Question

transformation summative activity chart
labeled point\trule:\tpre - image\ttranslation\treflection\trotation\tbonus: dilation
a\t\t(3, 14)\t(x + 6, y - 12)\t\t(-y, x)\tk =?
b\t\t(8, 19)\t\t\t\t
c\t\t(14, 14)\t\t\t\t
d\t\t(5, 14)\t\t\t\t
e\t\t(5, 10)\t\t\t\t
f\t\t(5, 6)\t\t\t\t
g\t\t(3, 7)\t\t\t\t
h\t\t(3, 10)\t\t\t\t
i\t\t(3, 12)\t\t\t\t
j\t\t(14, 12)\t\t\t\t

Explanation:

To solve the transformation for each pre - image point, we will apply the given transformation rules (translation: \((x + 6,y-12)\), rotation: \((-y,x)\)) one by one. We will take point A \((3,14)\) as an example to show the process. The same process can be repeated for other points.

Step 1: Apply the translation rule \((x + 6,y - 12)\) to point A \((3,14)\)

For the \(x\) - coordinate: We add 6 to the original \(x\) - coordinate. So, \(x_{new}=3 + 6=9\)
For the \(y\) - coordinate: We subtract 12 from the original \(y\) - coordinate. So, \(y_{new}=14-12 = 2\)
After translation, the point becomes \((9,2)\)

Step 2: Apply the rotation rule \((-y,x)\) to the translated point \((9,2)\)

Here, \(x = 9\) and \(y = 2\) from the translated point.
For the new \(x\) - coordinate: We use \(-y\), so \(x_{final}=-2\)
For the new \(y\) - coordinate: We use \(x\), so \(y_{final}=9\)
After rotation, the point becomes \((-2,9)\)

If we want to find the transformed points for all the given pre - image points:

For point B \((8,19)\)

• Translation: \(x=8 + 6=14\), \(y = 19-12=7\), translated point \((14,7)\)
• Rotation: \(x=-7\), \(y = 14\), rotated point \((-7,14)\)

For point C \((14,14)\)

• Translation: \(x=14 + 6=20\), \(y = 14-12 = 2\), translated point \((20,2)\)
• Rotation: \(x=-2\), \(y = 20\), rotated point \((-2,20)\)

For point D \((5,14)\)

• Translation: \(x=5 + 6=11\), \(y = 14-12=2\), translated point \((11,2)\)
• Rotation: \(x=-2\), \(y = 11\), rotated point \((-2,11)\)

For point E \((5,10)\)

• Translation: \(x=5 + 6 = 11\), \(y=10 - 12=-2\), translated point \((11,-2)\)
• Rotation: \(x = 2\), \(y=11\), rotated point \((2,11)\)

For point F \((5,6)\)

• Translation: \(x=5 + 6=11\), \(y=6-12=-6\), translated point \((11,-6)\)
• Rotation: \(x = 6\), \(y=11\), rotated point \((6,11)\)

For point G \((3,7)\)

• Translation: \(x=3 + 6=9\), \(y=7-12=-5\), translated point \((9,-5)\)
• Rotation: \(x = 5\), \(y=9\), rotated point \((5,9)\)

For point H \((3,10)\)

• Translation: \(x=3 + 6=9\), \(y=10-12=-2\), translated point \((9,-2)\)
• Rotation: \(x = 2\), \(y=9\), rotated point \((2,9)\)

For point I \((3,12)\)

• Translation: \(x=3 + 6=9\), \(y=12-12 = 0\), translated point \((9,0)\)
• Rotation: \(x=0\), \(y = 9\), rotated point \((0,9)\)

For point J \((14,12)\)

• Translation: \(x=14 + 6=20\), \(y=12-12 = 0\), translated point \((20,0)\)
• Rotation: \(x=0\), \(y = 20\), rotated point \((0,20)\)

If we assume the question is to find the transformed point of point A after both translation and rotation, the answer is \(\boldsymbol{(-2,9)}\) (for point A). If we consider all points, we can present the transformed points as above.

Answer:

To solve the transformation for each pre - image point, we will apply the given transformation rules (translation: \((x + 6,y-12)\), rotation: \((-y,x)\)) one by one. We will take point A \((3,14)\) as an example to show the process. The same process can be repeated for other points.

Step 1: Apply the translation rule \((x + 6,y - 12)\) to point A \((3,14)\)

For the \(x\) - coordinate: We add 6 to the original \(x\) - coordinate. So, \(x_{new}=3 + 6=9\)
For the \(y\) - coordinate: We subtract 12 from the original \(y\) - coordinate. So, \(y_{new}=14-12 = 2\)
After translation, the point becomes \((9,2)\)

Step 2: Apply the rotation rule \((-y,x)\) to the translated point \((9,2)\)

Here, \(x = 9\) and \(y = 2\) from the translated point.
For the new \(x\) - coordinate: We use \(-y\), so \(x_{final}=-2\)
For the new \(y\) - coordinate: We use \(x\), so \(y_{final}=9\)
After rotation, the point becomes \((-2,9)\)

If we want to find the transformed points for all the given pre - image points:

For point B \((8,19)\)

• Translation: \(x=8 + 6=14\), \(y = 19-12=7\), translated point \((14,7)\)
• Rotation: \(x=-7\), \(y = 14\), rotated point \((-7,14)\)

For point C \((14,14)\)

• Translation: \(x=14 + 6=20\), \(y = 14-12 = 2\), translated point \((20,2)\)
• Rotation: \(x=-2\), \(y = 20\), rotated point \((-2,20)\)

For point D \((5,14)\)

• Translation: \(x=5 + 6=11\), \(y = 14-12=2\), translated point \((11,2)\)
• Rotation: \(x=-2\), \(y = 11\), rotated point \((-2,11)\)

For point E \((5,10)\)

• Translation: \(x=5 + 6 = 11\), \(y=10 - 12=-2\), translated point \((11,-2)\)
• Rotation: \(x = 2\), \(y=11\), rotated point \((2,11)\)

For point F \((5,6)\)

• Translation: \(x=5 + 6=11\), \(y=6-12=-6\), translated point \((11,-6)\)
• Rotation: \(x = 6\), \(y=11\), rotated point \((6,11)\)

For point G \((3,7)\)

• Translation: \(x=3 + 6=9\), \(y=7-12=-5\), translated point \((9,-5)\)
• Rotation: \(x = 5\), \(y=9\), rotated point \((5,9)\)

For point H \((3,10)\)

• Translation: \(x=3 + 6=9\), \(y=10-12=-2\), translated point \((9,-2)\)
• Rotation: \(x = 2\), \(y=9\), rotated point \((2,9)\)

For point I \((3,12)\)

• Translation: \(x=3 + 6=9\), \(y=12-12 = 0\), translated point \((9,0)\)
• Rotation: \(x=0\), \(y = 9\), rotated point \((0,9)\)

For point J \((14,12)\)

• Translation: \(x=14 + 6=20\), \(y=12-12 = 0\), translated point \((20,0)\)
• Rotation: \(x=0\), \(y = 20\), rotated point \((0,20)\)

If we assume the question is to find the transformed point of point A after both translation and rotation, the answer is \(\boldsymbol{(-2,9)}\) (for point A). If we consider all points, we can present the transformed points as above.