Aula de Matemática 3 com o professor Diego Melo - Pré-ENEM - 03/06

Added: 2026-06-06

[00:00:02]Good evening to all students of Préen Popular and Área Velberg, Polo Nacional.

[00:00:09]How are you all doing? Uh, uh, sorry for the delay. The professor ended up having an unexpected problem.

[00:00:19]He 's just finishing up sorting things out. He's about to go in. So, in order not to delay things any further than they already are, I decided to get started here.

[00:00:32]And I'd like to ask if you have any questions regarding the content covered in the last class or any previous class by Professor Diego Melo, while he's not online yet, so I can chat with you and maybe clear up some doubts so we can start the class better as soon as the professor is online.

[00:01:09]Hey, if anyone has any questions about anything, good evening to everyone who's arriving. Bia, Ana Clara, Clara Ribeiro, Madu. If anyone has any questions about any of Diego's material, or anything he covered in class previously, feel free to send them in the chat. While the professor is on his way, I'll chat with you all and we can try to clear up any doubts to make understanding the lesson easier, in case anything was missed.

[00:02:42]Very good. I see that the people there aren't having any doubts. Excellent. Okay, great. M.

[00:03:50]Opa, the professor just came in, everyone. Good evening, Diego. All is well?

[00:03:55]Good evening, everyone.

[00:03:57]Sorry for the delay, I had a little problem here and the computer clock froze. I said, "Guys, this time is dragging on. When I look at my phone, I even see a message from Sabak.

[00:04:08][laughs] I'm going to project it here for you. I'll get everything going soon, okay?

[00:04:20]Uh, everyone, first of all, good evening.

[00:04:24]Those who followed Monday's class saw that we gave lesson two, Professor Reinaldo gave lesson three, and now we're catching up on those lessons, right, everyone?

[00:04:40]We were very focused on lesson zero, and now we're trying to balance these lessons that were a little behind schedule compared to the other subjects.

[00:04:51]Lesson two was about ratio, proportion, and the rule of three. And now we're on lesson three, which was mean, median, and mode. And now we're getting to lesson four, which is a very important lesson, which is what?

[00:05:09]It's a lesson about first-degree functions. And why is it so important? It's one of the subjects that appears most often not only on the ENEM exam, but also on the ERGE exam for those who are taking that test this Sunday.

[00:05:26]So, why do I Why am I talking so much about linear functions? First, I have to tell you what a function is, because if I don't tell you what a function is, you'll think it applies to everything, or only to a part of things.

[00:05:48]So first, I'm going to define what a function is. And after I define what a function is, then I'll talk about what a linear function is.

[00:05:57]Before you think it's rocket science, it's not very complicated.

[00:06:09]I'm trying to open it here on the tablet, but [clearing throat] everyone, did you have any questions about any past class? We'll have class 00 on Friday too.

[00:06:37]It's not satanic content, nothing like that.

[00:06:41]It's a beautiful subject. You'll see today. I'm opening it now, it worked.

[00:07:07]I'll share it here, Sabac, see if it shows up for you now.

[00:08:15]So, everyone, today's class is basically about first explaining what a function is so I can then talk about linear functions.

[00:08:25]And as Antônio very well said... Here in the comments, our beloved physics professor, functions, are intimately linked, especially with physics. Physics has the sine function, cosine function, trigonometric functions, wave equation. So, when you throw an object, gravity is a second-degree function. When you have velocity, time, the graph is a first-degree function, generally when it's a movement with constant acceleration and constant deceleration. So, knowing this, to understand functions, helps you solve math problems. Yes, but it also helps you in physics, chemistry, I think there's something about graphs, I won't remember because it's just a negation in chemistry. Thaís, correct me if I'm wrong.

[00:09:16]And to start talking about functions, first, where does the function originate? A function is a relationship between two quantities, where the second depends on the first, that is, it 's something that is related, which we will call first X and Y.

[00:09:30]But this idea didn't appear ready-made; it was built up throughout history. But who started this story of functions?

[00:09:39]It starts first in antiquity. It all begins with Mathematics in antiquity. Few things are recent.

[00:09:44]Greek mathematicians like Cleis, the one with the little book that's the second most translated book in the world, *The Elements*, were already studying what? Geometric relationships and proportions. And a function is nothing more than, depending on the function, like a first-degree function, it's intimately linked to proportion. Because if you increase one quantity, the other increases. If you increase one, the other decreases, decreases, decreases, if it's decreasing or remains constant, as we'll see later.

[00:10:12]These relationships were used to measure what? Land, to build buildings, to understand the cosmos, things they were discovering at the time. There wasn't yet a formal idea of function, only correspondence between values. It's as if they had little tables; they didn't know what a function was, what each one did. They created a table that was something like this: "Ah, for one this is worth three, for two this is worth five, for three this is worth seven." And that over there, ah, but what is this? It was a table that worked, and there was a correspondence between values that worked because that's what they wanted, right, guys? Sega is there to collect taxes, whether it's for selling land or for whatever reason.

[00:10:54]Here comes René Descartes, a name that shouldn't be unfamiliar to you. Descartes comes from the same source as Cartesian, our Cartesian plane.

