I likewise have needed such a thing—to converse coherently about transformation properties (actually symmetries) of power, exponential, and logarithmic functions. I did not find anything in textbooks or anything commonly used otherwise. I considered using $\rho$ for "ratio", but settled on $\sigma$ (and $\sigma_y$ as needed), because in conversing, "scaling" was more used than "ratio." Following are examples of summary statements.

An exponential function $f(x) = A b^x$ scales by the same factor over equal increments of the input. The base $b$ gives the factor over a unit increment. The base to the power of an input increment $\Delta x$ gives the output factor: $$\sigma_y = b^{\Delta x}.$$ The graph is invariant under the family of transformations $$(x,y) \rightarrow (x + \Delta, b^\Delta y)$$ where $\Delta$ can be any real number.

We use $\Delta x$, but "$\sigma y$" is clumsy, so I use a subscript. I try to avoid the subscript. I would write the transformation property ("equivariance") of the function as $$f(x + \Delta) = b^\Delta f(x),$$ provided students were ready for such symbolic stuff. (Usually some are, some aren't—a chance to "differentiate" instruction.) In making tables of values, often a column would be labeled with a $\sigma_y$ (or $\sigma_x$ for logarithmic and power functions).

I likewise have needed such a thing—to converse coherently about transformation properties (actually symmetries) of power, exponential, and logarithmic functions. I did not find anything in textbooks or anything commonly used otherwise. I considered using $\rho$ for "ratio", but settled on $\sigma$ (and $\sigma_y$ as needed), because in conversing, "scaling" was more used than "ratio." Following are examples of summary statements.

An exponential function $f(x) = A b^x$ scales by the same factor over equal increments of the input. The base $b$ gives the factor over a unit increment. The base to the power of an input increment $\Delta x$ gives the output factor: $$\sigma_y = b^{\Delta x}.$$ The graph is invariant under the family of transformations $$(x,y) \rightarrow (x + \Delta, b^\Delta y)$$ where $\Delta$ can be any real number.

We use $\Delta x$, but "$\sigma y$" is clumsy, so I use a subscript. I try to avoid the subscript. I would write the transformation property ("equivariance") of the function as $$f(x + \Delta) = b^\Delta f(x),$$ provided students were ready for such symbolic stuff. (Usually some are, some aren't—a chance to "differentiate" instruction.) In making tables of values, often a column would be labeled with a $\sigma_y$ (or $\sigma_x$ for logarithmic and power functions).

With exponential functions I mostly avoid this $\sigma_y$ notation. If we have the exercise of finding the parameters for an exponential curve going through $(1,6)$ and $(3,30)$, say, I would encourage putting the data in a table and writing $\Delta x = 2$ on the $x$ side and $b^2 = 5$ somewhere, just saying 5 is the output scaling associated to an input increment of 2, rather than writing $\sigma_y = 5$. Solve for the base, and go from there.

I likewise have needed such a thing—to converse coherently about transformation properties (actually symmetries) of power, exponential, and logarithmic functions. I did not find anything in textbooks or anything commonly used otherwise. I considered using $\rho$ for "ratio", but settled on $\sigma$ (and $\sigma_y$ as needed), because in conversing, "scaling" was more used than "ratio." Following are examples of summary statements.

An exponential function $f(x) = A b^x$ scales by the same factor over equal increments of the input. The base $b$ gives the factor over a unit increment. The base to the power of an input increment $\Delta x$ gives the output factor: $$\sigma_y = b^{\Delta x}.$$ The graph is invariant under the family of transformations $$(x,y) \rightarrow (x + \Delta, b^\Delta y)$$ where $\Delta$ is any real number.

We use $\Delta x$, but "$\sigma y$" is clumsy, so I use a subscript. I try to avoid the subscript. I would write the transformation property ("equivariance") of the function as $$f(x + \Delta) = b^\Delta f(x),$$ provided students were ready for such symbolic stuff. (Usually some are, some aren't—a chance to "differentiate" instruction.) In making tables of values, often a column would be labeled with a $\sigma_y$ (or $\sigma_x$ for logarithmic and power functions).

I likewise have needed such a thing—to converse coherently about transformation properties (actually symmetries) of power, exponential, and logarithmic functions. I did not find anything in textbooks or anything commonly used otherwise. I considered using $\rho$ for "ratio", but settled on $\sigma$ (and $\sigma_y$ as needed), because in conversing, "scaling" was more used than "ratio." Following are examples of summary statements.

An exponential function $f(x) = A b^x$ scales by the same factor over equal increments of the input. The base $b$ gives the factor over a unit increment. The base to the power of an input increment $\Delta x$ gives the output factor: $$\sigma_y = b^{\Delta x}.$$ The graph is invariant under the family of transformations $$(x,y) \rightarrow (x + \Delta, b^\Delta y)$$ where $\Delta$ can be any real number.

We use $\Delta x$, but "$\sigma y$" is clumsy, so I use a subscript. I try to avoid the subscript. I would write the transformation property ("equivariance") of the function as $$f(x + \Delta) = b^\Delta f(x),$$ provided students were ready for such symbolic stuff. (Usually some are, some aren't—a chance to "differentiate" instruction.) In making tables of values, often a column would be labeled with a $\sigma_y$ (or $\sigma_x$ for logarithmic and power functions).