## anonymous one year ago In an experiment a petri dish with a colony of bacteria is exposed to cold temperatures and then warmed again. Find a quadratic model for the data in the table. Show work

1. anonymous

Getting the table, one moment

2. anonymous

Time (hours) 0 1 2 3 4 5 6 Population (1000s) 5.1 3.03 1.72 1.17 1.38 2.35 4.08

3. anonymous

how are you supposed to do this (just asking because you have to "show work")

4. anonymous

I have no clue lol I typed everything the question asked and the table

5. DecentNabeel

The general formula for quadratic eqn is y = ax^2 + bx + c take any 3 points from the table (0, 5.1) (1, 3.03) (5, 2.35) plug these points into the above equation and you will get 3 different equations use substitution or any method that you know to solve these 3 eqns with 3 unknowns

6. anonymous
7. DecentNabeel

a point is written in the form (x,y) so plug 0 for x and 5.1 for y to get 1st eqn. follow same method to get other 2 eqns

8. anonymous

@DecentNabeel i am not sure that would work, since it is not exact

9. anonymous

im confused

10. anonymous

So this part is correct? The general formula for quadratic eqn is y = ax^2 + bx + c take any 3 points from the table (0, 5.1) (1, 3.03) (5, 2.35) plug these points into the above equation and you will get 3 different equations use substitution or any method that you know to solve these 3 eqns with 3 unknowns

11. DecentNabeel

@satellite73 many way for solve i solve another but this is true

12. anonymous

if it is exact you can do it using only three points $(0,5.1),(1,3.03),(2,2.35)$ or any other 3

13. DecentNabeel

yes is correct

14. anonymous

also you do not have to solve a system of three equations, only two

15. anonymous

you want $ax^2+bx+c$ but sine $$(0,5.1$$ is on the graph, you have $$c=5.1$$ right away, only need $$a$$ and $$b$$

16. anonymous

So lets write step one so i can start writing it so i can show our work lol

17. anonymous

the part you wrote above is correct

18. DecentNabeel

5.1 = a*0^2 + b*0 + c 3.03 = a*1^2 + b*1 + c 2.35 = a*5^2 + b*5 + c solve of a, b and c

19. anonymous

what @DecentNabeel said solve that system but really it is not as hard as it looks since the first one tells you $$c=5.1$$

20. anonymous

I dont know how to plug all that in

21. anonymous

oops second one is $3.03=a+b=5.1$ so $a+b=-2.07$

22. anonymous

do you see how @DecentNabeel got this one $5.1 = a*0^2 + b*0 + c$

23. anonymous

if the answer is "no" that is fine, just say so

24. DecentNabeel

yes c=5.1 @satellite73

25. anonymous

no i dont get it

26. anonymous

ok do you get that you are trying to find $$a,b$$ and $$c$$ to make up $y=ax^2+bx+c$?

27. anonymous

yes

28. anonymous

ok and from the table you see that $$(0,5.1)$$ is on the graph right?

29. anonymous

correct

30. anonymous

that means if $$x=0$$ then $$y=5.1$$ right ?

31. anonymous

correct

32. DecentNabeel

What @LynFran ????

33. anonymous

@satellite73 are you back? The serve kicked me off

34. anonymous

yea me too

35. DecentNabeel

@EllenJaz17 what are you said @LynFran

36. anonymous

they interrupted me learning lol

37. anonymous

lol

38. DecentNabeel

lol

39. anonymous

ok so to resume our story, were were here: we want $$a,b,c$$ so make $$y=ax^2+bx+c$$ and we know if $$x=0$$ then $$y=5.1$$ plug them in and get $5.1=a\times 0^2+b\times 0+c$

40. anonymous

is that ok?

41. anonymous

yes i get that one because its easy because x is 0 lol

42. anonymous

right so that tells us right away that $$c=5.1$$ yes?

43. anonymous

correct

44. anonymous

ok now $$a$$ and $$b$$ to go

45. anonymous

@LynFran send him a message, im trying to get hw help, not get interrupted

46. anonymous

47. anonymous

the next point we have is $$(1,3.03)$$ right ?

48. anonymous

yes

49. anonymous

now don't forget, we already know $$c=5.1$$ so we have $y=ax^2+bx+5.1$ put $$x=1$$ and $$y=3.03$$ to get $3.03=a\times 1^2+b\times 1+5.1$

50. anonymous

this also cleans up nicely, since $$1^2=1$$ we have $3.03=a+b+5.1$

51. anonymous

good so far?

52. anonymous

yes

53. anonymous

now to make it neater, subtract $$5.1$$ from both sides that gives $-2.07=a+b$ or $a+b=-2.07$ and we put that aside for the moment

54. anonymous

how we doing so far?

55. anonymous

good

56. anonymous

|dw:1435898261868:dw|

57. anonymous

ok now one more equation to go you can pick any point you like but i will pick $$(2,1.72)$$

58. anonymous

Sounds good

59. DecentNabeel

good @satellite73

60. anonymous

again we have $y=ax^2+bx+5.1$ put $$x=2,y=1.72$$ and get $1.72=a\times 2^2+b\times 2+5.1$

61. anonymous

ok with that?

