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We are given two functions: 
\[ f(x) = \sqrt[3]{x} \]
\[ g(x) = \sqrt[3]{x + 4} - 5 \]

### Analysis of transformations:
1. **Starting with \( f(x) \):** This is the basic cubic root function \( \sqrt[3]{x} \), which we can assume is plotted normally.
   
2. **Transformation for \( g(x) \):** 
   - \( g(x) = \sqrt[3]{x + 4} - 5 \) represents a transformation of \( f(x) \):
     - The term \( x + 4 \) means a horizontal shift **left by 4 units**.
     - The \( -5 \) means a vertical shift **down by 5 units**.

### Now, we analyze the given options:
- **First option:** "The graph of \( g \) is a translation 5 units left and 4 units up" is incorrect since the horizontal shift is 4 units left, not 5, and there is no vertical translation up.
  
- **Second option:** "The graph of \( g \) is a translation 4 units left and 5 units down" is correct based on the transformation observed.

- **Third option:** "The graph of \( g \) is a translation 5 units left and 4 units down" is incorrect for the same reasons as the first option.

- **Fourth option:** "The graph of \( g \) is a translation 4 units right and 5 units down" is incorrect because the shift is to the left, not right.

Thus, the correct description is the **second option**.

### Choosing the correct graph:
We need to identify which graph matches the transformations: \( g(x) \) should be shifted 4 units left and 5 units down compared to \( f(x) \).

Would you like me to go into detail regarding the graphs, or have any specific questions on transformations? 

Here are five related questions:
1. How do we determine if a function is shifted horizontally or vertically?
2. What is the effect of adding or subtracting constants inside the function versus outside?
3. How would the transformation change if we had \( g(x) = \sqrt[3]{x - 4} + 5 \)?
4. What is the general shape of a cubic root function?
5. How do we confirm the direction of shifts by inspecting the equation?

**Tip:** For horizontal transformations, focus on changes inside the function's argument. A shift left occurs when the constant is added, and a shift right when it's subtracted.

Let f(x) = ∛x and g(x) = ∛(x + 4) - 5. Describe the transformations.

Understand how to describe the transformations between the cubic root functions f(x) = ∛x and g(x) = ∛(x + 4) - 5. Learn about horizontal and vertical shifts and how they affect the graph. Suitable for high school students, Grades 9-12.

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