I recently worked with a student who mentioned this "standard form" of an exponential function taught to him by his teacher: $$f(x)=a\cdot b^{c(x-h)}+k.$$ (Contrast this with the form $f(x)=a\cdot b^x+k$.) He was given a graph of an exponential function and told to determine the equation of the graph. I explained that one can reduce to the form $f(x)=a\cdot b^x+k$ via the mappings $a\cdot b^{-ch}\mapsto a$ and $b^c\mapsto b$, at which point one can easily read off the equation given two points and the horizontal asymptote. But I'm curious why the first form of the exponential equation was given to him in the first place.
- Has anyone ever seen this so-called "standard form" before? I didn't find anything after a quick Google search.
- If so, what are the interpretation of the variables $h$ and $k$ here?