Question

determine the coordinates of the preimage given the image and composition of transformations.

1. reflection in the x-axis and then a reflection in the y-axis
2. rotation 180° about origin and then a translation 3 units up

Explanation:

To solve these problems, we first need to identify the coordinates of the image vertices. Let's assume the image vertices (from the left graph) are:

• Vertex 1: \((-4, 1)\)
• Vertex 2: \((-2, 5)\)
• Vertex 3: \((4, 2)\)

Problem 7: Reflection in \(x\)-axis, then reflection in \(y\)-axis

To find the preimage, we reverse the transformations (apply inverse transformations in reverse order).

Step 1: Reverse the \(y\)-axis reflection (inverse of \(y\)-axis reflection: \( (x, y) \to (-x, y) \))

• Vertex 1: \( (-(-4), 1) = (4, 1) \)
• Vertex 2: \( (-(-2), 5) = (2, 5) \)
• Vertex 3: \( (-4, 2) \)

Step 2: Reverse the \(x\)-axis reflection (inverse of \(x\)-axis reflection: \( (x, y) \to (x, -y) \))

• Vertex 1: \( (4, -1) \)
• Vertex 2: \( (2, -5) \)
• Vertex 3: \( (-4, -2) \)

Problem 8: Rotation \(180^\circ\) about origin, then translation 3 units up

Reverse the transformations: first reverse the translation (subtract 3 from \(y\)-coordinate), then reverse the \(180^\circ\) rotation (inverse of \(180^\circ\) rotation: \( (x, y) \to (-x, -y) \)).

Step 1: Reverse the translation (subtract 3 from \(y\))

• Vertex 1: \( (-4, 1 - 3) = (-4, -2) \)
• Vertex 2: \( (-2, 5 - 3) = (-2, 2) \)
• Vertex 3: \( (4, 2 - 3) = (4, -1) \)

Step 2: Reverse the \(180^\circ\) rotation (\( (x, y) \to (-x, -y) \))

• Vertex 1: \( (-(-4), -(-2)) = (4, 2) \)
• Vertex 2: \( (-(-2), -2) = (2, -2) \)
• Vertex 3: \( (-4, -(-1)) = (-4, 1) \)

Final Answers

Problem 7 Preimage Coordinates:

\( (4, -1) \), \( (2, -5) \), \( (-4, -2) \)

Problem 8 Preimage Coordinates:

\( (4, 2) \), \( (2, -2) \), \( (-4, 1) \)

(Note: If the image coordinates differ, adjust the calculations using the actual image vertices from the graph.)

Answer:

To solve these problems, we first need to identify the coordinates of the image vertices. Let's assume the image vertices (from the left graph) are:

• Vertex 1: \((-4, 1)\)
• Vertex 2: \((-2, 5)\)
• Vertex 3: \((4, 2)\)

Problem 7: Reflection in \(x\)-axis, then reflection in \(y\)-axis

To find the preimage, we reverse the transformations (apply inverse transformations in reverse order).

Step 1: Reverse the \(y\)-axis reflection (inverse of \(y\)-axis reflection: \( (x, y) \to (-x, y) \))

• Vertex 1: \( (-(-4), 1) = (4, 1) \)
• Vertex 2: \( (-(-2), 5) = (2, 5) \)
• Vertex 3: \( (-4, 2) \)

Step 2: Reverse the \(x\)-axis reflection (inverse of \(x\)-axis reflection: \( (x, y) \to (x, -y) \))

• Vertex 1: \( (4, -1) \)
• Vertex 2: \( (2, -5) \)
• Vertex 3: \( (-4, -2) \)

Problem 8: Rotation \(180^\circ\) about origin, then translation 3 units up

Reverse the transformations: first reverse the translation (subtract 3 from \(y\)-coordinate), then reverse the \(180^\circ\) rotation (inverse of \(180^\circ\) rotation: \( (x, y) \to (-x, -y) \)).

Step 1: Reverse the translation (subtract 3 from \(y\))

• Vertex 1: \( (-4, 1 - 3) = (-4, -2) \)
• Vertex 2: \( (-2, 5 - 3) = (-2, 2) \)
• Vertex 3: \( (4, 2 - 3) = (4, -1) \)

Step 2: Reverse the \(180^\circ\) rotation (\( (x, y) \to (-x, -y) \))

• Vertex 1: \( (-(-4), -(-2)) = (4, 2) \)
• Vertex 2: \( (-(-2), -2) = (2, -2) \)
• Vertex 3: \( (-4, -(-1)) = (-4, 1) \)

Final Answers

Problem 7 Preimage Coordinates:

\( (4, -1) \), \( (2, -5) \), \( (-4, -2) \)

Problem 8 Preimage Coordinates:

\( (4, 2) \), \( (2, -2) \), \( (-4, 1) \)

(Note: If the image coordinates differ, adjust the calculations using the actual image vertices from the graph.)