https://file.aiquickdraw.com/mathbot/file/1730473586402wsdcjgwd.jpg

To analyze this question, let's examine the transformation applied to the function.

The function given is:
\[
h(x) = \sqrt[3]{x + 1} - 2
\]

The parent function is:
\[
g(x) = \sqrt[3]{x}
\]

### Step 1: Determine the Transformations
The function \( h(x) = \sqrt[3]{x + 1} - 2 \) can be broken down into transformations from \( g(x) = \sqrt[3]{x} \):

1. **Horizontal Shift**: The term \( x + 1 \) inside the cube root function represents a horizontal shift. Specifically, it shifts the graph **1 unit to the left**.

2. **Vertical Shift**: The \( -2 \) outside the function represents a vertical shift, moving the graph **2 units down**.

### Conclusion
The graph of the parent function \( g(x) = \sqrt[3]{x} \) is translated **1 unit left and 2 units down** to obtain \( h(x) \).

### Answer
The correct choice is:
- **A) Left 1 unit and down 2 units**

Would you like a more detailed explanation or have any questions?

Here are some related questions to explore:
1. What would happen if the function was \( h(x) = \sqrt[3]{x - 1} + 2 \)?
2. How would the graph change if \( h(x) = \sqrt[3]{x} - 2 \) (no horizontal shift)?
3. What is the effect of shifting the graph \( g(x) = \sqrt[3]{x} \) up by 3 units?
4. How do transformations affect the domain and range of functions?
5. How would a vertical stretch or compression affect the graph of \( g(x) = \sqrt[3]{x} \)?

**Tip:** When dealing with transformations, remember that shifts inside the function affect horizontal movement, while shifts outside affect vertical movement.

The figure above shows the graph of h(x) = √[3]{x + 1} - 2, a translation of the parent function g(x) = √[3]{x}. How is the graph of the parent function translated?

Explore the function transformation of the cube root function h(x) = √[3]{x + 1} - 2 from the parent function g(x) = √[3]{x}. Understand horizontal and vertical shifts, suitable for high school students studying function transformations.

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