 In this video we'll talk about graphing quadratics. So we have a table here that we need to finish and then we want to graph the points. So 1 is my x and my y is going to be x squared. So 1 squared would be 1, 2 squared would be 4, and 3 squared would be 9. Negative 3, 9 is going to be this point right here. Positive 2, 4 will be right here. Negative 1, 1, 0, 0, positive 1, 1, positive 2, 4, and positive 3, 9. Looks like I have all those points and if we were to connect them all it would look something like that. So what is the vertex? The vertex is the very lowest or the very highest point on our graph and that happens to be right here. In our case it's the point 0, 0. What's the y-intercept? Where it crosses the y-axis is also 0, 0. What's the x-intercept? Again, where it crosses the x-axis is also 0, 0. That doesn't always happen. And then the axis of symmetry is an imaginary line that splits the parabola in half. If I were to draw that perfectly and fold my paper on the green side, this point over here would go over here and this point would end up over here and this point over here, everything would land on the other side of the graph. So it's a vertical line that means it's x and what does x equal where that line goes through? Well it goes through the origin so x equal 0. Let's try again. Negative 3 squared would be 9 and then we went the opposite. So 1 squared would be 1 and then we went the opposite so that would be negative 1. 2 squared would be 4 but we went the opposite so negative 4. And 3 squared would be 9 but we went the opposite so negative 9. So negative 3 and negative 9 would be this point. Nope, nope, this one. And negative 2, negative 4 would go here, and negative 1, negative 1, 0, 0, 1, negative 1, 2, negative 4, 3, negative 9. Looks like our same parabola, except that this time it's upside down. So the vertex is still the highest, this time it's the highest point, but in this case it's still at the origin. And so are our intercepts, and our axis of symmetry is that line that goes through our vertex. We'll always go through our vertex, so it's x, and what's the x of our vertex? Zero. So now we have one graph. It's x squared minus 2x minus 3, and we want to know what the vertex is. Well if we're looking at a graph, we can tell easily what that is. It looks like it's negative 4 and 1, negative 4 and y, x would be 1, so 1 negative 4. What are a and b? And that's referring back to the equation. So a is 1, and b is negative 2. So if I do negative b over 2a, negative b would be the opposite of negative 2. So 2 over 2 times 1, which is just 2, and 2 over 2 is equal to 1. Plug this value in for x. So 0 is equal to 1 squared minus 2 times 1 minus 3. 1 minus 2 minus 3. 1 minus 2 is going to be negative 1 minus 3, which is negative 4. And where do you see 1 and negative 4? This is a y value, and this is because we weren't really solving, so y was what it was equal to. Well x is 1, y is negative 4, that's our vertex. Now it says, set the equation equal to 0 and find the solutions. So we have 0 equal x squared minus 2x minus 3. And we can factor that. It's a is 1, so it factors very nicely. Factors of negative 3, so we know we have opposite signs, that will add up to 2. Well there's only one way to factor 3, and that's a 3 and a 1. But it needs to be negative 2, so the bigger number has to be negative. So x plus 1 and x minus 3 would be our factors. And then remember, set your factors equal to 0 to solve. So if we have x plus 1, when we subtract 1 from both sides, x will be equal to negative 1. And when we have x minus 3 equal to 0, if we add 3 to both sides, x will be equal to positive 3. And if you look on your graph, here's your negative 1 and here's your x equal 3. So x intercepts our solutions to the quadratic. Okay, nice little factor remember. If you're looking at a graph and you want to know what the solutions to the quadratic is, just find the x intercepts. So let's put it all together. Negative x squared plus 4x plus 5, set it equal to 0 and find the solutions. We'll just use a quadratic formula. So 0 is equal to this thing. So a equal negative 1, b equal 4, and c equal 5. And most of you at this point now have the quadratic formula in your calculator. So we can do program, enter. Mine looks a little bit different, but the answers are still the same. Yes, that's what I want to do. A is negative 1, b is 4, and c is 5. And it tells me that my answers are x equal negative 1 and positive, and negative 5 I think it was. Get my calculator back up here. Negative 1 and positive 5. So where do those go on my graph? x equal negative 1 and x equal positive 5. So now I already have two points on my graph as I'm trying to graph this thing. Negative b over 2a. Negative b, well b is 4, so that's going to be negative 4. And a is negative 1, so 2 times negative 1 means that I have negative 4 over negative 2, which ends up just being 2. And this is the x of my vertex. That's what that little b down there means. Plug that value in. And find y. So negative the opposite of 2 squared plus 4 times my 2 plus 5. This is the opposite of 4. 4 times 2 is 8 plus 5. Negative 4 plus 8 would be positive 4 plus 5 would give me 9, and that's my y. So my vertex then is 2 and the x, 9 and the y. So 2, 6, 7, 8, 9 is going to be somewhere down here. And then what is the y-intercept? We didn't talk about this a whole lot, but the y-intercept is always equal to the constant. If x is 0 for a y-intercept, so when I put 0 in for the squared term, I get 0. And I put 0 in for the x term, I get 0, and all I'm left with is the 5. So 5 is my y-intercept. I did 2, 9 and it should be 2, negative 9. See how nice it is though? I knew something was wrong because those weren't all going to go the right way. So 2, 9, somewhere up here. And now when I look at my points, I'm going to be able to go to this one and go to that one and then back down to this one. And now we've graphed a quadratic.