Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Graphing quadratics Quadratic graphs F of X equal AX squared plus BX plus C If a is greater than 0, the graph's going to be going up, and hence it's going to have a minimum value of whatever the vertex is, and the vertex is going to occur at negative b / 2 a comma F of negative b / 2 a, so it's going to occur at F negative b / 2 A. A different way to think about this would be the negative b ^2 + 4 AC all over 4A, and we would get that if I put negative b / 2 AN and computed it. So if we had a negative B over two a ^2 + b * -b / 2 A plus C, simplifying that up, we get that. Now if a is less than 0, it's going to go down. Hence it's going to have a maximum value at the same location. So the axis of symmetry is X equal negative b / 2 A, and that negative b / 2 a actually comes from the quadratic formula. If we thought about finding our X intercepts, that means our y = 0. So our Y is our F of X also. And then we would have a quadratic formula. And to solve that quadratic formula, X equal the opposite of B plus or minus the square root b ^2 -, 4 AC all over 2A. Now for it to be an axis of symmetry, this piece back here would need to be 0 because it would need to have a double root negative b + 0 and negative b -, 0. So that's where this negative b / 2 a comes from for the vertex. And then to get the Y value, we literally just stick that in. We'll do some examples in just a moment and remember that our Y intercept is similar to our X intercept, but this time we're just going to put zero in for our XS. So our Y intercept is going to be F of 0 equaling a * 0 ^2 + b * 0 + C or the 0 C when it's in this format. Now it might be in standard form, and standard form says Y equal a times the quantity X -, H ^2 + K. If a is greater than 0, it's going up, so it has a minimum value, the smallest height of K. If a is less than 0, it's going down, so it has a maximum height of K. Again, the vertex is HK and the axis is symmetry. As X equal K We'd find our X&Y intercepts the same way. To find the X intercept, we set 0 equal to Y and solve. To find the Y intercept, we put zero as X and solve. So graph. So F of X = X ^2 - 6 X plus 11. Our vertex is going to be negative b / 2 a, so in this case it's going to be the opposite of -6 / 2 * 1. Six over 2 is going to give me 3, so my vertex X values three. If I put 3 in here, nine 3 * 3 - 6 * 3 + 11, so 9 - 18 + 11 so 9 - 18 negative 9 + 11 so 2. Now we might have chosen to do a completing the square on this in order to easily find our other values. So if we wanted to do completing the square, we could say F of X equal X ^2 -, 6 X half of -6 ^2 is 9. So I need a nine to get these three to complete the square. I had 11 originally, so I actually have two more down there. So X -, 3 ^2 + 2. So in the standard form we can see negative or +3 positive. 2 was our vertex. Now if I put in two, 2 -, 3 is -1 negative, 1 ^2 is 1 + 2 is 3. Due to symmetry. Those should be the same. If I put in one, 1 -, 3 is -2 ^2 is 4 + 2 is 6. So my axis is symmetry is X equal 3. This is going to go up because the a was positive. If it's going up, it's a minimum and the value is going to be two. So if we graph this, we go 3/2. We'd have one being 61234562 being 3123. Oops, I'm off by one there. We go 2, not that one. So there and here and then when we graph, we get this. If we put in our axis of symmetry, it's the line that if I fold along the left side equals the right side. Now to find the X intercepts and Y intercepts X intercept, we hopefully see there isn't going to be any. But if we did this, we'd have 0 equaling X ^2 -, 6 X plus 11. So X would be the opposite of B plus or minus the square root b ^2 -, 4 * A * C all over to A. When we look at this discriminate part down here 36 -, 44. This is going to give me an imaginary number which will tell me that it's not going to cross the X axis. If I wanted to find my Y intercept, it does look like it's going to cross, and in fact it always will. We put zero in for our X's, so F of 0 would give us 0 - 0 + 11 or the .0 eleven. So not to scale here, but if we had kept going at 11 way up there, it would intersect. Let's look at another example. So this one negative two X ^2 - 4 X -6 X equal negative b / 2 A so negative of -4 / 2 * -2 so negative and negative is a positive over a negative so -1. So when X is -1 F of -1 negative 2 * -1 ^2 - 4 * -1 - 6, so that's going to give me a -2 + 4 - 6 or -4 No, Yes. Now, sometimes it's easier to go ahead and complete the square. So if we have F of X, we'd have -2 in order to find our other points. So we'd have X ^2 + 2 X half of two is 1, so we'd have a plus one if I distribute this -2 and one that'd be -2. But I started with -6 so I'd have a -4 there -2 + -4 is -6 so F of X would equal -2 times the quantity X + 1 ^2 - 4, so our vertex -1 negative 4, which is what we did find. The other method axis is symmetry X equal -1. Our a is a -2, so we know it's going down. If it's going down, it's a maximum at -4. We're going to put in a couple values bigger, couple values smaller. If I put in one here, 1 + 1 is two 2 ^2, 4 negative eight -4, negative 12, and -12. If I put zero in, I get 0 + 1 ^2 * -2 so -6 so one -12 which is actually off my graph. Zero negative 6123456, negative one -4, negative two -6. Plot those points. Put in our axis of symmetry. We can see that we're not going to have an X intercept. So if I tried to do the quadratic formula again, it wouldn't give me a real number, it'd be imaginary. And we actually already have our Y intercept zero -6. Let's do another one here. X equal negative b / 2 a. So X is going to be the opposite of -5 / 2 * 1 or five halves. And if we stick in five halves, we'd get 5 halves squared -5 * 5 halves. So 5 halves is 25 fourths -25 halves, 25 fourths -25 halves, so -25 fourths. By getting a common denominator there, this would be -50 fourths. So we could also do the completing the square methods. So in this case we'd have F of X equaling X ^2 -, 5 X half of five is -5 halves squared is plus 25 fourths, and I didn't have a number here. So if I add 25 fourths to keep it balanced, I have to subtract 25 fourths. So then this is going to give me X -, 5 halves, quantity squared -5 halves or 25 fourths. Sorry. So there's my 5 halves and my -25 fourths. If I add one, I'd get 7 halves and 9 halves, and if I take away I'd get 3 halves and 1/2. If I put 9 halves, 9 halves -5 halves is 4 halves, which is 22 squared is four 4 -, 25 fourths, so 16 fourths -25 fourths or -9 fourths. And if I do that one, I get this one also. Seven halves -5 halves is two halves, which is 1. So 1 -, 25 fourths, 1 -, 25 fourths. So 4 fourths -25 fourths would be -21 fourths. So our vertex is five halves -25 fourths. Our axis is symmetry X equal 5 halves. This is going up, so it's going to be a minimum at X equal -25 fourths. So when we graph this, 15 halves is 2 1/2 and four goes into -25 six times with one leftover, so 123456 and a quarter ish. Right about there, it's an approximate 7 halves. Four goes into -21 so -5 and a quarter due to symmetry there and there. Four goes into 9/2 and 1/4, so -2 and a quarter, 1-2 and a quarter, two and a quarter. Whoops, 2 1/4 over there. So when we connect those, we get something that looks like that we put in our axis of symmetry through the five halves. Now we need to find our X intercepts and Y intercepts also. So if X is 0, we can see 0 ^2 -, 5 * 0. So when X IS0Y is 0, so 00 is going to be one of our intercepts. It's going to be our X intercept. It's actually also going to be AY intercept, so it's actually both an X and AY in our sub. So when the Y is 0, that's our X in our sub, we're going to factor and we can see that X is 0 and X is five. So we get the .00 and we get the .50 and we expect that due to symmetry. So right there. So the last ones we're going to let you try. Thank you and have a wonderful day.