[00:11:04]Why is it so important for us to talk about the Cartesian plane? First, our functions are represented on Cartesian planes, which are what?

[00:11:16]Here, where my horizontal axis is X, the vertical axis is Y, and they are orthogonal.

[00:11:27]Professor, what does orthogonal mean?

[00:11:29]Orthogonal means it forms a 90º angle, right, everyone?

[00:11:35]And he creates analytical geometry, which we will study later on, that uses coordinates to connect water and geometry and shows that curves can be represented by equations linking two variables, which will become functions later on.

[00:11:49]This formed the basis for us to begin thinking about functions as mathematical relationships.

[00:11:59]Then comes libon, and calculus is intimately linked to functions.

[00:12:07]They create the calculus by studying how one variable changes in relation to another. And what was its function, nothing more than that? A variable that is changing, moving, flowing. Because the function, the original name of the function was fluent.

[00:12:22]Why? Because the function flowed.

[00:12:26]because it varied through that fluency. And it starts to give the idea of dependent and independent variables.

[00:12:34]In Portuguese, "como" refers to a dependent relationship, meaning it is dependent on another. And the independent woman, she has nothing to do with that person who is dependent on her. Ah, ah, are you an addict? I am not, I am independent.

[00:12:49]And the function begins to take shape as a rule that associates values. Remember when I mentioned in the previous slides that there was a small table, that there was an association of values, but that they didn't have a rule of formation? No, if you changed the function, they wouldn't understand. They had a table that worked. It was only with Labnis and Newton that we began to put the idea of ​​function into practice.

[00:13:15]Then Euler comes along and uses the notation system that we know today, which is what?

[00:13:21]Leonard Euler gives a clearer definition of function. He first introduces the notation f(x), as highlighted here, which reads as: F dex. Professor, why f? F because the function starts with f, but I have g of x, h of x, a of x, x of x. We usually refer to the function as f(x) because that's the easiest way to associate it with the function. And for him, the function was an expression that relates x and y, where y is what we'll call our f(x), which is our result.

[00:14:00]Okay, everyone? Any doubts so far?

[00:14:07]So far I've only told stories. I'm just pretending to be Will Marcos here. And now comes this guy called Diri who talks about functions in general. That's where it begins, the child cries, the mother doesn't see. He generalizes the concept that a function is any rule that associates each x with a unique y.

[00:14:51]Julio, which part should I repeat? When you ask for a repeat, to clarify a doubt, you can pick up the part and say it so they can, otherwise we have to wait for you to answer and it takes a long time. This allows the inclusion of functions that are not given by formulas, even discontinuous functions.

[00:15:07]What is a discontinuous function?

[00:15:10]It's a function that can be like this, a step function that jumps in small increments.

[00:15:15]And that's an important step for mathematics. Or it could be a function like the tangent function, which is like this: that it is a discontinuous function. He simply... what does he just say?

[00:15:31]The clearest definition of function and what does it relate to? Two letters that represent the variables X and Y. X is what you will put inside the function, and Y is the result of the function, which is what we will call f(x).

[00:15:49]Guys, the XY confusion doesn't need to happen.

[00:15:53]Generally, X is the value you're taking and putting inside. Y is the one who receives the full amount of the bill, understand?

[00:16:13][clearing throat] And I can't talk about function without talking about sets. What does the whole thing consist of?

[00:16:25]A set is a collection of well-defined and distinct elements. In other words, what is the set? It's like a little box.

[00:16:39]A little box, and what do I put inside? The things I want to use, the things I want to keep.

[00:16:51]And the elements can be numbers, letters, objects, or anything that can be identified. It's like me thinking, oh, in my living room at home, it's a living room set, there's the TV, the sofa, the china cabinet, the table, six chairs, one chair with a broken leg, you know, and the coffee maker.

[00:17:18]So, everything there belongs to the living room set.

[00:17:22]If I go to the bedroom set, I'll find an object from the living room. Only if I take the one from the living room set and put it in the bedroom set. Does everyone understand this?

[00:17:31]An object is only considered part of a set if it is also part of that set. If he is outside, he does not belong to that group.

[00:17:41]It's like me saying, "Oh, I'm going to buy a car from the BID brand. Will I be able to buy a brand new BID car at a Ford dealership?" I am not going. I'm not going to the GM dealership. Not at the competing dealership in GL either. Where can I find the zero BID?

[00:18:02]He needs to be here at the BID dealership, which is the Bad car dealership. Where is it? At Bia's dealership, which is the whole complex.

[00:18:13]And the elements, as I said, can be numbers, letters, objects, or anything that can be identified.

[00:18:20]But when we're talking about functions, we're only going to use numbers. But I'm talking about sets in general. First, you need to understand what a set is before I can show you where the set comes into play in the function.

[00:18:38]For example, here's an example of a set, set A: 1 2 3 4. If I ask, is 5 in A? No, it's only in this A here, look.

[00:18:51]The set B, A, B, C.

[00:18:56]The set C, red, blue, and green. If I ask, does yellow belong to set C? No, it's not within the set, so it's not defined, therefore it doesn't exist in set C.

[00:19:08]Better, Lidlândia.

[00:19:13]So, what exactly is the function, everyone? A function is a special relation between two sets.

