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Added Mar 22, 2022 •

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Functions

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alrighty guys welcome back to another hiset math lesson three were talking about functions lesson ahead of us is going to beat identify function and key characteristics of functions all right let's get started let's start by talking about ordered so they letting us know that in order for us to understand what the function is to understand what is 1, 30 seconds so that's what those numbers represent x value on the left and the Y value on the way Daniel I cannot hear you Daniel another one all right all right you guys aren't any better now are you better now all right okay so you guys should be able to give me a little better now okay alright so continue so we're talking about relations and functions so we're still we still haven't gotten to talkin about functions that we were talking about everything else I need to understand first in order to understand the function so stay here sets of ordered pairs are often written in set notation which means the individual points are separated by commas and the clothes by set of brackets so these are all individual points right this one is 1 points the second one is another point in a third one is another point right this is a set of ordered pairs written in set notation a set of ordered pairs it's called a relation so this is one of the new term relation right now so get the definition of the relation is every set of ordered pair and here's the catch some relations are also functions if a relation is a function each unique x value has a single unique y value so meaning that each each number as a unique output meaning for every number I put in in ft lb the X I have a unique output meaning why so they don't share numbers for example like this one down here is not a function because we have the same input corresponding to two different outfits for the same input would be the number to the number to corresponds to 4 and the number to course 7 we can't have that right each x value can only correspond to one y value talk about another concept related to function this is called domain and range so this is basically the spectrum that is covered either horizontally or vertically show all of the X values refer to the domain and again the domain and that's obvious those are the first numbers in the pair right domain runs left to right the range refers to all of the wives I'm the white valleys are the second number in the pair range runs up and down it says the domain and range are also commonly written in set notation such as ear and here I am since the domain and range are not ordered pairs there will be no parentheses so you notice the difference between the ordered pairs up top versus the domain and range write the ordered pairs have parentheses because they are representing a specific point on the coordinate grid the domain and range do not because they're not representing points on the grid representing only the X values for the domain in the Y values for the range I typically $2 listen to domain and range are written in increasing numerical order so meaning smallest to largest okay is this a function does each x-value correspond to only one y value what do you say to each axle you can respond to only one y value I said no why why you say no so go ahead. yes because the reason is x value only corresponds to one y value zero corresponds to -1 Course once before in for corresponds to to write we would have a problem if if 0 correspondent and four right a function is basically you know the way that that I wanted was think of a function as a machine right and Andrew X values as something you put into them and your white Valley or something that comes out of that machine right so he checks all you do you put you can only get one output right I can't put in the number three and then get like 6 and then put the number three again and then get an EIN right. That won't work each input can correspond only to one output right now you can have the opposite you can have the same output correspond to different inputs that's okay what is okay so here we have two written twice right and they correspond to different two different X values right that's okay we're concerned about is that every text value corresponsal only one y value okay what about this is this a function what do you guys say is hemp is a function or not oh yeah it's not a function right this one is not a because look at the x value - 3 - 3 corresponds to the number 2 and -3 also corresponds to the number for so we have the same x value -3 corresponding to two different Bally's 2 and 4 so we can't have that you see that -3 when we put into machine you can't give us two separate answers and give his two and four okay so what we're looking at is a set of ordered pairs right here on top these are all pears right ordered pairs now we need do is we need to select an answer that has the domain and range for these ordered pairs remember the domain is the first number of every pair the range is the second number so which letter has the Dominion range correctly displayed and mind you it's going to be in order from smallest to largest for both of the main and the range so what letter alright so let's why don't we go one by one and you guys tummy we'll take care of the domain first so we're going to put a red circle over the Domaine so let's look at the first order pear - 34 which one of these numbers belongs to the domain -3 or 4 -3 alright - 1 + 5 which one of these belongs to the domain -6 and 7 which one belongs to the domain -6 3 and 2 which one belongs to the domain do we identify all of the numbers that belongs to The Domain right so now by default the other numbers do you have to belong to the range so find the letter the domain and range despite correctly remember it's always going to