I understand where the following matrices come from and how they geometrically rotate a point (x, y) counterclockwise/clockwise by $θ$
$\begin{bmatrix}\cosθ&-\sinθ\\\sinθ&\cosθ\end{bmatrix} $ for counterclockwise and
$\begin{bmatrix}\cosθ&\sinθ\\-\sinθ&\cosθ\end{bmatrix} $ for clockwise
For example, I want to rotate the graph ${y} = x^{2}$ by 45° counterclockwise. By appying this matrix to any point (x, y),
$\begin{bmatrix}\cos45°&-\sin45°\\\sin45°&\cos45°\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}\frac{1}{\sqrt{2}}x-\frac{1}{\sqrt{2}}y\\\frac{1}{\sqrt{2}}x+\frac{1}{\sqrt{2}}y\end{bmatrix}$
So, $x$ moves to $(\frac{1}{\sqrt{2}}x-\frac{1}{\sqrt{2}}y)$ and $y$ moves to $(\frac{1}{\sqrt{2}}x+\frac{1}{\sqrt{2}}y)$
If I plug these values into ${y} = x^{2}$, it yields the following graph:
As you can see, despite plugging in the counterclockwise-rotated points, the graph was rotated clockwise by 45°.
So my question is: Why does plugging in the counterclockwise-rotated points result in a graph that is rotated clockwise?
I know that in order to rotate the graph counterclockwise you just have to input the points rotated counterclockwise, but I am having trouble understanding why, conceptually.
The rotation is properly applied to the original axes $(x,y)$ counterclockwise and for this reason, setting the new axes in the original position, apparently the graph rotates clockwise, according to the following sketch