Question

rotations
graph the image of the figure using the transfor

1. rotation 180° about the origin
2. rotation 90° counterclockwise about the origin

Explanation:

Step1: Recall rotation rules

For a 180 - degree rotation about the origin, if a point $(x,y)$ is rotated, the new point is $(-x,-y)$. For a 90 - degree counter - clockwise rotation about the origin, if a point $(x,y)$ is rotated, the new point is $(-y,x)$.

Step2: For 180 - degree rotation of first figure

Let's assume the coordinates of points in the first figure are $(x_1,y_1),(x_2,y_2),\cdots$. The new coordinates after 180 - degree rotation about the origin will be $(-x_1,-y_1),(-x_2,-y_2),\cdots$. Then plot these new points to get the rotated figure.

Step3: For 90 - degree counter - clockwise rotation of second figure

Let the coordinates of points in the second figure be $(x_3,y_3),(x_4,y_4),\cdots$. The new coordinates after 90 - degree counter - clockwise rotation about the origin will be $(-y_3,x_3),(-y_4,x_4),\cdots$. Then plot these new points to get the rotated figure.

Since this is a graphical task, we cannot provide a numerical answer. The general method to graph the rotated figures is as described above. To actually graph them, one would need to identify the coordinates of the vertices of the original figures from the graph paper, apply the rotation rules, and then plot the new points and connect them to form the rotated figures.

Answer:

Step1: Recall rotation rules

For a 180 - degree rotation about the origin, if a point $(x,y)$ is rotated, the new point is $(-x,-y)$. For a 90 - degree counter - clockwise rotation about the origin, if a point $(x,y)$ is rotated, the new point is $(-y,x)$.

Step2: For 180 - degree rotation of first figure

Let's assume the coordinates of points in the first figure are $(x_1,y_1),(x_2,y_2),\cdots$. The new coordinates after 180 - degree rotation about the origin will be $(-x_1,-y_1),(-x_2,-y_2),\cdots$. Then plot these new points to get the rotated figure.

Step3: For 90 - degree counter - clockwise rotation of second figure

Let the coordinates of points in the second figure be $(x_3,y_3),(x_4,y_4),\cdots$. The new coordinates after 90 - degree counter - clockwise rotation about the origin will be $(-y_3,x_3),(-y_4,x_4),\cdots$. Then plot these new points to get the rotated figure.

Since this is a graphical task, we cannot provide a numerical answer. The general method to graph the rotated figures is as described above. To actually graph them, one would need to identify the coordinates of the vertices of the original figures from the graph paper, apply the rotation rules, and then plot the new points and connect them to form the rotated figures.