rotation: 180 counterclockwise about origin translation: (x,y) (x-1,y+1)

Let's break down the transformation step by step:

### 1. **Rotation by 180° Counterclockwise About the Origin**
When a point \((x, y)\) is rotated \(180^\circ\) counterclockwise about the origin, its new coordinates become:
\[
(x', y') = (-x, -y)
\]

### 2. **Translation by the Rule \((x, y) \to (x-1, y+1)\)**
After the rotation, the translated coordinates are computed as follows:
\[
(x'', y'') = (x' - 1, y' + 1)
\]

### Combining Both Transformations
Substituting the first transformation into the second:
\[
x'' = -x - 1
\]
\[
y'' = -y + 1
\]

Thus, the combined transformation maps:
\[
(x, y) \to (-x - 1, -y + 1)
\]

### Example
If the original point is \((2, 3)\):
1. After \(180^\circ\) rotation:
   \[
   (x', y') = (-2, -3)
   \]
2. After the translation:
   \[
   (x'', y'') = (-2 - 1, -3 + 1) = (-3, -2)
   \]

Let me know if you need more examples or further details!

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#### Questions for Practice:
1. What are the coordinates of \((4, 5)\) after this transformation?
2. How does the transformation affect points on the x-axis (e.g., \((x, 0)\))?
3. How does the transformation affect points on the y-axis (e.g., \((0, y)\))?
4. What is the inverse transformation of this combined operation?
5. How would this transformation change a geometric figure, such as a square?

#### Tip:
When combining transformations, apply them step-by-step and substitute systematically for clarity!

Understand how to perform a 180° counterclockwise rotation about the origin followed by a translation (x,y) → (x-1,y+1). Learn key geometry concepts with step-by-step solutions and examples suitable for students in Grades 8-10.

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