Question

complete this table, and determine the rule for this transformation.
type your answers in the boxes.

input	output
(4, 5)	(5, 4)
(?, 4)	(4, 0)
(1, 5)	(?, ?)
(?, ?)	(-1, -2)
(x, y)	(?, ?)

determine the rule for this transformation.
type your answer in the box.

(x, y) → (?, ?)

Explanation:

Step1: Analyze first pair

Input \((2, -3)\), Output \((-3, 2)\). Notice \(x = 2\) becomes \(y=-3\) (output \(x\)), \(y = -3\) becomes \(x = 2\) (output \(y\))? Wait, no: output \(x\) is original \(y\) (but sign? Wait \(2\) to \(-3\)? Wait no, wait \((2,-3)\) to \((-3,2)\): original \(x = 2\), output \(y = 2\); original \(y=-3\), output \(x=-3\). Wait, swap \(x\) and \(y\) and... Wait, no: \((x,y)\) to \((y, x)\)? Wait \((2,-3)\) would be \((-3,2)\) – yes! Because \(x=2\), \(y=-3\); swap to \((y,x)=(-3,2)\). Check next pair: \((4,5)\) to \((5,4)\)? Wait the output is \((5,4)\)? Wait the image shows output \((5,4)\) for input \((4,5)\). Yes! So \((4,5)\) swapped \(x\) and \(y\) gives \((5,4)\). Third pair: input \((?, 4)\) output \((4, 0)\). So if rule is \((x,y)\to(y,x)\), then input \(y = 4\) becomes output \(x = 4\), input \(x\) becomes output \(y = 0\). So input \(x\) must be \(0\), so input is \((0,4)\). Fourth pair: input \((1,5)\), apply rule \((y,x)=(5,1)\). Fifth pair: output \((-1,-7)\), so input is \((-7,-1)\) (since \((x,y)\to(y,x)\), so \(y=-1\), \(x=-7\) gives output \((-1,-7)\)). The rule is swapping \(x\) and \(y\) coordinates, so \((x,y)\to(y,x)\).

Step2: Verify with second pair

Input \((4,5)\), output \((5,4)\) – which is \((y,x)\) (since \(x=4\), \(y=5\); swap to \((5,4)\)). Correct. Third pair: input \((0,4)\) (since output \((4,0)\) is \((y,x)\) where \(y=4\), \(x=0\)). Fourth pair: input \((1,5)\) gives output \((5,1)\). Fifth pair: output \((-1,-7)\) means input is \((-7,-1)\) (since \((y,x)=(-1,-7)\) implies \(y=-1\), \(x=-7\)).

Step3: Determine the rule

From the pairs, the transformation rule is swapping the \(x\)- and \(y\)-coordinates, so \((x, y) \to (y, x)\).

Answer:

\((y, x)\)