[00:19:22]So I'll have a set here that I'll call A. There's a set B here, and F is what takes something from set A and brings it to set B. But can I put this set any way I want?

[00:19:41]No, we'll see that each set will have a name, a function, and, let's say, a function, bam. One function turned out great. Who's talking about function? I'm talking about a function, a mission, let's say. He has a mission, which is what? to store those objects that are inside it. Each element of the domain, professor, what is a domain? Domain is where you are in control. What is the domain? It's set A.

[00:20:23]When I say "dominio" in Portuguese, it comes from "dominar," meaning to dominate, to control. It's who you control to get into your role. It's as if group A were the doorman at a party. And the party is the function.

[00:20:38]Only those whom the doorman, who is in group A, allows can enter the position. He dominates whoever enters there. So the domain is the one authorized to play with the function. Does everyone understand what a domain is?

[00:20:58]I think people are sleeping in class, there are only four people online. I'll even open YouTube here to see how many people are online.

[00:21:29]Guys, there are only 32 people watching.

[00:21:31]You don't want to fail math, a subject of such great importance.

[00:21:39]But let's go. We have 32 people wanting to pass.

[00:21:46]And each element of the domain is linked to only one element of my codomain.

[00:21:53]In other words, the domain will have an arrow pointing to someone on the counterdomain. Professor, what is a codomain?

[00:22:03]Counterdomain. If here, within the set, we have the domain, then the codomain is what opposes it, which is where my function leads. The domain is where the aircraft takes off from, and B is where it lands. In other words, F is the function, it's the little airplane that goes from one place to another.

[00:22:27]In other words, the function associates an input value with a single output value.

[00:22:35]For example, in the function f(x) = 2x + 1, for each number x there is a unique result f(x). This class goes until 7:30 and then we'll have class with Professor Pat, if I'm not mistaken, right Sab?

[00:23:00]It goes until 7:20 and then there's the part. But as you mentioned, there was a problem, which caused a slight delay. If you need to send it so you don't miss the explanation, that's fine, right? Whatever happens, I'll ask fewer questions, and then we can clear up any doubts with them. But I'm trying to get things moving here in a rush, don't worry.

[00:23:21]In other words, for each example of the function f(x) = 2x + 1, we have a single result f(x). If I change my f, the result changes. If I change x, f(x) changes. The f of 1 is different from the f of 2, which is then different from the f of 3. Each x you put there will return a different value.

[00:23:43]Functions are used to model various real-world situations where one variable depends on another. A good example is when you go to buy a savory snack. One savory snack costs R$ 8, two savory snacks cost R$ 16, and three savory snacks cost R$ 24. This is a function. And what is that function? f de, sorry, f dex = 8x, where x is the quantity of savory snacks and eight is the price of each snack.

[00:24:07]Ah, but then you stop and think: "Ah, but I'm not going to use a function for that in life. You're using it in ways you do n't even know, that you don't even feel."

[00:24:17]Mathematics is present in your daily lives in an incredible way, and you don't even realize it's right there, close to you, embracing you from behind, and you don't even see it. he.

[00:24:31]But who is this so-called domain, counterdomain, image?

[00:24:37]Domain is like the menu, the list of options available for you to choose from.

[00:24:44]In other words, the domain is when you arrive at the restaurant and you have the option of choosing from the dishes.

[00:24:53]Then you'll choose a dish from the menu. At a Japanese restaurant, you can choose a dish that isn't on the menu.

[00:25:01]No, because it's not within your control.

[00:25:04]You can't use it. The counterdomain is the group of customers who are in that restaurant.

[00:25:11]The image shows the group of customers who received a dish. Professor, but I can order, for example, picanha at a steakhouse, and another professor will choose picanha. It will also be two identical dishes. Yes, I ordered picanha and I ordered picanha.

[00:25:28]But to have the same dish coming out of the kitchen, you would have to duplicate that dish perfectly. He would have to use the same meat, the same cut of the same cow, with the same dish. He makes two identical dishes.

[00:25:44]Picanha for Diego, picanha for Rodrigo.

[00:25:47]But the two dishes were ordered differently; they have the same configuration, but for example, if you add 10g more meat, it becomes a different dish. Are you all making sense? And what is its function? The waiter who will get the dish from the kitchen and take the customer's order. Each dish on the menu that comes out of the kitchen can be ordered by multiple customers, but each customer will only receive one dish at a time.

[00:26:18]To change the plate, he takes the plate off the table, puts it back, and gets another plate. That's where the formation law will change.

[00:26:27]I just had lunch, folks, so you have an idea, but I'm talking about food, because food, money, is what you can understand.

[00:26:40]Everyone can understand what the domain, the counterdomain, and the image are.

[00:26:45]The domain is who I am here, look, who will leave for the counterdomain.

[00:26:50]The image, oh, that smaller yellow set is the one that has a connection to the domain.

[00:26:58]In other words, the codomain can have objects here, as you can see, that are not related to the domain, but they can still belong to the codomain.

