Hello wonderful mathematics people, this is Anna Cox from Kellogg Community College. Graph quadratics in vertex form graphing F of X equal AX squared. When we have this kind of a formula, it's a parabola with the axis of symmetry being X = 0. It's vertex is the origin. If A is greater than 0, the parabola opens upwards. We sometimes think of it as a bowl shape. If A is less than 0, it opens down, and we sometimes think of that as a bell shape. If the absolute value of A is greater than 1, the parabola is well, if we look at if it equals one, just say this is a rough sketch. If that absolute value of A is greater than one, it's going to be a narrow graph or closer to the Y axis, closer to Y. If the absolute value is in between zero and one, it's going to be a wider graph or closer to the X axis. So we'd like you to grab your Caraphing calculator and try some of these and graph them all at the same time and see what's happening. The next thing we're going to talk about is if we have a value with the X and that's going to make it a shift to the right or to the left. So if H is positive, X -, H = 0, we'd get X equal H and this is going to be a shift to the right. If our H is negative, we'd get X -, a negative value. So we'd have X + H, and when we solve that, we'd get a negative H which would tell us it's going to shift to the left. The vertex is H 0, and the axis of symmetry is X equal H So we're going to have you grab your graphing calculators and practice shifting left and right. The last type of graph. Remember the a tells us up or down. The H tells us left or right. The K is going to tell us up or down. If K is positive, we're going to shift the whole graph up. If K is negative, we're going to shift the whole graph down. So then the vertex is HK and the axis of symmetry is X equal H By adding K, we just increase the function by that value amount. So once again, grab your graphing calculator and try some. And now we're going to actually do some by hand. It's important that we understand how to do them by hand. So graph and turn in the following graph. OK, so here our vertex is going to be whatever makes that parenthesis go to 0 in this case -4 and there is no plus or minus at the end, so it's going to be -4 zero. Our axis is symmetry is X equal -4 so I'm going to bake AT chart and -4 zero. I'm going to put in a few that are smaller and a few that are bigger to do some symmetry. So I'm always going to put the vertex right in the middle. If I stick in -2 negative 2 + 4 ^2, so that's 2 ^2 or that's going to be 4. Now, if I've done it right -6 should give me the same value out because -6 + 4 ^2 would be -2 ^2, and we know that -2 * -2 is indeed a +4. When I stick a -3 negative 3 + 4 is 11. Squared is one and one. So if we graph -2 four -3 one -4 oops, doing this wrong, let's see -2 four right there -3 is one. Negative 4 is 0 -5 is one -6 is 4. We can see right here is the axis of symmetry at X equal -4. If I folded this graph right along that axis of symmetry, and if I could draw straight, the left side would fall right on top of the right side. So this is going to have a lowest value, hence it's going to have a minimum and that lowest value, the height of that lowest value would be 0. So for this next example, our vertex is going to be negative a half zero. I'm going to put a negative 1/2, I'm going to put in zero. I'm going to add 1, going to add 1, going to subtract 1 and subtract 1. My axis of symmetry is going to be X equal, negative 1/2. This -2 tells me it's going down South. This time we're going to have a highest value or a maximum, and it's going to be at 0 because it's the value for the Y of the vertex. So if I put in three halves here, three halves plus 1/2 is 4 halves which is 22 squared is 4/4 times -2 is -8. If I've done it correctly, I should have symmetry so that -5 halves should give me the same value. Negative two 1/2 + 1 half 1/2 + 1/2 is 11 squared is 1 so -2 and -2. Feel free to check the -3 halves and -5 halves. So at three halves were at -8 12345678 negative 3 half or +3 halves right there. I think at 1/2 we're at -2. At negative 1/2, we're at 0. Here we're going to be at negative 2345678. So my axis of symmetry is going to be this line that I draw in, and then my graph should be symmetrical once again. If I could draw, it would be symmetrical along that line. So let's look at a few more and then we'll give you some to work. So on this next one, my vertex is going to be 7 three. My axis is symmetry X = 7 negative 4 sevenths tells me it's going to be going down. So it's going to have a maximum at the value 3 this time. So if I put in seven three and I'm going to put in a couple that are smaller, couple that are bigger, If I put -4 sevenths times 5 -, 7 ^2 + 3, this is -2 ^2 is 4, so I get -16 sevenths plus 3, which is 21 sevenths. So I'm going to get 5 sevenths. And if I do this correctly, my 9 should give me the same value. If I stick in six, I'm going to get -4 sevenths, 6 -, 7 ^2 + 3. So -4 sevenths plus 21 sevenths is going to give me 17 sevenths, 17 sevenths, 17 sevenths. We don't always get pretty numbers and that's OK. So 37 gives us 3. Let's see if we can scooch over a little bit. 12345673 right there. 6 is going to give me 17 sevenths. Well, seven goes into 17 two times because that's fourteen with three sevenths leftover. So 2 and 3 sevenths, we're going to approximate to be right about there, there and there because of my symmetry. And then 5 sevenths is less than one. So we're going to be about here and about here. I connect those points, I get something that looks like that if I put in my axis of symmetry, which we should always put in our graphs so that we can see the line that if we reflected it, left side, right side would be the same. OK, next one we're going to get -3 negative 2 axis asymmetries X equal -3. This time the a is an understood 1, so it's going to go up, which gives it a smallest value or a minimum at -2 so -3 negative 2. Put a -2 negative one -4 negative 5 SX oops -1 plus 3 ^2 - 2. So that's 2 ^2. 4 - 2 is 2. I've done it right. That should be the same for both of those -2 + 3 ^2 - 2 negative one -1. We put these on the graph, we get -1 being positive two -2 being -1 -3 negative 2 due to symmetry, it should look like. So we connect our points and we put in our axis of symmetry. One last example, and then we'll let you try some. So here we're going to have -1 positive 4. We're going to have X equal -1 as our symmetry. It's going down because of the -2 when it goes down. That means it's a maximum point at 4. So -1 four. I'm going to put in 01 negative two -3. If I put in one, I get 1 + 1 ^2 + 4. So 2 ^2 4 four times -2. Negative 8 + 4 is -4 if I stick in 0, I'm going to get 2. And then I'm just going to plot these points. So one -4 zeros 2, negative one is +4 two and negative oh, positive two. Well, glad I caught that. OK. One negative 4, but zero was positive 2, not -2 -1 was four. There we go -2 is 2. And then here we connect our points, we get something like that. We put in our axis asymmetry and that would be our graph. And we've got quite a few for you to try. Thank you and have a wonderful day.