Answer for the first box is -x
Answer for the second box is y-8
The rule is [tex](x,y) \to (-x, y-8)[/tex]
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Explanation:
Look at the bottom right corner of figure 1 given in your diagram. This point's location is at (-1,1). We'll track this point as we apply the two transformations.
First we reflect over the vertical y axis. This takes (-1,1) and moves it to (1,1). Note how the x coordinate sign flips from negative to positive. The y coordinate stays the same. The reflection rule is [tex](x,y) \to (-x,y)[/tex]
Next we shift the reflected figure 8 units down so that (-1,1) moves to (-1,-7). Whatever the y coordinate is, subtract off 8 to go from y to y-8. Overall, the translation rule is [tex](x,y) \to (x,y-8)[/tex]. The x coordinate stays the same.
Combining the two rules (reflection + translation) leads to the overall rule of [tex](x,y) \to (-x,y-8)[/tex]
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Let's use that rule to see if (-1, 1) moves to (1, -7)
[tex](x,y) \to (-x,y-8)\\\\(-1,1) \to (-(-1),1-8)\\\\(-1,1) \to (1,-7)\\\\[/tex]
which works out. The other points on figure 1 will follow the same pattern as the point (-1,1) does.
All of this only applies if we want to go from figure 1 to figure 3.
