To obtain the graph of y = x2 - 6 shift the graph of y = x2

You would have to shift x^2 down 6 units.

Each different kind of shift in the writing of the equation corresponds to a different kind of shift in the graph. Shifts in the y-intercept, which is the last number in the equation, are reflected by shifts up and down in the parabola. Since you are subtracting 6, you move the graph down 6.