Lesson 3 1 Video

going to finally start unit 3 which is all about linear relationships and most of this will be brand new information especially a lot of vocabulary for you you've touched on a couple of these topics earlier like input output tables you may have heard of those rate of change you did someone in seventh grade proportional relationships and that kind of stuff but moving it up to more of a high school level we're going to be specifically looking at what functions are defined as and how to identify different functions in fact the rest of this year honestly is going to be about us learning different types of functions and we're going to start off with linear functions linear comes from the root word line okay so that means that when we graph these functions they're going to make a straight line okay we'll do functions later in the year that are non-linear which makes different types of curves okay but the one thing that you're going to notice about this since it's all mostly brand new information is you're going to learn lots of new terminology lots of vocabulary we've already talked briefly about domain and range and we're going to talk about that some more today and then i just threw out some other things some may look familiar to you things like rate of change which we're going to define as slope y equals mx plus b hopefully somewhat looks familiar to you uh linear regression is something we're going to spend a lot of time talking about so just my point is there's lots and lots and lots of information in this unit referring to lines okay so what we're going to do today is talk about what is a relation so that we can then translate that into the definition of a function because a function as you'll see tomorrow is literally defined as a special type of relation well if you don't know what a relation is we can't hope to understand what a function is so let's uh very quickly talk through that so you've got your hand out in front of you follow along with me do some highlighting and such just like i do uh so we talked last yesterday i gave you a short little uh slideshow and when i showed you that a relation is basically a set of ordered pairs okay so anybody can create their own relation you just use these little braces and then inside there you put different ordered pairs ordered pairs meaning right a point on a graph an x comma y coordinate so i could do two comma three that's one ordered pair negative one comma five okay and so forth you can put as many points or ordered pairs in a relation as you want okay now the whole purpose of a relation is that it shows a relationship right so i've given you an example here below so let's quickly talk through that it says in many naturally occurring phenomenon two variables may be linked by some type of relationship okay hence the name relation okay some type of relationship the example that i've provided talks about an archaeologist and he finds the bones of a woman at an excavation site and one of the bones is a femur the femur is the large bone in the thigh attached to the knee and hip and the table below shows a correspondence correspondence is the same thing as a relationship okay between the length of a woman's femur and her height so when he collected his data he saw that if the femur was 45.5 centimeters the height was 65.5 inches and he listed all this data and then he wrote those uh that data as a set of ordered pairs which basically created a relation okay because he is showing the relationship between femur and height so each data point may be represented as an ordered pair in this case the value represents the x value the first value represents the length of the femur and the second value the y value would represent her height okay you can see that if we write it as a relation using our braces it would look as the following okay so again the point is that the ordered pairs are collected from some data that show a relationship between two things for example i could go and collect data on how much you studied and the corresponding test grade okay and see if i can because what i'm eventually looking for are are there patterns right are there rules that i could write to represent those relations okay all right so moving to the next slide all right talks about what is the mathematical definition of a relation so a relation is simply what we just talked about a set of ordered pairs now we need to be more specific in identifying the different parts of a relation the first components in the ordered pairs are called the domain and the domain is you're going to make this note on your in your notes or your guided notes hand out the domain are basically all of the x values in the relation okay the range of the relation is the set of all of the y values okay so you have to do this today on your assignment um it says if i give you a relation it says find the domain and range of the relation this is the one from previous where it links the length is the x value of the femur and the y value of each ordered pair is her height and they want us to simply identify the domain and the range well the domain you write as such you use braces and you simply go and you identify each of the x values okay each of the x values from each ordered pair okay all of these are going to be our domain okay so let's look at that uh the domain in this case would be what is that 45.5 is that the first one 48.2 is in the second one 41.8 46. and 50.4 and then close your braces all right so there's five ordered pairs one two three four five and so i should have five x values now there are situations where the the number of elements values in your domain or range set may not match exactly the number of ordered pairs that are in the relation because as we discussed yesterday we don't have to repeat if there's the same value you don't want to write it down twice right that's just repetitive okay but in this case they were all different so i have five values in my domain