Worksheet for practicing geometry transformations, featuring two problems with pre-images, transformation sequences, and coordinate grids for plotting.

Problem Analysis:

The worksheet involves applying a sequence of geometric transformations to given points and recording the resulting coordinates. We will solve each part step by step.

---

Part 1:

#### Given:
- Pre-image points: \( A(0, 0) \), \( B(8, 1) \), \( C(5, 5) \)
- Transformations: \( R_{x\text{-axis}} \circ r_{180^\circ} \)

#### Steps:
1. Understand the transformations:
- \( r_{180^\circ} \): Rotation by \( 180^\circ \) about the origin.
- Formula: \( (x, y) \rightarrow (-x, -y) \)
- \( R_{x\text{-axis}} \): Reflection over the x-axis.
- Formula: \( (x, y) \rightarrow (x, -y) \)

2. Apply the transformations in order:
- First, apply \( r_{180^\circ} \).
- Then, apply \( R_{x\text{-axis}} \).

#### Step-by-Step Calculation:
##### Point \( A(0, 0) \):
1. Apply \( r_{180^\circ} \):
\[
A(0, 0) \rightarrow (-0, -0) = (0, 0)
\]
2. Apply \( R_{x\text{-axis}} \):
\[
(0, 0) \rightarrow (0, -0) = (0, 0)
\]
- Final coordinates: \( A''(0, 0) \)

##### Point \( B(8, 1) \):
1. Apply \( r_{180^\circ} \):
\[
B(8, 1) \rightarrow (-8, -1)
\]
2. Apply \( R_{x\text{-axis}} \):
\[
(-8, -1) \rightarrow (-8, 1)
\]
- Final coordinates: \( B''(-8, 1) \)

##### Point \( C(5, 5) \):
1. Apply \( r_{180^\circ} \):
\[
C(5, 5) \rightarrow (-5, -5)
\]
2. Apply \( R_{x\text{-axis}} \):
\[
(-5, -5) \rightarrow (-5, 5)
\]
- Final coordinates: \( C''(-5, 5) \)

#### Results for Part 1:
- After \( r_{180^\circ} \):
\[
A'(0, 0), \quad B'(-8, -1), \quad C'(-5, -5)
\]
- After \( R_{x\text{-axis}} \):
\[
A''(0, 0), \quad B''(-8, 1), \quad C''(-5, 5)
\]

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Part 2:

#### Given:
- Pre-image points: \( Q(3, 3) \), \( R(-1, 4) \), \( S(2, 6) \)
- Transformations: \( r_{90^\circ} \circ R_{y\text{-axis}} \)

#### Steps:
1. Understand the transformations:
- \( R_{y\text{-axis}} \): Reflection over the y-axis.
- Formula: \( (x, y) \rightarrow (-x, y) \)
- \( r_{90^\circ} \): Rotation by \( 90^\circ \) counterclockwise about the origin.
- Formula: \( (x, y) \rightarrow (-y, x) \)

2. Apply the transformations in order:
- First, apply \( R_{y\text{-axis}} \).
- Then, apply \( r_{90^\circ} \).

#### Step-by-Step Calculation:
##### Point \( Q(3, 3) \):
1. Apply \( R_{y\text{-axis}} \):
\[
Q(3, 3) \rightarrow (-3, 3)
\]
2. Apply \( r_{90^\circ} \):
\[
(-3, 3) \rightarrow (-3, -3)
\]
- Final coordinates: \( Q''(-3, -3) \)

##### Point \( R(-1, 4) \):
1. Apply \( R_{y\text{-axis}} \):
\[
R(-1, 4) \rightarrow (1, 4)
\]
2. Apply \( r_{90^\circ} \):
\[
(1, 4) \rightarrow (-4, 1)
\]
- Final coordinates: \( R''(-4, 1) \)

##### Point \( S(2, 6) \):
1. Apply \( R_{y\text{-axis}} \):
\[
S(2, 6) \rightarrow (-2, 6)
\]
2. Apply \( r_{90^\circ} \):
\[
(-2, 6) \rightarrow (-6, -2)
\]
- Final coordinates: \( S''(-6, -2) \)

#### Results for Part 2:
- After \( R_{y\text{-axis}} \):
\[
Q'( -3, 3 ), \quad R'( 1, 4 ), \quad S'( -2, 6 )
\]
- After \( r_{90^\circ} \):
\[
Q''( -3, -3 ), \quad R''( -4, 1 ), \quad S''( -6, -2 )
\]

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Final Answer:

\[
\boxed{
\begin{array}{c|c}
\text{Part 1} & \text{Part 2} \\
\hline
A''(0, 0), B''(-8, 1), C''(-5, 5) & Q''(-3, -3), R''(-4, 1), S''(-6, -2) \\
\end{array}
}
\]