split from smallest to largest which letter has the domain listed correctly in the range list of treasure continue Oakton so can be displayed as a graph and also has a table cigarettes a functions a relation wherever unique x value is paired with a single unique y value can also be represented as graphs and tables linear relationship shown in the graph is a linear function each total weight value there is a difference total cost to know the cost of the item you must know how much it weighs the cost is dependent on the way you guys see that relationship right there it's not the other way around right the weights doesn't depend on the cost it's the cost that depends on the weight room so you some of these relationships only go one way iFrogz a when you go to sleep depends on the time of day the time of day doesn't depend on when you go to sleep so if I were to go to sleep now you wouldn't make it night time right so it only it's a relationship that only works one way right one of the two things is dependent on the other so it says in this instance the way where the x value is the independent variable the cost or the Y value is the dependent variable do you guys see that relationship that the cost is always going to depend on how much wait you're purchasing right whatever this is this is some being announced but the cost always depends on how much it weighs right so if you ask me know how much is it going to cost how much do you have right now but it's not the other way around you wouldn't ask someone how much does this weigh and they say oh well depends how much it cost the wait is is Independence right will always wave the same regardless but the price will change depending on the weight is he the relationship who says notice that the weights and costs are related each point on the line is an ordered pair of the form wait, cost so for example if you look at this blue. What does that represent in terms of weight and cost if we're looking at a chart right here right we're looking at this chart and we see that blue dot on the road and let's say I'm 3 oz $6 3 oz cost $6 perfect thank you Andrew alright now look at the following table shows a set of six ordered pairs that make a function when Grafton connected the ordered pairs on the table create a problem run about that yesterday probably has a U-shaped graph notice that each axis alley has unique y value all right so here each number corresponds to only one y value one of the things that we talked about you know where the characteristics that we talked about was whether a problem was positive or negative and then we also talked about the concept of maximum and minimum so miss you guys this does anyone know if this problem is positive or negative or the positive because send that right price I'd oh okay by thinking it's negative because it opens down but yeah you see how the arrows Point down yeah it's negative but that that was a really good point that justification is really smart yeah the quadrant that we're looking at all of these points are in the positive right but the thing is that this problem is going to continue infinitely into the negatives so because it opens downwards it's a negative problem but that was a great thing you noticed that we're looking at the positive quadrant and then knowing now that it's negative two negatives have minimums maximum what do you guys think or if you remember for Mysteries lesson Tootsie yeah you're right maximum so because it's an egg his problem is going to have a highest point there's no lowest point because it continues into negative Infinity right there's no end towards the negative or the positive there's a Max right here in the maximum is that at the 9 and also learn about the line of symmetry that would be here at the 5x equals 5 that's the line of symmetry okay is this a function what are you is this a function Azitra that the technical term asking is basically just take guide you in answering the first does yeah I know right so what why why no 1 and 0 it's because of because of 0 Instagram different answers right here we got a 540 and here we go to 140 so not this isn't a function yes you do we put in the same thing that we're getting something different or supposed to get if I put 0 in the supposed to get the same thing out okay what about this is this a function free SMS Abenaki just play lady about punching your looking for is that every x value corresponds to only one y value so here are the nine the only correspond in this case to the 0 will we need to be on the lookout for is them putting the same x value but have any correspond to different y Valdez so for example on the previous we saw we saw the number zero repeated twice right that's not a problem that's okay but we're supposed to be when we input 0 we're supposed to get the same thing out right but the input 80 twice in the first time they got a 5 second time they got one so that's not a function you're supposed to if you print something in right whatever number you put in your supposed to get the same thing out every single time for that number if you keep putting in that number so where do 4 is is to make sure that each Exile you only corresponds to one y value so is that the case here does he check Sally correspond to only one y value yes I think the concept of think of a function as a machine is wonderful right because it's like imagine Legend of like you had a machine right listen to a vending machine and it only took a coin right it took a penny a nickel a dime and a quarter and that's it right next time you put the penny you're supposed to get the same thing but let's just say a penny gets you aren't a water right you said you put a penny you got it you're supposed to get a water and then you put a nickel and you get a soda before the time you get milk you for the quarter and you get wine each time you put that coin you're only supposed to get that specific beverage but if I put in a penny