[00:27:12]One more thing, look at this example. And now 2x. What does this function do? Double the value of my function of my x. Zero, double zero, zero. Double one is 2, double two is 4. Double three is 6, double four is 8. But one, three, five, and seven may belong to my domain, but they don't belong to my image set. Image is only for those who have what? Something connecting him to the domain. If nothing binds him to it, and the function hasn't affected him, then to whom will he belong? To mine, to my image. It belongs only to the codomain. He's lost over there.

[00:27:59]He didn't enter the VIP area of ​​the condominium, which would be what? The image. It's like going to a party where everyone can get in, but there, the image is of a VIP group, where it's more selective, with people who have a relationship with the domain, saying: "Hey, he's my best friend and he's doing me a favor." Oh, beautiful? Something that everyone can understand.

[00:28:29]And this applies to first- degree functions, second-degree functions, trigonometric functions, sine functions, cosine functions, constant functions, functions, functions, functions, trigonometric functions— everything will depend on the domain, codomain, and range.

[00:28:47]If anyone has a question, speak up now.

[00:28:51]She solves it similarly to the equation, Giovana. It's only slightly different, but I'll show you that the equation is quite straightforward. If you want to find the value of x in the function, you usually already have it; you just plug it in to be used for the calculation.

[00:29:26]So what is a function and what is not a function?

[00:29:40]A function is when I have a relationship within my set. Oh, always remember where the arrow starts from? Yes, I can, Gabriele. " Dominion" refers to who you will control in order to get into the party.

[00:29:55]It's kind of like there's a waiting line for you to get in. You're in, you're in control, you're part of the party.

[00:30:04]Whoever is having fun at the party will receive a little arrow, which is part of a function, and go to the VIP area, which is what? The image. Whoever is just dancing there, they entered the party and are waiting for something, they're in the wrong place, but they're not in the picture.

[00:30:23]The image is the one that has a relationship with the one in the domain, with the function arrow here, see, the blue arrow.

[00:30:31]This guy's connected to this one, those guys in this little group here, look, that 's the image. What is the image? It's the person who has a connection to the domain. And who is this connection? The function. And can anyone be in the opposite domain? It depends on who I defined as the codomain.

[00:30:56]Did you understand, Gabriele?

[00:31:33]Okay, Gabi, so what is a function and what is not a function according to the diagram? The function is when group A, which is my domain, is linked to only one.

[00:31:50]I can't have it like it is here, look. See?

[00:31:54]This one is connected to two, look. That's not possible, that ceases to be a function, professor. But this guy has two calls to this side. I usually say that it's a relationship that keeps growing.

[00:32:10]A domain can only have one link to a counterdomain, but a domain can receive whomever it wants.

[00:32:18]He's like this: "Look, I'm a bit of one, a bit of another, maybe two, maybe three, it all depends." Hey everyone!

[00:32:28]In other words, what function is it? What can't happen?

[00:32:32]No one can be left in the domain, and no one from my domain can go to two places at once.

[00:32:40]Right? Remember when I said that the element has a single X- Y bond? The letter Y can have more than one connection.

[00:32:50]Not X. X is monogamous. He only has one relationship.

[00:32:57]Can everyone understand this visual part? It 's not the other way around, Marquinho.

[00:33:07]A is monogamous, B can be polygamous. In other words, the domain is monogamous, and the codomain is polygamous.

[00:33:31]Alright, guys?

[00:33:34]And how will I be able to classify my role?

[00:33:39]First, we're going to classify it in four ways: injective, surjective, bijective, simple function.

[00:33:49]What is a surjective function?

[00:33:52]When all elements of the codomain, my second set, are affected by at least one element of the domain, that is, there are none left over.

[00:34:02]In other words, my image set is equal to the codomain. In other words, both sets have the same size, the same elements, everyone is being used. A codomain element can be associated with various, sorry, several elements of my domain.

[00:34:19]As you can see here, there's nobody left here on Y. Look, there's more than one connection, look, with rail men 3. That doesn't cause any problems.

[00:34:31]And what's the detail? There can't be anyone left behind. Is there anyone left over? No.

[00:34:38]What is the injective function? Each element of the domain has a unique and distinct image, monogamous.

[00:34:46]This means that different elements of the domain have different images. So I won't have repeated numbers there.

[00:34:58]In other words, no element of the codomain is the image of more than one element of the domain. In other words, it won't be a case where there was a nine with an image containing two domain elements. It's one on one.

[00:35:12]Tarzinho, everyone's jealous. No, you only go out with me, I only go out with you.

[00:35:22][cough][clearing throat] What is a bijective function?

[00:35:29]It's when it's both injective and surjective at the same time. In other words, each element in the domain has only one element in the codomain, and there are no elements left over in its codomain.

[00:35:49]A simple function is one that is neither injective nor surjective. In other words, there will be leftover people in your counter-domain.

[00:35:59]So she won't appear on the injector machine. There will be some element like -3 and tr going to prove it. In other words, it wo n't be able to be injected because I have one image for two elements. So then the argument about the injection molding machine falls apart, and since that's what's left, we're left with the argument about the over-injection molding machine. In other words, it is neither injective nor surjective, and certainly not bijective, because to be bijective, it has to be both injective and surjective at the same time. So what is a bijective equation?

[00:36:30]I am injective, I am, I am injective, I am surjective, I am, therefore I am binjective. And when they come together, it's as if they form a magical combo, becoming more powerful.

