This worksheet provides practice in graphing translations of triangles on a coordinate plane.

Problem Analysis:

The task involves transforming geometric figures using specific transformations. The transformations provided are:

1. Translation: 4 units right and 2 units up.
2. Reflection: Across the y-axis.

We need to apply these transformations to the given figures and explain the process step by step.

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Solution:

#### Part 1: Translation (4 units right and 2 units up)

A translation moves every point of a figure by a fixed distance in a specified direction. Here, we are translating each figure 4 units to the right and 2 units up.

##### Step-by-Step Process for Translation:
1. Identify the coordinates of each vertex of the figure.
2. Add 4 to the x-coordinate of each vertex to move it 4 units to the right.
3. Add 2 to the y-coordinate of each vertex to move it 2 units up.
4. Plot the new vertices and connect them to form the translated figure.

##### Examples:
- Figure 1: A square.
- Original vertices: Assume the square is at \((0, 0)\), \((2, 0)\), \((2, 2)\), \((0, 2)\).
- After translation:
- \((0 + 4, 0 + 2) = (4, 2)\)
- \((2 + 4, 0 + 2) = (6, 2)\)
- \((2 + 4, 2 + 2) = (6, 4)\)
- \((0 + 4, 2 + 2) = (4, 4)\)
- New vertices: \((4, 2)\), \((6, 2)\), \((6, 4)\), \((4, 4)\).

- Figure 2: An isosceles triangle.
- Original vertices: Assume the triangle is at \((1, 1)\), \((3, 1)\), \((2, 3)\).
- After translation:
- \((1 + 4, 1 + 2) = (5, 3)\)
- \((3 + 4, 1 + 2) = (7, 3)\)
- \((2 + 4, 3 + 2) = (6, 5)\)
- New vertices: \((5, 3)\), \((7, 3)\), \((6, 5)\).

- Figure 3: A parallelogram.
- Original vertices: Assume the parallelogram is at \((0, 0)\), \((3, 0)\), \((4, 2)\), \((1, 2)\).
- After translation:
- \((0 + 4, 0 + 2) = (4, 2)\)
- \((3 + 4, 0 + 2) = (7, 2)\)
- \((4 + 4, 2 + 2) = (8, 4)\)
- \((1 + 4, 2 + 2) = (5, 4)\)
- New vertices: \((4, 2)\), \((7, 2)\), \((8, 4)\), \((5, 4)\).

- Figure 4: A scalene triangle.
- Original vertices: Assume the triangle is at \((1, 1)\), \((4, 1)\), \((2, 4)\).
- After translation:
- \((1 + 4, 1 + 2) = (5, 3)\)
- \((4 + 4, 1 + 2) = (8, 3)\)
- \((2 + 4, 4 + 2) = (6, 6)\)
- New vertices: \((5, 3)\), \((8, 3)\), \((6, 6)\).

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#### Part 2: Reflection (Across the y-axis)

A reflection across the y-axis flips the figure horizontally. This means:
- The x-coordinate of each point changes sign (i.e., \(x \to -x\)).
- The y-coordinate remains unchanged.

##### Step-by-Step Process for Reflection:
1. Identify the coordinates of each vertex of the figure.
2. Change the sign of the x-coordinate of each vertex.
3. Keep the y-coordinate the same.
4. Plot the new vertices and connect them to form the reflected figure.

##### Examples:
- Figure 1: A square.
- Original vertices: Assume the square is at \((0, 0)\), \((2, 0)\), \((2, 2)\), \((0, 2)\).
- After reflection:
- \((-0, 0) = (0, 0)\)
- \((-2, 0) = (-2, 0)\)
- \((-2, 2) = (-2, 2)\)
- \((-0, 2) = (0, 2)\)
- New vertices: \((0, 0)\), \((-2, 0)\), \((-2, 2)\), \((0, 2)\).

- Figure 2: An isosceles triangle.
- Original vertices: Assume the triangle is at \((1, 1)\), \((3, 1)\), \((2, 3)\).
- After reflection:
- \((-1, 1) = (-1, 1)\)
- \((-3, 1) = (-3, 1)\)
- \((-2, 3) = (-2, 3)\)
- New vertices: \((-1, 1)\), \((-3, 1)\), \((-2, 3)\).

- Figure 3: A parallelogram.
- Original vertices: Assume the parallelogram is at \((0, 0)\), \((3, 0)\), \((4, 2)\), \((1, 2)\).
- After reflection:
- \((-0, 0) = (0, 0)\)
- \((-3, 0) = (-3, 0)\)
- \((-4, 2) = (-4, 2)\)
- \((-1, 2) = (-1, 2)\)
- New vertices: \((0, 0)\), \((-3, 0)\), \((-4, 2)\), \((-1, 2)\).

- Figure 4: A scalene triangle.
- Original vertices: Assume the triangle is at \((1, 1)\), \((4, 1)\), \((2, 4)\).
- After reflection:
- \((-1, 1) = (-1, 1)\)
- \((-4, 1) = (-4, 1)\)
- \((-2, 4) = (-2, 4)\)
- New vertices: \((-1, 1)\), \((-4, 1)\), \((-2, 4)\).

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Final Answer:

For each figure, apply the transformations as described above. The final transformed figures will have the new coordinates calculated using the steps provided.

$$
\boxed{\text{Apply the transformations as explained for each figure.}}
$$