see it. 4 squared 16. 8 x 4 32 + 8. Therefore, it is called F4. It's like 24 - 32 is -8. So the hill Our point has an abscissa of 4 and an ordinate of -8. The answer to the question is our ordinate. Our value is -8. Let's look at question 16. In question 16 The axis of symmetry of the function is x = 4. The axis of symmetry is r. So it gives me the letter 'r'. Here. Why did he give the letter R? function the unknown m in the equation So that I can find it. So we will say that 4 = -b - our b, that is, the coefficient of x, is 5 + m/2 pieces a. Our 'a' was -1. 2 times -2. From here I found the value of m to be 3. Therefore, the general equation of the function I found. I'll write it down right away. -x² to m Because I wrote 3, it became +8x +9. What He asked? Its greatest value. Arms Because it's downwards, it already... He could have asked, couldn't he? N is the smallest value He couldn't ask. So how did he get to that K? were we reaching? Take R away and replace it with... We were writing. Our R-value is also 4. Well He wants me to find F4. Replace X with 4 I wrote. I also have a +9. 16 and 9 I added them up, it was 25. So our peak Its abscissa is 4, and its ordinate is 25. The problem The answer was 25. The highest value peak Because the ordinate of the point is asked. Let's look at example 17. It is said in 17 that this the maximum value of the function is 13 According to whom did he give it to me? The peak He gave the ordinate, right? He gave it. He didn't give the abscess. I am in his apse. ryeim. Let it stay like that. Yes. Accordingly the sum of the possible values ​​of M How much is it? Now my priority is always R. pursue. Let me find the letter R. Even if I can't find the answer, let me state it in this question: Look at R, its numerical value. I won't be able to find it, but that's okay. - B. Our B is 4B/ 2 A's which is the coefficient of X. Our 'a' was already -2. From here, the value of R 4 are gone. R is equal to M. So the hill Express point M as 13 I was able to. Whether it's the peak or something else point. Isn't this point related to the function? because it belongs to the function It satisfies the equation. f(m) will be 13. In the function equation, replace x with m. I am writing. 4m x m + 2m + 9 is 13 It was supposed to be that way. 4m² - 2m²edden 2 m² arrived. + both 'm's on the left I threw it to the side. From 9 to 13, it became -4. All It can be simplified to 2. Let me simplify it. m² + m - 2 = 0. From here, m's I will find the values ​​it will receive. Actually I don't need to find it, but we'll find it anyway. Let's find it. Because why don't I need to find it? M He doesn't ask what might happen. M's The sum of the possible values ​​it can take. Quickly We can just say this and move on. Actually this We will call the roots of the equation m1 and m2. If I become that, then the sum of the roots of this equation -b/a is being asked. Here's what I found: -1 we could have said. But to its factors Because I saw that it was detachable, one factoring and from there m's The values ​​it can take are as follows, separately: Let me find it. It could be -2 or 1. It seems it's possible. Here's the sum of them. We can still say it's -1. Yes, below. A warning has been issued. Let's read it right away. Let's see. He says that FN is off. a function defined within a range. This When finding the smallest value of the largest pin, fm, For fn and r = -b/2a, these values ​​are... It is found and evaluated. What does that mean now? Does he want it? Look, here, uh, normally this... different from what we have done so far What's wrong? So far, the definition of the function always assume the set to be real numbers we had done. But this time he says, if function within a specific range If defined, give you the biggest or the smallest. When asked about its value or anything like that, they immediately go and buy this. Just like you've been doing until now, here's what I need. The question is about the k-value of the peak point, there He's trying to say you can't just say you were asked. Why can't we say that? Let's talk about it. Now Let me visualize a parabola. Like this Let me visualize a parabola and the x-axis Don't cut it. Let me put it more simply. Like this If it were a parabola. Now, of course... Our parabola, or rather our function, is real. If it is defined in numbers, the smallest it can take The value is the number that corresponds to this point. But but Imagine that even these are numerical. Let's give some examples. Let this be 3, then 0. Let this be number 2. The name of our function Let f(x) be the variable. Okay, let's try this function. the function x's are 4 and greater than 4 If only they had defined it when they were older. Now, under normal circumstances, the graph first... It was as I drew it, but the x's were 4 and 4's. if it was defined when it was large, even between 4 and 10 Look at the part within that range and comment on it. if they had said so. And now look at the 4 hills. A piece located to the right of the point. Top To the right of that point, let's say this place... Let's make this one 10 as well. What's the continuation of this? Actually, what we did a