A chemist is researching different sustainable fuel sources. She is currently working with benzene, which must be in liquid form
for her to successfully conduct her research. The boiling point of benzene is 176° F, and the freezing point is 42° F
1 answer:
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Short Answer: C
Remark
The first thing you really ought to do is get a graph from one of the graphing programs on the internet. Most of the time, I use Desmos. That will give you the answer. I will put the graph at the bottom of the page.
The second thing you could do is just set up the given points to see which one gives you the answer.
The third thing (and I'll do this one) is to actually solve the problem by completing the square.
Givens
a = -2
b = 6
c = -1
Solution
Put brackets around the first two terms.
y = (- 2x^2 + 6x ) - 1 Take out the common factor of - 2
y = -2(x^2 - 3x ) - 1 Complete the square inside the brackets.
y = -2(x^2 - 3x + (3/2)^2 ) - 1 Add what it took to complete the square after the - 1
y = -2(x^2 - 3x + (3/2)^2 ) - 1 + 2(3/2)^2 * See remark at end to see why its +
y = -2(x^2 - 3x + 9/4) - 1 + 2 * (9/4)
y = -2(x^2 - 3x + 9/4) - 1 + 18/4
y = -2(x^2 - 3x + 9/4) + 14/4 Show the brackets in square form.
y = -2(x - 3/2)^2 + 14/4 but 3/2 =1.5 and 14/4 = 3.5
y = -2(x - 1.5)^2 + 3.5
The axis of symmetry is x = 1.5 The vertex is (1.5, 3.5)
Discussion
You may wonder why you are adding 18/4 outside the brackets when you are completing the square and all you have inside the brackets is 9/4.
The reason is because first of all there is a - 2 outside the brackets. That gets you 18/4. Because you have got a minus 2, you have to compensate the result so that you get 0. The only way to do that is to add 18/4.
Remark
The first thing you really ought to do is get a graph from one of the graphing programs on the internet. Most of the time, I use Desmos. That will give you the answer. I will put the graph at the bottom of the page.
The second thing you could do is just set up the given points to see which one gives you the answer.
The third thing (and I'll do this one) is to actually solve the problem by completing the square.
Givens
a = -2
b = 6
c = -1
Solution
Put brackets around the first two terms.
y = (- 2x^2 + 6x ) - 1 Take out the common factor of - 2
y = -2(x^2 - 3x ) - 1 Complete the square inside the brackets.
y = -2(x^2 - 3x + (3/2)^2 ) - 1 Add what it took to complete the square after the - 1
y = -2(x^2 - 3x + (3/2)^2 ) - 1 + 2(3/2)^2 * See remark at end to see why its +
y = -2(x^2 - 3x + 9/4) - 1 + 2 * (9/4)
y = -2(x^2 - 3x + 9/4) - 1 + 18/4
y = -2(x^2 - 3x + 9/4) + 14/4 Show the brackets in square form.
y = -2(x - 3/2)^2 + 14/4 but 3/2 =1.5 and 14/4 = 3.5
y = -2(x - 1.5)^2 + 3.5
The axis of symmetry is x = 1.5 The vertex is (1.5, 3.5)
Discussion
You may wonder why you are adding 18/4 outside the brackets when you are completing the square and all you have inside the brackets is 9/4.
The reason is because first of all there is a - 2 outside the brackets. That gets you 18/4. Because you have got a minus 2, you have to compensate the result so that you get 0. The only way to do that is to add 18/4.