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Comparing maximum points of quadratic functions

Given several quadratic functions represented in different forms, Sal finds the one with the lowest maximum value. Created by Sal Khan.

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Video transcript

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 going 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 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 minus 9-- or negative 9-- plus 1 is negative 8. 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. 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 think about the negative-- that is always going to be a positive value. 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 first term right over here is 0. The only thing that this part of the expression could do is subtract from the 8. 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-- let me do that in a color that you can actually read-- this has a max value of 8. So which has the lowest maximum value? h of x.