Transformations of straight lines and other graphs
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Find the equation of the image of the graph of \(y=2x+3\) under a reflection in the \(y\)-axis followed by a dilation of factor \(2\) from the \(x\)-axis.
$$\begin{aligned}&\text{A reflection in the y-axis} \rightarrow\left[\begin{array}{cc}-1 & 0 \\ 0 & 1\end{array}\right]\\
&\text{A dilation 2 from the a-axis} \rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right]\\
&\text{Combining}\ \rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right]\left[\begin{array}{ll}-1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}-1 & 0 \\ 0 & 2\end{array}\right]\\
&\begin{array}{l}
{\left[\begin{array}{l}
x^{\prime} \\
y^{\prime}
\end{array}\right]=\left[\begin{array}{rr}
-1 & 0 \\
0 & 2
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]=\left[\begin{array}{c}
-x \\
2 y
\end{array}\right]} \\
x^{\prime}=-x \rightarrow x=-x^{\prime} \\
y^{\prime}=2 y \rightarrow y=\frac{1}{2} y^{\prime} \\
y=2 x+3 \rightarrow \frac{1}{2} y^{\prime}=-2 x^{\prime}+3 \\
\therefore y=-4 x+6
\end{array}\end{aligned}
$$
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