anonymous one year ago 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.

1. 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.

2. SolomonZelman

that is the rule.....

3. anonymous

Ok so the first one would be that it opens down correct?

4. anonymous

5. SolomonZelman

yes, the first one opens down. Right;)

6. SolomonZelman

And the second one opens where?

7. SolomonZelman

(and yes, you can ask another question right here, after you feel that you are done with this question.)

8. anonymous

It opens us since it's poitive

9. SolomonZelman

yes

10. SolomonZelman

ok i guess you are done with this question... r u?

11. SolomonZelman

12. anonymous

so the first one, is opening down w/ the vertex at an absolute maximum? I'm just checking

13. SolomonZelman

yes

14. SolomonZelman

the vertex in function 1 is absolute maximum

15. anonymous

Ok thank-you:)

16. 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

17. SolomonZelman

yw

18. anonymous

i don't understand this one

19. SolomonZelman

you don't understand the question, or don't know how to solve it?

20. anonymous

I don't understand how to solve it

21. SolomonZelman

oh....ok...

22. 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?

23. anonymous

yes. Yes I do

24. 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

25. 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

26. 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.

27. SolomonZelman

the last option is going to be obviously smaller than or greater than the 3rd option ?

28. SolomonZelman

can you tell me?

29. anonymous

so the answer would be c2+d2

30. SolomonZelman

hold on...

31. SolomonZelman

32. SolomonZelman

(you are trying to jump ahead to quickly:o)

33. anonymous

smalleer :) Sorry

34. SolomonZelman

yes (no need for apologies).

35. 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).

36. 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)

37. 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?

38. anonymous

)c+d)^2

39. SolomonZelman

Yes, (C+D)² is the largest, and this it is the answer you need:)

40. SolomonZelman

41. anonymous

No. :)

42. SolomonZelman

OK.....

43. 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

44. SolomonZelman

ok, this is a last question for this post. Alright>

45. anonymous

Yea:)

46. SolomonZelman

h(x) = ?

47. anonymous

h(x): graph parabola (1,-3) (3,5) (-1,5)

48. SolomonZelman

for h(x), if you were to graph the points:|dw:1435801328063:dw|

49. SolomonZelman

you can tell that (1,-3) is the vertex od the parabola.

50. anonymous

ys it is the vertex

51. SolomonZelman

And that way we know that $$\Large y=\color{red}{\rm a}(x-1)^2-3$$ part.

52. SolomonZelman

we know everything besides the scale factor (or the coefficient) we will use.

53. SolomonZelman

(btw, if you don't undertsnad something completely, you are welcome to ask anything that you would like)

54. anonymous

Ok. I understand everything so far. No worries:)

55. SolomonZelman

Ok, yes... so now we need to use one of your point. We know that this parabola should satisfy the point (3,5)

56. SolomonZelman

don't forget that this parabola that we are finding is the h(x)

57. 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: $$\large\color{black}{ \displaystyle \color{blue}{5}=\color{red}{\rm a}(\color{blue}{3}-1)^2-3 }$$

58. SolomonZelman

Can you solve for a, please?

59. anonymous

ok hold on

60. anonymous

a=1/5

61. anonymous

5=a(4)-3 5=a1 a=1/5

62. anonymous

hello?

63. anonymous

@SolomonZelman is it g(x), f(x), and then h(x) ? I'm sorry

64. 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

65. SolomonZelman

this a that we found, we found for h(x).....

66. 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)

67. anonymous

I see, I see

68. SolomonZelman

and x-axis of summetry (in this case) is just the x-coordinate of the vertex.

69. SolomonZelman

just "axis of summetry" (not "x-axis of summetry")

70. SolomonZelman

you need to complete the square for g(x)

71. anonymous

So, h(x) axis of symmetry would be 1?

72. anonymous

@SolomonZelman I have no idea how to do that

73. SolomonZelman

yes for h(x) it is 1

74. SolomonZelman

can you identify the axis of summetry for f(x) ?

75. anonymous

-4

76. SolomonZelman

yes

77. 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?

78. anonymous

So i don't get g(x)

79. SolomonZelman

we will do g(x)... do you undertsand my previousppost?

80. SolomonZelman

previous post*

81. anonymous

yes

82. SolomonZelman

Ok. $$\large\color{black}{ \displaystyle g(x)=2(x^2-8x)+15 }$$ what would you want to have added inside the parenthesis ?

83. anonymous

The x's?

84. 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 ?

85. anonymous

2

86. SolomonZelman

when you have $$x^2-bx$$ the number that you would like to add is $$(b/2)^2$$

87. SolomonZelman

in this case (8/2)² --> 4² --> 16

88. 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....

89. SolomonZelman

How would we add the number without changing the value of the function? like this....

90. SolomonZelman

$$\LARGE \color{black}{ \displaystyle g(x)=2(x^2-8x\color{blue}{+16}\color{red}{-16})+15 }$$

91. SolomonZelman

So far I haven't changed anything, I just added a "magic zero"

92. anonymous

ok I see

93. SolomonZelman

now, expand the -16 (in red) out of parenthesis $$\LARGE \color{black}{ \displaystyle g(x)=2(x^2-8x\color{blue}{+16})+2\cdot (\color{red}{-16})+15 }$$ $$\LARGE \color{black}{ \displaystyle g(x)=2(x^2-8x\color{blue}{+16})-32+15 }$$ $$\LARGE \color{black}{ \displaystyle g(x)=2(x^2-8x\color{blue}{+16})-17 }$$ $$\LARGE \color{black}{ \displaystyle g(x)=2(x-4)^2-17 }$$

94. SolomonZelman

so what is the axis of summetry for g(x) ?

95. anonymous

4

96. anonymous

so would be f(x), hx, gx

97. SolomonZelman

yes, now you need to list all axis of summetry in order from smallest to greatest.

98. SolomonZelman

99. anonymous

fx, hx, gx

100. anonymous

yes? No? Maybe so?

101. SolomonZelman

yes. f , h, g. -4, 1, 4.

102. anonymous

Thank you so much:)

103. SolomonZelman

yw