Choosing between two parametric models for a time series

This may be a bit theoretical, but here goes.

I am looking to decide between two different models to describe a curve. One (the reduced model) is simply a line

y = m*x + b.

The second (full model) is exponential growth on top of a line

y = mx + b + A * exp(S*x).

I appreciate that the models are nested and I know how to do the F-test for nested models. However what troubles me is whether the full model is really 2 extra parameters (A and s). This is because if A is not significant, then it doesn't matter what s is. This same question would apply more generally for any full model of the form

y = m*x + b + A * f(s1,s2,s3,...).

In this case, once A is non-significant, the other parameters don't "matter', whatever that means.

Gene