moment ago... The function itself in animation. But Which part do I need to focus on? will it be? This will be the only part. When we only look at that part, the function... instead of the smallest value, it's 4. It is the value that will be found when we write. Most when we write the larger value instead of 10 We can say that it is the value that will be found. Therefore, the peak point in every situation the largest or smallest ordinate It doesn't specify its value. What to pay attention to I will? That was given at the beginning of the problem. I will pay attention to the definition range. Him I will give my opinion based on that. Ha definition range Even if it has been given, perhaps again The determining factor is also the peak. It was possible. Think about the function, for example. I want it to be between 2 and what range? 8 If I had defined it within that range, that is If I were to look at only this part... again, the smallest value instead of 3 That would be the answer that would appear when I typed it, I mean... there is either a graphic animation we will do it or right at these extreme points We will find the values ​​that the function takes. And also finding the peak point and the peak point Look, the peak point of that minute is closed. If it is an element of the range, then replace it with it. We will find the result when we type it. Here is the biggest of these results. It is the maximum value of the function. Most We'll say that even the smallest one has the smallest value. Let's see an example of its application right away. Let's see. Yes, here's the 18th example function. given. He asks which function this is. Does it take values ​​within a certain range? Now pay attention. What are we going to do? Look. Function -1 It was between 3 and 5 closed. TO R, one of the determining factors Let me illustrate. -b/2a From 8/4 Its 'R' is 2. Pay attention. 2 to me an element of the given definition range. So, only those between -1 and 3? I'll think of part 'e' and 'R' will go there. It fell. I will be careful. Now Let's quickly find these things. First, let me find f(2). F2 I'm writing 2 instead of x in the function. 2 x 2 square - 8 x 2 + 3. So the value of f(2) is 8 - I added 3 to 16 -8. It was -5. And then Whom will I select? End of December the images of the points in the function. I'm after f-1 now. -1 instead of X I wrote. The value of F-1 is 13. I found. For the other extreme, 3. what value did the function receive? Let me find it. I'm writing 3 instead of X. -24 + 3 It happened. I added 18, 24, 6, and 3 as well. F3's I found its value to be -3. Now Among these resulting values, one is -5. One is 13, and the other is -3. Here are these the smallest inside the function It is the smallest value it receives. So at least -5 It seems it's a possible function. These values the largest of them, for this example 13. This is the maximum that the function can take. It is of great value. Then the function is -5d+ It takes values ​​within a closed range of 13. we can say. Of course, this is a function. I could have done it by imagining the graph. I didn't want to draw the graph right now. Because the next subheading is about parabolas. How to draw a graph is already a special question. We will take care of it. And then it was the same again. questions in this format on a graph We'll show it and then we'll talk again. A warning is given below. He says, "A parabola is separated by two axes of symmetry." functions when decomposed "It will be exactly the same." Now, look, there's a concept... Here. What does it mean? What does "one-to-one" mean? It will be a perfect match. Remember the functions Let's remember. Exact function. HE In the equation of the function, substitute a different x for x. The more values ​​we give, the more different answers we get. We were buying. So, uh, two different elements The image shouldn't have been the same. Even the cluster Their pairings are always like this: The animations are done here, three of them. Let's provide staff. Here's how to get from A to B. a set of defined functions Let it have an appearance with its matching. And this Let this be the person he's going to. This is where it went Let this be the employee. If this is also... We were saying that if it's going in one direction, it's not exactly the same. For example, this function in this animation. It's not a one-to-one function. Because two There are different elements, but the images are the same. dead. How about another animation like this? If I had done that, I would have brought another person here. If I had added more and posted it here... Look, this time it would be a one-to-one function. He got a different answer for each different x. It would be. So, what about function graphs? Is a function one-to-one based on looking at it? Isn't that right? How did we find this? Him Let's remember that too. There, immediately We were doing a simulation like this. Let's say we have a function like this: We have a graph. Is this function possible for an individual? Is it one or the other? To the x-axis We were drawing parallel lines. To the x-axis I've drawn the parallel lines. I drew it. You can