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12-66.

We want to find a linear transformation that is equivalent to reflecting across the line $y = 2x$

1. Sketch the graph of $y = 2x$ and find the image of the point $\left(5, 0\right)$ when reflected across the line $y = 2x$.

Move the red point so that it is reflected.
Note, the segment needs to be perpendicular to the line.

2. Find a convenient point on the $y$-axis so that its image when reflected across the line will give integer values for the coordinate of the image. Find the image.

Move the red point at $\left(0, 5\right)$ so that it is a reflection.

3. Use the points and images you found in parts (a) and (b) to find the linear transformation for reflecting across the line $y = 2x$.

$( 5 , 0 ) \left[ \begin{array} { c c } { a } & { b } \\ { c } & { d } \end{array} \right] =$

$( 0 , 5 ) \left[ \begin{array} { c c } { a } & { b } \\ { c } & { d } \end{array} \right] =$

Use the eTool below to view the reflection across the line.
Click the link to the right for full version. 12-66 HW eTool