Question

1. copy this pentagon on grid paper.

write the coordinates of each vertex.
after each transformation:

• write the coordinates of the image

of each vertex.

• describe the positional change of the

vertices of the pentagon.
a) draw the image after a translation 3 units left
and 2 units up.
b) draw the image after a reflection in the x - axis.
c) draw the image after a reflection in the y - axis.

Explanation:

To solve this problem, we first identify the coordinates of the vertices of the pentagon from the grid. Let's assume the coordinates of the vertices are:

• \( P(3, 3) \)
• \( Q(-2, 2) \)
• \( R(1, 0) \)
• \( S(-1, -3) \)
• \( T(4, -4) \)

Part (a): Translation 3 units left and 2 units up

A translation 3 units left subtracts 3 from the \( x \)-coordinate, and 2 units up adds 2 to the \( y \)-coordinate.

Step 1: Apply translation to each vertex

• For \( P(3, 3) \): \( (3 - 3, 3 + 2) = (0, 5) \)
• For \( Q(-2, 2) \): \( (-2 - 3, 2 + 2) = (-5, 4) \)
• For \( R(1, 0) \): \( (1 - 3, 0 + 2) = (-2, 2) \)
• For \( S(-1, -3) \): \( (-1 - 3, -3 + 2) = (-4, -1) \)
• For \( T(4, -4) \): \( (4 - 3, -4 + 2) = (1, -2) \)

Positional Change:

Each vertex moves 3 units left (decrease \( x \)-coordinate by 3) and 2 units up (increase \( y \)-coordinate by 2).

Part (b): Reflection in the \( x \)-axis

A reflection in the \( x \)-axis changes \( (x, y) \) to \( (x, -y) \).

Step 1: Apply reflection to each vertex

• For \( P(3, 3) \): \( (3, -3) \)
• For \( Q(-2, 2) \): \( (-2, -2) \)
• For \( R(1, 0) \): \( (1, 0) \) (since \( y = 0 \), it remains the same)
• For \( S(-1, -3) \): \( (-1, 3) \)
• For \( T(4, -4) \): \( (4, 4) \)

Positional Change:

Each vertex is mirrored over the \( x \)-axis. The \( x \)-coordinate remains the same, and the \( y \)-coordinate is multiplied by \( -1 \).

Part (c): Reflection in the \( y \)-axis

A reflection in the \( y \)-axis changes \( (x, y) \) to \( (-x, y) \).

Step 1: Apply reflection to each vertex

• For \( P(3, 3) \): \( (-3, 3) \)
• For \( Q(-2, 2) \): \( (2, 2) \)
• For \( R(1, 0) \): \( (-1, 0) \)
• For \( S(-1, -3) \): \( (1, -3) \)
• For \( T(4, -4) \): \( (-4, -4) \)

Positional Change:

Each vertex is mirrored over the \( y \)-axis. The \( y \)-coordinate remains the same, and the \( x \)-coordinate is multiplied by \( -1 \).

Final Answers (Coordinates After Transformations)

(a) Translation:

• \( P'(0, 5) \), \( Q'(-5, 4) \), \( R'(-2, 2) \), \( S'(-4, -1) \), \( T'(1, -2) \)

(b) Reflection in \( x \)-axis:

• \( P'(3, -3) \), \( Q'(-2, -2) \), \( R'(1, 0) \), \( S'(-1, 3) \), \( T'(4, 4) \)

(c) Reflection in \( y \)-axis:

• \( P'(-3, 3) \), \( Q'(2, 2) \), \( R'(-1, 0) \), \( S'(1, -3) \), \( T'(-4, -4) \)

(Note: To draw the images, plot these new coordinates on grid paper and connect them in the same order as the original pentagon.)

Answer:

To solve this problem, we first identify the coordinates of the vertices of the pentagon from the grid. Let's assume the coordinates of the vertices are:

• \( P(3, 3) \)
• \( Q(-2, 2) \)
• \( R(1, 0) \)
• \( S(-1, -3) \)
• \( T(4, -4) \)

Part (a): Translation 3 units left and 2 units up

A translation 3 units left subtracts 3 from the \( x \)-coordinate, and 2 units up adds 2 to the \( y \)-coordinate.

Step 1: Apply translation to each vertex

• For \( P(3, 3) \): \( (3 - 3, 3 + 2) = (0, 5) \)
• For \( Q(-2, 2) \): \( (-2 - 3, 2 + 2) = (-5, 4) \)
• For \( R(1, 0) \): \( (1 - 3, 0 + 2) = (-2, 2) \)
• For \( S(-1, -3) \): \( (-1 - 3, -3 + 2) = (-4, -1) \)
• For \( T(4, -4) \): \( (4 - 3, -4 + 2) = (1, -2) \)

Positional Change:

Each vertex moves 3 units left (decrease \( x \)-coordinate by 3) and 2 units up (increase \( y \)-coordinate by 2).

Part (b): Reflection in the \( x \)-axis

A reflection in the \( x \)-axis changes \( (x, y) \) to \( (x, -y) \).

Step 1: Apply reflection to each vertex

• For \( P(3, 3) \): \( (3, -3) \)
• For \( Q(-2, 2) \): \( (-2, -2) \)
• For \( R(1, 0) \): \( (1, 0) \) (since \( y = 0 \), it remains the same)
• For \( S(-1, -3) \): \( (-1, 3) \)
• For \( T(4, -4) \): \( (4, 4) \)

Positional Change:

Each vertex is mirrored over the \( x \)-axis. The \( x \)-coordinate remains the same, and the \( y \)-coordinate is multiplied by \( -1 \).

Part (c): Reflection in the \( y \)-axis

A reflection in the \( y \)-axis changes \( (x, y) \) to \( (-x, y) \).

Step 1: Apply reflection to each vertex

• For \( P(3, 3) \): \( (-3, 3) \)
• For \( Q(-2, 2) \): \( (2, 2) \)
• For \( R(1, 0) \): \( (-1, 0) \)
• For \( S(-1, -3) \): \( (1, -3) \)
• For \( T(4, -4) \): \( (-4, -4) \)

Positional Change:

Each vertex is mirrored over the \( y \)-axis. The \( y \)-coordinate remains the same, and the \( x \)-coordinate is multiplied by \( -1 \).

Final Answers (Coordinates After Transformations)

(a) Translation:

• \( P'(0, 5) \), \( Q'(-5, 4) \), \( R'(-2, 2) \), \( S'(-4, -1) \), \( T'(1, -2) \)

(b) Reflection in \( x \)-axis:

• \( P'(3, -3) \), \( Q'(-2, -2) \), \( R'(1, 0) \), \( S'(-1, 3) \), \( T'(4, 4) \)

(c) Reflection in \( y \)-axis:

• \( P'(-3, 3) \), \( Q'(2, 2) \), \( R'(-1, 0) \), \( S'(1, -3) \), \( T'(-4, -4) \)

(Note: To draw the images, plot these new coordinates on grid paper and connect them in the same order as the original pentagon.)