62. anonymous

yes

63. anonymous

now we clean it up actually you can what do you get in terms of $$a,b$$ and a number ?

64. anonymous

$-2.07^{2}+1.72+5.1$ ? Im sure that's wrong

65. anonymous

quite wrong, it is much easier than that

66. anonymous

haha oh jeez. I needed this class apparently very much lol and your help

67. anonymous

lets go back to $1.72=a\times 2^2+b\times 2+5.1$

68. anonymous

your equation will have an $$a$$ and a $$b$$ in it all you have to do is compute $$2^2$$

69. anonymous

So a=4?

70. anonymous

no just $$a\times 4$$ or $$4a$$

71. anonymous

lets back up a second

72. anonymous

remember that when we replace $$x$$ by $$1$$ and $$y$$ by $$3.03$$ we ended up with an equation with both an $$a$$ and a $$b$$ in it it was $a+b=-2.07$

73. anonymous

we are trying to accomplish the same thing here with $$x=2$$ and $$y=1.72$$ first step is to write $1.72=a\times 2^2+b\times 2+5.1$

74. anonymous

so don't do too much work just see what the numbers give you your equation should still have an $$a$$ and a $$b$$ in it

75. anonymous

if you are still confused, say so and i will spell it out

76. anonymous

Yeah im confused

77. anonymous

thought so i am just going to rewrite $1.72=a\times 2^2+b\times 2+5.1$

78. anonymous

since $$2^2-4$$ this is the same as $1.72=a\times 4+b\times 2+5.1$ clear?

79. anonymous

yes

80. anonymous

i meant "since $$2^2=4$$

81. anonymous

ok now we just make it look better $1.72=4a+2b+5.1$ go that ?

82. anonymous

yes

83. anonymous

again we subtract $$5.1$$ from both sides and arrive finally at $4a+2b=-3.38$

84. anonymous

ok

85. anonymous

now lets get those two equations together $a+b=-2.07\\ 4a+2b=-3.38$

86. anonymous

damn $a+b=-2.07\\ 4a+2b=-3.38$ thats better

87. anonymous

we have to solve this "system of equations" to find $$a$$ and $$b$$

88. anonymous

sounds good

89. anonymous

do you know how to solve this system?

90. anonymous

no

91. anonymous

i hate decimals, at this point i would cheat, but you can do whatever you like

92. anonymous

do you have to show ALL your steps?

93. anonymous

no it just says show work

94. anonymous

well we have come this far, lets grind it out

95. anonymous

probably the easiest thing to do is rewrite $a+b=-2.07$ as $b=-2.07-a$ and substitute that in the second equation get $4a+2(-2.07-a)=-3.38$

96. anonymous

ok im just writing as you go

97. anonymous

multiply out, combine like terms, work all the magic, get $a=.33$ i will let you grind that out for yourself, but the answer is $$a=.33$$

98. anonymous

so all thats left to find is b, than we do the equation, correct?

99. anonymous

now we know that $$a=.33$$ and $a+b=-2.07$ you have $.33+b=-2.07$ so $b=-2.4$

100. anonymous

yeah that is it , find $$a$$ and $$b$$ we get $$a=.33$$ and $$b=-2.4$$

101. anonymous

and we recall that $$c=5.1$$

102. anonymous

dang i made a mistake somewheres $$b=-2.45$$ actually but whatever

103. anonymous

So .33^2-2.45+5.1

104. anonymous

oh damn damn damn i made a stupid typo

105. anonymous

everywhere you see $$.33$$ replace it by $$.38$$

106. anonymous

i told you i hate decimals when you solve $4a+2(-2.07-a)=-3.38$ you get $$a=.38$$

107. anonymous

and since $$a=.38$$ we get $$b=-2.45$$ final answer $y=.33x^2-2.45x+5.1$

108. anonymous

your math teacher must hate you to give you such a mess

109. anonymous

at any rate, all the steps are there you can copy them down and look at them at your leisure took a while, hope you learned something

110. anonymous

should it say y=.38 or y=.33 lol

111. anonymous

Final Answer $\huge \color{red}{y=.38x^2-2.45x+5.1}$

112. anonymous

lol great thank you so much!

113. anonymous

|dw:1435900099694:dw|

114. anonymous

Do you wanna help me with another question that relates to this table lmao @satellite73

115. anonymous

hell no not without a shot of tequila

116. anonymous

ok post i will take a look, maybe it will be easier

117. anonymous

ooh to

118. anonymous

THIS TABLE ok

119. anonymous

|dw:1435900241301:dw|

120. anonymous

There you go lmao

121. anonymous

good, no lime or salt (for sissies)

122. anonymous

bahaha

123. anonymous

what is the question i am about to turn in to a pumpkin

124. anonymous

@satellite73 sorry are you still there