welcome ladies and gentlemen today we're going to talk about piecewise functions writing equations our learning objective today is that if I give you a graph you will be able to write the equation of a pie wise function so we're going to do three examples today and we're going to talk about some tools you need to get all of that into your notes please here's our first example we're going to write the equation of the line for the pie wise function so here we have two different functions so we have we'll talk about this one right here that I'm highlighting in red first so we know this is a line so it's going to be y equals and our line is mx + b so y equals if we look at our slope we just have a rise of a run of one and it's a negative line so it's going to be -1x plus our B where it crosses the Y AIS is three so there's our first equation our second equation right here the one I'm highlighting in green so we're going to have y equals our slope for this one if we look here it's going to be 2 over one it's negative so it's -2X plus we don't know where it meets the y axis but we could keep going we go up two over one up two over one and ironically it would end up at three so now we have our two equations I'm going to write it in our piecewise notation so y equals we have our two equation -1x + 3 and -2x + 3 now we have to put our restrictions on what our X values are so if we look at our first one the one in red it includes -2 and then it goes on forever in the negative Direction so X is less than or equal to -2 our other one does not include so it's just going to be X is greater than also two oh excuse me this one was a positive two sorry guys and positive two our x value is at two and it's going to be any number that's greater than two all right there's our first example now before we get into the rest of our examples ladies and gentlemen I want you to write down these useful tools when we're working with peie wise function writing the domain of a pie wise graph that's going to be really important because we have to be able to put our restrictions on our graphs equations that are vertical are going to say x equals the horizontal lines are going to be y equals and they're just going to be numbers so this could be like xal 3 this could be y equals -3 something like that okay it's very important that we can identify vertical and horizontal lines now general information When We're translating functions this is going to be so useful our linear and absolute value now if you look at this this is the way we're going to think about linear so that we don't always have to find where it crosses our y AIS okay like we did on that last problem so this is going to help us out a little bit and our absolute value we know when we translate when we're moving we use this we're going to be using these a lot so make sure you have these both in your notes highlight star whatever you need to do is make sure you can find these okay now we're going to move on to our last two examples here's example number two if we see here we're going to have three different equations so we're going to have y equals and we're going to have three equations here I'm going to start with this one on the left and I'm going to work my way right so open circle here highlighting this in Red so we know this is a line so we are going to use the Y = MX - H + K here okay so we can find our slope first my slope is going to be down one over 2 so my slope is a 12 parentheses x X I went to the left one so I'm going to put plus one and then I also went from the center I went down one so I'm going to put minus one there is my equation I don't need to multiply it out I can leave it just as it is there the next equation I have I'm going to highlight in green right here this one it's highlighted in green it's a horizontal line so y just equals a number and if we look we go up two so y just equals two here and our last one I'm going to highlight in blue so we're going from here all the way down we have another line if we look here we need to find our slope first so we're going down one over one so my slope is going to be a -1 put my parentheses x if we look here I'm going going to the right from my Center I'm going to the right three so I'll put minus three and then I'm going up 1 2 3 4 so I'll put plus 4 on the end now that we have our three equations written now we need to figure out our domains where X values are at on our first one the one in red over here it's not equal to but we're going from negative one all the way to negative Infinity so X has to be less than -1 not equal to but less than now where we have Y = 2 our green one here we go from one all the way to three and it includes both so we're going to put our X in the middle less than or equal to the furthest one on our left is -1 and it goes all the way to three and includes1 and it includes -3 that's why we have less than and equal two okay our last one in blue we see our x value is at three and it's going greater than so X is going to be greater than three but not equal to because we have this open circle there okay so we've come up with our three equations and the domains of our three equations all right moving on to our last example ladies and gentlemen here we go write the equation so if we look here we have three three different equations again so I'm going to say f ofx which is the same as y so we can put it either way is equal to our three equation our first one here our blue one is an absolute value okay we're looking here absolute value we're going to say it includes that part right there so our absolute value we're going to use the equation that we know which is a absolute value x - H + K where a represents our slope right so if we look at our slope it's up one over two and it's right side up so we're going to have two for our slope there absolute value of x we went to the left 1 2 three so it's going to be plus three and then we went up two so we're going to put plus 2 there's our equation for that our second equation in red is a horizontal line at 1 2 3 four so all we're going to have to put here is four then our final equation here in the purple which I'm going to highlight in green is going to be a line so when we're working with lines we're going to work with the Y = a parentheses x - H + K so our slope first down 3 over 1 so -3 parentheses x I went to the right three so I'm going to put minus three and then I went up four so I'm going to put plus four and I can leave it just like that so now I've come up with my three equations now I have to come with my restrictions so this one is including it so and it's at x equals -2 so X when is less than or equal to -2 our next one our X is going to be in the middle they're not including so I'm going to have just less than not equal to it's going from -2 all the way to three so we're going to -2 to three for our X values and our last one we're including it starts at three goes on till Infinity so X has to be greater than or equal to three for our restriction on that one all right ladies and gentlemen that is all I have for you today um make sure you had all three examples and those tools written down and I hope you have a fabulous day and I'll be seeing you soon bye