anonymous
  • anonymous
The functions f(x) = –(x + 4)2 + 2 and g(x) = (x − 2)2 − 2 have been rewritten using the completing-the-square method. Is the vertex for each function a minimum or a maximum? Explain your reasoning for each function.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
SolomonZelman
  • SolomonZelman
If your quadratic has a negative leading coefficient, then it opens down, and its vertex is the aboslute maximum. If your quadratic has a positive leading coefficient, then it opens up, and its vertex is the absolute minimum.
SolomonZelman
  • SolomonZelman
that is the rule.....
anonymous
  • anonymous
Ok so the first one would be that it opens down correct?

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anonymous
  • anonymous
thank-you:) Can I ask anither?
SolomonZelman
  • SolomonZelman
yes, the first one opens down. Right;)
SolomonZelman
  • SolomonZelman
And the second one opens where?
SolomonZelman
  • SolomonZelman
(and yes, you can ask another question right here, after you feel that you are done with this question.)
anonymous
  • anonymous
It opens us since it's poitive
SolomonZelman
  • SolomonZelman
yes
SolomonZelman
  • SolomonZelman
ok i guess you are done with this question... r u?
SolomonZelman
  • SolomonZelman
do you have any questions about this question ?
anonymous
  • anonymous
so the first one, is opening down w/ the vertex at an absolute maximum? I'm just checking
SolomonZelman
  • SolomonZelman
yes
SolomonZelman
  • SolomonZelman
the vertex in function 1 is absolute maximum
anonymous
  • anonymous
Ok thank-you:)
anonymous
  • anonymous
Suppose C and D represent two different school populations where C > D and C and D must be greater than 0. Which of the following expressions is the largest? Explain why. Show all work necessary. (C + D)2 2(C + D) C2 + D2 C2 − D2
SolomonZelman
  • SolomonZelman
yw
anonymous
  • anonymous
i don't understand this one
SolomonZelman
  • SolomonZelman
you don't understand the question, or don't know how to solve it?
anonymous
  • anonymous
I don't understand how to solve it
SolomonZelman
  • SolomonZelman
oh....ok...
SolomonZelman
  • SolomonZelman
I will also add, that C and D must be integers (since you can't have 3.5 people of anything like that).... do you agree with me?
anonymous
  • anonymous
yes. Yes I do
SolomonZelman
  • SolomonZelman
Now, C is at least 1. D is at least 2. (So, C+D is at least 3) Which is the largest? Explain why. Show all work necessary. (C + D)² 2(C + D) C² + D² C² − D²
SolomonZelman
  • SolomonZelman
(C+D)² when C=1 & D=@ is going to be (1+2)²=3²=9 2(C+D) 2(1+2)=2(3)=6
SolomonZelman
  • SolomonZelman
2(C+D) is a linear growth, and (C+D)² is a "quadratic" growth. And the bigger "C+D" we have the greater will (C+D) get when compared to (C+D) that will not grow even nearly as rapidly.
SolomonZelman
  • SolomonZelman
the last option is going to be obviously smaller than or greater than the 3rd option ?
SolomonZelman
  • SolomonZelman
can you tell me?
anonymous
  • anonymous
so the answer would be c2+d2
SolomonZelman
  • SolomonZelman
hold on...
SolomonZelman
  • SolomonZelman
pliz answer my last question...
SolomonZelman
  • SolomonZelman
(you are trying to jump ahead to quickly:o)
anonymous
  • anonymous
smalleer :) Sorry
SolomonZelman
  • SolomonZelman
yes (no need for apologies).
SolomonZelman
  • SolomonZelman
So, the last option falls off... and in competition is the 1st option (C+D)² which is larger than the 2nd option (as i showed), and the 3rd option (which is obviously larger than the 4th option).