and then I got to water and I print a penny and then I get milk that's not a function write each each thing that I'm inserting to correspond with one item that's coming out and what I'm inserting is the X values so it's a specific number that I put in and I have to get a specific number out and I should always get that same number out I put that same number in and put the put in the number 5 and then I'll comes at 4 and then I put in the number five again and then out comes a 9 I can't have that it has to match you know 1 to 1 I have a good day I have a question what I'm looking for is is not to have a repeating number no we can have a repeating number as long as he gives the same thing so for example if we had them but they said 12:05 3 12 0 it's okay because 12 corresponds to zero what we don't want is having 12 correspond to two different numbers okay that the input the input I can I can written many many times right I can put in a penny Penny Penny Penny doesn't matter but I have to get the same thing out every single time that I put that number in if I start getting difference in a different things each time I put the same thing in right that's not it it's kind of like a better example think about when you go up to the soda machine right when you go to the soda machine you press coke coke comes out you press for spray comes out in freshwater water comes out that's a function right you can press Coke over and over and over and cook is going to come out right that works but if you pressing Coke one time and you do Coke and then press again and you get Sprite that's not a function right. as as you know it has to match with one thing only you can't keep pressing it and different things are coming out of the Coke machine right it's always has to be sprayed it always has to be Sprite that's what I mean it has to correspond like each dispenser can only correspond to one drink each input can only correspond one output thank you all right now we talked about you know identifying you on a graph on a table right Saint a TV we look to make sure that each X input on the course on 21 y output now if we want to identify if it's a function on a we have what is called the vertical line test that means you put up a vertical line through any point of your grass and if it only hits one time then it's a function right so here they doing the vertical line test to do it 4 times right you don't need to do four times you only need one and if in order to pass a vertical line should only cross one time if I have something like this like a C shaped graph and I put a line through it that doesn't pass the vertical line test right cuz it's Crossing twice at the top and the bottom okay so this does not pass the vertical line test right I put a vertical line in any of these is causing multiple times to the vertical line test in order to pass it you can work like holy cross one time on the on the ground it does North have a vertical line of the relation at three points as it passes through x equals 4 so you know every function is a relation but not every relation is a function right think of it this way everyone who's on Dean's List is a student everyone who's a student is on Dean's List is that make sense it only works one way okay is this a funk I remember back to the vertical line test you know you can hold of a pencil to screen and put at any point on this graph and see if it only process one time is that possible Atlantis yes yeah and again you know you can grab a pencil and is run it through running from side like this right at every single place he would only across one time okay is this a function grab your pencil if you were to put a vertical line would it hit only at one spot yes yes okay is this a function take a pencil and see if it only hit that one spot no no right classes at two spots I power to put a line through it I'm crossing towards the top and the bottom circles that sound Okay so we'll talk about function notation so when it says when functions are Bruno's equation they're written in the following notation so instead of the why you know we're so used to watch equals MX plus b that's the last thing we've learned you know we're going to be replacing the Y with f of x that's how you pronounce this f of x but again just like when we switch variables like what they were were accustomed to using the variable X right but for example when were looking at a formula and has a different variables such as the formula for area of a wreck which is a equals l x w it it doesn't really change much right kind of like if you were to change your name okay you change your name but you still the same person right that's really what's happening here so in terms of you know terms of usage it's not going to affect us much we really just changing how we represent it so now instead of writing why we write f of x that's when we switch from you Asian to a function all right so when were the same blacks that does not mean f x x f is the name of the function and accept the variable ask states which variable is the independent variable together as of X represent why do people we decide on the previous page right they represent they took out the Y and they put a fax and you know just to see how we can use the variable X I could use another letter they're showing how how you can use different letters to represent that function right these are again going to be highly on comment you know you'll see you'll see G of x but this I mean totally possible highly unlikely okay so it says Define ethics when given a valley 4X complete the following steps to hear they're just really all there telling you do is replace x with the number so if you have if f of x equals 4x - 1 find three digit mean replace the X with a 3 so we do that here for time series 12 12 - 1 is 11 and then we have our answer him so what does this mean this means that when x equals 3 the value of the function is 11 so the ordered pair 