https://file.aiquickdraw.com/mathbot/file/1730737304102nzz2wqbe.jpg

To reflect the triangle across the line \( y = -2 \) and the \( y \)-axis, let's break down the transformations step-by-step.

### Step 1: Reflect Across \( y = -2 \)
1. Identify the coordinates of each vertex of the triangle. Let's assume they are:
   - \( A(x_1, y_1) \)
   - \( B(x_2, y_2) \)
   - \( C(x_3, y_3) \)
   
   By looking at the graph, I will estimate the coordinates. You might need to adjust these coordinates based on precise grid values, but for illustration, I will use approximate values.
   
2. To reflect a point across the line \( y = -2 \), take the following steps:
   - Calculate the vertical distance of each point from \( y = -2 \).
   - Mirror this distance on the other side of \( y = -2 \).
   
   The formula for the reflected \( y \)-coordinate \( y' \) is:
   \[
   y' = -2 - (y + 2)
   \]
   
   Repeat this for each vertex to get the new coordinates.

### Step 2: Reflect Across the \( y \)-Axis
1. After finding the new coordinates from the reflection over \( y = -2 \), reflect each point across the \( y \)-axis.
2. To reflect a point across the \( y \)-axis, change the sign of the \( x \)-coordinate:
   - If a point is \( (x, y) \), it will become \( (-x, y) \) after reflection.

After completing both reflections, plot these new points to form the reflected triangle. 

Would you like me to calculate specific points based on my estimated coordinates, or do you have exact coordinates for each vertex? 

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### Follow-up Questions:
1. What is the final position of each vertex after both reflections?
2. How do you determine the exact distance when reflecting across a line that is not the x-axis or y-axis?
3. Can we reflect an entire shape in one step over two different axes?
4. What changes if the line of reflection is diagonal instead of horizontal or vertical?
5. How would reflecting the triangle over the x-axis differ in this case?

### Tip:
When performing reflections, always start with one line at a time and verify the coordinates after each step to ensure accuracy.

Reflect the triangle across the line y = -2 and the y-axis. Use the Polygon Tool to plot the image.

Learn how to reflect a triangle across the line y = -2 and the y-axis with step-by-step instructions. This problem covers essential concepts in coordinate geometry, including transformations and reflections, and provides formulas to guide the solution. Suitable for students in Grades 8-10.

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