Use transformations of the graph of ​f(x)equals
xcubed
 
to determine the graph of the given function.
h left parenthesis x right parenthesis equals negative left parenthesis x minus 3 right parenthesis cubed

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.

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

Explore the transformation of the cubic function f(x) = x^3 into h(x) = -(x - 3)^3 through graph shifts and reflections. Learn how to horizontally shift by 3 units and reflect across the x-axis, suitable for high school students (Grades 10-12).

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