randomly select an x ​​from anywhere you want. that you drew lines parallel to the axis Think about it. If that random thing you drew graph of a given function if the function always intersects at a point It's an exact replica. But what I did as in animations If there are places where it intersects at that point, then We were saying it wasn't exactly the same. For example Let me draw the graph of a linear function. Here's the drawing. Is this an exact match or not? Is it? Where did that fantasy come from? If you draw the lines, then draw them, of course. It's an exact replica. Or one with a graph like this: I drew a polynomial function. Let's see. I'm checking the horizontal line. Always one It cuts off at that point. In one place. Only at that point. Yes. This is also identical. Alright Let's draw a parabola under normal conditions. Look, take a parabola. Arms down Draw a correct parabola. Is this exactly the same? isn't it? It's not an exact match. Because Look, he cut it in two places. Not Really? We have that function, that parabola of ours. with an appropriate definition range If we play, it should be one-on-one. We can provide it. This is what I'm talking about. Let's say we have such a parabola. There is. Let's say its 'R' corresponds to 8. If my function is for all real numbers If it's defined, it's not an exact match. But I defined our function as follows: Think about it. To be in the range of 8 to infinity I defined it as follows: So what I'm saying is... Replace x with 8 or greater than 8. Acknowledge that you are able to provide the values. E o The graph you'll be interested in is the one in pink. It will revert to the graph I drew. And now this Looking at the graph I drew in pink, yes. I can say this function is identical. This is what they are trying to convey to us above, That's what's being tried to be explained. He says from the axis of symmetry, he says, two separate When you separate it into functions, it's the same thing, of course. The property I gave is 8 plus infinity. not the range but between negative infinity and 8 If I define my function within the range... Again, they managed to make it exactly the same. I would be. Below, we will implement it. It was expected. Here is the equation of the function. I looked. And here's a rough, rough graph of it. Let me draw it. I won't draw a precise graph. But the letter R is absolutely necessary. -b/2a I found that r is equal to -2. TO a parabola with arms pointing upwards I will draw it. R is -2. Even the y-axis The point where it intersected was the final constant term, 14. Let me just take a look at all of this, Let me put it all together in my mind. Here are the 14. It cuts. With arms raised upwards, r Its value also turned out to be -2. Suitable for this He says find a gap. It is identical and covers everything. It happened. The broadest, most extensive definition Find the intervals. When it comes to E or R I'll take the part up to that point or from R. I'll take the next piece. So, two of them You can answer. You could say that It is either a closed interval between negative infinity and -2 or is the infinitely closed interval between -2 and +2. You can say that. Our 20th question. Question 20 also involves a diary. It's presented as if it were a life problem, like in the book. in the later parts completely this We will also be dealing with questions in that format. From the cows that are now on a farm x + 2 liters of milk are purchased per day. What per day means? Look, he says in one day. In one day x + 2 liters of milk are purchased, and this milk... from sales at 1 per liter, 14 per liter. - A profit of 2x lira is made. Beautiful. From the milk sold at the farm over 8 days What is the maximum profit that can be made in Turkish Lira? Now, if x + 2 liters of milk are consumed per day, then 8 How many liters of milk are purchased per day? First, one of them Let's find it. And naturally 8 x x + 2 l milk It is taken. That's how much my wife earns per liter. If so, for 8 x x + 2 L, then also 8 x x + 2 x 14 - Profit of up to 2x can be achieved. Well The profit mentioned in the question is based on x. in some way, in the form of x, like this We can write. Now look at what I've created. Doesn't anything in the statement catch your attention? Either this is a quadratic function. I've counted this before - 2x² and even 8. I multiplied it by -16x, squared, etc., etc. a continuing second-degree It would be a function. So what did I tell you? He asks? The biggest thing this can get What is the value, or the maximum profit that can be made in Turkish Lira? Is it done? The function that expresses E profit. So, what's the maximum possible value? he asks. What is the maximum possible value of this? It's surprising that such a question could even be asked. Really? No. Because these arms are down It will represent a straight line parabola. Isn't it? This is the peak of it. The ordinate, the ordinate, this is the largest It will give us its value. Of course To find the ordinate, first we need to know what Will I need to find it? His apse. E apse Look, we'll find it without any further delay. Like this I've distributed it now. Ri -b/2a. None There's no need to bother. We already have this We can easily draw its graph. Because I see its roots. One root of this -2. The other root is this The one that resets is 7. -2 and 7 are my roots. Let's make this point -2. Let this be a 7. A parabola with arms pointing downwards. will be. Okay, I just did my animation. So, is the 'r' in this always x? the places where it intersects the axis, that is, these two Wasn't it exactly in the middle of the number? 