Transformations of Graphs with Matrices
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Given the transformation matrix \(\left[ {\begin{array}{*{20}{c}}2&0\\0&{ - 2}\end{array}} \right]\), find the image of \(y = {x^2} + 2.\)
\begin{align}
&\begin{bmatrix}
2 & 0 \\
0 & -2
\end{bmatrix}\begin{bmatrix}
x \\
y
\end{bmatrix}=\begin{bmatrix}
x^{\prime} \\
y^{\prime}
\end{bmatrix} \\
&x^{\prime}= 2x \rightarrow x=\frac{1}{2} x^{\prime} \\
&y^{\prime}=-2y \quad y=-\frac{1}{2} y^{\prime}\\
&\begin{aligned}
y=x^{2}+2\rightarrow
-\frac{1}{2} y^{\prime}&=\left(\frac{1}{2} x^{\prime}\right)^{2}+2 \\
-y^{\prime}&=\frac{1}{2}\left(x^{\prime}\right)^{2}+4 \\
y^{\prime}&=-\frac{1}{2}\left(x^{\prime}\right)^{2}-4 \\
\therefore y&=-\frac{1}{2} x^{2}-4
\end{aligned}
\end{align}
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