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CCSS.Math: , ,

it might not be obvious when you look at these three equations but they're the exact same equation they've just been algebraically manipulated they are in different forms this is the equation and sometimes called standard form for a quadratic this is the quadratic in factored form notice this has been factored right over here and this last form is what we're going to focus on in this video this is sometimes known as vertex form and we're not gonna focus on how do you get from one of these other forms to vertex form in this video we'll do that in future videos but what we're going to do is appreciate why this is called vertex vertex form now to start let's just remind ourselves what a vertex is so as you might remember from other videos if we have a quadratic if we're graphing Y is equal to some quadratic expression in terms of X the graph of that will be a parabola and it might be an upward-opening parabola or a downward-opening parabola so this one in particular is going to be an upward-opening parabola and so it might look something like this so it might look something something like this right over here and for an upward-opening parabola like this the vertex is this point right over here you could view it as this minimum point you have your x coordinate of the vertex right over there and you have your y coordinate of the vertex right over here now the reason why this is called vertex form is it's fairly straightforward to pick out the coordinates coordinates of this vertex from this form how do we do that well to do that we just have to appreciate the structure that's in this expression let me just rewrite it again we have Y is equal to 3 times X plus 2 squared minus 27 the important thing to realize is that this part of the expression is never going to be negative no matter what you have here if you square it you're never going to get a negative value and so this is never going to be negative and we're multiplying it by a positive right over here this whole thing right over here is going to be greater than or equal to zero so another way to think about it it's only going to be additive to negative twenty-seven so your minimum point for this curve right over here for your parabola is going to happen when this expression is equal to zero when you're not adding anything to negative 27 and so when will this equal zero well it's going to be equal to zero when X plus two is going to be equal to zero so you could just say if you want to find the x-coordinate of the vertex well for what x value does X plus 2 equals zero and of course we can subtract two from both sides and you get X is equal to negative two so we know that this x coordinate right over here is negative two and then what's the y coordinate of the vertex you could say hey what is the minimum Y that this curve takes on well when X is equal to negative two this whole thing is zero and Y is equal to negative twenty-seven Y is equal to negative twenty-seven so this right over here is negative 27 and so the coordinates of the vertex here are negative two comma negative 27 and you were able to pick that out just by looking at the quadratic in vertex form now let's get a few more examples under our belt so that we can really get good at picking out the vertex when a quadratic is written in vertex form so let's say let's pick a scenario where we have a downward-opening parabola where y is equal to let's just say negative two times X plus five let me make it X minus five X minus five squared and then let's say plus ten well here this is going to be downward-opening and let's appreciate why that is so here this part is still always going to be non-negative but it's being multiplied by a negative two so it's actually always going to be non positive so this whole thing right over here is going to be less than or equal to zero for all X's so it can only take away from the ten so where do we hit a maximum point well we hit a maximum point when X minus five is equal to Z we're not taking anything away from the ten and so X minus five is equal to zero well that of course is going to happen when X is equal to five and that indeed is the x-coordinate for the vertex and what's the y-coordinate for the vertex well if X is equal to five this thing is zero you're not going to be taking anything away from the ten and so y is going to be equal to ten and so the vertex here is x equals five which and I'm just gonna eyeball it maybe it's right over here x equals five and Y is equal to ten if this is negative twenty seven this would be positive 27 ten would be something like this not using the same scales for the x and y axis but there you have it so it's five comma 10 and our curve is gonna look something it's gonna look something like this I don't know exactly where it intersects the x axis but it's going to be OB a downward-opening parabola let's do one more example just so that we get really fluent at identifying the vertex from vertex form so let's say and I'm just gonna make this up we have Y is equal to negative pi times X minus 2.8 squared plus I don't know plus seven point one what is the vertex of the parabola here well the x-coordinate is going to be the x value that makes this equal to zero which is two point eight and then if this is equal to zero then this whole thing is going to be equal to 0 and y is going to be seven point one so now you hopefully appreciate why this is called vertex form it's quite straightforward to pick out the vertex when you have something written in this way