[00:36:41]That's mostly theory. Why am I talking about this? Because An sometimes says: "Ah, an injective function, a surjective function". This is just so you understand what each thing is and don't get lost during the ENEM exam, guys. Quiet.

[00:37:22]Alright, guys?

[00:37:26]And how does the function work? I really like to joke that she's a meat grinder.

[00:37:33]I think there are classes on this holiday, yes. It's just not available in person, it's available online.

[00:37:37]Sabaque, there's class on Friday, right?

[00:37:40]As far as I know. There's school on Friday, and tomorrow is a holiday.

[00:37:48]So, what exactly are functions? A way to connect input values ​​to their output values. In other words, I'm joking that it is what? It's a meat grinding machine. You put the meat in the X there, the machine grinds the meat here, and out comes the F of X, which is the ground meat. Because, for example, if you put a duckling out, what should come out on the other side? ground duck.

[00:38:14]If you put in bilberry, you'll get ground bilberry. If you put in muscle, you'll end up with ground muscle. I can't, for example, put in a penis and expect it to come out; I mean, I can't put in a chest and expect it to come out the other side. She will, she will process the information you put into the domain. So the domain is the secret to making it work, because only those who are on the domain will be able to access the machine.

[00:38:38]Because if you don't have an object in the domain, it won't be able to be manipulated by the function f.

[00:38:45]For example, ordering a coffee for 5. One coffee is R$ 5.

[00:38:49]10 25. Ah, 50 coffees. 50 x 5, 250. That is, the input affects the output in a predictable way. There is a predictability.

[00:39:03]And who provides this predictability? The function, does everyone understand it?

[00:39:41]So, folks, what exactly is the graph of a function in a table? Remember when I talked about René Descartes? I picked it up and talked about the Cartesian plane.

[00:39:53]Our Cartesian plane is this one right here.

[00:39:59]Let's remember. What does the ENEM exam sometimes talk about?

[00:40:06]From abscissa to ordinate. What is abscissa? The abscissa is X. The ordinate is Y.

[00:40:14]Professor, how am I supposed to remember which is the abscissa and which is the ordinate? My head is spinning, I can't remember.

[00:40:23]My tip for you is to list them in alphabetical order: who comes first?

[00:40:28]X or Y? X. Which one comes first in alphabetical order? Abscissa or ordinate?

[00:40:34]Abscissa. So, X is abscissa.

[00:40:39]Y is ordered.

[00:40:49]Okay, everyone.

[00:41:16]Yes. And this concept of function that I've been talking about so far applies to all functions: linear functions, first-degree functions, second-degree functions, third-degree functions, exponential functions, and so on. As Marquinho said, it applies to everything because the abscissa is what's on the X-axis, and the ordinate is what's on the Y-axis. Okay, everyone?

[00:41:44]And for every value of x, I will have a corresponding value of y. Professor, but how am I supposed to guess the value of f(x)? [clearing throat][cough] You're not going to guess.

[00:41:58]You're going to play him within the role.

[00:42:00]In this case, [clearing throat] what is my role? [cough] [snoring] What is my job? Y = 2x. In other words, it takes the value of x and does what?

[00:42:18]Double it. So, x = -1 will become -2. x = 0, y = 0.

[00:42:26]x = 1, y = 2. x = 2, it 's 4. It will always double. If I set x = 10, then y would be 20. If x = 80, then y would be 160. Does everyone understand how to find this little table of values?

[00:43:35]Guys, did everyone understand or did my audio come out muted, man? I think everyone's already in holiday mode.

[00:44:07]And what happens? This will all turn into an ordered pair. What is my ordered pair?

[00:44:19]So, Alessandra, what is this part?

[00:44:21]You're going to take the value of x and put it where x is. So here, when I set y = -1, what do I do? 2 x -1. Who will this lead to? -2.

[00:44:35]2 x 0 2 x 1 2 2 x 2 4 Ah, but if I changed my function to 10x, then you would take the -1, it would become -10, the 0 would become zero, the 1 would become 10, the 2 would become 20. Can everyone understand that?

[00:45:04]If I change the function or change x, the values ​​will change. So what do I do now? Every point here will become an ordered pair. For example, the first one will be -1 - 2 0 0 1 2 2 4. Professor, can I change the position? Never, because an ordered pair follows a rule of formation, which is what? First comes X, then comes Y.

[00:45:51]First comes X, then comes Y.

[00:45:55]Why? Because first you're marking on the X-axis, then you're marking on the Y-axis of your Cartesian plane.

[00:46:09]Okay, everyone?

[00:46:12]And what happens? When I'm going to mark the point, I'll start here at 0, first point -1 - 2, -1. I'm going to walk to my left, come here. And now I'm going down two.

[00:46:28]So I'm going to mark this first point here, see.

[00:46:34]And point 00, look, here, look, point 1 2. I'm going to walk here to one, I'm going to go up two, look.

[00:46:47]And point 2.4 is this one here. Anda 1 2 1 2 3 4.

[00:46:56]It's simply marking a point on the Cartesian plane. And after you score that point, what do you do?

[00:47:07]Connect them, and that will be the graph of the function you're working on. So whose chart is this? From y = 2x.

