Question

1. what transformation do you get by rotating a shape 90° ccw around the origin, then reflecting it over the x - axis?rotate, then translate.reflect, then translate.translate, then rotate.reflect, then rotate.

Explanation:

Step1: Analyze the transformation rules

• Rotation: Changes the orientation of a shape.
• Reflection: Flips a shape over a line (in this case, the \(x -\)axis).
• Translation: Slides a shape without changing its orientation or size.

Let's consider the order of operations:

• If we first rotate a shape \(90^{\circ}\) counter - clockwise (\(ccw\)) around the origin, and then reflect it over the \(x -\)axis.
• Let the original point be \((x,y)\). After a \(90^{\circ}\ ccw\) rotation around the origin, the point becomes \((-y,x)\). After reflecting \((-y,x)\) over the \(x -\)axis (the transformation \((a,b)\to(a, - b)\)), the point becomes \((-y,-x)\).
• If we first reflect a shape over the \(x -\)axis ( \((x,y)\to(x, - y)\)) and then rotate it \(90^{\circ}\ ccw\) around the origin (using the rotation rule \((a,b)\to(-b,a)\)), the point \((x,-y)\) becomes \((y,x)\) which is not the same as the result of rotate - then - reflect.
• If we first translate and then rotate, the translation (a non - orientation - changing, non - flipping operation) followed by a rotation will not be equivalent to rotate - then - reflect.
• If we first reflect and then translate, the reflection (flipping) followed by a translation (sliding) will not be equivalent to rotate - then - reflect.

Answer:

Rotate, then translate.