311 is a point on this function remember all of these functions can be graph is equation part the table part and then the grass three different ways of displaying the same information but again all we're doing switch out that letter for the number and solve and once you have that you know this is really just even though this is a more advanced concepts write a function you know you can ask you know almost anyone what is 4 * 3 okay what is minus take one away and they can tell you it's 11:00 right so New Concept but then again it's also you know not really Newan in the way that we're doing familiar things right and all of these new steps really just repeating the basics of multiplication division subtraction and addition all right so when functions are written as equations they are written in function notation equations that are also functions written as y equals 2x + 3 and also be written as f of x equal to x + 3 values for X can be substituted into function equation to determine the value of f x all right we knew that from the beginning right we've been doing that when we were solving for X and Y remember who we would soften that back in the lesson of system of linear equations we were solving for x and we're solving for y so we've been doing this you know this process hasn't are solving for x and ethics what's different when I we don't call it why we call it after effects why because it's not an equation it's a function so yeah same same Concepts so let's see what they have here so I hear they giving us the function 2x + 3 so that means whatever number they give you your going to multiply 2 * 2 and then at 3 to it really what's the number - 20 - 2 * 2 is -4 + 3 is -13 that's how they get this ordered pair -2, -1 next the inputs the number zero to 1000 + 3 is 3 so here they get zero, three and then finally the input for 4 * 2 is 8 + 3 is 11 so they get 4, 11 okay so let's evaluate this function for each of these inputs so as of 2 means I'm going to replace x with two what's 2 * -5 what is 2 * -5 -10 what is -10 + 20 all right let's do an action what is a -5 * 0 all right and if I had 20 20 what is a -5 * 4 -20 28. What do I get 0 all right cool I'll get right Play No multiplication and addition all right sis is there a mini key features of functions the fussing will describe the following key features all right while we have a lot unit rates and slope intercept function values function Behavior symmetry asymptotes minimums and maximums and end behavior got a lot of stuff you know some of the stuff we already what's a familiar with examples we've already talked about slow prints Lopez how steep something is right it's basically referring to the incline of the graph intercepts we've already talked about that's a location on the coordinate grid your ax and y intercepts like if you were to give instructions to someone on how to get to to a store and you see all go five blocks go five blocks straight and three blocks to the right symmetry you know we've talked about the line of symmetry in the minimums and maximums we talked about that yesterday right I asked if your problem is positive you have a minimum if negative you have a maximum so a lot of this stuff is review and then some of the stuff will be in YouTube all right. With the first topic unit rates and slope so it says there are many distinct features of functions linear functions have a slope the slope is considered a unit rate because it describes the change in the Y value a one-unit increase in the x value okay so you know this me looking you to you but it it's really not you may remember it as rise over run right we're looking for slope and we would say what is the rise over run meaning how many units do I go up how many minutes do I go to the right all right sugar they given us a real life example considered this proportional relationship very six eggs used in a cookie recipe the baker must also have two cups of flour the relationship can be written as 6/2 cups of flour this relationship can also be written by its unit rates by dividing the numerator and the denominator okay so you know here they just reduced right if they would have divided the numerator by the denominator they would just have three right but in reality with the is divided both of these by 2 the thing is you know I don't know if this will always be the case where you can just take the denominator and divided divided everything but the denominator yeah everything got / 226 that / 2 and the two got / 2 so you know I think the more appropriate way of referring to this is they reduce the fraction that won't change they're going to reduce the fat how to find the unit rate in any circumstance right so do they have three eggs over one cup of flour for every 3 eggs we put one cup of flour it says this position can be plotted on a plane to make a line with a slope the slope of a linear function as a unit rate or how much amount of change when the other changes all right so here they graft the graph the line using the slope right maybe can at the origin Point went up three units over one now noticed the increments on the x value for every in it increases by but on the on the f of x cuz it's a function that side effects for every inch it goes up by three units even though it's the same distance they have a different scale they're both they both increase by one but look at the scale on on the X you know it's increasing by one but each increment is about one in space and then for the free sfx the Austin priest I want but they have increments of about half an inch right so then they went ahead and bought a line by putting at least two points rent all right it says when given a graph of a linear relationship between two quantities the unit rate can be found look at this look at this find miles driven compared to miles of gallons of gas used okay so here what is the unit rate here take a look at it and see if anyone can tell me what do unit rates