y=-1/2x^{2}

I think you want to graph -1 divided by 2 x squared. It is easiest to think of the numerator -1 and denominator 2x squared separately. The numerator is negative. What sign will the denominator be? Since 2 is poitive and x is squared, x squared is positive and 2 x squared is positive. So this is a graph of a fraction formed by a negative numerator and a positive denominator. The quotient is then negative. So y is equal to a negative number and whole graph appears in quadrants 3 and 4. Think next about end behavior of the graph. As x gets really large or really negative, x squared gets huge (the denominator gets big). For example, if x is 100, then x squared is 10,000. And then y = -1/ 20,000 or -.00005. So as you see, the graph approaches y=o as you move far left and far right. Can x equal 0? No because then divsiion by zero occurs. So the graph has the asymptote x=0. Lastly, as x is really close to zero, the denominator is so small that the fraction becomes huge negative number. The graph goes toward negative infinity then near x = 0 on both sides of the y axis (it is symmetric).