Step 1 :The function $y=x^{2}$ is a parabola that opens upwards with its vertex at the origin (0,0).
Step 2 :When we shift this graph to the right by 6 units, the vertex of the parabola will be at (6,0).
Step 3 :This means that the x in the equation $y=x^{2}$ is replaced by $(x-6)$ to represent this shift.
Step 4 :\(\boxed{y=(x-6)^{2}}\) is the function that represents the graph of $y=x^{2}$ shifted to the right by 6 units.