now the range is going to be all of the y values in each of the ordered pairs okay so let's list our sorry our range so our range would consist of the elements 65.5 comma 68.0 comma 62.2 comma go down here somewhere amount of room 66.0 comma and 70.0 and then i'll close my braces okay and again since they're all different there are five elements in my range okay so that's very simple right we did that last night and just again clarify the domain is the set of x coordinates and the range is the set of y coordinates okay so when i say set that means you put them in the braces and group them together as a collection right that's what we mean by set all right so this is a really cool example because i know you know in math we think everything has to be numerical but i just wanted to give you an example where the data may not necessarily be numerical meaning it can be words it doesn't have to be numbers all right so what i've got listed here in this table are five states and the corresponding number of representatives that are in the house of representatives as of july 20 2005 because again that change is based on population right as population they do the census that can affect the number of uh representatives in the half in the congress okay for each state so anyway uh if i give you a set of data like this and i simply say state the relation what that means is and i want you to do this another set of wordpress because we're going to learn later that there's different ways to write a relationship i want you to use a set of ordered pairs right now so that's when you use your braces and i simply write this data as sets of ordered pairs right this is your x here the state and this is our y so i mean i'm going to use abbreviations i'm going to use what a l for alabama i don't even know if these are right for the ones for the state but that's what miss felix is going to use today i'm going to use ca for california oh let's do co for colorado fl for florida and i don't know k a for kansas those may be right okay so uh alabama has seven so i'm going to write that literally as an ordered pair a l for alabama comma seven close parenthesis okay so my point here is that uh yes most of the ones we do are going to be all numbers but i just wanted you to see that in this case the x values are state names okay so it doesn't have to be numerical okay so alabama goes with seven right the next one would be california corresponds to 53 house of numbers of representatives in the house the next one would be colorado co has seven florida fl goes 25 and then i guess i'll go up here kansas as full okay then close your braces make sure if you open a set you close this head that's very very important i know it seems like a minor detail but again little details like that in mathematics we make a huge difference we have to pay attention to details all right so that is that data written as a relation now let's quickly identify the domain well the domain right is always your x's or your input values which we'll talk a lot about later so my domain put me some braces consists of alabama comma california comma colorado comma florida comma and kansas close your braces so your domain is always your x's okay and they correspond to the first value in each of the ordered pairs the range okay comes from the y's right so this this is your range and of course that's all the second values in each of your ordered pairs so race and then my range would consist of 7 comma 53 comma oh so i got 7 again see how i have two 7's here i'm not going to write that down twice i've already got 7 so i don't need to repeat it so 7 53 then 25 comma 4. okay now you'll notice that there's only four elements in my range yet there's five ordered pairs in my relation but again that's because you don't have to repeat values okay now obviously you can't have more values in the range than you did in the relation that doesn't make any sense but you could have less if you have some that repeat okay all right um let's look at the different ways we can represent relations okay and there are four main ones that we're going to look at these are very very important so pay close attention the first one is as a set of ordered pairs which is the most common that's why we've been doing it so far if i ask you to represent a relation as a set of ordered pair then you use your braces okay and then you have your little groups of ordered pairs inside here there's an example okay that's the way we've been doing it uh so far however the next one is i like a lot it says it may be defined by a correspondence now i refer to this as a mapping make this note on your guided note sheet okay this is a what we call a mapping and basically what it does is it visually shows almost in table form notice your domains are all grouped together but instead of just a chart they're grouped in almost like a venn diagram a little circle and then you put all your range values over in another circle and then what we mean by correspondence or mapping is you literally show with arrows which x matches with which y to make an ordered pair and what's cool about this is you can see why you don't need to repeat for example i can see that this one matches to this two so that makes this ordered pair but then i can see that that one also is mapped to negative four okay which is why um you know you'll see sometimes where you'll have more ordered pairs in a relation than you will values in your domain or range okay and then i could put what negative 3 goes to positive 4 here and then positive 3 also goes to positive 4. okay so you'll notice if our braces around this to write a relation i've got four points in my relation but only three values in the domain and range and again that's because several of them repeat and you can see that by the way the arrows are going okay so mappings are cool another one is a graph all right and a graph is just what it sounds like i got points on a graph okay now this is often