[00:47:17]Can everyone understand how I create a graph from a table, or can I do the reverse as well?

[00:48:00]There are 39 people online. Only two people are responding. Three now.

[00:48:06]Man, people are pretending they're paying attention in class.

[00:48:11]The graph, guys, what am I doing? I'm going to replace the values ​​here, which is...? -1 0 1 2.

[00:48:24]And then I'll substitute it into the function.

[00:48:26]What is my role in this situation?

[00:48:30]2x.

[00:48:31]So what am I going to find out?

[00:48:34]Who is -1 connected to? With the -2. 0 with 0, 1 with 2, 2 with 4. And they will form these ordered pairs here. Everyone managed to understand ordered pairs up to this point in the ordered pairs graph, forgetting about this part here.

[00:49:09]So now, starting from " Isso aí," you don't even have to start with the old exam. Then write down which test you've already taken so you don't get lost. Now, let's mark the point. Scoring the point is the easy part, oh. I 'm here, at this point, -1 - 2.

[00:49:29]I'm going to look for the -1 on the x-axis and the -2 on the y-axis. And, look, you see? Where they meet is at this point right here. The zero here.

[00:49:45]1 2 1 2 here, look.

[00:49:51]2 4, oh, 2 4. And where they meet, mark another dot. Did everyone understand how to find the points?

[00:50:49]Now that you have the points, what are you going to do? They'll call them.

[00:50:56]And when you connect them, this graph here will be the graph of the function, which in this case is...? The graph of 2x.

[00:51:08]And that's all, folks. So far so good?

[00:51:50]Yes, Silas will be there.

[00:51:54]And there's a vertical line test to determine if it's functional or not. If you draw vertical lines, what are they? Straight impact lines. What is vertical or horizontal? What does horizontal always remind you of? Skyline. You look at it lying down, vertical line standing up.

[00:52:12]When you draw vertical lines like this, it will only cut off your function in one place.

[00:52:23]If this happens for any vertical line you draw, then your function is the function. Now, if this happens here, look, I cut it in two places, it's not a function, look. Ah, professor, but if I cut here, it will only cut at one point.

[00:52:39]But if a straight line is cut in two, it ceases to be functional; it's already broken.

[00:52:52]Alright, guys?

[00:53:05]It's a simple test. You'll have the graph, you'll draw a vertical line.

[00:53:08]Where you see that it will have two little pieces that, when you draw a straight line, will meet, it ceases to be functional.

[00:53:19]Marquinho, it's like, for example, my job title is here. If you have a piece of land that has two positions, and I draw a line that connects to the top and bottom, then it's considered a function. Why?

[00:53:30]Remember that she's monogamous? The same x cannot have 2 y's.

[00:53:36]If she cuts it in one piece, it's still functional. But if a point somewhere in the entire function intersects two, it's off.

[00:53:42]It lost. It's as if she broke the rules of the game.

[00:54:03]Here's the little mind map. Then I'll send this material to Sabac to put on your drive, which is what? I'm saying, summarizing everything I've said, that it's the relationship between sets A and B, that the function connects from A to B, that it's an injective function, that it 's surjective, that it's invariable, that it 's composite.

[00:54:30]And the image, oh, the domain here, oh, is what you grab here, oh, in X. The image is Y. It's like I'm taking it and thinking, what?

[00:54:47]If I pick up and drop my function here, look, where it's falling, look, in this little piece here.

[00:54:57]If I drop it this way, look, where is it falling?

[00:55:13]Guys, you need to study a little at home, right? Just attend the classes. It 's not difficult. I'll show you.

[00:55:28]And the first-degree function, it has a face, a characteristic, which is what?

[00:55:34]It is of the type f(x) = x + b. That's all, meaning it's someone who follows x. No, that's just any function, Marquinho, not necessarily an exponential one. That's just an example, where the coefficient of X is what we'll call the slope, which is A.

[00:56:15]And B is the constant term. Remember what I said?

[00:56:20]Yes, the function can be a weird, curved function, folks. I know I'm going to switch here to show them how a first-degree function works, okay? I'm going to remove this presentation. I 'll pull another one here.

[00:57:09]Can you project it onto the screen for me? Sabac, put my little face down there.

[00:57:14]Oh, folks, this is it, a linear function is just a straight line. I didn't say it was an affine function. I just gave an example of what it is there. A linear function, a first-degree function, will always be a straight line. He looked at the graph; it's a straight line. It's an affine function, as Marquinho said.

[00:57:36]Notice that here I'll have Ax, and I'll have B.

[00:57:49]Let's see what each element does in the function. First, I'm going to work with B.

[00:57:54]See if you can figure out what B does to my function.

[00:58:22]Did anyone notice what B is doing?

[00:58:57]That's right, Marquinho. Guys, take the advice that Marquinho gave. B is where it intersects the Y-axis. If I set B = 3, it will intersect at 3. If I set B = 2, it will intersect at 2.

[00:59:09]If I set B = -10, it will intersect at -10.

[00:59:24]So what happens?

[00:59:28]We know that the letter B already gives you a hint.

[00:59:35]So, Marquinho, now is where I'm going to give you the inside scoop. What does A do? The 'A' in my function, remember that it was the slope? I'm going to shake things up and we'll see if you can spot who's shaking it over there.

