 In this video, we're going to talk about how to graph the function h of x equals the square root of 1 minus x plus 2. We're also going to talk about the domain and range of this function. So the important thing to remember here is that this is going to be based upon, well, the basic graph here. We have to identify what's the basic graph without any of the transformations going on here. Like if I give rid of this 1 and the minus sign, I give rid of the plus 2, the basic graph in play here is going to be the square root of x. And so that's kind of like step zero when it comes to graphing these things. We want to look at the basic graph y equals the square root of x. And then we start identifying transformations we've done to that. And the square root of x is going to look like this graph right here. It goes to the right, something like that. And so then what do we do? Well, we can very quickly see things like the negative in front of the x is going to tell us that there's a reflection of some kind. A coefficient of negative 1 tells us there's a reflection. A negative one in front of the x is going to be a horizontal reflection, aka a reflection across the y-axis. If you saw a negative sign in front of the square root, which was the basic function, that would be a vertical reflection across the x-axis. We don't have such a thing here. So we're going to be reflecting across the y-axis. So our standard graph is going to reflect over, let me clear it out now, it's going to be reflected across the y-axis. And so it's going to then appear this way. So it's the same square root, but now it points to the left instead of points to the right. Then we want to shift things vertically by 2, right? If you look at this plus 2 that's outside of the square root, the plus 2 we're adding to the function f of x here. And so this is going to move the entire graph up by 2 units. And so if we then take a look at that picture, our square root now has been shifted up by 2, and it would look something like that. The final step here is to figure out what does this plus 1 do right here? The plus 1 is actually going to be a shift to the right by a factor of 1, which you can see here, our final graph. We could shift it to the right by 1. It should go 1 to the right. But why to the right? Why isn't it a shift to the left? Isn't it a plus 1? Well, this is the issue we have to worry about, right? When we talked about the order of operations, we always did reflection first, then shift, then scaling, then shifting, right? And an important thing you have to have here is, if ever, this is, again, the weird consequence of the horizontal zone. If ever the coefficient of x is anything other than 1, you should factor, you should factor the coefficient of the x. Otherwise, you might be getting the wrong transformations here. So the fact that we have this negative sign in front, what I want you to do is factor it out. So rewrite the function as the square root of negative x minus 1 here. Let me write that. So we factor the negative 1 from the x, but we have to also subtract away from the positive 1, like so. So we really want to think of this as the function square root of negative x minus 1 plus 2. And you'll see here that when you look at this factored form, you'll still see a negative sign. So that means reflect across the y-axis, because you have a negative in front of the x. You'll still see this plus 2 outside of the square, which will still mean shifting up by 1, shifting up by 2, excuse me. No difference there. But now when you look inside of the horizontal zone, you see a negative 1. And this correctly indicates to us that we want to shift things to the right. And so the order of operations matters here. This function, the original form, actually was doing a shift then a reflection, but we agreed earlier that we're going to reflect and then we're going to shift. So to avoid those confusions, always factor the coefficient away from the x. And if you're adding something to the x, that you'll need to factor away from that as well. So you prefer to see the horizontal zone factor. The horizontal zone should be factored. Let me even write this on the screen. The horizontal zone should be factored. If it's not factored, you're going to probably get a lot of headache there. Life will be easier if you factor the horizontal zone. So we've correctly identified the transformations in play here. What does this have to do with the domain, right? One thing we could see here is that this is a square root function. The horizontal zone is going to affect the domain of this thing, right? We could then look like one minus x is greater or equal to zero. We then solve and get one should be greater or equal to x or x is less or equal to one. So notice you can very quickly identify that the domain of our function h here is going to be all numbers less than one. And you can see that right here, right? You have one and then you get everything to the left, but you don't get anything to the right. That's the correct domain. We can solve that algebraically, but I should also mention that we can find the domain of this thing perfectly geometrically because if we think of the original graph, y equals the square root of x here, y equals the square root of x, its original domain is going to be all non-negative numbers, zero to infinity. And so if we pay attention to the transformations we did, reflected across the y-axis, that changes the domain to be negative infinity to zero because now the graph, instead of being on the left-hand side, the right-hand side of the y-axis, it got shifted over, reflected over to the other side. So you switch the domain when you did that horizontal transformation. In terms of vertical transformations, you shift it up, that didn't change the domain. Then when you shifted everything to the right by one, you move everything over by one, what that does is you didn't get a picture over here, that's going to shift the domain one more time, giving us the current picture we see right there. And so every time you do a horizontal, oops, I got way much backspace too much here, every time you do any type of horizontal transformation, it changes the domain. And so with this one reflecting across the x-axis, you do standard graph, remember, the domain switched from zero to infinity, that's the standard one. This horizontal transformation switched the domain to be negative infinity to zero, and then shifting it right switched the domain to be negative infinity to one, which is the final domain. So by changing the graph with horizontal transformations, you change the domain because isn't the domain, the set of all possible x-coordinates, which the x-coordinate is the horizontal component, the horizontal coordinate for the graph. So every time you change the horizontal, you change the possibilities for the horizontal, which is what the domain is. Could we do the same thing for range, right? Because we have algebraic ways of finding the domain, we don't need to do this idea of transformations, but we're not very good at finding the range of functions yet. What if we did the same game for the range? Because the standard function, the square root of x right here, it could go up and up and up and up and up, it goes off towards infinity there. So its range, the range here, is likewise going to be zero to infinity. So that's the default range for this basic graph, zero to infinity. Well, what happens if we shift everything up by two? Well, if we shift everything up by two, we go up, we get a picture like this, and then the horizontal transformations come into play. If you shift everything up by two, the range would then change to be, it's not going to be this anymore, it's going to be two to infinity, right? The range goes up when you shift it up by two, and so the range would then go from zero to infinity to two to infinity. So if we know the domain and range of the original function, the original graph, and then if we apply the appropriate transformations, horizontal transformations affect the domain and vertical transformations affect the range, we can then actually determine without any algebra whatsoever the domain and range of the final function. And I mean, there is a little bit of hiccup there. You have to know the domain and range of the original graph, but knowing that we can find the domain of the range purely geometrically by identifying the transformations. And this actually gives us a very powerful way of finding the range of a function for which at the moment we don't have really a good algebraic way of finding the range. The range is in this situation best studied by looking at the geometry. And so that's going to bring us to the end of lecture five here where we've talked about graph transformations. Graph transformations are deeply connected with function composition, right? This function right here, we could decompose it. Each of these steps we do is a decomposition here. We see that there's this plus two on the left-hand side. There's these negative and negative one. Decomposing the function is actually identifying graph transformations. And we can see in this example that even if we're only interested in the function algebraically, the graph transformations help us understand better the domain and range of the function based upon these transformations. 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