 And welcome back. Today we're going to talk about graphing quadratic inequalities. Basically what we're going to do is we're going to graph a combination of inequalities, so your dashed lines and solid lines and shading up or down and that kind of stuff. And then also graphing quadratic, so your quadratic, your parabolas, your u-shaped curves that are either opening upwards or downwards or how I like to say, either your smiley faces or your fronty faces. Okay, that's an easy way to remember that. So anyway, a couple of things we're going to do. So this is kind of the gist of how we are going to graph these quadratic inequalities. Okay, first thing, graph the parabola. Remember to use either a dashed line or a solid line. Whenever you graph inequalities, what you're doing is you're graphing a boundary line. Not necessarily just a line, just a boundary. Okay, so the shaded area tells you where all your solutions are. And so when you graph this boundary, this tells you, okay, I have answers on this side and then I don't have any answers on that side. So once you get a picture, it'll make a little bit more sense. Okay, so we want to graph the parabola. Remember to use either dashed lines or solid lines. Dashed lines is when we have greater than or less than symbols. Solid lines, we have greater than or equal to or less than or equal to. It's actually really easy to remember if it's dashed or solid. If you have a solid line underneath, it's going to be a solid line. If you don't have anything, it's going to be a dashed line. So it's actually relatively easy to remember if it's dashed or solid. Okay, now after we've graphed it, we want to shade inside or outside the parabola, depending on what the inequality symbol tells us. So in this case, for this example, we have a greater than symbol. And once we get the example, once I go through one of these, I'll show you, okay, kind of logically. When I shade up or down, do I shade inside the bowl? Do I shade outside? I'll show you here in a minute. Okay, so the first thing we want to do is graph the parabola. Okay, I'm going to do this kind of quickly. You should know how to graph parabolas before this. But I'm just going to kind of very quickly go over this. All right, so the first thing I want to do is I want to find the vertex. X equals negative B over 2A. This little formula here tells us the X coordinate of my vertex. So in this case, my B number is negative 7. Okay, don't make that mistake of not getting that negative in there. This is a negative, negative 7. Okay, so I got two negatives there. And then two times my A number is 1. So in this case, I have a positive 7 over 2. So in this case, my X is equal to 3.5. Now you can either use 3.5, 7.5, whatever you want to. For this video, I'm going to use decimals because those are more widely used. Personally, I like fractions better, but it doesn't really matter what I want. Anyway, so I'm going to make a little table. Easy to graph with a table. Easy, easy. Okay, in the middle of this table, I'm going to put my vertex. Okay, I already know the X coordinate for that vertex. It's going to be 3.5 something. Okay, I don't quite know what it is quite yet. But what I'm going to do now is I'm going to find numbers that are around this vertex to give me a good idea of what my parabola is going to look like. Now, if you have choices of what numbers to use, don't use decimals and fractions. Use something easier. So in this case, I'm going to use numbers like 3 and 2, 4 and 5. I'm going to use numbers like that. Okay, so now my vertex is right here at 3.5 something. And then these are the other numbers that I'm going to plug into my function to figure out what the Y coordinates are. Okay, now I've already figured out what these numbers are rather quickly. So I won't waste anybody's time. So here you go. 2, 0 is what this Y coordinate is. 3, negative 2. Okay, and then 4, what's this one? 4, negative 2, and then 5, 0. Okay, now again, to find these Y coordinates, all we have to do is take these X coordinates, plug them into the function, and then whatever number we get out of it is going to be what we have here. And I forgot my vertex here, which is actually, you know what? I don't know what it is, but I can find out real quick because calculators are awesome. If I plug in this number, it's going to give me negative 2.25 right there. Let's do that again. I don't want my numbers covered up. Here we go. Negative 2.25. There we go. That's a little better. Okay, decimal sometimes, what I was using for a calculator, I didn't have my decimal set up, but that's okay. Anyway, so now we have what the vertex is. We have a couple of points to graph, so here we go. So 3 and a half, my vertex is 3 and a half, 1, 2, 3 and a half, and then negative 2.25. That's negative 1, 2, and about a quarter, so it's about right there. Sometimes those can be hard to graph, but just put it as close as you can. So 3 and a half, 1, 2, 3 and a half. So there's your half part right there. Down 2 and a quarter, 2.25. So 1, 2, and then a quarter. Get as close as you can. Okay, so my other numbers are 3, negative 2, and 4, negative 2. So 1, 2, 3, negative 2 right here. And 1, 2, 3, 4, negative 2 right here. You can already start to see the smiley face here. 2, 0, and 5, 0. So 1, 2, 0, and then 1, 2, 3, 4, 5, 0. You can really start to see the smiley face going on now. Okay, so now before I start doing my lines, I need to figure out, okay, do I have a dash line or solid line? In this case, I have greater than or equal to, so it's going to be a solid line. Okay, so I'm just going to graph my smiley face here. Graph my smiley face there. Okay, make sure you put arrows. Make sure you put arrows on the end of your graphs. That way, the teacher knows that you know that these lines go on forever. That's the picky little thing there. Okay, so that's the first part of it. We have now graphed our parabola, and now what we want to do is we want to shade inside or outside the parabola, depending on our inequality symbol. So in this case, y is greater than this line. So this is kind of how I rationalize it to myself. This is where I want to shade. The y's, this is where I want to shade. The y's that are bigger than this line. We use all these numbers here to create this line over here. Okay, so what we want to do is, excuse me, my voice is going, so what we want to do is say to ourselves logically, the y's that I want are bigger than this line. The y's that I want are bigger than this line. So as I look at this line here, we'll start right here at the vertex. Where are the bigger y's? Well, the bigger y's are going to be up here. So if I start this point on the line, if I want the y's that are bigger, the bigger y's are going to be up here. Okay? If I start over here on this point, that I want are going to be bigger than this line. They're going to be up here. So notice each time I did that, even though it's on different sides of the curve, I went in the middle, I went in the left, I went on the right, notice every time I did that, I ended up being inside the bowl here. So this right here, this right here is where I'm going to shade. That's where all of my solutions for this inequality, this quadratic inequality are going to be. And that's it. That's all we have to do. We graph the parabola. Either to dashed or solid lines, and then we shaded inside or outside depending on what we needed for this equation. In this case, we wanted to shade inside. Now again, it depends if it opens up or down, whether it's going to be inside or outside. There is kind of no set rule unless you define rules for parabolas to open up and parabolas to open down. But separate rules, I don't want to remember rules. I want to think logically here. Let's think rationally here. The y's that I want are bigger than this line that I created. y's are bigger than this line I created. So the y's that are bigger are going to be up. They're going to be up here. So inside this little bowl that we have here, that's where we're going to shade. That's graphing inequalities. Really short and simple there. Just kind of a combination of graphing quadratics and also graphing inequalities, solid and dashed lines and shading and all that kind of good stuff. I hope you enjoyed this video. Thank you for watching and we'll see you next time.