Consider the map $f: \mathbb{R}^2 \rightarrow \mathbb{R^2}$ given by $f(x,y)=(3x,2y)$. This map scales the plane horizontally by a factor of $3$ and vertically by a factor of $2$.
For the function given by $g(x)=x^2$, the image of its graph under $f$ has the equation $y=2g(\frac{1}{3}x)$.
It happens that, under a dilation centered at the origin by a factor of $\frac{9}{2}$, the graph of $g$ has the same image. That is, $2g(\frac{1}{3}x)=\frac{9}{2}g(\frac{2}{9}x)$. However, this is not true for the function given by $h(x)=\sin x$. Clearly, in general, $(3x,2y) \not = (\frac{9}{2}x,\frac{9}{2}y)$
My specific question is if $f$ is not equivalent to a dilation of the plane in general, why does it "seem" to be for $g$?
My general question is about vocabulary. When two figures can be mapped to each other by a sequence of reflections, rotations, and dilations, they are "similar." Is there a word for two figures that can be mapped to each other by a general linear transformation?