Forms & features of quadratic functions

I have a function here defined as x squared minus 5x plus 6 and what I want us to think about is what other forms we can write this function in if we say wanted to find the zeros of this function if we wanted to figure out where does this function intersect the x axis what form would we put this in and then another form for maybe finding out what's the minimum value of this we see that we have a positive coefficient on the x squared term this is going to be an upward-opening parabola but what's the minimum point of this or even better what's the vertex of this parabola right over here so if the function looks something like this we could find we could use one form of the function to figure out where does it intersect the x axis so where does it intersect the x axis and maybe we can manipulate it to get another form to figure out what's the minimum point what's this point right over here of this function I don't even know if the function looks like this so I encourage you to pause this video and try to manipulate this into those two different forms so let's work on it so in order to find the roots the easiest thing I could think of doing is trying to factor this funk or factor this quadratic expression which is being used to define this function so we could think about well let's think of two numbers whose product whose product is positive 6 and whose sum and whose sum is negative 5 so since their product is positive we know that they have the same sign and if they have the same sign but we get to a negative value that means they most they both must be negative so let's see negative 2 times negative 3 is positive 6 negative 2 plus negative 3 is negative 5 so we could rewrite f of X and so let me write it this way we could write f of X as being equal to X minus 2 times X minus 3 now how does this help us find the zeros well in what in what situations is this right hand expression is this expression on the right hand going to be equal to 0 what's the product of these two expressions if either one of these is equal to 0 0 times anything is 0 0 times anything else is 0 so this whole thing is going to be 0 if X minus 2 is equal to 0 or X minus three is equal to zero add two to both sides of this equation you get X is equal to two or X is equal X is equal to three so those are the two zeros for this for this function I guess you could say and we could already think about it a little bit in terms of graphing it so if we wanted to so let's try to graph this thing so this is x equals one this is x equals 2 this is x equals 3 right over there so that's our x-axis that you could say is our y is equal to f of x axis and we're seeing that we intersect we intersect both here and here when X is equal to 2 this f of X is equal to 0 when X is equal to 3 f of X is equal to 0 and you could substitute each either of these values into the original expression you'll see it's going to get you to 0 because that is the same thing as that now what about the vertex what forum could rewrite this original thing in order to pick out the vertex well we're already a little familiar with completing the square and when you put it in kind of a in that form or when you complete the square with this expression that seems to be a pretty good way of thinking about what the minimum value of this function is so let's just do that right over here so I'm just going to rewrite it so we get f of X is equal to x squared minus 5x and I'm just going to throw the plus 6 right over here and I'm giving myself some real estate because what I need to do what I want to think about doing is adding and subtracting the same value so I'm going to add it here I'm going to subtract it there and I can do that because then I've just added 0 I haven't changed the value of this right-hand side but I want to do that so that this part that I've underlined or in this magenta color so this part right over here is a perfect square and we've done this multiple times when we've completed the square I encourage you to watch those videos if you need a little bit of a review on it but the general idea is this is going to be a perfect square if we take if we take this coefficient right over here we take negative 5 we take half of that which is negative 5 halves and we square it so we could write this as plus negative what's negative five-halves squared so I could write this negative five-halves squared well if we square a negative number it's just going to be a positive so it's going to be the same thing as 5 halves squared 5 squared is 25 2 squared is 4 so this is going to be plus 25 over 4 plus 25 over 4 now once again if we want this equality to be true we either have to add the same thing to both sides or if we're just operating on one side if we added it to that side we could just subtract it from that side and we haven't changed the total value on that side so we added 25 over 4 and we subtracted 25 over 4 so what is this part right over here what does this become the part that I've underlined in magenta well this is going to be the whole reason why we engineered it in this way is so that this could be X minus 5 over 2 squared and I encourage you to verify this and we go into more detail about why that you know taking half the coefficient here and then squaring it adding it there and subtracting there why that works we do that in the completing the square videos but these two things you can verify that they are equivalent so that's that part and now we just have to simplify 6 minus 25 over 4 so what is so 6 could be re-written as 24 over 4 24 over 4 minus 25 over 4 is negative 1/4 so minus minus 1/4 just like that so we could we've rewritten our original function as f of X is equal to X minus five-halves squared minus 1/4 now why is this why is this form interesting well one way to think about it is this part is always going to be non-negative the minimum value of this part in magenta is going to be 0 because why because we're squaring this thing if you're taking something like this and we're just dealing with real numbers and you're squaring it you're not going to be able to get a negative value at the minimum value this is going to be 0 and then or or and then it obviously could be positive values as well so if we want to think about when does this thing hit its minimum value well it hits its minimum value when you're squaring zero and when are you squaring zero well you're squaring zero when X minus five-halves is equal to 0 or when X is equal to 5 halves if you just want to add 5 halves to both sides of that equation so this thing hits its minimum value when X is equal to 5 halves and then what is y or what is fun f of X when X is equal to 5 halves F of 5 halves and once again you could use any of those forms to evaluate 5 halves but it's really easy in this form when X is equal to 5 halves this term right over here becomes 0 0 squared 0 you're just left with negative 1/4 you're just left with negative 1/4 so another way to think about it is our vertex is at the point so we could say our vertex is at the point x equals 5 halves y equals negative 1/4 so x equals 5 halves that's the same thing as 2 & 1/2 so x equals 5 halves and y is equal to negative 1/4 so if that is negative 1 1/4 would be something like that so that right over there is a vertex that is the point make it clear that's the point 5 halves comma negative 1/4 negative 1/4 and what's cool is we've just used this form to figure out the minimum point to figure out the vertex in this case and then we can use the roots to kind of as two other points to get a rough sketch of what this parabola rough sketch of what this parabola will actually will actually look like so the interesting or I guess the takeaway from this video is just to realize that we can rewrite this in different forms depending on what we're trying to what we're trying to understand about this function