Symmetrical

Does y = -x2 = y = -(-x)2 = -x2
Using a numerical example let x = 2 and then x = -2  Substitute these values into y = -x2
y = -(2)2 = - 4   y = -(-2)2 = - 4

The answers are the same and will be for any values you substitute in for x so yes
y = -x2 is symmetrical to the y axis.  

y = -x2 is a parabola as is y = x2  The negative sign in front of -x2 in y = -x2 tells us that the parabola opens downward.

y = x2 will open upward.

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