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given that $\rho:\mathbb{R}^2\to\mathbb{R}^2$ be the composition of $\rho=R\circ \sigma_A$, where $R$ is the reflection in the x axis and $\sigma_A$ is the rotation around the point (0,b) through $\pi$

could anyone help me to derive formula for $\rho$?

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    Would this composition be a rotation or a reflection, to begin with? – 2017-02-27

1 Answers 1

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When you rotate a point $P$ by angle $\pi$ around the centre $C=(0,b)$, the image $P'$ will be in the line joining $P$ and $C$, with $C$ as the mid-point of the line segment $PP'$. Using this fact for any $P=(x,y)$ you can write the formula for $P'=\sigma_A(P)$. And the function $R$ is easy, it simply changes the sign of $y$ co-ordinate keeping $x$ co-ordinate unchanged.

Now compose these two functions in the desired order.