Problem 1
One method Olivia could have used is to reflect over the x axis and then translate 10 units to the right.
Let's apply these set of steps to point C.
C = (-2,-4)
C' = (-2,4) after reflecting over the x axis; y coordinate flips in sign
C'' = (8,4) after shifting to the right 10 units; add 10 to the x coordinate
Point C(-2,-4) moves to C''(8,4) which is shown in the diagram. I'll let you check the other four points.
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Another method Olivia could have used is to shift to the right 10 units first, then reflect over the x axis. In this case, the order doesn't matter. Though for some other combinations of transformations, order does matter.
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Problem 2
Reflecting over the y axis will have the point (x,y) turn into (-x,y). Only the x coordinate changes, and we flip the sign here. The y coordinate stays the same.
A point like C(-2,-4) becomes C'(2,-4) after the y axis reflection. Shifting up 8 units means we add 8 to the y coordinate to arrive at C''(2,4) but this does not match what the diagram shows. Therefore, Angelica's method is incorrect.
You should find other inconsistencies for the other four points as well.
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Problem 3
Michael's method isn't correct either. He does not apply an x axis reflection and instead relies entirely on translations to move ABCDE to A''B''C''D''E''
At a quick glance, it looks like Michael may be correct. But upon a much closer look, you'll see that things don't line up perfectly.
Instead of point C, we'll pick on point D this time.
D is located at (-6,-2). If we shift up 8 units, then we add 8 to the y coordinate to get to D'(-6,6). Then add 10 to the x coordinate to shift 10 units to the right. This leads us to D''(4,6).
However, the diagram shows D'' is actually at (4,2)
I'll let you check the other points. You should find that only points C and E go to the correct locations, but the other points do not. Again, this is because Michael did not apply an x axis reflection. His figure of A''B''C''D''E'' is upside down compared to what it should be.
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To summarize, the methods described in problem 1 are the correct way to get ABCDE to move to A''B''C''D''E''. If Olivia followed either of those two methods, then she is correct. Both Angelica and Michael have incorrect methods.
. Now we can complete the square on the polynomial by taking half the linear term, squaring it, and adding it to both sides. Our linear term is 4. Half of 4 is 2, and 2 squared is 4, so we add 4 to both sides: 
and simplify to get 
. In this process we have created a perfect square binomial on the left, which happens to be 
. Now move the -6 back over by addition to get 
. The vertex is found at (-2, 6), the third choice down.