A polar coordinate function f(x) can be rotated around the axis by h with the shift f(x - h). However, this rotates the graph.
Is there a way to shift a polar coordinate function by x and y measurements? If so, what is it? Can it be done for all functions or only a subset?
In principle, every continuously differentiable regular plane curve that crosses each ray through the origin at most once can be expressed as a polar graph $r = f(\theta)$ over some interval of angles. The shifted polar graph $$ x = x_{0} + f(\theta)\cos\theta,\qquad y = y_{0} + f(\theta)\sin\theta, $$ can be decomposed into such pieces, again in principle.
In practice, this may be a Real Nuisance. For example, a shift of the polar graph $r = \cos(4\theta)$ may require a number of polar graph pieces for a complete description. For the curve shown, any "drawable arc" is delimited by a pair of rays through the origin that are tangent to the blue curve. (I haven't tried to count these rays carefully.)