7 with Exactly in the middle of -2. I divide the balls in half. I added them up (5) and divided them by 2 (5/2). Now, let's immediately assign this 5/2 to the function value. writer. I will find the answer to the question. So x I'm writing 5/2 instead. There are 8 x 5/2 + 2 next to you. X 14 - that's two 5/2s. 4 plus 5 equals 9, so 8 x 9/2 is the total. The twos over there gone. 14 - 5 of 9 came up. These 8 and 2 I simplified it. That makes 4. So that's 4 x 81. From there, we can say it's 324. Look, here's a parabola, roughly speaking. vertex axes in what is called The places it cut, oh my, the R... K is the axis of symmetry, the largest and the smallest. value. Are your arms pointing upwards or downwards? is it true? How to learn R quickly Can we find it? A on a parabola The point satisfies the parabola equation. This is it and such second-order expressions We call these functions parabolas. More And then, as I said before... The concept of the axis of symmetry relates to the state of delta. Does it intersect the x-axis or not? like all that general information about parabolas We had a conversation. What now, right now? are we doing? Tests at the back of the book We're starting to solve it. Yes, now it's my turn. Here are our tests at the end of the topic. to solve. Now I'll start with question 1. See, it was stated in question 1 that this... The graph of the function is a parabola. According to what he stated, e is a parabola. According to this, being second degree It is necessary. When I look at the given statement... There's something with x cubed here. This is it The term shouldn't be there. This should not happen For this to work, m -5 needs to be equal to 0. So its value should be 5. This We found it. Okay, I got rid of the x-cube. E one of something with x squared by degree 2 It was supposed to be. That's why the value of n - 2 above x here It should be equal to 2. So, of Its value should also be 4. This In this case, the sum of m and n is also 9. will be. In question 2, this function... Since the graph does not show a parabola He said it couldn't be a certain number. Now this parabola He said he wouldn't mention it. Well, actually, it's like this... We can do that. The options one by one You can try it. Look, here's option A. Imagine you wrote 4 in place of A. If A is 4, the expression becomes 2x³. It will continue indefinitely. And in this case... It wouldn't represent a parabola. Okay, so 5 If I give A a 5, then I get a 6. - It would be 1 out of 5. x cubed dot dot And so it went on. Parabola He wouldn't mention it. If A is 6, then e This time, the cubed expression 6 - ax would be 0. TO we would leave and what was left was 4x²are It would be something that went on and on. In this situation It indicated a parabola. So that means a 6 It wasn't possible. In question 3, the arms of the parabola are pointing downwards. TRUE. The arms of the parabola are pointing downwards. If that's true, then what's actually being done here... What is it that we can understand from the emphasis? would it be? The arms of a parabola whether it is downwards or upwards What was the determining factor? Leading coefficient. So x was the coefficient of the squared term. X The coefficient of the squared term is negative. When the result is negative, the arms are pointing downwards. When the result is positive, the arms should be raised upwards. It was happening. E is the coefficient of the squared term. a - 3 will be less than 0. Well 'a' needs to be less than 3. The largest integer value less than 3 The question was asked. Here is the answer, 2. In question 4 The equation of the function is given. Top The question was, "What is the point?" First, R. I will find it. Replace R with K. I will find it. I'm calculating its r right away. I was calculating it with -b/2a. -6/ Our value for 'a' is 1. Twice that is 2. Well With r being 3, I found the abscissa of the vertex. It was 3. Something for 3. That thing, you know... To find K, press F3 immediately. I will calculate it. I wrote 3 instead of X. 9 - It's now 18 + 1. -9 + 1 became -8. Means and the abscissa of our apex is 3, Its ordinate is -8. So, 3 to -8 That was the point. Let's see what's asked in question 5. Here Let's enlarge it a little bit. The equation of the parabola is given, and it says: "One of the points where it intersects the x-axis" Which one is it?" Or the x-axis of a parabola How did we find the points where it intersected? y We were giving 0 instead. So, what we call f(x) It was a thing. I wrote 0 again. Actually, the question is x² We've returned to solving the equation - x - 6 = 0. It happened. The expression can be factored. I separated it as x - 3x + 2. Therefore, from