 In this video, we're going to discuss how to graph the function y equals x plus three quantity squared minus five. So the first thing to recognize here is that there's graph transformations involved with this thing, but we have to first identify what is the basic graph in play here? What is the basic graph? If we ignore all of the transformations, what's left? Like if we ignore this minus five, if we ignore this plus three, our function looks just like y equals x squared. And so that's the basic graph here, y equals x squared. And so in order to start graphing this formula, this function here, we would start off with this basic function y equals x squared. If you don't know what that graph looks like, that's okay. Many of you might have seen this before, but it's not a big deal. y equals x squared is just our standard parabola, which looks something like the following. We might talk more about how to graph these basic functions very later on in this series here, but y equals x squared is just a standard parabola. It goes through zero, zero, one, one, two, four, three, nine, four, sixteen, et cetera here. And so on the screen, it's illustrated as this dashed curve in white. Okay. And so then once you've identified the basic function, then you want to start looking for transformations that are applied to it. So for example, notice that I replaced the x with an x plus three. What does that do to the graph x plus three? That x plus three causes the graph to be shifted. We shifted to the left. We might have to reference the table we had from the previous video. We want to shift left by a factor of three. And so you see that's what's happened here as we go from the white graph, one to the green graph, which is also a dashed curve right here. This function, y equals x plus three squared is a shift of the graph left by three units. And I mean, this is this is exactly three units right here. Like this, the point zero zero moves over to negative one zero to negative two zero to negative three zero. Right. This point got shifted over by three units. And this happens for all of the points on the graph. Like if we take the point one one, it's going to move over one, two, and then three, we get this point right here, which is going to be negative two comma one. This came from one comma one. What did we do here? We just take the x coordinate one and we subtract from it three, which is a shift to the left and then we get negative two. So each of these points we can do this for each point on the graph, right? The graph itself is just a collection of all the points we shift each graph to the left by three. We just subtract three from its x coordinate. That's where this green graph comes from right here. And if we want a formula for it, it'll just be y equals x plus three squared like so. But then what else is happening here? This is negative five. What is the negative five do? Notice that the x plus three is inside of the function. A good indicator of this is the parentheses, right? Our function is x squared. And so everything inside of the square is part of the horizontal zone. And so that's why the x plus three causes the graph to move to the left. On the other hand, the negative five is outside of the horizontal zone. The parentheses usually give you a good indicator of that horizontal zone. That negative five is outside of the horizontal zone. And this will thus affect the graph as a shift. This is going to be a vertical shift. So it's going to shift down by a factor of five. And so taking this point we had before negative three zero, we're going to shift it down by five. One, two, three, four, five. I'm just kind of counting the marks here on the y-axis. We shift everything down by five. Oops, put that point back there. And so now we get this new point negative three comma, comma negative five. It went down. And we did that for this point that started at the origin. If you take this point that was at one one, right? That was this point where it moves to here negative two one. We'll move that down. You're going to get this point negative two comma negative four. Where did negative four come from? It's just one minus five. So to move things to the left, you should subtract three from the x-coordinate. A shift to the right would just be adding three to the x-coordinate. A shift down just means you're going to take the y-coordinate and subtract five from it. A shift up you would add to the y-coordinate. So what I want to do is switch over to the graph of this function. So this is the little Desmos page I created for this lecture. You can find the link to this in the video description here. I'm going to switch our function. We're going to switch it to x squared. This is now our, whoops, this is going to be the graph you see right here. And so now let's change things, right? We didn't do any vertical or horizontal stretching or compressing. So I'm going to leave that number A and B at one. But the h value, if we have x plus three, that means you're taking x minus negative three. So switch h to be negative three. You see the graph is then moving to the left by three units, just like we predicted there. And then for k, k is going to control the move up and down. So we had a negative five right there. So we're going to move this thing down by five. Up that's moving it up, moving it down by five. So we should have a negative five right there. Went off the screen a little bit. And I'll zoom out. And so you can see then how did this graph change? The point that was at the origin moves. So the point that was at the origin moves down to be negative three, negative five. And the point at one, one moved down to negative two, negative four, just like we saw here. And so the graph that we produced is identical to the one that the calculator produces. And so you see that here on the screen. And so that's what we're trying to be able to do in this section. How do we, how do we produce the same graph that the graph calculator produced without the need of a calculator? We want to understand the connection between the algebraic formula of the function with its graph that we see here on the screen. And we saw on this one that we just had a vertical and horizontal shift.