Let $S$ and $T$ denote transformations of the $x y$-plane $$S ( x , y ) = ( x + 1 , y ) , \quad T ( x , y ) = ( - y , x )$$ We will write, for example, $T S$ to denote the composition of applying $S$ then $T$, that is $$T S ( x , y ) = T ( S ( x , y ) )$$ and write $T ^ { n }$ to denote $n$ applications of $T$ where $n$ is a positive integer. (i) Show that $T S ( x , y ) \neq S T ( x , y )$. (ii) For what values of $n$ is it the case that $T ^ { n } ( x , y ) = ( x , y )$ for all $x , y$ ? (iii) Show that applications of $S$ and $T$ in some order can produce the transformation $$U ( x , y ) = ( x - 1 , y )$$ What is the least number of applications (of $S$ and $T$ in total) that can produce $U$ ? Justify your answer. (iv) Show that for any integers $a$ and $b$ there is some sequence of applications of $S$ and $T$ that maps $( 0,0 )$ to $( a , b )$. (v) The parabola $C$ has equation $y = x ^ { 2 } + 2 x + 2$. What is the equation of the curve obtained by applying $S$ to $C$ ? What is the equation of the curve obtained by applying $T$ to $C$ ?
x & = \frac { 1 } { 2 x ^ { 1 / 2 } }
\section*{2. For ALL APPLICANTS.}
Let $S$ and $T$ denote transformations of the $x y$-plane
$$S ( x , y ) = ( x + 1 , y ) , \quad T ( x , y ) = ( - y , x )$$
We will write, for example, $T S$ to denote the composition of applying $S$ then $T$, that is
$$T S ( x , y ) = T ( S ( x , y ) )$$
and write $T ^ { n }$ to denote $n$ applications of $T$ where $n$ is a positive integer.\\
(i) Show that $T S ( x , y ) \neq S T ( x , y )$.\\
(ii) For what values of $n$ is it the case that $T ^ { n } ( x , y ) = ( x , y )$ for all $x , y$ ?\\
(iii) Show that applications of $S$ and $T$ in some order can produce the transformation
$$U ( x , y ) = ( x - 1 , y )$$
What is the least number of applications (of $S$ and $T$ in total) that can produce $U$ ? Justify your answer.\\
(iv) Show that for any integers $a$ and $b$ there is some sequence of applications of $S$ and $T$ that maps $( 0,0 )$ to $( a , b )$.\\
(v) The parabola $C$ has equation $y = x ^ { 2 } + 2 x + 2$.
What is the equation of the curve obtained by applying $S$ to $C$ ?\\
What is the equation of the curve obtained by applying $T$ to $C$ ?