How to express transformations of exponential functions, "stretches" and "compressions", in terms of "dilations"?

Question

When referring to the transformations of exponential functions, how do you refer to vertical "stretches" and "compressions" in terms of "a dilation of $z$ from the y-axis or x-axis"?

From my understanding,

For any factor $a > 0$, the function $f(x) = ab^x$

• is stretched vertically by factor $a$ if $|a|>1$

• is compressed vertically by factor $a$ if $0<∣a∣<1$

e.g. if you have $2b^x$, the function is that of $b^x$ stretched vertically by factor $2$.

What would be the equivalent of this in dilation terms - e.g. (for the above example) the function has dilated FROM the $x$-axis by factor $2$? Is this correct? Are there any other ways to express this?

Answer 1

We can understand what kind of action would correspond to what kind of transformation with respect to a given variable by isolating the variable we want to affect. However, everything I'll write is intuitive and non-formal.

Consider $y=f(x)$, and some $|a|>1.$ By looking at $y=af(x),$ we have increased the size of $y$ for any given $x$ - which corresponds to the graph of $f(x)$ stretching vertically. Easy stuff.

Now, had we wanted to translate or stretch $f(x)$ along the horizontal part of the graph, i.e. the domain, input, range of definition, etc. we'd like to do some action to the variable - "doing stuff" inside the function.

Let's look at $y=f(ax).$ For $|a|>1$, our graph would shrink along the $x$-axis - this happens, in an intuitive fashion, because we've increased the size of our input, and hence the function reaches any point of output earlier - which is to say more to the left, and again not at all formally, under $ax.$ Thus, multiplication of the input $x$ by $|a|>1$ would squish our function along the vertical axis, and multiplication by $|a|<1$ would correspond to a stretch of the function along the $x$ axis - imagine now the function ranges over smaller inputs and takes longer to reach a given output.

Finally, when it comes to taking the graph and translating it, the same lines of reasoning (what axis=variable do I want to affect and how <-> what is the variable associated with that axis and what is the action associated with my transformation) should make it quite clear what to do.

Hope this clarifies things.