**Justin, a 10 grader, studies reflections with matrices**

(see Don's book "Changing Shapes With Matrices")

Justin worked on different transformations with matrices. He found the following 2x2 matrix
that would take the ** original dog (blue)** into its
**image**, the mirror being the vertical y-axis.

The big breakthrough came when Justin asked "What matrix
would reflect the dog, if the mirror (the **purple**
line) was at x=3.5??" !!! Don had never done this before, so this was very
exciting- he was going to learn something new.

Justin and Don worked on this separately and together. Don looked at the x coordinates and found that Newx = 7- Oldx or 2*3.5 - Oldx . They both realized that the Newy=Oldy. Then it was a matter of multiplying x by something to get 7-x. They figured this out to be 7/x - 1 or 2*3.5/x - 1. This was the number that goes in the first row and first column of the transformation matrix. The final result was:

Justin checked various vertical lines of symmetry and generalizing, where the mirror or line of symmetry is at x=a, wrote:

Justin then decided to do reflections with a horizontal line of symmetry at y=a.. He knew the 2x2 transformation matrix that reflects the dog in the x-axis is Then he found the 2x2 transformation matrix that reflects the dog in the mirror or line of symmetry at x=a is:

fine work Justin!

Justin then decided to do a reflection in the line y=x ! He's working on this.

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