okay so we just want to take a look at how we write kind of some equations linear equations for cost functions um and some other functions from information and problems so um this is kind of like one of your homework problems um a plant can manufacture and one of the best kind of I mean one possible problem solving Str strategy is to take notes so you'll see me doing that so it can manufacture 50 tennis rackets per day for a total daily cost of $3,855 then at 60 tennis rackets per day we have a daily cost I'm going to put that in there it's a daily cost of $4,240 daily assuming that daily cost and production are linearly related meaning I think cost is equal to um some like MX plus b where X is the number of brackets that I produce this is a b right then find the total daily cost of producing X tennis rackets so I want to write this equation and if you recognize like this kind of our linear related and recognize you really are just trying to find like slope in the Y intercept then we can find this okay so instead of C of X which can be um a little bit um I don't know what the word is bulky to work with I'm going to work with Y for a little bit okay then at the end I'll say that that's my cost function so first of all I know that m is the slope right now the question is I from this information I have two points up here 50 is an x value and cost is a y value right do you see how Y is equal to cost and X over here here is the number of rackets so one of the points I have is 50 comma 3855 and the other point I have is 60 comma 4245 and I just want to find the slope and the equation of these two okay so if I know that then I have slope which is Y2 - y1 over X2 - X1 is 4245 minus 38 55 all over 60 - 50 I'm going to put that in my calculator 4245 - 3855 I get 390 over 10 which will give me a slope of 39 okay um now I don't really want to find B so I'm going to use this point slope form to get that and so then I'll just plug in Yus [Music] 3855 is equal to 39 * x - and this is 50 okay um and then I want to then put it into this MX plus b so I'll multiply 35 I mean 39 x - and what's 39 * 50 1950 - 1950 and then I have to add 3855 to both sides okay and so I'll get Y is equal to 39x and then I got to do 3855 - 1950 + 1905 now this is the cost function so like I told you I'm going to come back in and say c c ofx is equal to this function okay okay now I forget let me scroll up and see we want to graph the total daily cost I'll probably remember that and interpret slope and Y intercept let's do the interpretation now and then we'll graph um C and interpret okay slope tells me rise over run right slope is rise over run or change in Y which we use this little triangle Delta Delta y over Delta X right okay so this is how the Y value is cost right how cost changes for every one unit change in number of rackets so if this number this total number m is 39 over 1 it means cost increases by $39 for each additional racket produced okay the Y intercept let me do this in a different color the Y intercept which is 1905 sept which is 1905 this this Y intercept is what happens when X is equal to zero right that's how I find the Y intercept is I set x equal to Z plug that in so the Y intercept being 1905 what is what happens cost at zero right so when X is zero it means the number of rackets produced is0 Z zero rackets so that means I have I have $ 1,95 of costs um if I produce zero rackets but this also what it means is that 1905 is the fixed cost right do you remember that we talked about this right this is the fixed cost here 1905 where are we fix cost and the 39x is the variable cost the cost that's dependent on how many rackets that I produced okay um if I want to graph this I'm going to write it again C of X is equal to 30 9 x + 1905 right um these can be kind of weird to graph and honestly one of the best ways to graph it is to actually maybe just use the original two points because then you have two solid points you don't have to find any more points and you know that it's a straight line so and now I have this Y intercept so maybe what I'll do is I have so I want this really to be a true origin zero um and then let's say I'm going to put 10 20 30 40 50 60 because I remember my values um were from like we're at 50 and 60 right 50 rackets to 60 rackets okay and then I also remember so the two points I had if you remember were 50 3855 and 60 4245 so I have to go up to about 4 or $5,000 so I might just make this $5,000 and then make that four 3 2 1 right so I really don't want to be graphing this like finding the Y intercept and then count over one and up 39 that's a terrible way to graph this function a much better way is probably to put this 1905 on here the Y intercept because you want that number and then to put 50 and then oh hold on for some reason I have 400 and that should be 4,000 50 and then go up to about 38 55 and 60 and go up to 42 maybe like this and I might try and just line this up and put these three points on and honestly that that is a good graph right so I don't continue it I I'll make some notes I don't continue it in this left Direction look here see how that's blank yeah I don't continue it that way because there's no cost function if I produce less than zero items right there's no meaning to producing one rackets so really The Domain in this function domain goes from zero out to however many is a maximum but at the very kind of least you could say it's just positive number of rackets produced okay