Let's look at graphing quadratic inequalities. Now, graphing quadratic inequalities is gonna be very similar to graphing a linear inequality we're gonna find Follow some of the same rules. Remember that. If it says less than or greater than but not equal to, we're gonna have a dotted line. If it says the equal to sign which the line below it, we're gonna have a solid line. We're also gonna have the shading above and shading below. So if it's less than we're gonna shade below the line And if it's greater than we're gonna shade above the lot So that's gonna be the same for, uh, quadratic inequalities and linear inequalities. All of that's gonna work the same. So we're gonna start off and we're going to create an X Y table for the equation. Y is greater than negative. X square minus six X minus seven. Well, we can go ahead and we can find a few things to decide what we wanna put Honor X y table. So, for instance, I can see that since my constant is negative seven when x zero, my wife is gonna be negative seven, so I can actually go ahead and drove through that point on my line. The other thing I confined is I can find my access, a symmetry. And this is when X is negative. Be over to a. So for this one, that would be negative. Negative six over two times. Negative one. So that would be positive. Six over to negative two, which would be negative. Three. So that means we're exes. Negative. Three. That's gonna be my axes of symmetry. So that means I'm going to be looking at points on both the left and the right off X is negative three. So let's go ahead and find out what it would be if X is negative. Three. So if excess negative three, I would plug those values in, and I would have negative negative three squared minus six times negative. Three minus seven. And that would actually give me positive, too. So we're excess native three. My wife would be positive, too, so I can pick some numbers between there. So let's pick numbers between zero and negative three. So we can do negative two and weaken Do negative one. And if we plug those numbers in place on our inequality to save it sometime. I'm going to go on and kind of tell you what the why would be. But you would plug those numbers in place of X for negative too. I would get positive one. And for negative one, I would get negative, too. So then this point, my is gonna start switching up. So let's kind of do some numbers. The right. So let's do negative four. If I put negative forward in place, I'm going to get positive one. We could even do negative five. And if I put negative five in place, I'd get negative too. And negative six. I would get negative seven. So here you can see my parabola. Now, before I draw my line, I need to decide what kind of line I'm going to draw. Well, this says greater than but does not say equal to. So that means I'm going to be drawing a dotted line. So there's my parabola with the dot in line. The other thing I'm going to see is where I shade. This is greater than so I'm going to shade above this line, so basically I'm going to shade the areas of my graph that are above this quadratic. Why? So at this point, this would be this entire area. I know it's not the neatest shading, but it kind of gives you a view with me and still be able to set a line. So now let's look at this. Inequality wise, less than or equal toe X squared plus two X plus four. We can go ahead and create an X Y table and we can find a couple of things. First thing we confined is our Y intercept. Why intercept is going to be four, because that's my constant. So we're X zero. Why is gonna be four so we can go ahead and do that? Point The other thing? We confined our axis of symmetry, which says, negative, be divided by two A. So that's gonna be negative. Two divided by two times one So negative two divided by two, which would be negative one. So that's gonna be my center is gonna be these this point right here. So that means that wherever my problem is, I want points on the left of negative one and the right, But let's go ahead and do negative one itself as well. So we're gonna put in wise, less than or equal to negative one squared plus two times negative one plus four. And when I do that, I'm going to get three, which would be that point right there. So we've got a point on the right side of our axis of symmetry, and we've got one owner axis of symmetry. So let's do one on the left side. Well, yeah, so let's do negative, too. So that would be wise. Less than or equal to negative. Two squared, plus two times negative two plus four. And if I worked that out, I'm going to get negative to. And why would be four, which would be that point right there. Now I can't even do another point. I could do positive one. And when I do positive one, I would get seven. And if I did negative three, I would get seven as well. All right, so now we've got some points for probably five. It's a good, really good number for a parabola to really see the direction it's going. So let's decide on our line. Well, this says less than or equal to equal to means we're going to be drawing a solid line. So here I'm gonna go ahead and draw our line and it's gonna be a solid line. It says it's less than or equal to, So this means I'm gonna go ahead and I'm going to be shading the area below my quadratic, so it's going to be in this area right there.

