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We are given the cosine function \( y = -4 \cos(4x - \pi) \), and we need to use the quarter points of the base function \( y = \cos x \) to find the corresponding quarter points for this transformed function.

### Key transformations:
- **Horizontal shift**: The \( (4x - \pi) \) represents a horizontal shift by \( \frac{\pi}{4} \) units to the right.
- **Horizontal stretch/compression**: The factor \( 4 \) compresses the period by a factor of 4. The new period is \( \frac{2\pi}{4} = \frac{\pi}{2} \).
- **Vertical stretch**: The factor \( -4 \) scales the amplitude by 4 and reflects the graph across the x-axis. 

Now, let's transform the quarter points of \( y = \cos x \):

1. **Original quarter points of \( y = \cos x \)**:
   - \( (0, 1) \)
   - \( \left( \frac{\pi}{2}, 0 \right) \)
   - \( (\pi, -1) \)
   - \( \left( \frac{3\pi}{2}, 0 \right) \)
   - \( (2\pi, 1) \)

### Transformation steps:

1. **Horizontal Compression by 4**:
   For each x-coordinate \( x_{\text{new}} = \frac{x_{\text{old}}}{4} \).

2. **Horizontal Shift by \( \frac{\pi}{4} \)**:
   For each x-coordinate \( x_{\text{new}} = x_{\text{new}} + \frac{\pi}{4} \).

3. **Vertical Stretch/Reflection**:
   For each y-coordinate, multiply by -4.

Let's calculate the new quarter points:

1. **First point**:
   - Original: \( (0, 1) \)
   - New x-coordinate: \( \frac{0}{4} + \frac{\pi}{4} = \frac{\pi}{4} \)
   - New y-coordinate: \( -4 \times 1 = -4 \)
   - Transformed point: \( \left( \frac{\pi}{4}, -4 \right) \)

2. **Second point**:
   - Original: \( \left( \frac{\pi}{2}, 0 \right) \)
   - New x-coordinate: \( \frac{\frac{\pi}{2}}{4} + \frac{\pi}{4} = \frac{\pi}{8} + \frac{\pi}{4} = \frac{3\pi}{8} \)
   - New y-coordinate: \( 0 \times (-4) = 0 \)
   - Transformed point: \( \left( \frac{3\pi}{8}, 0 \right) \)

3. **Third point**:
   - Original: \( (\pi, -1) \)
   - New x-coordinate: \( \frac{\pi}{4} + \frac{\pi}{4} = \frac{2\pi}{4} = \frac{\pi}{2} \)
   - New y-coordinate: \( -4 \times (-1) = 4 \)
   - Transformed point: \( \left( \frac{\pi}{2}, 4 \right) \)

4. **Fourth point**:
   - Original: \( \left( \frac{3\pi}{2}, 0 \right) \)
   - New x-coordinate: \( \frac{\frac{3\pi}{2}}{4} + \frac{\pi}{4} = \frac{3\pi}{8} + \frac{\pi}{4} = \frac{5\pi}{8} \)
   - New y-coordinate: \( 0 \times (-4) = 0 \)
   - Transformed point: \( \left( \frac{5\pi}{8}, 0 \right) \)

5. **Fifth point**:
   - Original: \( (2\pi, 1) \)
   - New x-coordinate: \( \frac{2\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4} \)
   - New y-coordinate: \( -4 \times 1 = -4 \)
   - Transformed point: \( \left( \frac{3\pi}{4}, -4 \right) \)

### Final result:
The corresponding quarter points for \( y = -4 \cos(4x - \pi) \) are:
- \( \left( \frac{\pi}{4}, -4 \right) \)
- \( \left( \frac{3\pi}{8}, 0 \right) \)
- \( \left( \frac{\pi}{2}, 4 \right) \)
- \( \left( \frac{5\pi}{8}, 0 \right) \)
- \( \left( \frac{3\pi}{4}, -4 \right) \)

Would you like further details or have any questions?

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**Related Questions**:
1. What is the general form of the transformation of cosine functions?
2. How do horizontal stretches and compressions affect the period of trigonometric functions?
3. Can you explain how vertical reflections work with cosine functions?
4. What is the impact of a horizontal phase shift on the graph of a function?
5. How would the quarter points change for the function \( y = 2 \cos(3x + \frac{\pi}{3}) \)?

**Tip**: Always apply horizontal transformations (stretch/compression and shift) before vertical ones when working with trigonometric functions.

Use the coordinates of the five quarter points of y = cos x to determine the corresponding quarter points on the graph of y = -4 cos(4x - π).

Learn how to graph transformations of cosine functions by calculating the quarter points for y = -4 cos(4x - π). This problem covers horizontal shifts, vertical stretches, and compression. Suitable for students in Grades 10-12.

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