 This video is going to deal with radicals and their graphs. So graphing radical functions is very similar to quadratics. So let's remember what we knew about quadratics. The general vertex form quadratic equation was a times x minus h quantity squared plus k. And a, if you'll remember, told us which way it opened. Remember, told us the left and right shifts, how far over to the left or how far over to the right that we did. And k told us how much we were going to go up or down. So we have the same thing with these radical equations. It's just a times the square root of x minus h plus k instead of the quantity squared. So we want to know what y equal the square root of x looks like. So second x squared and x, I'm over in my calculator. Let it graph it for me. Standard window should be good. And I'm making a rough sketch and it looks something like this. So where is the starting point? This is called the minimum value. Well, if we go and look at our table and we come back this way, you can see that we get zero, zero, but at negative one there's an error. And if we go anywhere beyond that backward, we'll have an error. So that means that our starting point right here is zero, zero. Okay, what is the domain? Remember, that's how far to the left and how far to the right does it go? Well, it starts at zero. That's how far to the left it goes. And it goes to infinity, that's what this arrowhead's telling me. So it would actually include zero and go to infinity. All right, let's think about the range. Remember, that's how far up and how far down did we go? Well, we started at zero and we went up from there. And we include the zero because we actually started at that point. And it increased from there. So the biggest value we could have would be infinity. All right, so let's look at this graph. I'm just going to second insert here at negative two, square root x. And now look at that graph. And if I look at that graph, I'm going to get a graph that looks something like this. And what is the starting point? Again, it looks like it's going to be zero, zero, but let's double check. And sure enough, zero, zero, negative one, we start seeing the errors happen. So it's at zero, zero. This is actually not the minimum value. I copied and pasted this one. This is actually the maximum value because if you look at that, that point is the highest point on our graph. So it's the maximum value. And the domain then starts at zero, but then it goes to the right forever. So the smallest value would be zero, and it goes to infinity, including zero to infinity. The range that starts at zero, but it goes down from there. So actually we start down at negative infinity. That's how low we go, and it goes all the way up to and including zero, but no farther. All right, so what happens if we just change this a little bit and put more underneath the radical? So second x squared, and now we're going to put x minus four in that graph. Remember that it started at zero, zero for the square root of x. And when we look at the square root of x minus four, it actually comes out here to four and then it starts. But we'll verify that. Does it really start at four? Sure enough, there it is. Four is when we can see that we have our minimum value. So what is the domain? The domain starts at four this time. Remember, that's our x values. And it's how far to the left and how far to the right. So it started at four and it goes to infinity. Again, these are x's and we're going left or right. Now the range, we're still on the x-axis to start with and then we move up. So we start at zero and we go up to infinity. So remember that these are my y's and we're talking about going up and down. How far up and how far down do we go? So one more thing we can do to our graph. What if we don't add or subtract underneath the radical? What if we take the square root of just plain x and outside the radical we add to? Remember, square root of x started at zero and when we look at this graph, it looks like it might have started at two on the y-axis. Still looks like the same graph, it just shifted up. So let's go and see what happens. Negative one is where I have my error. Zero is where I start but it's at two. So my minimum value here is zero two. So what is my domain? Where do I start in the x's? My x's start at zero and where do they go? They're increasing all the x's to the right are going to be included. So we have two infinity. Well, where does my range start? It starts at the y value of two and goes up. So two is the smallest value up to infinity. All right, so going back here now we're going to put it all together. And let's see if we can figure out what does this part in here do? That moves it left and right. And what does the constant out here do? The constant moves it up and down. So let's see if we can sketch it first before we verify with the calculator. So it's a plus two. Remember those are always backwards so we're actually going to be at negative two. But then this one tells me that I go down three because it's minus three. So one, two, three. So this should be where I'm starting right here. And it's a positive radical so it should increase. And let's see what happens if we verify that with our graph. The square root of x plus two under the radical to close the parenthesis and then minus three. And we want to look at our graph and our graph looks very good. So we did it right. So what's our starting point? That point negative two, negative three. What's my domain? Now I'm starting back here at negative two. Here's where my x's start and go this way forever. So I start at negative two and I go to infinity. Where do my y's start? My y's start down here and go up because the graph is getting bigger. It's increasing. So we go start at negative three, include it, and it goes to infinity.