https://file.aiquickdraw.com/mathbot/file/1728512063907g1ydh8gx.jpg

Let's go through the questions shown in the image step by step:

### Question 8:
**The function that describes the reflection is:**

Reflection across the x-axis involves changing the sign of the y-coordinate. The correct transformation rule is:
\[
(x, y) \to (x, -y)
\]
So, the correct answer is **D. (x, -y).**

---

### Question 9:
**An up-pointing arrow is drawn inside the square. In the image, the arrow points in which direction?**

Based on the reflection across the x-axis, any up-pointing arrow would be flipped vertically. Therefore, the up-pointing arrow would point **down** after the reflection.
So, the correct answer is **B. down.**

---

### Question 10:
**Describe a reflection of the square that would map it onto itself.**

The square can be reflected either across the x-axis, y-axis, or both (a reflection across both axes would be a 180-degree rotation). In this case, reflecting the square across the **y-axis** would map it onto itself because the vertices would coincide.

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### Question 11:
**Right triangle ABC is reflected across line ℓ, which is parallel to but distinct from AB, one of the legs of the triangle. Prove that points B, C, B', and C' are collinear.**

This question involves reflection across a line parallel to one of the triangle's sides. When reflected, the image of triangle ABC will have B' and C' positioned such that they align on a straight line with B and C, forming a line segment. This is because reflecting across a parallel line preserves the slope and the direction of the original segment. Thus, B, C, B', and C' will be collinear.

Would you like more details or clarification on any of these answers? Here are 5 related questions that build on this material:

1. How does a reflection across the y-axis differ from a reflection across the x-axis?
2. Can the orientation of a shape change after a reflection? Why or why not?
3. How can you find the equation of a line of reflection between two points and their reflected images?
4. What transformations (other than reflections) can map a square onto itself?
5. How can you determine the collinearity of points after a geometric transformation?

**Tip:** When reflecting shapes, always focus on how each vertex's coordinates change to ensure you're applying the transformation rule correctly.

The function that describes the reflection, an up-pointing arrow, and triangle ABC reflection questions.

Explore how to solve problems related to reflections across the x-axis and y-axis, including mapping squares and proving collinearity in reflected triangles. Perfect for students in Grades 7-9.

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