Comparing maximum points of quadratic functions

which quadratic has the lowest maximum value so let's figure out the maximum value for each of these and they're defined in different ways and then see which one is the lowest and I'll start with the easiest so H of X we can just graphically look at it visually look at it and say what's the maximum point and the maximum point looks like it's right over here when X is equal to 4 and when X is equal to 4 y or H of X is equal to negative 1 so the maximum for H of X looks like it is negative 1 now what's the maximum for G of X and they've given us some points here and here once again we can just eyeball it and say well what's the maximum value they gave us well 5 is the largest value it happens when X is equal to 0 G of 0 is 5 so the maximum value here is 5 now f of X they just give us an expression to define it and so it's gonna take a little bit of work to figure out what the maximum value is the easiest way to do that for a quadratic is to complete the square and so let's do it so we have f of X is equal to negative x squared plus 6x minus 1 I never like having this negative here so I'm going to factor it out this is the same thing as negative times x squared minus 6x and plus 1 and I'm going to write the plus 1 out here because I'm fixing to complete the square now just as a review of completing the square we essentially want to add and subtract the same number so that part of this expression is a is a perfect square and to figure out what number we want to add and subtract we look at the coefficient on the x-term it's a negative 6 you take half of that that's negative 3 and you square it negative 3 squared is 9 now we can't just add a 9 that would change the actual value of the expression we have to add a 9 and subtract a 9 and you might say well why are we adding and subtracting the same thing if it doesn't change the value of the expression and the whole point is so that we can get this first part of the expression to represent a perfect square this x squared minus 6x plus 9 is X minus 3 squared so I can rewrite that part as X minus 3 squared and then not minus 9 or negative 9 plus 1 is is negative 8 so - let me do that in a different color so we can keep track of things so this part right over here is negative 8 and we still have the negative out front still have the negative out front and so we can rewrite this as if we distribute the negative sign negative X minus 3 squared plus 8 now let's think about what the maximum value is and to understand the maximum value we have to interpret this negative X minus 3 squared well X minus 3 squared before we thinking about before we think about the negative that is always going to be a positive value but then when we make it negative or it's always going to be non-negative but then when we make it negative it's always going to be non positive think about it if X is equal to 3 this thing is going to be 0 and you take the negative of that it's going to be 0 X is anything else X is anything other than 3 this part of the expression is going to be positive but then you have a minus sign so you're going to subtract that positive value from 8 so this actually has a maximum value when this term when this first term right over here when this first term right over here is 0 the only thing that this this part of the expression can do is subtract from the 8 so you want this if you want to get a maximum value this should be equal to 0 this equals 0 when X is equal to 3 when X is equal to 3 this is 0 and our function hits its maximum value of 8 so this has a max do that in a color that you can actually read this has a VC max value of 8 so which has the lowest maximum value H of X