here it will either be x - 3 0 So x could be 3 or x + x could be 0. It might be -2. Well Our function is 3 to 0 and -2 to 0. It intersects the x-axis at certain points. X One of the points where it intersects is 3 to 0. That's the point. Question 6. It has been said that the apex of the parabola The point is on the y-axis. Look at this The emphasis on the peak is important, but more There is an important point to emphasize. Any The point being on the y-axis. Y for every point that lies on the axis We can say this. The abscissa of that point It is 0. Points on the Y-axis Their x-coordinates, or abscissas, are 0. And here too the apex on the y-axis Since it was stated that it is, its R automatically It is 0. R0 is the abscissa of the apex. will be. To find R, use -b/2a. I was benefiting from it. - Our B value is 4 - m/ 2a. For this to be 0, m must be... Its value should be 4. Our 7th question The equation of the parabola is given. Symmetry axis. What was it? The axis of symmetry is from r. That was true last time. So you'll find R. R To find it, immediately from -b/ 2A I am benefiting from it. Therefore, 8/4 minus 2 I found it as... Here is the line x = 2. He defined our axis of symmetry. In question 8, the axis of symmetry of the parabola is x - 2. The line x = 0 means x = 2. Well The value of r is 2. So how do we represent R as -b/2a? We were finding it. There's a minus sign in the formula. One There's a minus sign in between. Those two minus each other It converted it to plus y. We have m + 2 / 2 of 'a's. It was 3. So this is number 2. From here, the denominator I threw the number 6 across. 12 m + 2 = 12, so m is He found that its value should be 10. We became. Let's see what happens in question 9. Hey, we'll do a quick cut there. Yes The graph of the function is given in question 9. The question concerns the axis of symmetry. So R We are asked to find it. In the given graph What things caught our attention? Look at X The points where it intersects the axis are given. Each The abscissa of the apex for time is this equal to the points where it intersects the x-axis It was far away. In other words It was somewhere in between the two. Therefore, -1 I add 5 to 2 and divide by 2. 4/2 It is 2. The value that R will take is 2. Naturally, the axis of symmetry is also x = 2 I can say that it's true. 10. Question. The axis of symmetry is the line x = 3. from the points where the parabola intersects the x-axis One is 5 to 0. Which is the other point? The question was asked. So he gave the letter 'r'. R'si It was 3. One of the points where E intersects x It was the 5 to 0 point. Look like this. Let's draw a symbolic picture. Here Let x be the axis. Here's how it cut x. There are two points. Our R value is equal to 3. He's coming. This is number 3. E cut x If one of the points is 5 and the other is 0, then this point is 5. Again, the information I used a moment ago... I will use it. Exactly in the middle of this. It will give 3. Here's something with 5. If I add them up and divide by 2, I get 3. From there, it's easy to see that the other one is 1. I'll find it. So the other point where it intersects x is 1e. It will be point 0. Question 11. The rule for the function is given. The smallest value is being asked for. Arms an upward-pointing parabola. Natural It's normal for them to ask about the youngest one. Before I'll find r. I'll substitute R. Because the smallest value is also the greatest value. What was he interested in? The peak with its ordinate. Immediately give the r value Let's calculate. -b/ 2a helps to make r a Let's find it. Its 'R' is 3. x in the function I'll write the number 3 everywhere I see it. Here are the three Its square is 9.6 x 3 = 18. And also at the end. We have -9. The value of F3 is -18 from here. I found it as... So the peak Its coordinates were 3 to -18. The biggest Its value is the ordinate of the peak point. the smallest value is also the peak Its ordinate is -18. Question 12 is again about the biggest and the smallest. A question about value. The function's most Since its greatest value is 6, that means to me He says the peak It says its ordinate is 6. The biggest and the smallest The ordinate of the peak value. Okay then. Can't I find its abscissa? I'll find it. -b/2a with the help of -b/ 2 a's. Our A was also -1. 6/2 to 3 It was found as such. We have the peak in our hands. Whether it's called the peak or something else. Let there be a point that belongs to the function. Since it is a point, the equation of the function provides. So I am f(3 is equal to 6) I know that. Now x in the equation When I write 3 everywhere I see it, the answer is... Since it will be equal to 6, it is actually 9 - 2. 