Translating a graph in the $xy$ plane given by,

$$f(x,y)=c$$

With $c \in \mathbb{R}$, $k$ units right and $h$ units up one gets,

$$f(x-k,y-h)=c$$

The best explanation as to why this is the one I got in Algebra, to see why this is we can consider, the origin. Suppose $f(0,0)=c$. Translating the origin $k$ units right and $h$ units up, gives $(h,k)$. This means the new coordinate is $x=h$ and $y=k$. So $f(k-k,h-h)=c$ works.

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> Do you have a more general, better, explanation? In the above explanation it doesn't really explain how we get the new equation from scratch , but it shows that once we get it, it works.
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> Also what happens to the equation,
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> $$f(x,y)=c$$
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> When we rotate the graph of this curve by $\theta$ counterclockwise, and why.
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I know to rotate we need $(x,y) \to (x\cos(\theta)-y\sin(\theta), x\sin (\theta))+y\cos (\theta))$. But what becomes the new equation of the new curve? My guess is it has something to do with the "inverse rotation=clockwise" as I see in the case of translating by $T$ the graph of $f(x,y)=c$ we get a new graph whose equation is given by $f(T^{-1}(x,y))$. This is what is intuitive to me, but I can't grasp how to explain my intuition  

mathematically.