what students miss because if the points aren't labeled like they are on my graph here when i say all right find me the domain and range okay well if the points aren't identified like i have them you have to first identify like i would have to first identify this point as negative 3 4 and then i would know that negative 3 goes in my domain right another value in my domain is positive 3 and this one would be positive 1 that's the x value and this one also has a 1 so i'm not going to repeat that right now if i did the range that's all the y values of these points so i've got a positive 4 here i've got another positive 4 so i'm not going to repeat that i've got a y value of 2 and a y value of negative 4. here all right so just be careful with that uh again those are the ones i most often miss just because you first have to identify what the points are before you can identify the domain and range now the last one i've given you a graph but what i wanted to really talk about here is an equation these are the hardest ones and i'm not going to spend a whole lot of time focused on this today we all know what an equation is but when i give you an equation you need to understand that when you graph an equation it is continuous and what i mean by that those are two words we need to talk about well continuous is a word versus discontinuous okay or we call discontinuous actually discrete all right so the graph that i've given you right here i'm going to put this word up here do this with me on your paper this is actually a discrete graph and what i mean by that is it's individual points okay i've got specific points okay and the domain and range are easy to identify in those scenarios because i can identify each specific x value at each specific y value but if i give you an equation think about the equations we've been graphing our lines when we've been graphing systems of equations and so forth right we plot the points right and then we connect them to make what is called a continuous like this f a continuous line or picture right and there's no way we could literally identify every single point on these types of graphs because how many points are on here well there's billions and millions of points well i want to write down the domain and range for all of that right that's free that would literally take me the rest of my life because it never ends okay so in those cases it's a little more difficult and um in fact what we're going to use for the domain and range for continuous graphs are inequalities okay so in this case i don't want to confuse you but in this case your domain if i think about all of the points on this graph all of the x values on this graph are going to be greater than 0 because i don't have anything back here right there's nothing back here so in this case my domain would be all x values greater or equal to 0. all right notice i'm using an inequality to represent the domain then the range would work the same way but the range is interesting in this case because yes even though the graph goes forever left and right it also is going up and down so when i think about the range and the range you think about all the y values the y values think about your y axis how far down does it go how far up does it go well it goes forever both directions so in this case we would say all real numbers and to make a little symbol for real numbers i'm going to use a little double bar r okay all real numbers and again i want you to stress about that right now we're going to have a big conversation about that a little bit later okay so just to review a relation can consist of a finite number of ordered pairs this the street is finite and what i mean by that is there are specific points two four six eight whatever okay or it can have an infinite number of ordered pairs and that would be continuous meaning it goes on forever never ends so there's no way you can list every point in the domain and range because it never ends okay so those get a little more complicated the other thing i just want to make sure you review you may want to highlight this on your paper with me okay the different ways to represent a relation we can use a set notation which is a list of ordered pairs the correspondence remember is the same thing as a mapping that's the little arrow thingies right the graph oops must be looks okay the graph and then the equation those are the four big types that we're going to talk about okay all right let's uh go on and do a couple of quick examples it's very similar to what you're going to do on your assignment today so i've given you different representations of relations and i simply want you to first write the relation in set format okay meaning with the braces and ordered pairs and then identify the domain and range okay so if you look at number one this is a mapping and correspondence because i can see by the arrows which x corresponds to which y so remember your domain is always here in your first little bubble and your range your y values are in the second so when i i can see that there's three ordered pairs in this relation because i can see that the x value of 3 is mapped to the y value of 9 that makes one ordered pair another one okay looks as though make sure i can read this correctly because my eyes are kind of bad yes 2 goes to oh is that nine negative i believe it is negative miss felix needs glasses on two negative nine and then uh the last x value is negative seven and they all map to the same y value negative nine okay so there we go close my braces so that's what it looks like as a set now the domain right it means i'm going to identify all of the x values and simply group those in a set so that this is my domain right here so 3 comma 2 comma negative 7. a lot of people will say put them in order from least to greatest it's nice if you do that i'm not going to count it wrong if you don't it's just an extra step of reordering okay um just i don't really don't want you to repeat if