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Let's analyze the transformations given for the function \( f(x) = \frac{1}{x^3} \) in each case and describe them verbally, along with drawing the graph transformations.

### Original Function:
\[ f(x) = \frac{1}{x^3} \]

### Case (a):
\[ t(x) = \frac{-2}{x^3} + 1 \]

**Transformations:**
1. **Vertical Stretch and Reflection:** The factor of \(-2\) indicates a vertical stretch by a factor of 2 and a reflection across the x-axis.
2. **Vertical Shift:** The \( +1 \) indicates a vertical shift upward by 1 unit.

### Case (b):
\[ t(x) = \frac{1}{3x^3} \]

**Transformations:**
1. **Horizontal Stretch:** The factor of \(3\) inside the function means a horizontal stretch by a factor of \(\frac{1}{3}\).

### Case (c):
\[ t(x) = \frac{1}{(4(x - 2))^3} \]

**Transformations:**
1. **Horizontal Compression:** The factor of \(4\) inside the function means a horizontal compression by a factor of \(\frac{1}{4}\).
2. **Horizontal Shift:** The \( x - 2 \) inside the function means a shift to the right by 2 units.

### Case (d):
\[ t(x) = \frac{1}{(0.5x + 1)^3} \]

**Transformations:**
1. **Horizontal Stretch:** The factor of \(0.5\) inside the function means a horizontal stretch by a factor of \(2\).
2. **Horizontal Shift:** The \( +1 \) inside the function means a shift to the left by 1 unit.

### Graphs:
I will describe the general appearance of the graphs given these transformations. 

1. **Original graph \( f(x) = \frac{1}{x^3} \):**
   - It has a vertical asymptote at \( x = 0 \).
   - It passes through \((1, 1)\) and \((-1, -1)\).
   - As \( x \) approaches 0, \( f(x) \) approaches \(\infty\) or \(-\infty\).

2. **Graph of \( t(x) = \frac{-2}{x^3} + 1 \):**
   - Stretched vertically by a factor of 2.
   - Reflected across the x-axis.
   - Shifted up by 1 unit.

3. **Graph of \( t(x) = \frac{1}{3x^3} \):**
   - Horizontally stretched by a factor of \(\frac{1}{3}\).

4. **Graph of \( t(x) = \frac{1}{(4(x - 2))^3} \):**
   - Horizontally compressed by a factor of \(\frac{1}{4}\).
   - Shifted to the right by 2 units.

5. **Graph of \( t(x) = \frac{1}{(0.5x + 1)^3} \):**
   - Horizontally stretched by a factor of 2.
   - Shifted to the left by 1 unit.

Would you like me to generate these graphs for visual representation? If you have any specific questions or need further details, please let me know.

Here are some example questions you might ask next:
1. Can you explain horizontal and vertical transformations in more detail?
2. How do you find the vertical and horizontal asymptotes of transformed functions?
3. What effect does a negative sign have in function transformations?
4. How do you determine the new critical points after a transformation?
5. Can you provide a step-by-step solution to a specific transformation problem?

**Tip:** When dealing with function transformations, always identify the type of transformation (e.g., shift, stretch, compression) and apply them step by step to understand the cumulative effect on the graph.

Explore function transformations through examples of vertical and horizontal shifts, stretches, and compressions. Learn how each transformation affects the graph of the function, using the example of f(x) = 1/x^3 and its variants. Suitable for high school students studying advanced algebra and function transformations.

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