7 - m + 7 equals 6, so the value of m is 1. I can say that. In question 13 Let's see what was asked. In question 13, one The function equation is given and various Given the premises, try to avoid these. The question was asked if any of them would be correct. X places where it intersects the axis, places where it intersects Y places Now, the places where x is cut To find it, we need to factorize the expression. Let me separate them. y = 35 is expressed as -5 to -7 I can separate them. So x - 5 became x - 7. Indeed, x is either 5 or 7. Well either at 5 to 0 or 7 to 0 that it intersects the x-axis at that point I can say. Therefore, this is true. The point where it intersects the y-axis becomes 0 in the equation for x. I'll give it. When you actually set x to 0, you get 0 - 0 + 35 becomes 35. So f(0) is 35. This too That's true. 3. Axis of symmetry. R. R. Let me determine that right away. -b/ 2A. 12/2 Its R value is 6. Yes. That's also true. Top The point is 6 to 1. We see that R is 6. To find its ordinate, I need to find F6. When I wrote 6 everywhere I saw X 36 -72 -36 -36 + 35 leaves -1. But this one said +1. This was a mistake. And f23 is equivalent to f-21. it was said. These two are equal. for it to be possible, the text inside half of their sum equals R is necessary. Now I'm comparing 2023 with 2011. When I add them up and divide by 2, now I collected them. What comes from inside? 12. 12/2 equals 6. Was our 'R' (right) 6? Yes, our R. It was 6. Therefore, this turned out to be true. Well Four of these statements were true. It happened. The graph of the function in question 14. given. Various tips, various Information has also been provided. So the x-axis cut. Now, the function is in question 14. where the function intersects the x-axis One of the points is given. Y-axis The point where it was cut is also given. Also below The equation of the function is like this: a and b. Two unknowns are given and these We were asked to find it. any located on the graph of the function the points form the equation of that function It provided. Now this point is 3 to 0. Because of this, the first thing we understand is... What's happening? Sometimes F3 is 0. And this because point 0a is the point -1 What do we understand? F0 to -1 Sometimes they are equal. Now immediately f(0) Let me bring it to life. x in the equation When I wrote 0 instead, 0 + 0 just canceled out. B remained. Now we've found the value of b. b is -1. Also from f(3) = 0 Let me benefit from it. Substitute 3 for x in the equation. Let me write. 9 + 3a + b = 0. And we anyway, the value of b is -1 We had found it. From this, 3a is -8, and a is... We found that it is -8/3. Now we have A Yes, B exists. Finding their sums It's easy. And question 15. This is the last test. the axis of symmetry of the function in the question given. From here, I'll just leave x alone. I gave up. x became -5. So its 'r' is -5. The way to find R was -b/2a. - Our value of b is the coefficient of x, so b/2 pieces a. Our A was also equal to -1. Here is -5 It seems it will happen. From this, the value that m will take What will happen? These would be a plus. 2 across I threw it. -10. I also sent the number 6 to the other side. m was found to be -16. Immediately Let me write down the equation of the function. M -16 isa 6 - 16 - 10x - 7. So what? has it been asked? The question was asked about the greatest value. Well The ordinate of the vertex is being asked. Er I knew its value. I know his apse for, that is, the abscissa of this apex It was -5. We are after the ordinate. Immediately I'll find the F-5. So instead of x, it's -5 I will write. - e -5in squared is 25. -10'ile -5 to +50 -7 25 -7 and can also be obtained from there They found that the greatest value is 18. We became. We've completed the first test. Now and also with the solutions to the second test Let's try. Let's go. Yes, we passed the test. Look, in question 1 there is one A function is given and it says that this Since the graph represents a parabola, e A parabola will be indicated here. There is a rational fraction expression. Like this That situation shouldn't happen. Once something with fractions or something at the bottom It shouldn't stay. So, it has an x ​​at the bottom. The term shouldn't be there. Once The first conclusion we can draw from this is 2n - m will be 0. So actually, m's It will be equal to 2n. The upper part is It should be something with x squared. E x squared For something to happen, m-2 is also necessary. It should be equal to 2. So m's The value it will receive is 4. E m also does this I'll take it and write it here instead. N2 will be I say. From this, we can also deduce m + n. The sum of the possible values ​​is 6 I would say it's possible. Question 2. The abscissa of the parabola's vertex is e Beautiful. R is being asked. Immediately with -b/ 2a I'll find it. - B itself is already -9/ There were two A's. The value of A was 3/2. 