is before they tell us remember to be a fraction it's going to be something over something from your number over another Little Caesar 1 / 30 30 / 1 rise over the run 30 miles per gallon internet 1 units the right 30 minutes don't jump hear that hear that you're making it right would be 30 miles per gallon your voice is breaking other name was only for me 4 m Yeah Yeah by Usher the world without you all right what is the unit rate what is the Danny I don't hear you very well it's like you're worth it breaking all right you guys are mean now it's been awhile okay so what is the unit rate 9 $9 per hour I really I mean if you see where the one those are you can see what numbers to left that's that's going to be okay we also talked about the intercepts remember we talked about all what's the y-intercept you talk about that couple lesson to go back when we were talking about slope intercept form slope intercept with the slope was that's how Steve and then we talked about intercept that was where does that line cross the y-axis in this case it's called the f of x in the Sea Mist room or the x-intercept rent the x intercept is where the line crosses the x-axis it's all visual there's no math involved with this calculation calculate anything you don't use any math operations we just look at the graph and see what does that line cost all right Interceptor locations where function crosses over and access in this craft the y-intercept is at the point 0 3 and there are two x intercepts right the one on the left and one on the right these are all points where the graph is crossing over any of that access over the x or the f of x so you just visually look at the Owens Crossing -3 / 2 that was it intercept just where does the line so if I were to ask you to look at the intercept or the wire where is it what number nothing is 0 not okay where does the graph cross the f of x 1 anyone else -2 what does it cost after tax if I were to ask where does it cross the x axis across is it at the one and it crosses up to 4 what has 3 intercepts one of the intercepts is the f of x let me make this little bit larger one of them is here what's one of them that's the alphabets crosses at 4 for the X across is at 1 at 4 the three places by crosses let's talk about our next concept this is called function value functions can have positive or negative it is positive when the function is below the x-axis is negative so that's something that I I'm sure brought up when I was asking who is this Alyssa or negative and it was it was negative because it opened downwards but I brought up a really good point and it's positive because she's all the positive numbers and I was really good observation because it's true that's probably what we're looking at that was in positive they're asking gum has to look at the different parts of the function right yes those are all negative points on the line right all of these points you know that's negative and above the line here that's where it's positive so you know it's going to have different areas right looks like it's somewhere to say oh is this will depends what part of the city every city has parts that are better than other parts that I'm just like in a problem well this is a positive problem because imma just highlighted in red this is the negative for everything above that is deposit all right okay what is the x-intercept so again this is visual take a look at the x-axis and see at what number does the red line across the x axis so now that we know that it's number three which one represents our answer correctly is it 80 or is it seat thieves the three comes first because the three or four times the x value right oh what is the values of X is the function of it okay take me to look at them at the x-axis right we're looking at the horizontal line is it when X is less than 0 is it when X is greater than 0 is it when is less than 3 or is it when X is greater than less than three that's about that would be all of these right all these are above be there on the positive basically it's going to continue down in the negative so the positive side is three on this way right when when X is less than 3 here is all the positive is really all exist all right so I'll talk about the next concert and decreasing function Behavior All right we know this as positive or negative slope steampunk just has a different name different name because because a category category as it's kind of like if you say 0 a hand or a paw right it's essentially the same thing just depends how you talking about a human or an animal referred to the body parts right it just depends is it human or animals on steps have different name are we talkin about expression and equation increase and decrease positive or negative slope on a chart how your savings account is doing right you want to see that arrow pointing up this way from left to right all right do we have a line that has no slope right you talked about that before this would be you know the equation for this line would be half of x equals that's it it would be no other acts you know how normally have y equals 2x + 5 or something there is no X why is there no X because there's no slow we just have the the y-intercept right which is right at the negative 3 so the equation for this line is f of x equals negative 3 and again this is considered a constant right stock going up or down in the past like I mean that we've known this as 0 okay here we have both increasing and decreasing function behavior in this case we're going to have to identify it by the intervals right because there is a portion where it's increasing and there's another Secret different cheese alright so here's how they express it right what we use is interval notation maybe you own a company right and you want to know how are your cells been for the past 10 years will maybe cells are going up and then they went down there remain constant went down down up you're not going to