I have a polar graph function
$$r=2\cdot\sin(\theta)$$
How would I translate (shift) graph $r=2\cdot\sin(\theta)$ with a vector $(^x_y)$?
What I am looking for is a change to the polar function that moves the graph along $x$ by $x$ units and $y$ by $y$ units
I realised this is confusing for people. However, I do not understand how. I have always been told that an $r=$ graph is called a polar graph, and that the transformation of shifting is called translating
On
General equation of a circle in polar coordinates:
$$r^2-2rr_0\cos (\theta-\phi)+r_0^2=R^2$$
Now
$$r=2\sin \theta \implies r(r-2\sin \theta)=0 \implies r^2-2r\cos \left( \theta-\frac{\pi}{2} \right)+1=1$$
Hence, $$ \left \{ \begin{align} r_0\cos \phi &= x \\ r_0\sin \phi &= y+1 \end{align} \right.$$
The translated circle is $$r^2-2r[x\cos \theta+(y+1)\sin \theta]+x^2+(y+1)^2=1$$
The two roots of $r(\theta)$ represent the same circle but with different senses of rotation.
$$\binom xy=\binom{r\cos\theta}{r\sin\theta}=\binom{f(\theta)\cos\theta}{f(\theta)\sin\theta}$$