Question

(x, y) → ( , )
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Explanation:

1. First, identify key - points and their coordinates:

• Let's assume the original points of the blue - colored figure are \(Y(-9,0)\), \(X(-4,0)\), \(W(-6, - 6)\). The corresponding points of the green - colored figure are \(Y'(0,5)\), \(X'(5,5)\), \(W'(1,0)\).
• To find the translation rule \((x,y)\to(x + a,y + b)\), we can use one of the points. Let's use point \(Y\) and \(Y'\).
• For the \(x\) - coordinate of the translation:
• The \(x\) - coordinate of \(Y\) is \(-9\) and the \(x\) - coordinate of \(Y'\) is \(0\). We set up the equation \(-9+a = 0\), and solve for \(a\).
• By adding \(9\) to both sides of the equation \(-9+a = 0\), we get \(a=9\).
• For the \(y\) - coordinate of the translation:
• The \(y\) - coordinate of \(Y\) is \(0\) and the \(y\) - coordinate of \(Y'\) is \(5\). We set up the equation \(0 + b=5\), and solve for \(b\).
• We find that \(b = 5\).

1. The translation rule is \((x,y)\to(x + 9,y + 5)\).

Step1: Analyze \(x\) - coordinate change

Find the difference in \(x\) - coordinates of a corresponding pair of points. For \(Y(-9,0)\) and \(Y'(0,5)\), we solve \(-9+a = 0\) for \(a\).
\(a=9\)

Step2: Analyze \(y\) - coordinate change

Find the difference in \(y\) - coordinates of a corresponding pair of points. For \(Y(-9,0)\) and \(Y'(0,5)\), we solve \(0 + b=5\) for \(b\).
\(b = 5\)

Answer:

\((x,y)\to(x + 9,y + 5)\)