Why is the dilation factor of a^kx

[autodidude]

...equal to 1/k? Why isn't it just interpreted as k?

 

[autodidude]

Or is that just the way it is?

 

[HallsofIvy]

I have absolutely no idea what you are asking. Can you please give an example?

 

[uart]

Had me baffled too, as OP talks about a specific function (e^x). But I think he's referring in general to calling f(x/a) a dilation of f(x). Not something I'm familiar with (by name) but I think I can guess what he's referring to.

Autodidude, say you have a function f(x) and you construct a new function g(x) = f(x/2). Can you see that if you plotted g(x) it would look identical to f(x) except that it would look like you'd stretched out the x-axis by a factor of two. Sketch some simple examples like f(x)=sin(x), g(x)=sin(x/2) and you'll soon see it. So f(x/2) would be a dilation because it has the same effect as stretching out the x-axis (by a factor of 2). On the other hand f(2x) would be a contraction. This is a general property of functions and not restricted to exponentials.

 

[autodidude]

Graphs of exponential functions:

The graph of a^kx has a dilation factor 1/k parallel to the x-axis. So if it's f(x) = 2^2x, then the dilation factor is 1/2, I was curious as to why it isn't just 2

 

[uart]

So if it's f(x) = 2^2x, then the dilation factor is 1/2, I was curious as to why it isn't just 2

There must be a problem with your browser not displaying things properly, because I'm pretty sure I just explained that in the previous post.