remain standing in the same direction at this function way she right you know we can identify that in the function by using interval so by saying things like oh from the year 2000 to 2005 sales are increasing from 2005 to 2007 from 2008 to 2010 and 2015, the past 20 years 2020 to 2000 about how cells have been rare and they haven't been constant so I pick interval in which the behavior is the same I said something like 2000-2005 sales went up thank you thank you I'll see you tomorrow and so let's talk about how we display this right so it says if the interval is from 1 to 5 including one in five people use brackets if the interval is from survive but not including 1 or 5 use parentheses right you know is it's basically a differentiating whether we include the num showing or if it's everything in between those numbers right is it everything greater than 1 and everything smaller than 5 or is it that include parentheses alright so here we have it right here we have a negative Parabola opening downward in the portion from 1 to 4 so when between the number one and the number 4 on the x-axis this is increasing from 124 with brackets because it's increasing the one all the way to the Forum now after we cross then we begin with decreasing for that have occurred across from the number 47 on the x-axis and they use brackets there as well include two numbers all right look at this function so a function might increase or decrease forever longest urgent part of the domain to show continuous increase or continuous decrease it is sometimes necessary to use infinity and negative Infinity in interval notation all right so look at these look at this graph right it looks like a n right like a capital letter n and look at the split it up they cut it up into three during the first purple left is from negative Infinity all the way to -0.8 to rent not sure why why did Industries phone number anyways that means that on the left-hand side it goes down into negative Infinity there's no lines do right in the pink part in the middle they have a line segment here it decreases from this point to this and then does the first part is increase right left right A B start all the way from - Cindy on all the way to -0.82 right if you look at the bottom left or the right please increase increases again this function spends almost all of its time increasing constantly except for that middle section that pink section that's the only part where they chop it up into three parts increases or decreases will you can increase like for example like your words to be increasing if you go from owing $500,000 to $200,000 right you're paying off your debt for your net worth is increase the negative but if you're in a better situation okay here we have a different function right we have the increase happening from -5 just this part right here all the way to -1 so that's that purple line the far left or right that's where it goes up then the negative one to the four I says because there are no arrows on the ends of the function is not continue to increase or decrease in either direction so this is very important and you know they haven't always been consistent with but there's a difference between a line and a line segment right in math a line is something that continues infinitely it has no n versus a line segment a line segment is a you have a life which has it start and then we differentiate the two is when we draw the line we put arrows at the end of the line with you to indicate that it continues into Infinity if you don't have arrows and we're drawing a line segment any has a start of a sentence if this had an arrow then it would be from negative Infinity all the way to -1 and from -1 all the way to positive but it doesn't have an arrow so that means if that's the end okay over what interval is the function on 32326 360 yeah so so we're looking from what what number does it start being constant at 6 that's being trouble starting at 3 and ending at 6 you know we didn't think of this you know we may look at this and we could say will look at this function represent right we can think of it as as daylight daylight varies depending on what part of the world read some places are in complete darkness for months and other places little sunlight for months so you know you can imagine maybe a place on Earth during a specific time of the year you may only have 3 hours of sunlight right and you can think of 20 from being like noon all the way till 10 p.m. at night right to think of this as from noon to 3 p.m. is rising from 3 to 6 p.m. the sun's location does not change from 6 to 10 the sun's location goes down so this is an example of something that it could represent okay what part of this function is increasing so from what number to what number is it increase and again we're just looking at the x-axis so we're on the X does its beginning fees mr. Zero all right that's that's the only option that begins at the zoo and I noticed they put in parentheses around the infinity symbol right including Infinity because you know it is a concept not a number right so you can include like Santo it ends at this concept this concept is infinite meaning it has no end so that's why okay symmetry we talked about this this is called the line of symmetry or the axis of symmetry that's basically if you were to cut a vertical line through this graph where would you cut it so that you have the same thing on both sides right as at this problem in the middle if you were to split body you have the same things on both sides to Arms one on each side when he died everything would be equal symmetrical left and right all right so this isn't a function oh here they tell you one of the ways we know that this isn't a function is it doesn't pass the vertical line test power to take a pencil and put it in any part of this crap in more than what is a bit misleading that they use off of exit instead of why right knowing full well that this is