[01:00:06]So, what does A do, guys? Did anyone manage to fix it?

[01:01:09]Comments are blocked.

[01:01:13]What does the letter A change? the slope of your line. If A is positive, its line increases. And the bigger the A, the faster it grows. If A becomes zero, my function becomes a constant function. For every x I multiply there by 0, 0 x 1, 0, 0 x 2, 0, 0 x 5, 0, 0 x 1000, only b will remain. So, when a equals 0, it's a constant function like this, see?

[01:01:45]Its value doesn't change. It's a dead function, let's say, when the little graph looks like this.

[01:01:55]And, look, if I have 'a' equal to 0 and change only 'b', look what happens, folks!

[01:02:02]That's exactly where she cuts, look. I remember, oh. If I put it here, -3 is at -3. If I put it at -2, at -2.

[01:02:11]If I put zero, look, it stays on the x-axis. If I put it on three, it stays on three. If I put it here, look, in number two, look, I'm going to move the A. I put the A as negative, look.

[01:02:23]The more negative the angle, the more inclined it will become.

[01:02:30]Then I'll make her more positive, you see.

[01:02:32]The more positive it is, the faster it rises.

[01:02:44]Did you all understand what the letter A does in a first-degree function?

[01:02:47]What does B do?

[01:02:50]B is where it cuts. That's what I'm saying, it's like an elevator, it goes up and down according to its function. Oh, if I change the B, notice, the slope of the line doesn't change.

[01:03:03]Oh, the line remains the same. All he does is what? He only walks up and down where she cuts, look.

[01:03:10]But the line itself retains the same slope. Can everyone see this?

[01:03:16]Yes. If your graph is cutting off, going up, it's positive. If it's cutting off, going down, it's negative. And B is where it intersects the Y-axis.

[01:03:29]Had you guys noticed that before?

[01:03:31]Marquinho, I've already seen this.

[01:04:02]Guys, X is the one lying down here.

[01:04:08]Y is the one standing, look, X is here, look. It's what you'll input into the function, and the result will be y.

[01:04:52]Let's do some exercises to see if it gets easier for you guys.

[01:04:56]THE.

[01:05:01]In a digital game, there are three characters: one hero and two villains. The programming is designed in such a way that the hero will always be attacked by the villain who is closest to him. One way to confuse the villains is to move the hero along trajectories that keep him equidistant. What equidistant? At the same distance.

[01:05:23]And that means equal, distant, distance, creating an ambiguity between them. In other words, I attack, you attack, I attack, you attack. Since they're banging their heads against the wall trying to figure out who's going to attack, you keep advancing.

[01:05:35]For programming one of the stages of this game, the programmer considered the square Stuv, which has all sides equal, on the Cartesian plane as the region of movement for the characters, where V and T represent the fixed positions of the villains. In other words, one villain is here, another villain is here.

[01:05:58]Then he wants to know the equation of the trajectory in which the hero can move without being attacked. For him to move, what does he need to be in?

[01:06:11]At the same distance. And what does that same distance even mean?

[01:06:19]He walks here because he will have the same distance on one side, the same distance on the other. He is walking diagonally across the square.

[01:06:28]Professor, but how am I going to find the equation of this line?

[01:06:32]You already have a point here, we just need to figure out what that U-shaped point is.

[01:06:42]I'm going to do it here in an easier way.

[01:07:04]No, it can only be A, B, or D, because your line is like this, look. Let's think about this. What's she doing?

[01:07:15]Decreasing. So what do I need to see then?

[01:07:20]It will intersect at -20, +16, or +20. Here's a tip: your line will intersect the y-axis upwards. So, it can't be this one here where it would have to be crescent-shaped, it can't be this one here where it would have to be cut down here, look.

[01:07:38]I'm standing between A and B.

[01:07:47]So how am I supposed to figure out where it cuts? You need to define where my role is.

[01:07:53]From here on, it turned into eight, look.

[01:07:57]So, how big is this thing? Two.

[01:08:00]Two became six. So, how big is this thing? Four. I walked two this way and climbed four. That's why you're in class today, Marquinho, so you don't kick, so you can learn.

[01:08:15]And everyone here will have the same triangle pattern, see? 2 4 2 4 2 4.

[01:08:24]So, to walk here to my U, I'm going to climb two on whom? Standing. It's y, here it's X. So, what will my U coordinate be? I was at point 8, so I went up 2, then 8, and I'll go back 4. I was at point 8, so it will be 4. Therefore, my line will have to pass through the points 4 and 8, 6 and 2. When I put this in, it has to be -x, because your line is decreasing, see? She's coming from the top down.

[01:09:15]Yes, Marquinho, it was personal. That's why it has to be negative.

[01:09:21]Remember what A does? Oh, the A is the one that goes with the X, oh.

[01:09:28]So I'll go, my job has to be of the first level. FX = x + b. Here is x, here is y. So what am I supposed to do?

[01:09:39]A x 4, which is x + b, has to equal 8.

[01:09:46]A x 6 + b has to equal 2.

[01:09:51]I'm going to subtract this from that. What will remain? 2ab cancels out with b - 6a = -6/2 which gives -3. In other words, my function is of the form f(x) = -3x. We still need to find out what b is.

