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I see the problem requires graphing the image of triangle \( \Delta A \) under a sequence of transformations. I'll break down each part step by step for clarity:

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### **1. Reflection across the y-axis, followed by a translation 2 units up and 3 units left**
- **Reflection across the y-axis**:
  Each point of \( \Delta A \) reflects across the y-axis by negating the x-coordinate while keeping the y-coordinate the same.

  - \( (2, 4) \to (-2, 4) \)
  - \( (6, 1) \to (-6, 1) \)
  - \( (8, 3) \to (-8, 3) \)

- **Translation 2 units up and 3 units left**:
  Subtract 3 from the x-coordinates and add 2 to the y-coordinates of the reflected points.

  - \( (-2, 4) \to (-5, 6) \)
  - \( (-6, 1) \to (-9, 3) \)
  - \( (-8, 3) \to (-11, 5) \)

The new points form triangle \( B \) with vertices:
\[
B = (-5, 6), (-9, 3), (-11, 5)
\]

---

### **2. 270° clockwise rotation around the origin, followed by a reflection across the x-axis**
- **270° clockwise rotation**:
  A 270° clockwise rotation swaps and negates the y-coordinate of each point:
  - \( (x, y) \to (y, -x) \)

  For triangle \( A \):
  - \( (2, 4) \to (4, -2) \)
  - \( (6, 1) \to (1, -6) \)
  - \( (8, 3) \to (3, -8) \)

- **Reflection across the x-axis**:
  Negate the y-coordinates of the rotated points:
  - \( (4, -2) \to (4, 2) \)
  - \( (1, -6) \to (1, 6) \)
  - \( (3, -8) \to (3, 8) \)

The new points form triangle \( C \) with vertices:
\[
C = (4, 2), (1, 6), (3, 8)
\]

---

### **3. Translation 1 unit down and 2 units right, followed by a 90° counterclockwise rotation around the origin**
- **Translation 1 unit down and 2 units right**:
  Add 2 to the x-coordinates and subtract 1 from the y-coordinates of triangle \( A \):
  - \( (2, 4) \to (4, 3) \)
  - \( (6, 1) \to (8, 0) \)
  - \( (8, 3) \to (10, 2) \)

- **90° counterclockwise rotation**:
  Swap and negate the x-coordinates:
  - \( (x, y) \to (-y, x) \)

  For the translated points:
  - \( (4, 3) \to (-3, 4) \)
  - \( (8, 0) \to (0, 8) \)
  - \( (10, 2) \to (-2, 10) \)

The new points form triangle \( D \) with vertices:
\[
D = (-3, 4), (0, 8), (-2, 10)
\]

---

### **Summary of Results**
- Triangle \( B \): \( (-5, 6), (-9, 3), (-11, 5) \)
- Triangle \( C \): \( (4, 2), (1, 6), (3, 8) \)
- Triangle \( D \): \( (-3, 4), (0, 8), (-2, 10) \)

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Would you like me to graph these transformations step by step or clarify any specific part?

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### **5 Related Questions**
1. What are the general formulas for rotations in the coordinate plane?
2. How do translations affect the coordinates of points in a shape?
3. Can you explain how reflections across the axes work geometrically?
4. What is the difference between clockwise and counterclockwise rotations?
5. How can transformation sequences be combined into a single operation?

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### **Tip**
When performing multiple transformations, always apply them in the given order step by step to avoid confusion!

Graph the image of triangle ΔA under the given sequence of transformations: 1) Reflection across the y-axis followed by a translation, 2) A 270° clockwise rotation followed by a reflection across the x-axis, 3) Translation followed by a 90° counterclockwise rotation.

Learn how to graph transformations of triangle ΔA under a sequence of reflections, rotations, and translations. This detailed guide explains the steps and formulas for each transformation, including reflections across axes, 270° clockwise and 90° counterclockwise rotations, and translations in the coordinate plane. Suitable for Grade 8 students.

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