2s It was simplified. From 9/3 I found that R is 3. Question 3. The graph of the function intersects the axes. The area of ​​a triangle whose vertices are the points. You know, a kind of animation like this. I had done it. Let's draw the graph of E. expression It can be factored. Look, x - 2 factoring the expression in the form x - 5 I'll separate them. Arms pointing upwards It will be a parabola. I see him. Because The sign of the leading coefficient is positive. X to 0 I can give it a try and find where it intersects Y. X I saw that when I gave 0, f(0) worked. I substituted 0 for x in the equation. Here is y. It cuts at 10. Let's do it this way. Here It was 10. This is number 2. This is number 5. What was asked of us? This The points where it intersects the axes are considered vertices. Eden triangle. Now, where it intersects the Y-axis. Here. These are the places where he cut the X, and here. Now, he says to form a triangle. Let's create it. That triangle is like this Let's show it symbolically. At work We need to find the area of ​​the triangle I created. It is requested. He/She will pay a little attention. If you are, this place is already between 2 and 5 the distance between the bases of 3 triangles length. Our height is also equal to 10. Therefore, base times height divided by 2 and the area of ​​this triangle is 15 square units. We can say that. In question 4, it asks for the vertex of the function. Given this, the abscissa of the apex not given but the function equation By looking at it and using -b/2a, we can find the value of r. So we can find the value of m. Like this Let me write. You know, usually it's like "r". We were naming them. R, which in this question is M - B. Our B is already -2 / 2 A's. 2s It was simplified. I found that it's 1/4. So the value of M is 1/4. I understand that. So in this case, our peak point is actually The coordinates were 1/4 by 3/4. A function whose vertex belongs to it. Since it is a point, the equation of the function provides. So f(1/4's) We know it's equal to 3/4. Now in the equality, x in the function equation I'll write 1/4 everywhere I see it. 1/4 of Its square automatically became 1/16. - 2 pieces I have 1/4 of it. + There is also the value of 'a'. At work This is 3/4. E 4/16 this is 1/4 -2/4 -1/4 -1/4 I threw it across. 3/4 + 1/4/4 So I found the value of 'a' to be 1. I know the letter A. I know M. These We can find their sums. Let's see what's asked in question 5. In question 5 the point where the parabola intersects the y-axis The ordinate is given. E cut off y How do we find the ordinate of a point? We? We were writing 0 instead of X. So actually That point is point 0a 12. A different The change is F0 12. In fact, you can quickly say this: I was saying we could even say that. Y The point where it intersects is where 'e' is the constant term at the end. Because when you write 0 instead of x, it already happens here. He will leave. This place will be gone too. So that's 4m -8 = 12. So the value of m is 5. will be. Question 6 states that the vertex of the parabola... Since the point is on the x-axis Okay, let's stop here for a moment. There's something very, very important and wonderful here. The vertex of a parabola is on the x-axis. It means "on". That's a very nice expression. Actually, and this is what it means. A box like this Let me include it. Just imagine. The apex has arms pointing upwards. It is on the parabola and the x-axis. E o time this parabola is tangent to the x-axis It is necessary. It is tangent to the x-axis. time, that peak point, uh... x It is located on the axis. So what was the condition for being tangent to the x-axis? The delta of the parabola was equal to 0. We understood that the delta of the parabola is 0. It seems it will happen. Delta = 0. So b² - 4ac Its value will be 0. From here m's We can find its value to be 9. Even In addition to that, I can say the following. TO if the vertex is on the x-axis Actually, the equation of that parabola is an integer. It must be a square equation. Actually, this And so it continues as x² - 6x... Because I saw it, either this x - 3 is in parentheses. It is a square. Let the square be in parentheses so that it is even. Let there be multiple roots. Let it be tangent to the x-axis. And in this way it will be tangent to the x-axis. and then he opened this place and again m's We could have said that its value would be 9. Question 7. This parabola originates from... It seems to be passing. And what exactly does "origin" mean? The origin is a point, i.e., 0 to 0. point. If the parabola passes through here... The point satisfies the equation of the function. f(0) is 0. I see 0 everywhere I see X. This will automatically disappear when you type it. This place will be gone too. What's left is m² - m - 12 will remain. I'll state it right here. Let's factorize it. This m -4 m + 3 It happened. Now, for this to become 0... The value of m will either be 4 or -3. will be. But there's one thing we need to pay attention to. If m is -3, then m being -3 In that case, look here instead. when you write There are no more squared terms. Then statement 1 It's not a parabola. For this reason, M-3 I can't be. The value of M will be 4. Question 8. The vertex of the parabola is g(x) Since it is on the parabola now First of all, the top of that one above Let's find that point. The peak I start by finding its abscissa. R'sini I'll calculate it right away. -b/2a. Now, because we do it so often... Let's just do the calculations mentally and move on. In 8/2, the 'r' is 4. To find its K I immediately wrote 4 instead of X. 16 - 32 + M. So it became 16 - 32 m - 16. So this The apex of the vertex is 4, and its ordinate is m. She's 16. Now, this is the point, this is the point. on the other given parabola What did it mean? His equation provides. So he's telling me the value of the G4... m is equal to 16. G4 x instead