so we have 5.3 we are going to be graphing these radical functions as we've been studying radicals so now there's two types of radical functions you have an even or you have an odd so i'm going to give you an example of any even radical so first of all you have to understand all evens will take this general pattern all odds will take the general pattern as well as we talk through these so i'm going to take the square root of 4 excuse me the fourth root of x so therefore if i draw this graph it's going to look kind of like what i have below it it's going to start at the origin and then it's going to go up and off to the right so then every single even graph will take this shape now the center which is at 0 0 here that can move under certain transformations when we talk transformations earlier they can go up down left right so we're going to have a couple examples of those two but if we talk about the domain remember the domain are the x values and here we are starting at the x is zero because this ordered pair is zero comma zero so the x is zero and the y is zero so here everything to the right of this x is a value for this particular function so everything to the right is greater than zero now if you want to say everything to the right of zero or everything more than zero i'm okay with that as well if you don't like the notation so now if you take a look at your y's everything is graphed above the x-axis which that's where y equals zero so everything is more than zero so you could put everything here as more than zero as well um every y you could say instead of everything you could put every x here or every y so that's how those roll out now if you talk about odds now we have a different scenario for the odds it takes a different shape it kind of takes like an s um but it has to once again go through the origin so it all always starts here and now the the higher the uh radical that you take so instead of three say it was five it would get closer to the x-axis but in this case i can have x's that go all the way to the left i can have x's that go all the way to the right so this is all real numbers because it can go both ways now if you think of your y's your y's are very similar if your y's are going above the zero zero point and they're going below the zero zero point they're not any different they're all reals real numbers as well actually any odd function will always have a domain and a range of all real numbers so let's try two that are not sitting right at the origin so now we have a function where we have some transformation so we're bringing back transformations of how we're moving so first of all we identify that this is an even function so i do know that in my graph here i do know that the parent function looks something like this so then i'm going to be moving that particular little little graph okay so i'm gonna do the same here for this one so the black is the parent function like where it originally starts but then when you throw these numbers in here like in example one i have x plus 2. well we got to remember that when we did our transformations earlier with quadratics this number right here made the graph go left 2 units and the 1 is going to make the graph go down 1 unit so therefore i take my vertex which is currently at 0 0 and i move it so that it is left two and down one so here is my new vertex but it still takes the same shape okay and it doesn't have to go through the origin mine just so happened to when i drew it so then if we're talking domain and range your x starts at the negative 2 but goes to the right hand side so if it goes towards the positive it's getting to be greater than or equal to if you go ahead and talk range range is your y's so that's related to your negative one in your ordered pair and here it's everything above here and when you go above or you go up that becomes positive so it's a greater value okay so looking at number two we have an odd root so we know that it's going to take the formation of the graph that i kind of have here we also should say hey you know what it's an odd root i know these answers right away all real numbers all real numbers that does not change ever for an odd root evens are a little picky but odds all real numbers all the time so let's talk about the transformations here so the 2 here that is going to be a vertical stretch by 2 so it gets steeper faster the 7 is going to go the opposite of what you think so it's going to go i better put opposite over here because we did that over here too it's going to go to the right 7. and then this 2 does exactly what you say or it does as it looks so it goes up 2. so my new vertex is going to go 1 2 3 4 5 6 7 and it's going to go up 2. so right here is where that curve is going to be instead of at the origin so therefore i'm gonna go like so still has the same shape it just moves that middle point and then like i said it's odd so all real numbers for the domain as well as the range let's try one more and we're going to put a couple another twist to it it's a little slow we got to be patient okay this is our example three so we're going to have y is equal to negative let's do the square root of x minus 4 plus 5. so now we have something interesting here this negative and if you remember what a negative does it reflects over the x-axis and this 4 does its opposite and that's going to go right 4. this 5 is going to go ahead and go up 5. so if we look at our general pattern or shape of our graph it normally goes like this and so now we have to go ahead and do our vertex it's no longer going to be at 0 0. it's going to go 1 2 3 4 up 5 1 2 3 4 5 okay but this reflection causes that graph to not go up towards the the positives it's going to make it reflect and go down because of the reflection so now i have this point that's at four comma five my domain is going to be the x values but all of the x values that are to the right so your x has all the x values to the right which that stays positive so greater than or equal to four your range on the other hand since your graph goes down down means less than or equal to so what's it less than or equal to the five because we have the y being five and that's how you do our graphs today