Transforming the square-root function

So let's think about the graph of y is equal to the principal root of x. And then we'll start playing around with this and see what happens to the graph. So y is equal to the principal root of x. Well, this is going to be undefined if we want to deal with real numbers. For x being any negative value. So the domain here is really x is greater than or equal to 0. When x is 0, y is going to be equal to 0. When x is 1, the principal root of 1 is positive 1. So it's going to be like that. When x is 4, the principal root of 4 is 2. When x is 9, the principal root of 9 is 3. So this is what it is going to look like. Which is going to look more like this. So it's going to look something like that. That's my best attempt at actually graphing it. Now let's think about what happens if we wanted to shift it in some way. So let's say we wanted to shift it up. Let's say we wanted to shift it up by 4. So how would we do that? Well, whatever value we're getting here, we want y to be 4 higher. So we could just add 4 to it. So we could just use y is equal to the square root of x plus 4. So that would be like taking this graph right over here. So let me copy and then let me paste it. So it's like taking this graph and we're shifting it up 1, 2, 3, 4. It would look that. Well, that was pretty straightforward. Let me do it in that same blue color so that you recognize that that's that one right over there. But what if we wanted to shift it, let's say, to the left. Let's say we wanted to do something like this. Let's say we wanted to shift it to the left by 3. Like that. So how would we define the function then? And I'll do this in this orange color. We want to shift by 3. So you're shifting to the left by 3. So think about it. This point, y equaled 0, right over here, where whatever you put under the radical was equal to 0. So what do you have to put under the radical here to get 0? Well, here x is negative 3. So if you put x plus 3 under the radical then you are going to get-- and you take the square root of that-- you're going to get 0. So this right over here, this orange function, that is y. Let me do it over here. y is equal to the square root of x plus 3. And once again it might be counter-intuitive. We went from square root of x to square root of x plus 3. When we added 4 outside of the radical that shifted it up. But when we add 3 inside of the radical instead of it shifting it to the right, instead of shifting it that way, it shifted it to the left. It made this point go from 0 to negative 3. And the important thing to realize is what makes y equal to 0. Over here y equals 0 when x is 0. Over here and over here, y is equal to 0 when x plus 3 is equal to 0, or x is equal to negative 3. And you could do that for other points to see that it does definitely shift to the left. And that's an important thing to realize. This isn't actually just for radical functions. This is actually for functions in general. If you're just going to add a 4-- add a number out here-- whatever you add is going to shift it up or down. If this was a minus 4 it would have shifted it down. But within, when you replace the x with an x plus 3, when you replace it with an x plus 3, this actually shifted it to the left by 3. If you wanted to shift it to the right by 3 you would put an x minus 3 over there. Well, that's all interesting. But let's say that I wanted to flip this thing over. So I wanted it to look like this. Let me see if I can draw it. I want this graph to look something like that. I'll try my best to-- to look something like that. So it's flipped around. I could do a better job than that. So we have that point on it. And then we're going to go 3 and then we're going to have that point on it. So I want it to look-- actually I did a decent job the first time I drew it-- I want to look something like that. So essentially I want to take its mirror image around the line x is equal to 3. How could I do that? Well, now my domain is different. Now my domain-- it should be undefined for anything where x is greater than negative 3. And it should be defined for any x that is less than or equal to negative 3. Or another way to think about it is we need to flip the sign of whatever we have under the radical. So this thing over here-- let me actually scroll over a little bit-- this thing over here in green could be y is equal to the square root of the negative of x plus 3. And I encourage you to try some x values here to try it out. What we've done is we've essentially flipped what happens under the radical. Now in order to get a positive value under the radical, now x plus 3 has to be negative. And the only way that x plus 3 is negative, or the only way that x plus 3 is, I guess you can say nonpositive, is if x is less than negative 3. Now what if we wanted to do something even more interesting? What if we wanted to flip this one right over here over the horizontal axis, over y equals 0? Well, then we're just flipping what root we take. So that would be y is equal to the negative square root of negative x plus 3. So it would look like this. It would look like that. And if we wanted to shift that thing, we could just add or subtract something outside of the radical. So let's say we wanted to do-- let me copy and paste this. So let's copy and then let's see. Let's say that we wanted to shift it down here. So instead of beginning at y equals 0 right over here, where y equals negative 4, then we would just subtract 4 outside of the radical. So this thing and this thing-- I'm running out of colors here-- this thing right over here would be y is equal to negative square root of negative x plus 3 minus 4. And so we could keep going on and on and on. But hopefully this gives you a sense of the different ways you could manipulate this thing. And we'll do some more examples to get a better understanding of it.