This simple problem became extremely confusing when I started to attempt understanding it in relation to linear transformations, change of basis, etc. Consider the linear transformation $$\newcommand\m[1]{\begin{bmatrix}#1\end{bmatrix}} \m{x'\\y'} = \m{x/2\\y}$$ and the function $y'=x'^2$.
The basis vectors in which $x', y'$ are described are respectively $\m{1/2\\0}$ and $\m{0\\1}$. So what I thought was that if the relationship between $y'$ and $x'$ is given, which is $y'=x'^2$, then obviously the graph in $xy$ system (in contrast to in $x'y'$ system) will look as if it's been shrinked in the $x$ direction by the factor of 2. See the diagram below.
But when I do try to derive the equation algebraically a problem occurs. Because $x'=x/2$ and $y'=y$ we can derive that $$y'=x'^2 \Leftrightarrow y=(x/2)^2,$$ but the graph of this of obviously stretched by a factor of 2 in the $x$ direction, not shirinked (green in the diagram below).
What is causing this contradiction?