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# Graphs of square-root functions

CCSS.Math:

## Video transcript

I think you're probably reasonably familiar with the idea of a square root but I want to clarify some of the notation that at least what I always found a little bit ambiguous at first I want to make it very clear in your head so when I write if I write a nine under a radical sign I think you know you'll read this as the square root of nine but I want to make one clarification when you just see a number under a radical sign like this this means this means the principal principal square root of nine and when I say the principal square root I'm really saying the positive square root of nine so this this statement right here is equal to three and I'm being clear here because you might already know that nine has two actual square roots nine also by definition square root is something school if that a square root of nine is a number that if you square it equals nine three is a square root but so is negative three negative three also a square root but if you just write a radical sign you're actually referring to the positive square root or the principal square root if you want to refer to the negative square root you would actually put a negative in front of the radical sign that is equal to negative three or if you want to refer to both the positive and the negative both the principal and the negative square roots you'll write a plus or minus sign you'll write a plus or minus sign in front of the radical sign and of course that's equal to plus or minus three right there so with that out of the way what I want to talk about is the graph is the graph of the function y is equal to the principal square root of x and see how it relates to the function y is equal to X let me write it over here because I'll work on it see how it relates to y is equal to x squared and then if we have some time we'll shift them around a little bit and he had a better understanding of how these what causes these functions to shift up down or left and right so let's make a little Value table before we get out our graphing calculator so this is for y is equal to x squared so we have x and y values and this is y is equal to the square root of x once again we have x and y values right there so let's let me just pick some arbitrary x values right here and I'll stay in the positive x domain so let's say X is equal to zero one let me stay make it color-coded when X is equal to 0 what's Y going to be equal to well Y is x squared so 0 squared is 0 when X is 1 Y is 1 squared which is 1 when X is 2 y is 2 squared which is 4 when X is 3 when X is 3 y is 3 squared which is 9 we've seen this before I could keep going maybe I'll add a little let me add 4 here so when X is 4 y is 4 squared or 16 we've seen all of this we've graphed our parabolas this is all a bit of review now let's see what happens when y is equal to the principal square root of x let's see what happens now I'm going to pick some X values on purpose just to make it interesting when X is equal to 0 what's Y going to be equal to the principal square root of 0 well it's 0 0 squared is 0 when X is equal to 1 the principal square root of x of 1 is just positive 1 it has another square root that's negative 1 but we don't we don't have a positive or negative written here we just have the principal square root when X is 4 what is y well the principal square root of 4 is positive 2 when X is equal to 9 what's y when X is equal to 9 the principal square root of 9 is 3 finally when X is equal to 16 principle square root of 16 is 4 so I think you already see how these two are related we've essentially just swapped the X's and the y's right you have well these are the same X and Y's but here you have X is 2 y is 4 your X is 4 y is 2 3 comma 9 9 comma 3 4 comma 16 16 comma 4 and that makes complete sense if you were to square both sides of this equation if you were to square both sides of this equation you would get Y squared is equal to X right there and of course you would want to restrict the domain of Y to positive wise because this can only take on positive values because this is a principal square root but the general idea we just swapped the X's and Y's between this function and this function right here if you assume a domain of positive X's and positive Y's now let's see what the graphs look like and I think you might already have a guess of what let me let me just graph them here let me do them by hand because I think that's instructive sometimes before you take out the graphing calculator so I'm just going to stay in the positive in the first quadrant here in the first quadrant so let me graph this first so we have the point 0 0 the point 1 comma 1 the point 2 comma 2 which I'm gonna have to do it a little bit smaller than that let me mark this is 1 2 3 actually let me do it like this let me go 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 that's about how far I have to go and have 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 that's about how far I have to go in that direction as well now let's graph it so if 0 0 1 1 2 comma 1 2 3 4 2 comma 4 right there 3 comma 9 3 comma 5 6 7 8 9 3 comma 9 is right about there and then we have 4 comma 16 4 comma 16 is going to be right right about there so the graph of y is equal to x squared we've seen this before it's going to look something like this we're just graph get in the positive quadrant so we get this upward-opening you just like that just like that now let's graph y is equal to the principal square root of x so here we have once again we have 0 0 we have 1 comma 1 we have 4 comma 2 1 2 3 4 comma 2 we have 9 comma 3 5 6 7 8 9 comma 3 right about there and we have 16 comma 4 16 comma 4 is right about there so this graph looks like that so notice they look they look like they're kind of flipped around the axes this one opens along the y-axis this one opens along the x-axis and once again it makes complete sense because we've swapped the XS and the Y's especially if you just consider the first quadrant and actually these are symmetric around the line y is equal to X and we'll talk about things like inverses in the future that are symmetric around the line y is equal to X and we can graph this better on a regular graphing calculator I found this on the web I just did a quick web search I want to give proper credit to the people I'm whose resource I'm using so this is my dot h RW comm slash math Oh 6 you could pause this video and