SolomonZelman
  • SolomonZelman
((remember D must be greater than C, and both C and D are natural numbers)) C²+D² vs (C+D)² DIFFERENCE ----------------------------------------------------------- C=1 & D=2 1²+2²=5 vs (1+2)²=9 (C+D)² exceeds by 4 ----------------------------------------------------------- C=1 & D=3 1²+3²=10 vs (1+3)²=16 (C+D)² exceeds by 6 ----------------------------------------------------------- C=2 & D=3 2²+3²=13 vs (2+3)²=25 (C+D)² exceeds by 12 ----------------------------------------------------------- C=2 & D=4 2²+4²=18 vs (2+4)²=36 (C+D)² exceeds by 18 (& twice)
SolomonZelman
  • SolomonZelman
The greater values for C and D we pick, the more (C+D)² exceeds, (outmatches if you will) the C²+D². So which one is the largest?
anonymous
  • anonymous
)c+d)^2
SolomonZelman
  • SolomonZelman
Yes, (C+D)² is the largest, and this it is the answer you need:)
SolomonZelman
  • SolomonZelman
Any questions about this problem?
anonymous
  • anonymous
No. :)
SolomonZelman
  • SolomonZelman
OK.....
anonymous
  • anonymous
Three functions are given below: f(x), g(x), and h(x). Explain how to find the axis of symmetry for each function, and rank the functions based on their axis of symmetry (from smallest to largest). f(x) g(x) h(x) f(x) = 3(x + 4)2 + 1 g(x) = 2x2 − 16x + 15
SolomonZelman
  • SolomonZelman
ok, this is a last question for this post. Alright>
anonymous
  • anonymous
Yea:)
SolomonZelman
  • SolomonZelman
h(x) = ?
anonymous
  • anonymous
h(x): graph parabola (1,-3) (3,5) (-1,5)
SolomonZelman
  • SolomonZelman
for h(x), if you were to graph the points:|dw:1435801328063:dw|
SolomonZelman
  • SolomonZelman
you can tell that (1,-3) is the vertex od the parabola.
anonymous
  • anonymous
ys it is the vertex
SolomonZelman
  • SolomonZelman
And that way we know that \(\Large y=\color{red}{\rm a}(x-1)^2-3\) part.
SolomonZelman
  • SolomonZelman
we know everything besides the scale factor (or the coefficient) we will use.
SolomonZelman
  • SolomonZelman
(btw, if you don't undertsnad something completely, you are welcome to ask anything that you would like)
anonymous
  • anonymous
Ok. I understand everything so far. No worries:)
SolomonZelman
  • SolomonZelman
Ok, yes... so now we need to use one of your point. We know that this parabola should satisfy the point (3,5)
SolomonZelman
  • SolomonZelman
don't forget that this parabola that we are finding is the h(x)
SolomonZelman
  • SolomonZelman
So, \(\large\color{black}{ \displaystyle h(x)=\color{red}{\rm a}(x-1)^2-3 }\) we know that point (3,5) should be on the parabola. So: \(\large\color{black}{ \displaystyle \color{blue}{5}=\color{red}{\rm a}(\color{blue}{3}-1)^2-3 }\)
SolomonZelman
  • SolomonZelman
Can you solve for a, please?
anonymous
  • anonymous
ok hold on
anonymous
  • anonymous
a=1/5
anonymous
  • anonymous
5=a(4)-3 5=a1 a=1/5
anonymous
  • anonymous
hello?
anonymous
  • anonymous
@SolomonZelman is it g(x), f(x), and then h(x) ? I'm sorry
SolomonZelman
  • SolomonZelman
\(\large\color{black}{ \displaystyle \color{blue}{5}=\color{red}{\rm a}(\color{blue}{3}-1)^2-3 }\) \(\large\color{black}{ \displaystyle \color{blue}{5}=\color{red}{\rm a}(2)^2-3 }\) \(\large\color{black}{ \displaystyle \color{blue}{5}=4\color{red}{\rm a}-3 }\) \(\large\color{black}{ \displaystyle 2=4\color{red}{\rm a} }\) a=½
SolomonZelman
  • SolomonZelman
this a that we found, we found for h(x).....
