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## Algebra 1

### Unit 14: Lesson 12

Comparing quadratic functions# Comparing maximum points of quadratic functions

CCSS.Math: , , , ,

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

## Want to join the conversation?

- i have literally no clue whatsoever what that last bit where he "completed the square" was. can anyone help me with this?(13 votes)
- Completing the square is a mathematical concept which was created to find the X intercepts of a function (as seen at3:50). It does the exact same thing as the quadratic formula, but is often easier to do, when the coefficients and whole numbers and you don't have a calculator. It's really impossible to explain how to do this in one comment, but take a look at this website that gives examples and shows the process of completing the square. http://www.purplemath.com/modules/sqrquad.htm(13 votes)

- Sal lost me in the last 30 seconds of the video. The negative: - (x - 3) seemed to disappear with no explanation. What happened, To me it seems that x-3 = 0 would produce x = 3, but the preceding negative would make x = -3 and y = 8. Is that what happened? Some of these videos assume that we can follow ideas with leaps of imagination that my current math knowledge will not permit. I need to see EACH STEP EXPLICITLY EXPLAINED!(6 votes)
- Haha, I'm answering this 7 years later...

Anyway, x = 3 because in order to find the y-value, -(x-3)^2 has to equal 0. And the only way that's possible is if x = 3. If you plug in x = 3, then -(3-3)^2 = 0. I guess that equals -0, but -0 and 0 are the same thing. If x = -3 like you thought, then -(-3-3)^2 = -36, and that doesn't help to find the y-value.

Hope this helped!(2 votes)

- For the minimum/maximum, is the vertex the same thing?(4 votes)
- Or to put it another way, for a given parabola, the parabola's maximum or minimum value will occur at its vertex.(5 votes)

- Where did the 6x go when he converted it to a perfect square?(1 vote)
- He factored the quadratic. (x^2 - 6x + 9) = (x - 3)(x - 3) = (x - 3)^2.(9 votes)

- I solved this problem without completing the square. I just tried out x=2 in f(x)

f(2) =-4 +12 -1 =7

If a point of f(x) is 7 then its maximum cannot be less than -1 so of the three functions the one with the lowest maximum is h(3 votes)- Well... that's a valid way to do it in this particular case, but consider the fact that you sort of got lucky there. If
`f(x)`

had been shifted down (and contained the lowest maximum), you wouldn't be able to used that method.(2 votes)

- Why did Sal use -1 as the maximum value? (at0:20) There is also 4 (x-coordinate). Can't he just take the average of the two? But if he does, it becomes a crazy fraction. Can somebody please clear this up for me? Thank you so much.(2 votes)
- He's looking for max. g(x) coordinate not the x coordinate(2 votes)

- At the beginning of the video we're given a quadratic function g(x) for which only a table of x & y values are provided. My question is how do we find the quadratic equation of such a function for which only a few x & y values are given.(2 votes)
- Hello Avishek, in this problem we only need the max value, not the equation. There is an interesting property of quadratics in that, every quadratic equation,when graphed, has either a maximum or a minimum. What do i mean by this? well with the -x^2+6x-1 for example, the max is 8, and no matter what x-value we choose, we will never find a higher value. Hence, since the table for g(x) shows that g(0) =5 and the values of both adjacent x-values are less, we can conclude that 5 is the maximum value for g(x). As for finding the equation for g(x)? It could be approximated accurately, but it would take a good bit of guess work, or calculus. Hope this helps! Side Note: after some calculations, g(x) appears to be -(x^2)+5 though this is just what i came up with.(1 vote)

- At1:00Sal begins using completing the square method in order to find the lowest maximum value.

Can I use X= -b/2a where X gives the lowest maximum value and b and a are the coefficients of x and x^2 respectively.

Thanks.

PS I did get the correct answer using this method. He showed it in finding the x coordinate of finding the vertex of a parabola.(1 vote)- By completing the square, Sal finds a quadratic (x - something)^2 that is shifted up or down by some other thing. X=-b/2a does not take into consideration the "c" term or constant. The quadratic shows explicitly the min or max value of this function.(3 votes)

- Are there any videos where we can do some exercises on getting the perfect square? I've seen it done on a few videos now but must have missed wherever that was.(1 vote)
- At1:37, why do we take half of -6?(1 vote)
- We are working with completing the square. It starts from the concept that (x + b) ^2 = x(x+b)+b(x+b) = x^2 + bx + bx + b^2 or x^2 + 2bx + b^2. Since the middle has to come from two middle terms, we divide by two. In the specific problem, we have to say that 2b = - 6, thus when we divide both sides, we get b = -3 and b^2 = (-3)^2 = 9.(2 votes)

## 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.