I hopefully you should be able to read this especially if you're looking at it in HD but let's graph these different things let's graph it a little bit cleaner than what I can do by hand and actually let me have some of what I wrote there so that should give you that should give you okay so let's first just graph y is equal to y is equal to x squared and then in green let me graph y is equal to the square root of x they have some buttons here on the right just so you know what I'm doing I have some buttons here on the right it's squared and the radical sign and all of that but I want you to let me just focus on this so let me just graph those so first it did x squared and then it did the square root of x and look when if you just focus on the first quadrant right here you see that you get the exact same result that I got oh over there although mine is Messier now just for fun and you know I really didn't do this yet with the regular quadratics let's see what happens what we need to do to shift the different graphs so with x squared well I'm going to do two things I'm going to scale the graphs and I'm going to shift them so that's x squared so let's just focus on the x squared and see what happens when we scale it and then I'll do it with the radical sign as well because I want to get this this will really work for anything let's see what happens when you get 2 times 2 times not not 2 squared 2 times x squared and let's do another one that is 1 0.5 0.5 times our point I could just do point 4 actually 0.45 times x squared let's graph these right there so x squared so notice our regular x squared is just in red if we scale it by 2 it's still a parabola with the vertex at the same place but we go up faster in both directions and if we have 0.5 times x squared we still have a parabola but we go up a little bit slower we have a wider we have a wider opening you because our scaling factor is lower than 1 so that's how you kind of decide how wide or how narrow the opening of our parabola is and if you want to shift it to the left or the right and I want you to think about why this is so that's x squared let's say I want to just take the graph of x squared and I want to shift it 4 to the right what I do is I say X minus 4 X minus 4 squared and if I want to shift it to to the let's say I want to shift it to to the left X plus 2 squared what do we get notice it did exactly what I said X minus 4 squared was shifted 4 to the right X plus 2 squared was shifted 2 to the left and it might be unintuitive at first this shifting and that I'm talking about but really think about what's happening over here the vertex is where X is equal to 0 when you get 0 squared up here now over here the vertex is when X is equal to 4 but when X is equal to 4 you stick 4 in here you get 4 minus 4 so you're still squaring zero right 4 minus 4 is 0 and that's what you're squaring over here when you when X is equal to negative 2 negative 2 plus 2 you are squaring 0 so if in other words whatever you're squaring that at 0 is equivalent to 4 here and where 4 is equivalent to 0 and negative 2 is equivalent to 0 over there so I want you to think about it a little bit another way you can think about it when X is equal to 1 where at this point of the red parabola but when X is equal to 5 on the green parabola you have 5 minus 4 inside of the parentheses you have a 1 just like X is equal to 1 over here up here so you're at the same point in the parabola so I want you to think about that a little bit it might be a little non-intuitive that you [ __ ] you say - and 4 to shift to the right and plus 2 to shift to the left but it actually makes a lot of sense now the other interesting thing is to shift things up and down and that's actually pretty straightforward you want to shift this curve up let's say we want to shift the red curve up a little bit you do x squared + 1 notice it got shifted up if you want this green curve to be shifted down by 5 put a minus 5 right there and then you graph it and got shifted down by 5 if you want it to be will open up a little wider than that maybe scale it down a little bit scale it down let's say point 5 times that so now the green curve will be scaled down and it opens slower it has a wider opening and the same idea can be done with the principal square roots so let me do that let me do the same idea and the same magic sure can be done with any function so let's do the square root of x and let's in green let's do the square root of x let's say - 5 so we're shifting it over to the right by 5 and then let's have the square root of x + 4 so we're going to shift it to the left by 4 and let's shift it down by 3 and so let's graph all of these so the square root of x then have a square root of x - 5 notice it's the exact thing same thing as the square root of x but I shifted it to the right by 5 five when X is equal to five I have a zero under the radical sign same thing as square root of zero so this point is equivalent to that point now when I have the square root of x plus four I've shifted it over the left by four when X is negative 4 I have a 0 under the radical sign so this point is equivalent to that point and then I subtracted 3 which also shifted it down 3 so this is my starting point if I want this blue square root to open up slower if I want it to open up slower so it'll be a little bit narrower I would scale it down so here putting a low number will scale it down and make it more narrow because we're opening along the x axis so let me do that and let me make let me make this green one let me open up wider so let me say it's 3 times the square root of x minus 5 so let's graph all of these so notice this blue one is now opens up more narrow and this green one now opens up a lot I guess you could say a lot faster it's scaled up then we could shift that one up a little bit by 4 and then we graph it and there you go and notice that we don't have when we graph these it's not a sideways parabola cuz we're talking about the principal square root and if you did the plus or minus square root it actually wouldn't even be a valid function because you would have two y values for every x value so that's why we have to just use the principal square root anyway hopefully you found this little a talk I guess about the relationships with X of with parabolas and or with the X Squared's and the principle square roots useful and how to shift them and that'll actually be really useful in the future when we talk about inverses and and shifting functions