SolomonZelman
  • SolomonZelman
So, \(\large\color{black}{ \displaystyle f(x) = 3(x + 4)^2 + 1 }\) \(\large\color{black}{ \displaystyle g(x) = 2x^2 − 16x + 15 }\) \(\large\color{black}{ \displaystyle h(x) = \frac{1}{2}(x -1)^2 -3 }\) (we foudn a=1/2)
anonymous
  • anonymous
I see, I see
SolomonZelman
  • SolomonZelman
and x-axis of summetry (in this case) is just the x-coordinate of the vertex.
SolomonZelman
  • SolomonZelman
just "axis of summetry" (not "x-axis of summetry")
SolomonZelman
  • SolomonZelman
you need to complete the square for g(x)
anonymous
  • anonymous
So, h(x) axis of symmetry would be 1?
anonymous
  • anonymous
@SolomonZelman I have no idea how to do that
SolomonZelman
  • SolomonZelman
yes for h(x) it is 1
SolomonZelman
  • SolomonZelman
can you identify the axis of summetry for f(x) ?
anonymous
  • anonymous
-4
SolomonZelman
  • SolomonZelman
yes
SolomonZelman
  • SolomonZelman
Now, for g(x). \(\large\color{black}{ \displaystyle g(x)=2x^2-16x+15 }\) \(\large\color{black}{ \displaystyle g(x)=2(x^2-8x)+15 }\) following so far?
anonymous
  • anonymous
So i don't get g(x)
SolomonZelman
  • SolomonZelman
we will do g(x)... do you undertsand my previousppost?
SolomonZelman
  • SolomonZelman
previous post*
anonymous
  • anonymous
yes
SolomonZelman
  • SolomonZelman
Ok. \(\large\color{black}{ \displaystyle g(x)=2(x^2-8x)+15 }\) what would you want to have added inside the parenthesis ?
anonymous
  • anonymous
The x's?
SolomonZelman
  • SolomonZelman
what number do you need to add \(\large\color{black}{ \displaystyle g(x)=2(x^2-8x~+\color{blue}{\bf here} )+15 }\) to make the part in the parenthesis a perfect square trinomial ?
anonymous
  • anonymous
2
SolomonZelman
  • SolomonZelman
when you have \(x^2-bx\) the number that you would like to add is \((b/2)^2\)
SolomonZelman
  • SolomonZelman
in this case (8/2)² --> 4² --> 16
SolomonZelman
  • SolomonZelman
So we would like to add 16, but we can't just add numbers, we are going to change the value of the function....
SolomonZelman
  • SolomonZelman
How would we add the number without changing the value of the function? like this....
SolomonZelman
  • SolomonZelman
\(\LARGE \color{black}{ \displaystyle g(x)=2(x^2-8x\color{blue}{+16}\color{red}{-16})+15 }\)
SolomonZelman
  • SolomonZelman
So far I haven't changed anything, I just added a "magic zero"
anonymous
  • anonymous
ok I see
SolomonZelman
  • SolomonZelman
now, expand the -16 (in red) out of parenthesis \(\LARGE \color{black}{ \displaystyle g(x)=2(x^2-8x\color{blue}{+16})+2\cdot (\color{red}{-16})+15 }\) \(\LARGE \color{black}{ \displaystyle g(x)=2(x^2-8x\color{blue}{+16})-32+15 }\) \(\LARGE \color{black}{ \displaystyle g(x)=2(x^2-8x\color{blue}{+16})-17 }\) \(\LARGE \color{black}{ \displaystyle g(x)=2(x-4)^2-17 }\)
SolomonZelman
  • SolomonZelman
so what is the axis of summetry for g(x) ?
anonymous
  • anonymous
4
anonymous
  • anonymous
so would be f(x), hx, gx
SolomonZelman
  • SolomonZelman
yes, now you need to list all axis of summetry in order from smallest to greatest.
SolomonZelman
  • SolomonZelman
go ahead...
anonymous
  • anonymous
fx, hx, gx
anonymous
  • anonymous
yes? No? Maybe so?
SolomonZelman
  • SolomonZelman
yes. f , h, g. -4, 1, 4.
anonymous
  • anonymous
Thank you so much:)
SolomonZelman
  • SolomonZelman
yw

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