Practice worksheet for graphing sequences of congruence transformations including rotations, reflections, and translations on coordinate planes.

Problem Analysis:

The task involves performing congruence transformations on geometric figures. These transformations include:
1. Rotation: Turning a figure around a point (e.g., the origin).
2. Reflection: Flipping a figure over a line (e.g., the x-axis or y-axis).
3. Translation: Sliding a figure to a new position without rotating or flipping it.

Each problem requires applying one or more of these transformations step-by-step and then graphing the resulting figure.

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Solution to Each Problem:

#### Problem 1: Graph the image of \( \triangle FGH \) after a rotation \( 90^\circ \) counterclockwise around the origin and a reflection over the y-axis.

1. Identify the vertices of \( \triangle FGH \):
- Assume the vertices are given as \( F(x_1, y_1) \), \( G(x_2, y_2) \), and \( H(x_3, y_3) \).

2. Perform the rotation \( 90^\circ \) counterclockwise around the origin:
- The formula for rotating a point \( (x, y) \) \( 90^\circ \) counterclockwise around the origin is:
\[
(x, y) \rightarrow (-y, x)
\]
- Apply this to each vertex:
- \( F(x_1, y_1) \rightarrow F'(-y_1, x_1) \)
- \( G(x_2, y_2) \rightarrow G'(-y_2, x_2) \)
- \( H(x_3, y_3) \rightarrow H'(-y_3, x_3) \)

3. Perform the reflection over the y-axis:
- The formula for reflecting a point \( (x, y) \) over the y-axis is:
\[
(x, y) \rightarrow (-x, y)
\]
- Apply this to the rotated vertices:
- \( F'(-y_1, x_1) \rightarrow F''(y_1, x_1) \)
- \( G'(-y_2, x_2) \rightarrow G''(y_2, x_2) \)
- \( H'(-y_3, x_3) \rightarrow H''(y_3, x_3) \)

4. Graph the final image:
- Plot the points \( F''(y_1, x_1) \), \( G''(y_2, x_2) \), and \( H''(y_3, x_3) \) and connect them to form the transformed triangle.

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#### Problem 2: Graph the image of \( \triangle ABC \) after a reflection over the x-axis and a translation 4 units right.

1. Identify the vertices of \( \triangle ABC \):
- Assume the vertices are given as \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).

2. Perform the reflection over the x-axis:
- The formula for reflecting a point \( (x, y) \) over the x-axis is:
\[
(x, y) \rightarrow (x, -y)
\]
- Apply this to each vertex:
- \( A(x_1, y_1) \rightarrow A'(x_1, -y_1) \)
- \( B(x_2, y_2) \rightarrow B'(x_2, -y_2) \)
- \( C(x_3, y_3) \rightarrow C'(x_3, -y_3) \)

3. Perform the translation 4 units right:
- The formula for translating a point \( (x, y) \) 4 units right is:
\[
(x, y) \rightarrow (x + 4, y)
\]
- Apply this to the reflected vertices:
- \( A'(x_1, -y_1) \rightarrow A''(x_1 + 4, -y_1) \)
- \( B'(x_2, -y_2) \rightarrow B''(x_2 + 4, -y_2) \)
- \( C'(x_3, -y_3) \rightarrow C''(x_3 + 4, -y_3) \)

4. Graph the final image:
- Plot the points \( A''(x_1 + 4, -y_1) \), \( B''(x_2 + 4, -y_2) \), and \( C''(x_3 + 4, -y_3) \) and connect them to form the transformed triangle.

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#### Problem 3: Graph the image of \( \triangle JKL \) after a rotation \( 90^\circ \) counterclockwise around the origin and a translation 5 units down.

1. Identify the vertices of \( \triangle JKL \):
- Assume the vertices are given as \( J(x_1, y_1) \), \( K(x_2, y_2) \), and \( L(x_3, y_3) \).

2. Perform the rotation \( 90^\circ \) counterclockwise around the origin:
- Use the formula:
\[
(x, y) \rightarrow (-y, x)
\]
- Apply this to each vertex:
- \( J(x_1, y_1) \rightarrow J'(-y_1, x_1) \)
- \( K(x_2, y_2) \rightarrow K'(-y_2, x_2) \)
- \( L(x_3, y_3) \rightarrow L'(-y_3, x_3) \)

3. Perform the translation 5 units down:
- The formula for translating a point \( (x, y) \) 5 units down is:
\[
(x, y) \rightarrow (x, y - 5)
\]
- Apply this to the rotated vertices:
- \( J'(-y_1, x_1) \rightarrow J''(-y_1, x_1 - 5) \)
- \( K'(-y_2, x_2) \rightarrow K''(-y_2, x_2 - 5) \)
- \( L'(-y_3, x_3) \rightarrow L''(-y_3, x_3 - 5) \)

4. Graph the final image:
- Plot the points \( J''(-y_1, x_1 - 5) \), \( K''(-y_2, x_2 - 5) \), and \( L''(-y_3, x_3 - 5) \) and connect them to form the transformed triangle.

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#### Problem 4: Graph the image of trapezoid \( TUVW \) after a translation 2 units left and a reflection over the y-axis.

1. Identify the vertices of trapezoid \( TUVW \):
- Assume the vertices are given as \( T(x_1, y_1) \), \( U(x_2, y_2) \), \( V(x_3, y_3) \), and \( W(x_4, y_4) \).

2. Perform the translation 2 units left:
- The formula for translating a point \( (x, y) \) 2 units left is:
\[
(x, y) \rightarrow (x - 2, y)
\]
- Apply this to each vertex:
- \( T(x_1, y_1) \rightarrow T'(x_1 - 2, y_1) \)
- \( U(x_2, y_2) \rightarrow U'(x_2 - 2, y_2) \)
- \( V(x_3, y_3) \rightarrow V'(x_3 - 2, y_3) \)
- \( W(x_4, y_4) \rightarrow W'(x_4 - 2, y_4) \)

3. Perform the reflection over the y-axis:
- Use the formula:
\[
(x, y) \rightarrow (-x, y)
\]
- Apply this to the translated vertices:
- \( T'(x_1 - 2, y_1) \rightarrow T''(-(x_1 - 2), y_1) = T''(2 - x_1, y_1) \)
- \( U'(x_2 - 2, y_2) \rightarrow U''(-(x_2 - 2), y_2) = U''(2 - x_2, y_2) \)
- \( V'(x_3 - 2, y_3) \rightarrow V''(-(x_3 - 2), y_3) = V''(2 - x_3, y_3) \)
- \( W'(x_4 - 2, y_4) \rightarrow W''(-(x_4 - 2), y_4) = W''(2 - x_4, y_4) \)

4. Graph the final image:
- Plot the points \( T''(2 - x_1, y_1) \), \( U''(2 - x_2, y_2) \), \( V''(2 - x_3, y_3) \), and \( W''(2 - x_4, y_4) \) and connect them to form the transformed trapezoid.

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Final Answer:

\[
\boxed{\text{See the detailed steps above for each transformation.}}
\]