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Sal graphs y=√x. Then, he shows a few examples of how we can shift and stretch the graphs of y=√x and y=x^2, and how the equations of those graphs look. Created by Sal Khan and CK-12 Foundation.
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, I always found a little bit ambiguous at first. I want to make it very clear in your head. If I write a 9 under a radical sign, I think you know you'll read this as the square root of 9. But I want to make one clarification. When you just see a number under a radical sign like this, this means the principal square root of 9. And when I say the principal square root, I'm really saying the positive square root of 9. So this statement right here is equal to 3. And I'm being clear here because you might already know that 9 has two actual square roots. By definition, a square root is something-- A square root of 9 is a number that, if you square it, equals 9. 3 is a square root, but so is negative 3. Negative 3 is 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'd actually put a negative in front of the radical sign. That is equal to negative 3. Or if you wanted to refer to both the positive and the negative, both the principal and the negative square roots, you'll write a plus or a minus sign in front of the radical sign. And of course, that's equal to plus or minus 3 right there. So with that out of the way, what I want to talk about 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 get a better understanding of 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. This is y is equal to the square root of x. Once again, we have x and y values right there. So 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 0, 1-- Let me make it color coded. When x is equal to 0, what's y going to be equal to? Well y is x squared. 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, y is 3 squared, which is 9. We've seen this before. And I could keep going. 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. And 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 have a positive or negative written here. We just have the principal square root. When x is a 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, the principal 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. Well, these are the same x and y's, but here you have x is 2, y is 4. Here 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, 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 y's 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-- 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. 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 going to have to draw 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 then I 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. And now let's graph it. So we have 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 above there. So the graph of y is equal to x squared, and we've seen this before. It's going to look something like this. We're just graphing it in the positive quadrant, so we get this upward opening u just like that. Now let's graph y is equal to the principal square root of x. So here, 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. Then we have 16 comma 4. 16 comma 4 is right about there. So this graph looks like that. So notice, 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 x's 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 whose resource I'm using. So this is my.hrw.com/math06. You could pause this video. And hopefully, you should be able 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-- OK. So let's first just graph 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: squared and the radical sign and all of that. 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. Look, if you just focus on the first quadrant right here, you see that you get the exact same result that I got 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, 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. This will really work for anything. Let's see what happens when you get 2 times-- no, not 2 squared --2 times x squared. And let's do another one that is 1.5 times our 0.-- I could just do 0.4 actually. 0.5 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 opening u 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 then 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 four to the right. What I do is I say, x minus 4. x minus 4 squared. And if I want to shift it two to the-- Let's say I want to shift it two to the left. x plus 2 squared, what do we get? Notice it did exactly what I said. x minus 4 squared was shifted four to the right. x plus 2 squared, was shifted two to the left. And it might be unintuitive at first, this shifting 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 0. 4 minus 4 is 0 and that's what you're squaring. Over here, when x is equal to negative 2-- negative 2 plus 2 --you are squaring 0. So, in other words, whatever you're squaring, that 0 is equivalent to 4 here. Or 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 could think about it, when x is equal to 1, we're 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 say minus 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 plus 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 it got shifted down by 5. If you want it to open up a little wider than that, maybe scale it down a little bit. Scale it down and let's say 0.5 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 idea actually, can be done with any function. So let's do the square root of x. And in green, let's do the square root of x. Let's say, minus 5. So we're shifting it over to the right by 5. And then let's have the square root of x plus 4. So we're going to shift it to the left by 4. Let's shift it down by 3. And so lets graph all of these. The square root of x. Then have the square root of x minus 5. Notice it's the exact same thing as the square root of x, but I shifted it to the right by 5. When x is equal to 5, I have a 0 under the radical sign. Same thing as square root of 0. So this point is equivalent to that point. Now, when I have the square root of x plus 4, I've shifted it over to the left by 4. 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, 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 to do that. 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 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 when we graph these, it's not a sideways parabola because 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 talk, I guess, about the relationships with parabolas, and/or with the x squared's and the principal square roots, useful. And how to shift them. And that will actually be really useful in the future when we talk about inverses and shifting functions.