Question

(a) the arrows below show that the coordinates on the left are mapped to the coordinates on the right. fill in the blanks to give the coordinates after the rotation. original coordinates \\( \
ightarrow \\) final coordinates \\( x(-8,1) \
ightarrow x(\square, \square) \\) \\( y(-6,7) \
ightarrow y(\square, \square) \\) \\( z(-2,-3) \
ightarrow z(\square, \square) \\) (b) choose the general rule below that describes the rotation mapping \\( \triangle x y z \\) to \\( \triangle x y z \\). \\( (x, y) \
ightarrow(-x, y) \\) \\( (x, y) \
ightarrow(x,-y) \\) \\( (x, y) \
ightarrow(y, x) \\) \\( (x, y) \
ightarrow(-y, x) \\) \\( (x, y) \
ightarrow(-x,-y) \\) \\( (x, y) \
ightarrow(y,-x) \\) \\( (x, y) \
ightarrow(-y,-x) \\)

Explanation:

Step1: Determine the coordinates of \(X'\)

From the graph, \(X(-8,1)\) is rotated. By observing the position of \(X'\) on the coordinate - axis, when we rotate a point \((x,y)\) according to the rule \((x,y)\to(-y,x)\), for \(x = - 8\) and \(y = 1\), we substitute into the formula: \(x'=-y=-1\) and \(y'=x=-8\). So \(X'(-1,-8)\)

Step2: Determine the coordinates of \(Y'\)

For \(Y(-6,7)\), using the rule \((x,y)\to(-y,x)\). Substitute \(x=-6\) and \(y = 7\) into the formula: \(x'=-y=-7\) and \(y'=x=-6\). So \(Y'(-7,-6)\)

Step3: Determine the coordinates of \(Z'\)

For \(Z(-2,-3)\), using the rule \((x,y)\to(-y,x)\). Substitute \(x=-2\) and \(y=-3\) into the formula: \(x'=-y = 3\) and \(y'=x=-2\). So \(Z'(3,-2)\)

Step4: Verify the rotation rule

Let's check the general rule.
For a point \((x,y)\), if we apply the rule \((x,y)\to(-y,x)\)

• For \(X(-8,1)\): \((-8,1)\to(-1,-8)\)
• For \(Y(-6,7)\): \((-6,7)\to(-7,-6)\)
• For \(Z(-2,-3)\): \((-2,-3)\to(3,-2)\)

Answer:

a) \(X'(-1,-8)\), \(Y'(-7,-6)\), \(Z'(3,-2)\)
b) \((x,y)\to(-y,x)\)