Worked example: graphing piecewise functions

so I have this somewhat hairy function definition here and I want to see if we can graph it and this is a piecewise function it's defined as a different essentially different lines you see this right over here even with all the decimals of the negative signs this is essentially a line it's defined by this line over this interval for X this line over this interval of X and this line over this interval of X let's see if we can graph it and encourage you officially if you have some graph paper to see if you could graph this on your own first before I work through it so let's think about this first interval if one negative 10 is less than or equal to X which is less than negative 2 then our function is defined by negative zero point one two five X plus four point seven five so this is going to be a line as downward sloping line and the easiest way I can think about graphing it is let's just plot the the end points here and then draw the line so when X is equal to 10 so when our sir when X is equal to negative 10 so we would have negative zero actually let me write it this way let me do it over here where I do the so we're going to have negative 0.125 times negative 10 plus four point seven five that is going to be equal to let's see the negative times a negative is a positive and then 10 times this is going to be it's going to be one point two five plus four point seven five that is going to be equal to six so we're going to have the point negative 10 comma six and that point and it includes so X is defined that's less than or equal to and then we go all the way to negative two so when X is equal to negative two we have negative 0.125 times negative two plus four point seven five is equal to see negative times negative is positive two times this is going to be point is going to be positive 0.25 plus four point seven five it's going to be equal to positive five now we might be tempted we might be tempted to just circle in this dot over here but remember this interval does not include negative two it's up to it including it's up to negative two not including so I'm going to put a a I'm going to put a little open circle there and then I'm going to draw the line and then I'm going to draw and I'm going to draw the line I am going to draw my best attempt my best attempt at the line now let's do the next interval the next interval this was a lot more straightforward we started x equals negative two when x equals negative two negative two plus seven is negative 2 plus 7 is 5 so negative 2 so negative 2 comma 5 so it actually includes that point right over there so we're actually able to fill it in and then when X is negative 1 negative 1 plus 7 is going to be positive 6 positive 6 but we're not including x equals negative 1 up to and including so it's going to be it's going to be right over here when X is negative 1 we are approaching or as X approaches negative 1 we're approaching negative 1 plus 7 is 6 so that's that interval right over there and now let's look at this last interval this last interval when X is negative 1 you're going to have well this is just going to be positive 12 over 11 because we're multiplying it by negative 1 plus 54 over 11 which is equal to 66 over 11 which is equal to positive 6 so we're able to fill in that right over there and then when X is equal to 10 you have negative 120 over 11 I just multiplied this times 10 12 times 10 is 120 and we have the negative plus 54 over 11 so this is the same thing this is going to be what is this this is negative 66 over 11 is that right let's see if you that is negative 66 over 11 which is equal to negative 6 so when X is equal to 10 our function is equal to negative negative 6 and so this one actually doesn't have any jumps in it it could've but we see so there we have it we have graphed this function that has been defined in a piecewise way