[01:10:13]Marquinho, I located the 4 and eight, and I know that from here to here it will go up two, look. Because here he walked two, here he walked four. Here he walks two, he walks four. Walk two, walk four. Move 2, move 4.

[01:10:33]So, if I'm here at Y6, I go up 2, it becomes 8. I'm at eight, I go back four, it becomes four.

[01:10:48]There's a pattern there. It's as if your square has been twisted.

[01:10:56]So now that you know, you can choose this equation or this equation here. I'll choose the first one, it's the prettiest.

[01:11:05]A x 4 + b = 8.

[01:11:08]So, -3 will be in place of a x + b = 8, which will give -1 + b = 8.

[01:11:17]b = 8 + 12 = a 20. Therefore, the answer can only be A.

[01:11:34]This is a tricky question because you have to remember geometry and look at the function.

[01:11:42]But why did I put it there? So you can understand how it works, and not be surprised during the test.

[01:11:57]Question two is easier.

[01:12:04]Unfortunately, there's no escaping the consequences of this role.

[01:12:07]Function is where Sabato and I get sad because we can't apply our religion with complete mastery.

[01:12:17]Yes, I'll ask Sab to put it on the drive. Guys, today is awful. I'm forgetting the names of everything. I lost track of time.

[01:12:30]Oh, this here, look. A school analyzed proposals from five companies to rent a photocopier that could meet the demand for 12,000 copies per month. Each company charges a fixed amount, which is this one here, a fixed amount for the rental plus an additional amount for the copy, which is what? She charges 40 cents for the copy. A copy costs 50 cents a copy, 20 cents a copy, 25 cents a copy, 30 cents a copy. So what are we going to do? To find out which one has the lowest cost? What is it? I'm going to take it and put 12,000 where the X is. It will add 500 here. Here's a tip I'll give you: when you do something, multiply this by 12,000, it's 1000. Think of this as this, see?

[01:13:31]You break down 12,000 as 100 x 120.

[01:13:35]Because when you take this here, it becomes... now that the firefighter has passed, I can continue speaking.

[01:13:50]What are you going to do? You're going to take it, move the decimal point two places, and then you take it and eliminate the zeros here.

[01:14:01]That'll be 40 x 120, which will give you 4800.

[01:14:12]Here it will be 2000 plus 2400.

[01:14:49]Marquinho, unfortunately there's no way to call the fire department because I'm using it, doing the calculation. Look how absurd that is.

[01:15:01]I was so religious that they had to call the fire department. So the chosen company will be company number four, which is the cheapest. All I had to do was put it there? What is this question? Just what? Place the value inside the function, calculate it, and see which one is the smallest.

[01:15:42]Okay, I'll just do this one for today so I can wrap things up and not interfere with the next class. A company produces school backpacks to order, blah blah blah, it has a fixed cost. Remember that a first- degree function, what is it always?

[01:16:01]ax + b.

[01:16:06]Giovana, what did we do then? It had a print run of 12,000 copies.

[01:16:12]Wherever it said X, I put 12,000 and did the calculations.

[01:16:17]Hey Giovana, how's it going? Do you understand?

[01:16:29]But depending on the situation, I'll see with Marco if we can do another class just on solving problems from this subject. Alright, guys?

[01:16:40]Edlândia, message me privately later and I'll answer your question from the previous post, okay? What are numbers? I repeated the function here, look, 500, and this is times 12,000, which gives 4,800.

[01:16:57]Those are the calculations. Did I understand, Giovana?

[01:17:18]No, right?

[01:17:27]But can you understand what that is?

[01:17:29]The first example is right there. 500 + 0.40x.

[01:17:34]What did I do? 500 + 0.40 x 12,000. This will give you 4800.

[01:17:48]So this will give you 5300.

[01:18:04]Do you understand now, Giovana?

[01:18:25]Guys, look, I'm going to talk to Marcos so we can do this somewhat complicated story. You have a lot of difficulty replacing them. Let's dedicate a lesson solely to taking and calculating, to doing these questions from list four. Let's not jump to the next topic, otherwise you'll get confused.

[01:18:44]If you have any questions, feel free to ask me.

[01:18:46]I'll post the list, and ask Sab to post list four in the group for you guys, so you can grab it and deal with it calmly. If you have any questions, feel free to contact me. But I want to teach a problem-solving class. I'll talk to Marcos because this matter is quite complicated. You have to give the theory first before talking about the issue.

[01:19:04]Alright, guys? If anyone has any questions, feel free to message me privately, I'll help, okay?

[01:19:11]And a thousand apologies for the delay, guys.

[01:19:17]Bye, everyone. Until the next class. I'll see when I can schedule that lesson with Marcos, now that I'll be more relaxed, he'll take a break, okay? Hey, I want to see the class fuller next time, okay? You all look tired today. They took the holiday and extended it, right?

[01:19:34]Bye everyone, see you in the next class. Sab, I wo n't keep you waiting too long so you can finish up and start the next class, okay?

[01:19:46]Okay, I'm already here waiting.

[01:19:50]Okay, bye, Sab. Ciao.

[01:19:52]Ciao.