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?

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.