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The given function is \( h(x) = -(x - 4)^3 \). We need to determine how the graph of this function is related to the graph of the basic cubic function \( f(x) = x^3 \) through transformations.

### Step-by-Step Explanation of Transformations:

1. **Horizontal Translation**:
   - The expression \( (x - 4)^3 \) indicates a **horizontal shift** of the graph of \( x^3 \). Specifically, it shifts the graph **4 units to the right**. This is because the \( x \)-term is replaced by \( (x - 4) \).

2. **Reflection**:
   - The negative sign in front of the cubic term, \( -(x - 4)^3 \), represents a **reflection across the x-axis**. This flips the graph upside-down.

### Transformation Summary:
- Start with the basic graph of \( f(x) = x^3 \).
- Shift the graph 4 units to the right.
- Reflect the graph across the x-axis.

### Visualization:
The new graph will look like a vertically flipped version of \( f(x) = x^3 \), shifted to the right by 4 units. The reflection causes the ends of the curve to reverse: instead of increasing as you move to the right, it will now decrease as you move to the right.

Would you like more details, or do you have any questions?

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Here are 5 related questions to deepen your understanding:
1. How would the graph change if the function were \( h(x) = (x + 2)^3 \)?
2. What is the effect of a vertical stretch or compression on the cubic function?
3. How would the transformation change if the negative sign were inside the cube, like \( (-x + 4)^3 \)?
4. What happens if the graph is shifted left instead of right?
5. How can we find the points of inflection for cubic functions?

**Tip**: When analyzing transformations, always identify the basic function first and then apply shifts, stretches, or reflections in the correct order.

Use transformations of the graph of f(x) = x^3 to determine the graph of the given function h(x) = -(x - 4)^3.

Learn how to apply transformations to the cubic function f(x) = x^3 to determine the graph of h(x) = -(x - 4)^3. This step-by-step guide will walk you through horizontal shifts and reflections. Suitable for high school students in Grades 9-12.

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