Question

rotations
graph the image of the figure using the transformation given.

1. rotation 180° about the origin
2. rotation 180° about the origin
3. rotation 90° counterclockwise about the origin
4. rotation 90° counterclockwise about the origin
5. rotation 270° counterclockwise about the origin
6. rotation 180° about the origin

Explanation:

Step1: Recall rotation rules

For a \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\), for a \(90^{\circ}\) counter - clockwise rotation \((x,y)\to(-y,x)\), and for a \(270^{\circ}\) counter - clockwise rotation \((x,y)\to(y, - x)\).

Step2: Identify original vertices

Determine the coordinates of each vertex of the given figure on the coordinate grid.

Step3: Apply rotation rules

Use the appropriate rotation rule to find the new coordinates of each vertex.

Step4: Plot new vertices

Plot the new - found coordinates on the same coordinate grid.

Step5: Connect new vertices

Connect the plotted new vertices to form the rotated figure.

Answer:

Since this is a problem about graph - ing the image of a figure after rotation transformation, the general steps for each rotation are as follows (taking a point \((x,y)\) in the original figure as an example):

1. For a \(180^{\circ}\) rotation about the origin:

• The transformation rule is \((x,y)\to(-x,-y)\). Identify the coordinates of each vertex of the original figure. For example, if a vertex of the original figure has coordinates \((x_1,y_1)\), its new coordinates after a \(180^{\circ}\) rotation about the origin will be \((-x_1,-y_1)\). Plot these new - coordinates and connect the points to form the rotated figure.

1. For a \(90^{\circ}\) counter - clockwise rotation about the origin:

• The transformation rule is \((x,y)\to(-y,x)\). Identify the coordinates of each vertex of the original figure. If a vertex has coordinates \((x_2,y_2)\), its new coordinates after a \(90^{\circ}\) counter - clockwise rotation about the origin will be \((-y_2,x_2)\). Plot these new coordinates and connect the points to form the rotated figure.

1. For a \(270^{\circ}\) counter - clockwise rotation about the origin:

• The transformation rule is \((x,y)\to(y, - x)\). Identify the coordinates of each vertex of the original figure. If a vertex has coordinates \((x_3,y_3)\), its new coordinates after a \(270^{\circ}\) counter - clockwise rotation about the origin will be \((y_3,-x_3)\). Plot these new coordinates and connect the points to form the rotated figure.