Let's follow the transformations that happen to A, to get to A' and A''.
Point A is at (-5, -2)
It moves to (-5, 2) which is where A' is located. Note the x coordinate stays the same while the y coordinate flips from negative to positive. This must mean we applied a reflection over the x axis.
That rule in general is [tex](x, y) \rightarrow (x,-y)[/tex]
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Now compare A'(-5,2) and A''(1,4). We can shift A' 6 units to the right and then 2 units up so we move from A' to A''.
Algebraically this is stated as [tex](x,y) \rightarrow (x+6, y+2)[/tex]
Whatever the x coordinate is, add 6 to it. For the y coordinate, we add on 2.
Applying that rule to B'(-1,2) gets us to
[tex](-1,2) \rightarrow (-1\textbf{+6}, 2\textbf{+2}) = (5,4)[/tex]
which is the proper location of B''
The same applies to moving C' to C''
[tex]C' = (-5, 0) \rightarrow (-5\textbf{+6}, 0\textbf{+2}) = (1, 2) = C''[/tex]
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In summary, we started off by reflecting over the x axis. Then we applied the translation rule of "shift to the right 6 units, shift up 2 units".
In terms of algebra, combining the rules [tex](x,y) \rightarrow (x,-y)[/tex] and [tex](x,y) \rightarrow (x+6, y+2)[/tex] will have us end up with [tex](x,y) \rightarrow (x+6, -y+2)[/tex]
