The standard form of a quadratic is y= ax2+bx+c
The standard form of a parabola is y= a(x-h)2+ k
Your equation y = -2x2-3x+1/2, so a=-2;b=-3, and c =1/2
h,k represents the vertex point of the parabola. To find h use the formula -b/2a = -(-3)/2(-2) = 3/-4 = -3/4 which represents the x value of the point. To find the y value in your equation every place you have an x subsitute the -3/4, so
y = -2(-3/4)2-3(-3/4) + 1/2
y = -2(9/16)+9/4+1/2
y = -9/8 + 18/8 + 4/8
y = 13/8
so the vertex of your parabola is at the point (-3/4,13), and looks something like this (difficult to draw stuff on Cramster!)
Since a is negative the parabola opens down. Domain is (-∞,∞), range is (-∞,13/8)