Question

translate the figure as indicated. label the image using prime notation.

1. translation: 3 units right and 3 units down
2. translation: 4 units left and 1 unit up

Explanation:

Problem 7 (Translation: 3 units right and 3 units down)

Step 1: Identify Coordinates of Original Points

First, we determine the coordinates of the vertices of the original figure (let's assume the grid has integer coordinates). Let's denote the points:

• Let \( W = (-5, 4) \) (assuming from the grid), \( N = (-5, 2) \), \( Y = (-5, 0) \), \( C = (-2, 3) \) (these are approximate based on the grid; actual coordinates depend on the grid's origin, but the translation rule applies regardless).

Step 2: Apply Translation Rule

The translation rule for 3 units right (add 3 to the \( x \)-coordinate) and 3 units down (subtract 3 from the \( y \)-coordinate) is:
For a point \( (x, y) \), the image \( (x', y') = (x + 3, y - 3) \).

• For \( W(-5, 4) \): \( W' = (-5 + 3, 4 - 3) = (-2, 1) \)
• For \( N(-5, 2) \): \( N' = (-5 + 3, 2 - 3) = (-2, -1) \)
• For \( Y(-5, 0) \): \( Y' = (-5 + 3, 0 - 3) = (-2, -3) \)
• For \( C(-2, 3) \): \( C' = (-2 + 3, 3 - 3) = (1, 0) \)

Step 3: Plot and Label the Image

Plot the new points \( W'(-2, 1) \), \( N'(-2, -1) \), \( Y'(-2, -3) \), \( C'(1, 0) \) on the grid and connect them to form the translated figure, labeling each vertex with prime notation.

Problem 8 (Translation: 4 units left and 1 unit up)

Step 1: Identify Coordinates of Original Points

Let the vertices of the original figure be \( E(1, -1) \), \( S(1, -2) \), \( H(3, -1) \), \( P(5, -3) \) (approximate based on the grid).

Step 2: Apply Translation Rule

The translation rule for 4 units left (subtract 4 from the \( x \)-coordinate) and 1 unit up (add 1 to the \( y \)-coordinate) is:
For a point \( (x, y) \), the image \( (x', y') = (x - 4, y + 1) \).

• For \( E(1, -1) \): \( E' = (1 - 4, -1 + 1) = (-3, 0) \)
• For \( S(1, -2) \): \( S' = (1 - 4, -2 + 1) = (-3, -1) \)
• For \( H(3, -1) \): \( H' = (3 - 4, -1 + 1) = (-1, 0) \)
• For \( P(5, -3) \): \( P' = (5 - 4, -3 + 1) = (1, -2) \)

Step 3: Plot and Label the Image

Plot the new points \( E'(-3, 0) \), \( S'(-3, -1) \), \( H'(-1, 0) \), \( P'(1, -2) \) on the grid and connect them to form the translated figure, labeling each vertex with prime notation.

Final Answer (Summary of Translations)

For Problem 7, the translated figure has vertices \( W'(-2, 1) \), \( N'(-2, -1) \), \( Y'(-2, -3) \), \( C'(1, 0) \) (after translating 3 right, 3 down).
For Problem 8, the translated figure has vertices \( E'(-3, 0) \), \( S'(-3, -1) \), \( H'(-1, 0) \), \( P'(1, -2) \) (after translating 4 left, 1 up).

(Note: Actual coordinates may vary slightly based on the exact grid origin, but the translation process follows the same rule.)

Answer:

Problem 7 (Translation: 3 units right and 3 units down)

Step 1: Identify Coordinates of Original Points

First, we determine the coordinates of the vertices of the original figure (let's assume the grid has integer coordinates). Let's denote the points:

• Let \( W = (-5, 4) \) (assuming from the grid), \( N = (-5, 2) \), \( Y = (-5, 0) \), \( C = (-2, 3) \) (these are approximate based on the grid; actual coordinates depend on the grid's origin, but the translation rule applies regardless).

Step 2: Apply Translation Rule

The translation rule for 3 units right (add 3 to the \( x \)-coordinate) and 3 units down (subtract 3 from the \( y \)-coordinate) is:
For a point \( (x, y) \), the image \( (x', y') = (x + 3, y - 3) \).

• For \( W(-5, 4) \): \( W' = (-5 + 3, 4 - 3) = (-2, 1) \)
• For \( N(-5, 2) \): \( N' = (-5 + 3, 2 - 3) = (-2, -1) \)
• For \( Y(-5, 0) \): \( Y' = (-5 + 3, 0 - 3) = (-2, -3) \)
• For \( C(-2, 3) \): \( C' = (-2 + 3, 3 - 3) = (1, 0) \)

Step 3: Plot and Label the Image

Plot the new points \( W'(-2, 1) \), \( N'(-2, -1) \), \( Y'(-2, -3) \), \( C'(1, 0) \) on the grid and connect them to form the translated figure, labeling each vertex with prime notation.

Problem 8 (Translation: 4 units left and 1 unit up)

Step 1: Identify Coordinates of Original Points

Let the vertices of the original figure be \( E(1, -1) \), \( S(1, -2) \), \( H(3, -1) \), \( P(5, -3) \) (approximate based on the grid).

Step 2: Apply Translation Rule

The translation rule for 4 units left (subtract 4 from the \( x \)-coordinate) and 1 unit up (add 1 to the \( y \)-coordinate) is:
For a point \( (x, y) \), the image \( (x', y') = (x - 4, y + 1) \).

• For \( E(1, -1) \): \( E' = (1 - 4, -1 + 1) = (-3, 0) \)
• For \( S(1, -2) \): \( S' = (1 - 4, -2 + 1) = (-3, -1) \)
• For \( H(3, -1) \): \( H' = (3 - 4, -1 + 1) = (-1, 0) \)
• For \( P(5, -3) \): \( P' = (5 - 4, -3 + 1) = (1, -2) \)

Step 3: Plot and Label the Image

Plot the new points \( E'(-3, 0) \), \( S'(-3, -1) \), \( H'(-1, 0) \), \( P'(1, -2) \) on the grid and connect them to form the translated figure, labeling each vertex with prime notation.

Final Answer (Summary of Translations)

For Problem 7, the translated figure has vertices \( W'(-2, 1) \), \( N'(-2, -1) \), \( Y'(-2, -3) \), \( C'(1, 0) \) (after translating 3 right, 3 down).
For Problem 8, the translated figure has vertices \( E'(-3, 0) \), \( S'(-3, -1) \), \( H'(-1, 0) \), \( P'(1, -2) \) (after translating 4 left, 1 up).

(Note: Actual coordinates may vary slightly based on the exact grid origin, but the translation process follows the same rule.)