Question

graph the reflected image or determine the points of the reflected image.

1. reflection across ( y = x )
2. reflection across the x - axis
3. reflection across ( y=-x )
4. reflection across the y - axis

Explanation:

To solve these reflection problems, we use the rules for reflecting points over different lines:

5) Reflection across \( y = x \)

The rule for reflecting a point \((x, y)\) across \( y = x \) is \((x, y)
ightarrow (y, x)\).

• Identify the coordinates of the original points from the graph.
• Apply the rule to each point to find the reflected points.
• Plot the reflected points to graph the image.

6) Reflection across the \( x \)-axis

The rule for reflecting a point \((x, y)\) across the \( x \)-axis is \((x, y)
ightarrow (x, -y)\).

• Identify the coordinates of the original triangle’s vertices.
• For each vertex \((x, y)\), compute \((x, -y)\).
• Plot these new points to graph the reflected triangle.

7) Reflection across \( y = -x \)

The rule for reflecting a point \((x, y)\) across \( y = -x \) is \((x, y)
ightarrow (-y, -x)\).

• Find the coordinates of points \( R \), \( S \), and \( T \) from the graph.
• For each point \((x, y)\), apply \((-y, -x)\) to get the reflected coordinates.
• Plot the reflected points to graph the image.

8) Reflection across the \( y \)-axis

The rule for reflecting a point \((x, y)\) across the \( y \)-axis is \((x, y)
ightarrow (-x, y)\).

• Identify the coordinates of points \( B \), \( C \), and \( D \) from the graph.
• For each vertex \((x, y)\), compute \((-x, y)\).
• Plot these new points to graph the reflected figure.

Example (for a typical triangle with vertices):

Suppose a triangle has vertices \( A(2, 3) \), \( B(4, 1) \), \( C(1, 2) \):

• Reflection across \( y = x \): \( A(3, 2) \), \( B(1, 4) \), \( C(2, 1) \).
• Reflection across \( x \)-axis: \( A(2, -3) \), \( B(4, -1) \), \( C(1, -2) \).
• Reflection across \( y = -x \): \( A(-3, -2) \), \( B(-1, -4) \), \( C(-2, -1) \).
• Reflection across \( y \)-axis: \( A(-2, 3) \), \( B(-4, 1) \), \( C(-1, 2) \).

To graph the reflected images, follow these steps for each problem:

1. Identify original coordinates.
2. Apply the reflection rule.
3. Plot the new points.

For specific coordinates (e.g., from the graph):

• Let’s assume the triangle in problem 6 has vertices \( A(-5, 3) \), \( B(-6, 1) \), \( C(-3, 2) \).
• Reflection across \( x \)-axis: \( A(-5, -3) \), \( B(-6, -1) \), \( C(-3, -2) \).

Final Answer (General Approach):

For each problem, apply the reflection rule to the original points and plot the results. The key is to use the correct transformation rule for each line of reflection.

• 5) \( y = x \): Swap \( x \) and \( y \) coordinates.
• 6) \( x \)-axis: Negate the \( y \)-coordinate.
• 7) \( y = -x \): Negate both \( x \) and \( y \), then swap.
• 8) \( y \)-axis: Negate the \( x \)-coordinate.

Graph the reflected points using these rules.

Answer:

To solve these reflection problems, we use the rules for reflecting points over different lines:

5) Reflection across \( y = x \)

The rule for reflecting a point \((x, y)\) across \( y = x \) is \((x, y)
ightarrow (y, x)\).

• Identify the coordinates of the original points from the graph.
• Apply the rule to each point to find the reflected points.
• Plot the reflected points to graph the image.

6) Reflection across the \( x \)-axis

The rule for reflecting a point \((x, y)\) across the \( x \)-axis is \((x, y)
ightarrow (x, -y)\).

• Identify the coordinates of the original triangle’s vertices.
• For each vertex \((x, y)\), compute \((x, -y)\).
• Plot these new points to graph the reflected triangle.

7) Reflection across \( y = -x \)

The rule for reflecting a point \((x, y)\) across \( y = -x \) is \((x, y)
ightarrow (-y, -x)\).

• Find the coordinates of points \( R \), \( S \), and \( T \) from the graph.
• For each point \((x, y)\), apply \((-y, -x)\) to get the reflected coordinates.
• Plot the reflected points to graph the image.

8) Reflection across the \( y \)-axis

The rule for reflecting a point \((x, y)\) across the \( y \)-axis is \((x, y)
ightarrow (-x, y)\).

• Identify the coordinates of points \( B \), \( C \), and \( D \) from the graph.
• For each vertex \((x, y)\), compute \((-x, y)\).
• Plot these new points to graph the reflected figure.

Example (for a typical triangle with vertices):

Suppose a triangle has vertices \( A(2, 3) \), \( B(4, 1) \), \( C(1, 2) \):

• Reflection across \( y = x \): \( A(3, 2) \), \( B(1, 4) \), \( C(2, 1) \).
• Reflection across \( x \)-axis: \( A(2, -3) \), \( B(4, -1) \), \( C(1, -2) \).
• Reflection across \( y = -x \): \( A(-3, -2) \), \( B(-1, -4) \), \( C(-2, -1) \).
• Reflection across \( y \)-axis: \( A(-2, 3) \), \( B(-4, 1) \), \( C(-1, 2) \).

To graph the reflected images, follow these steps for each problem:

1. Identify original coordinates.
2. Apply the reflection rule.
3. Plot the new points.

For specific coordinates (e.g., from the graph):

• Let’s assume the triangle in problem 6 has vertices \( A(-5, 3) \), \( B(-6, 1) \), \( C(-3, 2) \).
• Reflection across \( x \)-axis: \( A(-5, -3) \), \( B(-6, -1) \), \( C(-3, -2) \).

Final Answer (General Approach):

For each problem, apply the reflection rule to the original points and plot the results. The key is to use the correct transformation rule for each line of reflection.

• 5) \( y = x \): Swap \( x \) and \( y \) coordinates.
• 6) \( x \)-axis: Negate the \( y \)-coordinate.
• 7) \( y = -x \): Negate both \( x \) and \( y \), then swap.
• 8) \( y \)-axis: Negate the \( x \)-coordinate.

Graph the reflected points using these rules.