Question

1. translate 2 units right and 4 units down. rotate 180° around the origin. 4. reflect across the x - axis. rotate 90° counterclockwise around the origin.

Explanation:

Step 1: Analyze translation for the second - shape

For the shape with vertices \(H, I, J, K, L\), the translation rule is \((x,y)\to(x + 2,y-4)\). Let's assume a vertex \((x_1,y_1)\) of the original shape. After translation, its new coordinates \((x_2,y_2)\) are given by \(x_2=x_1 + 2\) and \(y_2=y_1-4\).

Step 2: Analyze \(180^{\circ}\) rotation for the second - shape

The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\). If a vertex has coordinates \((x_3,y_3)\) after translation, its coordinates \((x_4,y_4)\) after \(180^{\circ}\) rotation are \(x_4=-x_3\) and \(y_4=-y_3\).

Step 3: Analyze reflection for the first - shape

The rule for reflecting a point \((x,y)\) across the \(x\) - axis is \((x,y)\to(x, - y)\). Let the vertices of the original shape be \((x_5,y_5)\), and after reflection, the vertices are \((x_6,y_6)\) where \(x_6 = x_5\) and \(y_6=-y_5\).

Step 4: Analyze \(90^{\circ}\) counter - clockwise rotation for the first - shape

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\). If a vertex has coordinates \((x_7,y_7)\) after reflection, its coordinates \((x_8,y_8)\) after \(90^{\circ}\) counter - clockwise rotation are \(x_8=-y_7\) and \(y_8=x_7\).

(Note: Since no specific coordinates of the vertices of the shapes are given, we have provided the general transformation rules. To fully solve, we would need to identify the vertices of the original shapes, apply the translation/rotation/reflection rules one - by - one to each vertex, and then graph the new shapes.)

Since we can't actually draw the final shapes here, you would follow these steps for each vertex of the given shapes on a graph paper. For example, if a vertex of the first shape is \((3,2)\):

1. After reflection across the \(x\) - axis, it becomes \((3,-2)\).
2. After \(90^{\circ}\) counter - clockwise rotation about the origin, it becomes \((2,3)\).

For a vertex of the second shape, say \((- 2,-3)\):

1. After translation (2 units right and 4 units down), it becomes \((-2 + 2,-3-4)=(0,-7)\).
2. After \(180^{\circ}\) rotation about the origin, it becomes \((0,7)\).

No specific numerical answer can be provided without the initial vertex coordinates of the shapes in the graph, but the above steps show how to perform the geometric transformations.

Answer:

Step 1: Analyze translation for the second - shape

For the shape with vertices \(H, I, J, K, L\), the translation rule is \((x,y)\to(x + 2,y-4)\). Let's assume a vertex \((x_1,y_1)\) of the original shape. After translation, its new coordinates \((x_2,y_2)\) are given by \(x_2=x_1 + 2\) and \(y_2=y_1-4\).

Step 2: Analyze \(180^{\circ}\) rotation for the second - shape

The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\). If a vertex has coordinates \((x_3,y_3)\) after translation, its coordinates \((x_4,y_4)\) after \(180^{\circ}\) rotation are \(x_4=-x_3\) and \(y_4=-y_3\).

Step 3: Analyze reflection for the first - shape

The rule for reflecting a point \((x,y)\) across the \(x\) - axis is \((x,y)\to(x, - y)\). Let the vertices of the original shape be \((x_5,y_5)\), and after reflection, the vertices are \((x_6,y_6)\) where \(x_6 = x_5\) and \(y_6=-y_5\).

Step 4: Analyze \(90^{\circ}\) counter - clockwise rotation for the first - shape

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\). If a vertex has coordinates \((x_7,y_7)\) after reflection, its coordinates \((x_8,y_8)\) after \(90^{\circ}\) counter - clockwise rotation are \(x_8=-y_7\) and \(y_8=x_7\).

(Note: Since no specific coordinates of the vertices of the shapes are given, we have provided the general transformation rules. To fully solve, we would need to identify the vertices of the original shapes, apply the translation/rotation/reflection rules one - by - one to each vertex, and then graph the new shapes.)

Since we can't actually draw the final shapes here, you would follow these steps for each vertex of the given shapes on a graph paper. For example, if a vertex of the first shape is \((3,2)\):

1. After reflection across the \(x\) - axis, it becomes \((3,-2)\).
2. After \(90^{\circ}\) counter - clockwise rotation about the origin, it becomes \((2,3)\).

For a vertex of the second shape, say \((- 2,-3)\):

1. After translation (2 units right and 4 units down), it becomes \((-2 + 2,-3-4)=(0,-7)\).
2. After \(180^{\circ}\) rotation about the origin, it becomes \((0,7)\).

No specific numerical answer can be provided without the initial vertex coordinates of the shapes in the graph, but the above steps show how to perform the geometric transformations.