not a function this is a relation so in this case the axis of symmetry the x-axis write the x-axis overlaps with the axis of symmetry if you were to cut that grass right along that line we have the mirror opposite top and bottom Kim has another key feature found in some functions is called an axis of symmetry or line of symmetry are we already know Maxim symmetry is a line that divides a function into two congruent house the function is a mirror image of itself on either side of axis of symmetry they can be represented by the equation when x equals a constant value you know that we already knew from yesterday's last night I think I think everyone pretty much understands this concept of you were given same on both sides sounds like if your sharing a meal right when I cut it right down the middle so that you and the other person have the exact same amount you and another person okay where is the axis of symmetry at what number could we draw our line 1 - 1 if we put that one right there but a right down the middle all right so some functions have minimum or maximum values and value is the lowest point on a function maximum is the highest point on Earth if it's a positive function you know we have a minimum positive larger numbers maximum no there's no maximum on a positive because you know look at these lines they have arrows meaning that it goes up to positive Infinity but there is a starting point if you open up words that's our minimum that's our lowest points that exist on the problem now the opposite is also true if we have a negative problem meeting opening downwards we're going to have a maximum it's kind of like a valley as a lowest point in the mountain as a highest point and think of how it opens up right makeup like a valley if you're at the very bottom you're at the minimum and it only goes up from there if you're at the top of a mountain at the maximum and it only goes down all right so if you have a function that looks something like this right in which case we have a portion that is positive or snot is negative you have what is called a palliative minimum and a relative maximum that has both in this case you know we can identify both parts and it's called the route to maximum clouds at middle all right what's the maximum gear what are the coordinate points for the highest point on this problem 11 Yes Rick Ross is that the one that has an ordered pair 0 - 1 I want isn't an option A show from the options we have our tree display number D remember RX 31 4 x 20 meaning you don't move at all Left Right a second one is a y or when we're talking about functions are f of x so that's the one this is the Loki 0, what if you start at the origin point zero let you know to remain they're going to write one tells you to move one up okay true about the following function okay so look at this w-shaped function so how many maximums would you say there are relative maximum one two three or four yes remember maximum is like the Mountaintop right that's like the highest point how many highest points do you see on it on the on the curvature we're looking for the curved part of the I mean mountaintops if you want to call that you see I don't understand this and maybe if I draw a straight line also I won't be tough free how many mountain tops this is the highest point one Mountaintop to Valdez 1 Mountain Top at the tippy-top point to Valley such this point so this this is our maximum there's only one point on the curvature that's the highest that's the Mountaintop right and then the minimums those would be like The Valleys those are the dips so there's two dips and one eye points that's your maximum that's your highest point right there right on the furniture and it really looks like so that's that's the max to the two little brown those are your minimums another key feature to know about the functions end behavior and behavior is a behavior of a function explain the x value approaches Infinity Necklace so obvious you a line goes in opposite direction as a moves to the right so let's see Its Behavior you know both of these cases it's going it's going downwards into the negatives both as it approaches negative and positive Infinity right on the x-axis or the right he continues to move downward as a so we continue to move down there the book both of those you know I think about you it has the same and behavior doesn't matter that they're going in opposite directions one for the last month they're both having the same direction it's kind of like you know it's kind of like you me and and another person who started you start at one point together right and then I told him okay you're going to head Northwest and I'll head Northeast right if we both had Northwest Northeast you know even though we're still both heading north the end both end up in Canada either sites what is the end behavior of this function so what is the nbk bear of the function as X approaches negative Infinity which way is negative Infinity left or right left left okay so look at the line as it goes left what happens decrease please keep fluctuating or remain onzas remain constant yeah I mean this one is a bit hard to say right between remains constant and decreasing because you know truly we can see it's decreasing right starts here and -3 goes down and then you know it's really decreasing but by such small amounts you know it may never hit the number to so yeah let's do it remains constant what about the positive this one's very clear What's Happening Here increases increase you know it is decreasing but then as it continues on it seems like it remains yeah this one you know Auburn because it is find out what this is what's called the an asymptote that's where you know the line gets closer and closer and closer to a certain number but it never touches it you know what you getting and like all the way from the Moon each year all right this one this one is a fair one for us what happens as as X approaches