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Let's solve the given problem step-by-step:

### Given Function:
\[ f(x) = x^3 \]

### Transformed Function:
\[ g(x) = a \left[ k(x - d) \right]^3 + c \]

We'll choose the following values for the parameters:
- \( a = 2 \) (vertical stretch by a factor of 2)
- \( k = 1 \) (no horizontal stretch or compression)
- \( d = 3 \) (horizontal shift to the right by 3 units)
- \( c = -1 \) (vertical shift downward by 1 unit)

### Transformed Function Equation:
\[ g(x) = 2[1(x - 3)]^3 - 1 = 2(x - 3)^3 - 1 \]

### Describe the Transformations:
1. **Vertical Stretch**: The graph is stretched vertically by a factor of 2.
2. **Horizontal Shift**: The graph is shifted 3 units to the right.
3. **Vertical Shift**: The graph is shifted 1 unit downward.

### Key Points Transformation:

Let's start with some key points from \( f(x) = x^3 \):
- Original points: \((-1, -1)\), \((0, 0)\), \((1, 1)\), \((2, 8)\)

Apply the transformations to these points:
1. **For \((-1, -1)\):**
   - Horizontal shift: \((-1 + 3, -1) = (2, -1)\)
   - Vertical stretch: \((-1) \times 2 = -2\)
   - Vertical shift: \(-2 - 1 = -3\)
   - New point: \((2, -3)\)

2. **For \((0, 0)\):**
   - Horizontal shift: \((0 + 3, 0) = (3, 0)\)
   - Vertical stretch: \(0 \times 2 = 0\)
   - Vertical shift: \(0 - 1 = -1\)
   - New point: \((3, -1)\)

3. **For \((1, 1)\):**
   - Horizontal shift: \((1 + 3, 1) = (4, 1)\)
   - Vertical stretch: \(1 \times 2 = 2\)
   - Vertical shift: \(2 - 1 = 1\)
   - New point: \((4, 1)\)

4. **For \((2, 8)\):**
   - Horizontal shift: \((2 + 3, 8) = (5, 8)\)
   - Vertical stretch: \(8 \times 2 = 16\)
   - Vertical shift: \(16 - 1 = 15\)
   - New point: \((5, 15)\)

### Graphing the Transformed Function:
1. Plot the new points: \((2, -3)\), \((3, -1)\), \((4, 1)\), \((5, 15)\)
2. Draw the curve passing through these points to represent the function \( g(x) = 2(x - 3)^3 - 1 \)

Here is a summary to fill in the table:

- \( a = 2 \)
- \( k = 1 \)
- \( d = 3 \)
- \( c = -1 \)
- Transformed function equation: \( g(x) = 2(x - 3)^3 - 1 \)

### Graph:
(You would need to plot this on the provided graph paper.)

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:
1. What would the graph look like if \( a = -2 \) instead of \( 2 \)?
2. How does changing \( k \) from \( 1 \) to \( 0.5 \) affect the graph?
3. What if \( d \) were a negative value?
4. How does the value of \( c \) influence the graph's position?
5. Can you describe the transformations if \( a = 1 \), \( k = 2 \), \( d = 0 \), and \( c = 3 \)?
6. What are the key points if the original function is \( f(x) = x^2 \) instead?
7. How do you determine the new coordinates of a point after transformation?
8. What are the effects of combining multiple transformations in one function?

### Tip:
When graphing transformed functions, always start with the base function and apply transformations step-by-step for accuracy.

Explore function transformations with a detailed step-by-step guide. Learn how to transform the function f(x) = x^3 into g(x) = 2(x - 3)^3 - 1, including vertical and horizontal shifts and stretches. Suitable for high school students.

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