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Let's look at graphing quadratic inequalities. Now, graphing quadratic inequalities is gonna be very similar to graphing a linear inequality we're gonna find Follow some of the same rules. Remember that. If it says less than or greater than but not equal to, we're gonna have a dotted line. If it says the equal to sign which the line below it, we're gonna have a solid line. We're also gonna have the shading above and shading below. So if it's less than we're gonna shade below the line And if it's greater than we're gonna shade above the lot So that's gonna be the same for, uh, quadratic inequalities and linear inequalities. All of that's gonna work the same. So we're gonna start off and we're going to create an X Y table for the equation. Y is greater than negative. X square minus six X minus seven. Well, we can go ahead and we can find a few things to decide what we wanna put Honor X y table. So, for instance, I can see that since my constant is negative seven when x zero, my wife is gonna be negative seven, so I can actually go ahead and drove through that point on my line. The other thing I confined is I can find my access, a symmetry. And this is when X is negative. Be over to a. So for this one, that would be negative. Negative six over two times. Negative one. So that would be positive. Six over to negative two, which would be negative. Three. So that means we're exes. Negative. Three. That's gonna be my axes of symmetry. So that means I'm going to be looking at points on both the left and the right off X is negative three. So let's go ahead and find out what it would be if X is negative. Three. So if excess negative three, I would plug those values in, and I would have negative negative three squared minus six times negative. Three minus seven. And that would actually give me positive, too. So we're excess native three. My wife would be positive, too, so I can pick some numbers between there. So let's pick numbers between zero and negative three. So we can do negative two and weaken Do negative one. And if we plug those numbers in place on our inequality to save it sometime. I'm going to go on and kind of tell you what the why would be. But you would plug those numbers in place of X for negative too. I would get positive one. And for negative one, I would get negative, too. So then this point, my is gonna start switching up. So let's kind of do some numbers. The right. So let's do negative four. If I put negative forward in place, I'm going to get positive one. We could even do negative five. And if I put negative five in place, I'd get negative too. And negative six. I would get negative seven. So here you can see my parabola. Now, before I draw my line, I need to decide what kind of line I'm going to draw. Well, this says greater than but does not say equal to. So that means I'm going to be drawing a dotted line. So there's my parabola with the dot in line. The other thing I'm going to see is where I shade. This is greater than so I'm going to shade above this line, so basically I'm going to shade the areas of my graph that are above this quadratic. Why? So at this point, this would be this entire area. I know it's not the neatest shading, but it kind of gives you a view with me and still be able to set a line. So now let's look at this. Inequality wise, less than or equal toe X squared plus two X plus four. We can go ahead and create an X Y table and we can find a couple of things. First thing we confined is our Y intercept. Why intercept is going to be four, because that's my constant. So we're X zero. Why is gonna be four so we can go ahead and do that? Point The other thing? We confined our axis of symmetry, which says, negative, be divided by two A. So that's gonna be negative. Two divided by two times one So negative two divided by two, which would be negative one. So that's gonna be my center is gonna be these this point right here. So that means that wherever my problem is, I want points on the left of negative one and the right, But let's go ahead and do negative one itself as well. So we're gonna put in wise, less than or equal to negative one squared plus two times negative one plus four. And when I do that, I'm going to get three, which would be that point right there. So we've got a point on the right side of our axis of symmetry, and we've got one owner axis of symmetry. So let's do one on the left side. Well, yeah, so let's do negative, too. So that would be wise. Less than or equal to negative. Two squared, plus two times negative two plus four. And if I worked that out, I'm going to get negative to. And why would be four, which would be that point right there. Now I can't even do another point. I could do positive one. And when I do positive one, I would get seven. And if I did negative three, I would get seven as well. All right, so now we've got some points for probably five. It's a good, really good number for a parabola to really see the direction it's going. So let's decide on our line. Well, this says less than or equal to equal to means we're going to be drawing a solid line. So here I'm gonna go ahead and draw our line and it's gonna be a solid line. It says it's less than or equal to, So this means I'm gonna go ahead and I'm going to be shading the area below my quadratic, so it's going to be in this area right there.

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