Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. Quadratic graphs. A quadratic function F of X equal AX squared plus BX plus C has a vertex at negative b / 2 AF of negative b / 2 a. So here's our X value and our Y value. If our A is positive, the graph has a minimum of whatever that Y value was at the location of the X. So if A is positive, this graph is going to be going up somehow, and the lowest point is going to be the minimum, and the minimum's going to be the Y value, IE how high or low it went at the location of the X value. If the A is less than 0, the graph's going to go down. If it's going down the graph then has a maximum, and the maximum occurs at the height, so the maximum of whatever this Y value is at the location of the X value. There is another form for quadratics and that's called standard form and it's Y equal a times the quantity X -, H ^2 + K The vertex in this form is HK. The axis of symmetry is X equal H If a is greater than 0, the graph has a minimum of the Y value, in this case K at X equal H, or if a is less than 0, it has A graph has a maximum of X at X equal H Let's look at a couple of examples. If we had it in standard form, F of X equal 2 times the quantity X -, 1 ^2 -, 8. If we wanted to graph this, we'd start with figuring out our vertex, and our vertex in this case is one -8. Frequently, it's easy to think about what makes that parenthesis go to 0. If X is one, that ( 0 zero squared 0 * 2 zero -8 so one -8. We also know the axis of symmetry because it is always a line that goes through the X value of the vertex. So X equal 1. We might find some information if we knew our X intercepts. So if we have our X intercepts, we set our Y equal to 0 to find our X intercepts. If we set our Y equal to 0, we'd add 8 to each side, we'd divide by 2, we'd take the square root remembering positive and negative, and then we would take and add the one across. So X is 3 when Y is 0, and X is also -1 when Y is zero. Another thing that might be useful is the Y intercept. And the Y intercept says set your X equal to 0. So 2 * 0 - 1 ^2 - 8 negative 1 ^2 is one, 2 - 8 is -6. So when our X is 0, our Y is -6. Now we also know that this graph, because the two is positive our A equal to and two is greater than 0, so it's going to be going up, which means we have a minimum at -8 minimum of -8. So if we put all this together for our graph, we would have a rough graph that had three zero as an intercept, zero -1 as an intercept, zero -6 as an intercept, one -8 as our vertex. We know that axis of symmetry always goes through our vertex. We also know that a parabola is symmetric. So if we go right one here, we would have the exact same point when we come the other direction. So here's a very rough sketch of that parabola. Let's look at one if it's not in this form, let's look at it if it was in the other form. Well, there are a couple different ways to do this. One, what is our negative b / 2 A and our F of negative b / 2 a? Well, our negative B -24 / 2 * -3, F of whatever. That all simplifies to so -24 / 2 / -3 negative divided by negative is positive. So we're going to get F of four. So when we put in four, we get -3 * 4 ^2 + 24 * 4 - 46, and that's my F of four value. So 64 squared is 16 negative 3 * 16 is -48 24 * 4 is 96 - 46. So that's going to give us 2SO4, 2. Now a different way that a lot of students prefer is to complete the square because you don't have as big a numbers. If I factor out that -3 I get X ^2 -, 8 X, and I'm going to leave that -46 at the end for a minute. If I complete the square, we're going to take half a -8 and we're going to square it. So half a -8 negative 4 ^2 is +16. Now, I didn't really add 16 because it had the distributive property here. I really had -3 * 16. So I really, really added, or actually, yeah, I added -48 So to keep it balanced, if I add a -48 to one side, I'd have to add a +48 to the same side. Or you could think about putting -48 on one side -48 on the other and then adding it across. So now if we look at this, we get -3 X -4 quantity squared +2. In this format, we just did an example. The vertex would be 4/2, which is the same thing that we found up here. We'd then find the X intercepts, Y intercepts, axis is symmetry, and whether the graph is going up or down. So our A is -3. The fact that -3 is less than 0 tells us it's going down. Thus, it's going to have a maximum of two at X equal 4. It's going to have an axis of symmetry at X equal 4, and then we just find our X and our X intercepts and Y intercepts the way we did before. If I put zero in for X and it doesn't matter, I can use either of the two equations. I'm going to use the top one just because it's a little easier. If I have zero times anything, we know that those are going to be 0. So in this case when X is 0, my Y is -46 and then when Y is 0 we have our X. And we could always use quadratic formula if we didn't want to factor it. So X equals the opposite of B plus or minus the square root b ^2 -, 4 * a * C all over 2A. And in reality, it might have been easier to use this equation here because if I had put a 0 here, I would have subtracted 2 / -3. So we can compute that out with our calculator. If we wanted to do it with this other formula, we would have had 0 equal -3 X -4 ^2 + 2 negative 2 negative three X - 4 ^2, divide by 3 so 2 thirds X - 4 ^2. So X should equal 4 plus or minus sqrt 2 thirds, which when we simplify 4 ± sqrt 6 / 3. Thank you and have a wonderful day. This is Anna Cox.