As mugen mentioned, $k$ represents the number of parameters estimated. In other words, it's the number of additional quantities you need to know in order to fully specify the model. In the simple linear regression model
you can estimate $a$, $b$, or both. Whichever quantities you don't estimate you must fix. There is no "ignoring" a parameter in the sense that you don't know it and don't care about it. The most common model that doesn't estimate both $a$ and $b$ is the no-intercept model, where we fix $b=0$. This will have 1 parameter. You could just as easily fix $a=2$ or $b=1$ if you have some reason to believe that it reflects reality. (Fine point: $\sigma$ is also a parameter in a simple linear regression, but since it's there in every model you can drop it without affecting comparisons of AIC.)
If your model is
the number of parameters depends on whether you fix any of these values, and on the form of $f$. For example, if we want to estimate $a, b, c$ and know that $f(c,x)=x^c$, then when we write out the model we have
with three unknown parameters. If, however, $f(c,x)=cx$, then we have the model
which really only has two parameters: $ac$ and $b$.
It is crucial that $f(c,x)$ is a family of functions indexed by $c$. If all you know is that $f(c,x)$ is continuous and it depends on $c$ and $x$, then you're out of luck because there are uncountably many continuous functions.