TITLE: Rotations/Reflections of a point
QUESTION [1 upvotes]: It's been $10$ years since I see some kind of geometry and I'm preparing for a test of these sort of questions. I need help figuring out the following problems:
1) A point $P(x,y)$ is rotated $180$ degrees about the origin, then reflected over the y-axis. What is the resulting image of $P$? 
My conjecture: $(-x,y)$ but I don't quite agree with it graphically. I'm having trouble with what exactly a 180 degree rotation about the origin simply is. 
2) A point $P(x,y)$ is reflected over the y-axis and then rotated 180 degrees about the origin. What is the resulting image? 
Is this the same thing as 1? 
3) A point $A(x,y)$ is reflected over the line $y=x$ and then reflected over the y-axis. What is the resulting image of A? 
My conjecture: $(-y,x)$
4) A point $A(x,y)$ is reflected over the lines $y=-x$ and then reflected over the y-axis. What is the resulting image of A? 
My conjecture: $(y,-x)$

REPLY [2 votes]: If a point $P(a,b)$ is rotated $180$ degree about the origin, then the resulting image of $P$  is $(−a,−b)$. 
1) $(x,y)\rightarrow (-x,-y)\rightarrow (x,-y).$
2) $(x,y)\rightarrow (-x,y)\rightarrow (x,-y).$
Hence, this is the same as 1.
3) $(x,y)\rightarrow (y,x)\rightarrow (-y,x).$
4) $(x,y)\rightarrow (-y,-x)\rightarrow (y,-x).$
Hence, your conjectures are correct.
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