Part 1) Convert the given equation to vertex form
y = -2x^2+2x+2
y-2 = -2x^2+2x
y-2 = -2(x^2-x) ... notice the x coefficient here -1
y-2 = -2(x^2-x+1/4 - 1/4) ... take half of -1 to get -1/2, then square it to get 1/4
y-2 = -2(x^2-x+1/4)-2(-1/4)
y-2 = -2(x^2-x+1/4)+1/2
y-2-1/2 = -2(x^2-x+1/4)
y-5/2 = -2(x-1/2)^2
y = -2(x-1/2)^2+5/2
y = -2(x-0.5)^2+2.5
Answer in fraction form: y = -2(x-1/2)^2+5/2
Answer in decimal form: y = -2(x-0.5)^2+2.5
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Part 2) Find the vertex
The answer from part 1) leads to the answer to part 2). The nice thing about vertex form that it's easy to read off the vertex without having to do any extra work.
Vertex form is y = a(x-h)^2 + k
Match that to y = -2(x-1/2)^2+5/2 and we see that a = -2, h = 1/2, k = 5/2
The vertex is (h,k) = (1/2, 5/2) in fraction form. That's equivalent to (h,k) = (0.5, 2.5) in decimal form
Answer in fraction form: (1/2, 5/2)
Answer in decimal form: (0.5, 2.5)
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Part 3) Find the y value when x = 6
Plug in x = 6, to get...
y = -2x^2+2x+2
y = -2(6)^2+2(6)+2
y = -2(36)+2(6)+2
y = -72+12+2
y = -60+2
y = -58
Answer: y = -58