negative detergent which way is it going I think it's going to left that's fine. This would be like someone going Northwest and then this would be the North we're heading up words all right asymptotes see this probably I've been better better placed before the other one right because you know the is a question about a concert they hadn't introduced so what we were looking at at the other ground you know it had an awesome so that's a line where a function continually approaches a curve but never reaches it eating gets closer and closer and closer you know it's because you can only get so close to something before you touch it closer and closer and closer you just keep expanding the number of zeros to the right of the number right making the number smaller and smaller you know closing the Gap slime without ever touching so weird asymptote of this line this is the one we looked at at the previous I want to go ahead and put the blue like you guys tell me where is the asymptote authentic meaning wedding stuff that the line is approaching but never touching what is it do you know was it a 20 your down here it was over here it's not even close to it's 2 now the question is is it x equals 2 or y equals 2 where does that line Blue Line cross the x-axis or the y-axis Hawaii and again it's not labeled as why it's labeled as f of x because it's a function but you know what's interchangeable referring to the same line alright let's talk about modeling says there are times when you will be asked to compare functions presented in two different ways speed function for two different cards are shop one showing how to break Lee and 1 as a graph which of the two traveling at a greater Speed 3 seconds after they take off okay so you know in different ways for the card number one plugging 3 for cheap that would be 75 write 25 * 375 rental car one really goes faster Target all right so here they are they're plugging in 3 seconds for the first car and they got 75 MI that's all we got now for the second car they just look at the graph they go to the three on the x-axis and then they move upwards at 60 miles per hour the function is represented in number format or on a graph all right okay what can be said about the amount of money earned after the first 4 hours who owns more money so tell me how much money does Alia earn after 4 hours get paid $8 an hour and work for 4 hours how much money do you make Somalia is going to earn $32 how much money is Brayden going to earn after 4 hours 26 2828. You are on the same they did not bother this evening and this you know what we did here for a Leah would be called evaluating a function where are evaluating a function meaning we're being told what she equals it's for you plug it into the function all right and a half lesson learned I said math history we broke a record all right let's do the review let's go over what we already know when the first things we talked about was ordered pair air consists of a y of an X and A Y value rent and they represent a point on the grid what we're looking for to determine if something is a function or not if it's a function each x value can only corresponds to one y value and the opposite is true if you wanted to try it if it's not a function and we're looking for the same x value for two different valleys that's how we can example here we have the point 24 and 27 that's not a function y because you have the x value to corresponding to the number for answer the number 7 because that example about the soda machine dispenser you go up to the sewing machine you have Coke Sprite Pepsi water represents one soda what if I go and I hit it when I go back and then I get Sprite can I get water that's not a function we also were able to tell based off of the table right you know we have our two we have our X and Y input and output chart this is a function this is okay in put corresponds to one y out so the number 10 and I'm course boxes your that's okay where we would have a problem is if we have the number one corresponding 20 and the number one also corresponding 24 I would not be okay and then we also looked at the vertical line test right if you graph this morning put a vertical line you can only cross through one point of that crap goes through multiple not a function here it is the vertical line test tonight you can grab a pencil or pen put it on your screen see does this pencil only cost at one point multiple not a function all right and we talked about all of these Concepts here unit rates that's the slope that's just the slope that's not a New Concept you know we've been working with slow Tawana intercepts also a familiar constant where does the line cross the y-axis where is across the x axis intercept minimum and maximum if your problem is you have a minimum if your problem is negative you have a maximum income of Valley and around the bottom of a valley that's your minimum you can only go up from there show up in a positive problem if if you have a maximum and could be not a Mountaintop if you're already at the Mountaintop you have only one direction to go down right you already at the highest point we also talked about relative minimum and relative maximum you know you're in an area where there's mountains and valleys right what's the highest point in that area what's the lowest point of Maximum and minimum axis of symmetry we talked about that yesterday too you know you're already familiar with the concept probably even before you heard the term right anytime you've gone out to you in person you're sharing a meal you or the other person you already have the concept in the notion of splitting it at the point in which you have the same for both people right we don't get a burger you know with 20% of burger for you and 80% for me you understand that concept even line of symmetry right so we're just applying it to the graphs miliar stuff already got

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