Reflections - Kuta Software - Free Printable

To solve the problem of graphing the image of the given figures using the specified transformations, we need to understand how reflections work in a coordinate plane. Here's a step-by-step explanation for each transformation:

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1) Reflection across \( y = -2 \)

#### Steps:
- A reflection across the line \( y = -2 \) means that each point \((x, y)\) in the original figure will be mapped to a new point \((x, y')\) such that the line \( y = -2 \) is the midpoint between \((x, y)\) and \((x, y')\).
- The formula for the reflected point is:
\[
y' = -2 - (y + 2)
\]
Simplifying:
\[
y' = -4 - y
\]

#### Application:
- For each vertex of the figure, apply the transformation \( (x, y) \to (x, -4 - y) \).
- Plot the new points and connect them to form the reflected image.

---

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

#### Steps:
- A reflection across the \( x \)-axis means that each point \((x, y)\) in the original figure will be mapped to a new point \((x, -y)\).

#### Application:
- For each vertex of the figure, apply the transformation \( (x, y) \to (x, -y) \).
- Plot the new points and connect them to form the reflected image.

---

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

#### Steps:
- A reflection across the line \( y = -x \) means that each point \((x, y)\) in the original figure will be mapped to a new point \((-y, -x)\).

#### Application:
- For each vertex of the figure, apply the transformation \( (x, y) \to (-y, -x) \).
- Plot the new points and connect them to form the reflected image.

---

4) Reflection across \( y = -1 \)

#### Steps:
- A reflection across the line \( y = -1 \) means that each point \((x, y)\) in the original figure will be mapped to a new point \((x, y')\) such that the line \( y = -1 \) is the midpoint between \((x, y)\) and \((x, y')\).
- The formula for the reflected point is:
\[
y' = -1 - (y + 1)
\]
Simplifying:
\[
y' = -2 - y
\]

#### Application:
- For each vertex of the figure, apply the transformation \( (x, y) \to (x, -2 - y) \).
- Plot the new points and connect them to form the reflected image.

---

5) Reflection across \( x = -3 \)

#### Steps:
- A reflection across the line \( x = -3 \) means that each point \((x, y)\) in the original figure will be mapped to a new point \((x', y)\) such that the line \( x = -3 \) is the midpoint between \((x, y)\) and \((x', y)\).
- The formula for the reflected point is:
\[
x' = -3 - (x + 3)
\]
Simplifying:
\[
x' = -6 - x
\]

#### Application:
- For each vertex of the figure, apply the transformation \( (x, y) \to (-6 - x, y) \).
- Plot the new points and connect them to form the reflected image.

---

6) Reflection across \( y = x \)

#### Steps:
- A reflection across the line \( y = x \) means that each point \((x, y)\) in the original figure will be mapped to a new point \((y, x)\).

#### Application:
- For each vertex of the figure, apply the transformation \( (x, y) \to (y, x) \).
- Plot the new points and connect them to form the reflected image.

---

Final Answer:

The solution involves applying the appropriate transformation formula to each vertex of the given figure and plotting the new points. The final images are obtained by connecting these transformed points.

\[
\boxed{\text{Apply the transformations as described above for each figure.}}
\]