To determine the graph of the function $h(x) = -(x - 3)^3$ using transformations of the base function $f(x) = x^3$, follow these steps:

Step 1: Start with the base function

The base function is $f(x) = x^3$, which is a cubic function. Its graph has the following key features:

• It passes through the origin $(0, 0)$.
• It is symmetric with respect to the origin (odd function).
• The general shape is an "S" curve, increasing steeply as $x$ moves away from zero.

Step 2: Horizontal Shift

The function $(x - 3)^3$ represents a horizontal shift. Since the subtraction of 3 occurs inside the cube, it shifts the graph 3 units to the right. The new graph will pass through the point $(3, 0)$ instead of $(0, 0)$.

Step 3: Reflection

The negative sign outside the cube, $-(x - 3)^3$, reflects the graph vertically across the x-axis. This means that the graph will be flipped upside down: where the original graph was increasing, the new graph will be decreasing.

Final Graph Description

The graph of $h(x) = -(x - 3)^3$ is:

• The graph of $x^3$ shifted 3 units to the right.
• Reflected vertically across the x-axis.

Thus, it will still have the cubic "S" shape, but the curve will open downwards instead of upwards, and its center will be at the point $(3, 0)$.

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Do you want further details or have any questions about these steps?

Here are five related questions for deeper understanding:

1. What effect does a vertical stretch or compression have on the graph of a cubic function?
2. How does the transformation $f(x + k)$ shift the graph horizontally?
3. What happens if you apply a horizontal reflection to the graph of $x^3$?
4. How does the graph of $x^n$ for different powers $n$ compare to the cubic function?
5. Can you describe how the graph of $x^3$ changes under a vertical shift?

Tip: A reflection across the x-axis changes the sign of all y-values, essentially flipping the graph vertically.