of 4 I wrote. In the equation of GX. From here too I found the value of m to be 32. Always Look at the things I've been emphasizing throughout the day. as a question model in questions It keeps appearing before us, doesn't it? On it or The peak point is somewhere in Tet, the double point. multi-story, single-story, arms up, arms down Or the places where it intersects the Y. The questions are always based on the same information, the same achievements It appears before us in an established form. 9. Our question. In question 9, within a specific range There is a function that has been defined. One There is a parabola, and the biggest and most The question asks for the sum of the smaller values. What What were we doing in this situation? First, this flight the values ​​the function takes at those points We were finding it. Then, immediately identify r. also the value that the function receives We were determining it. I saw that R was 1. I saw it with -b/2a. Then these values ​​together We were finding just one and comparing them. F3 instead of X When I write 3, it's 9 - 6 + 5 = 8. The value of F3. I wrote 4 instead of x in F4. 16 - It is 8 + 5. So 8 + 5 is 13. So what about this? The question also includes F1 coming from R. Shall we calculate it? We won't calculate it. Why? Because 1 is an element of 3 and 4. Is it? So, one of the 3 or 4 closed intervals. He's not an employee. So that peak point even if it occurs, the peak point is formed We are not interested in that part. His right We were working on something on that side. Therefore, 8 and 13 are smaller than these. That is our minimum value. 13 This is our maximum value. The total of these is 21. Question 10. There were two dots. He says this symmetry of the parabola passing through the points axis. Pay attention to the points given. Those locations have one thing in common. The ordinate of those two given points is the same. isn't it? So, what we actually mean is this We know the answer to this question. F-2 and F12 They were equal to each other. They are equal to each other Like, both of them are equal to 9. What is equal to It doesn't matter that it is. But with the F-2 It's equivalent to the F12. In the parabola E, F-2 and F12 If they are equal, what is our interpretation? would it be? R is exactly in the middle of these two. So, 12 - 2 = 10. I divided 10 by 2, which is 5. So... This means the line x = 5 is our symmetry. It is our axis. Question 11. In the FX parabola, f is equal to something. The sum of the possible values ​​it can take. Two It was a possibility, wasn't it? From these The first one is this. Simple logic. E. What's written inside They are equal to each other. So f3 is equal to f3. It's like saying F5 equals F5. From here, 2m = 3m I found that its value could be 3/2. Secondly, and more importantly, What kind of comment will we make? If F is equal to the other R is exactly in the middle of these two. Both of them When we add them up and divide by 2, we get the value of r. We are arriving. m + 1 + 3m - 2/2 is equal to r. will be. So how do we find r by looking at this equation? Couldn't I find it as -b/2a? I'll find it. 3/4 = It was simplified from 4m - 1/2 around here. 1'i I threw it across. I divided 5/2 by 4. I found the value of M to be 5/8. Or It was either 3/2 or 5/8. 3/2 with 4 Let me expand on it. I added 12/8 and 5/8. m's The sum of the possible values ​​is 17/8 I found it as... Question 12. Look, this question... That's a good question. He says 24 is one. I divided it into two parts. The maximum this expression can take Find the great value. So now there is one here There is a division operation. There's a number at the top. Below is the result depending on the value x takes. There is a changing expression. The result is the biggest I say, let it be. The result of the division operation is E. To make it the biggest, the e-pay part is already there. Because it's fixed, the bottom part is possible. We try to keep it as small as possible. Well For this expression to be the largest possible taking the smallest possible value of the denominator is necessary. Then the question became this. At work What is its smallest value? And this too a parabola with its arms pointing upwards Therefore, the smallest value is the peak. It is the ordinate of the point. First, R I found. -b/2a means R is 2. I saw. Immediately substitute 2 for x in the function. I wrote. 4 - 8 + 12 from f(2) I saw that its value is 16 - 8 + 8. TO The top was 24. The bottom can take The smallest value was 8. Therefore the greatest value the expression can take There are 3. That was a good question. One for question 13. Let's see. Question 13 also says that this also requires a... Let's give it a star. Another arm in the future with many questions similar to this We'll try. Look at this parabola the sum of the coordinates on it is at least What is the ordinate of the point? Now There are infinitely many points on it. TO There are a lot of points to consider, of course. What is the condition? This will satisfy the equation. At work those coordinates that satisfy this equation among the points that it has the one whose sum of coordinates is the least