you have any repeating values all right now my range oh like here my range only has one number in it so i'm not going to write negative 9 negative 9 negative 9 that's crazy so i'm just going to write negative 9 once and move on okay so that's what that would look like as a set domain and the range all right so let's look at number two so number two i've given you a discrete graph meaning there are what one two three four five individual points here so when i go to write my relation in set format i should have five ordered pairs here and i strongly suggest when you do this highlight them as you do them so you don't skip any all right so for example this point right here what would that be negative 2 negative 3 close your parentheses this point looks to be negative one zero this would be zero one this would be positive one zero and this would be positive 2 negative 3. i'll go right here and close my braces okay oops close that parenthesis too all right so one two three four five ordered pairs in my relation all right domain the domain is simply all of the x values okay so it would be these numbers right here they're all different so i'm going to list each of those negative two comma negative one comma zero comma one comma two close your brace and my range is my y values and i know that some of my y values repeat okay because i've got negative three here zero 1 but then 0 repeats and negative 3 repeats so i only need to list those three okay all right now the last one is the tricky one okay because this is a continuous graph and what i mean by that see how number two is specific individual points this is again what we call discrete those are much much easier to do than the continuous ones okay but i did want to just talk briefly one more time about continuous continuous uh there's no way we can write this relation as a set of ordered pairs because there are a billion ordered pairs on this continuous graph so we're not even excited for that but we should be able to at least identify the domain and range and the domain again you look at the x values right so what you want to do when we're doing the domain is think about every ordered pair that you that could possibly be somewhere on this picture what could the x values be well that means you need to find the value for others to the left and the value furthers to the right and it looks like right this is the value furthers to the left and then they the numbers get bigger bigger bigger bigger and then once they get here they're going to get smaller smaller smaller so it looks like my x values fall in between those two points the one feathers to the left and feathers to the right and i believe that's noted on your paper from this is a negative eight here and this is a positive eight so i'm going to write it as this all the x's have to be greater or equal to negative eight right because that this point right here the x value is going to be like negative seven which is bigger than negative eight okay this x value is going to be like i don't know what uh two or something okay which is greater than negative eight but i can't just stop there because that would imply that the x value any point on this graph could have an x value of 100 but that's not right because i don't have any that go over positive 8. so what i'm going to do is i'm going to say all right yes the x's have to be greater or equal to negative 8. and they must also be less or equal to positive eight okay so i could write it like that now you could also write it as a compound inequality which says negative eight is less or equal to x which is less or equal to positive eight this and this are the same thing now you may say with a sign switched right here okay well when you say something is between this and this these two symbols here have to always go the same direction and it's always going to go the same direction as the second one that you wrote okay we're going to talk a lot about that later so don't stress about that right now i just wanted to introduce that now talking about the range again it's going to be very similar but with the range remember you're looking at the y values and the y values go up and down your x values you go side to side so when i think about the range i'm thinking well the lowest point on the y axis is here the highest one is here and i believe they tell you that that's a negative 5 down here and a positive 5 here so when i go to write that as an inequality i would say okay well that means all the points on this little graph can't go below negative 5 so i'm going to say well the y values have to be greater or equal to negative 5 but again that implies that they can go on forever and they can't so we'll put out the word and and say well wait they must also be less than or equal to positive five so they're going to fall between negative five and five okay and i can write that as a compound you put your smallest one on the left the biggest one on the right and then you put your little symbols in between okay that's what it would look like that's what we call a compound inequality and again that's kind of a preview of things to come on that last example okay i'm not going to give any that hard tonight speaking of tonight what i want you to do is you're going to go into your quizzes account so you can click on the quizzes icon from our homepage or go to joinmyquiz.com and i have already linked for you in your account the little assignment called lesson 3-1 domain and range and i give you different representations like mappings and graphs and whatnot tables and ask you to identify the domain and the range okay so hopefully that you find that pretty easy get that done submit it by midnight tonight and have a great day please i couldn't find it where did it go it never really asked me to stop let's see what it says

Channel: Julie Felix

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Id: W291r2GscSU

Length: 27min 21sec (1641 seconds)

Published: Thu Nov 19 2020