how to scale a function?

Question

My intuitive understanding of scaling a function $f(x)$, is that $$f'(sx) = f(x),$$ where $s$ is the scaling factor.

For example, $f(x) = x^2$, and $s = 2$, then plot them out, the blue one is $f(x)$, green $f'(sx)$.

Am I right about the scaling? Further, I usually see some other transformations, which I'm not sure whether it is scaling or not.

For example, still $f(x)=x^2$, and $$g(x) = \exp(\frac{wt^2}{2})f(x),$$ where $w,t$ are parameters, which can be chosen at will. So if I tune $t$ from $[0,1]$, can I say I'm scaling $f(x)$?

UPDATE

If $$g(x) = \exp(\frac{-x^2}{2\sigma^2})*f(x),$$ which means $g(x)$ is the result of convolving $f(x)$ with Gaussian function. Then is there any scaling in it by tuning $\sigma$?

Answer 1

You can scale a function horizontally or vertically (in terms of its graph). The first equation you wrote is scaling the graph horizontally. When you scale vertically, you get the function $g(x) = cf(x)$ which stretches the graph of $f$ vertically by a factor of $c$.

In your example, you stated that $w$ and $t$ are just constants. So $\exp\left(\frac{wt^2}{2}\right)$ is just a constant (let it be denoted by $c$). Then $g(x) = cf(x) = cx^2$ which is just scaling in the vertical direction.

Answer 2

The function y=f(Cx) where C is a multiplication factor, means to compress horizontally by a factor of C. So, for example, if the function is sin(theta) then sin(2*theta) compresses the function such that its period is half than what it was before compression (from two pi to pi).