in the equation -2x2+2x+1,

a= -2, b=2, and c=1

the axis of symmetry is given by the line:

x = -b/2a, so substitute in 2 for b and -2 for a

x = -2/(2*-2) = -2/-4 = 2/4 = 1/2

so x = 1/2 is the axis of symmetry line.

the vertex will be on this line, since it is always on the line of symmetry, so substitute this into the equation for x to find the value for y of the vertex. 

y = -2(1/2)2+2(1/2) + 1

y = -2(1/4) + 1 + 1

y = -1/2 + 2

y = 3/2

so the vertex is the point (1/2, 3/2)

3/2 is also the maximum value of the function; the vertex is the maximum point on the graph

this is because the value of a is negative, meaning that the graph opens downward.