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The problem involves the transformation of the function \( h(x) = -2 \cos \left( 8x - \frac{\pi}{4} \right) + 5 \) to \( j(x) = -2 \cos \left( 8x - \frac{\pi}{4} \right) - 5 \).

### Analysis of Transformations:
The function \( h(x) \) is of the form \( A \cos(Bx - C) + D \), where:

- \( A = -2 \) affects the amplitude and the reflection across the x-axis (since it is negative).
- \( B = 8 \) affects the period of the cosine function (period = \( \frac{2\pi}{B} = \frac{\pi}{4} \)).
- \( C = \frac{\pi}{4} \) represents a phase shift to the right by \( \frac{\pi}{4} \).
- \( D = +5 \) is the vertical shift, moving the graph up by 5 units.

In the transformation from \( h(x) \) to \( j(x) \), the only change is in the vertical shift, from \( +5 \) to \( -5 \). Therefore, the transformation is a **vertical translation** down by 10 units.

### Conclusion:
The transformation from \( h(x) \) to \( j(x) \) is a vertical translation down by 10 units.

Would you like more details or have any questions? Here are five related questions:

1. What effect does changing the amplitude have on the graph of a trigonometric function?
2. How does changing the frequency parameter \( B \) affect the period of a cosine function?
3. What is the impact of phase shifts on trigonometric functions?
4. Can you describe how reflections across the x-axis are represented in cosine functions?
5. What other transformations can be applied to cosine functions?

**Tip:** Vertical translations affect only the y-values, shifting the graph up or down without changing the shape or orientation.

Consider the function h(x) = -2cos(8x - π/4) + 5. What transformation results in j(x) = -2cos(8x - π/4) - 5?

Explore the transformation of the function h(x) = -2cos(8x - π/4) + 5 to j(x) = -2cos(8x - π/4) - 5. This problem covers vertical shift transformations in trigonometric functions, with a detailed explanation of the cosine function and how it is shifted vertically by 10 units.

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