The function $$ g(x)=2.5 \sqrt[3]{x-3}+5 $$ is a transformation of the cube root parent function. How can these transformations be described? Select each correct answer. Select 3 correct answer(s) The function is translated 3 units right. The function is translated 3 units left. The function is translated 5 units up. The function is translated 5 units down. The function undergoes a vertical stretch. The function undergoes a vertical compression.

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UpStudy AI · Accepted Answer

To analyze the transformations of the function $$ g(x) = 2.5 \sqrt[3]{x - 3} + 5 $$ compared to the parent cube root function $$ f(x) = \sqrt[3]{x} $$, let's break down each component:

1. **Horizontal Translation ($$ x - 3 $$)**:
   - **Interpretation**: The term $$ x - 3 $$ inside the cube root indicates a horizontal shift.
   - **Direction and Magnitude**: This translates the graph **3 units to the right**.
   
2. **Vertical Stretch ($$ 2.5 $$)**:
   - **Interpretation**: The coefficient $$ 2.5 $$ outside the cube root affects the vertical scaling.
   - **Effect**: Since $$ 2.5 > 1 $$, this results in a **vertical stretch** of the graph by a factor of 2.5.

3. **Vertical Translation ($$ +5 $$)**:
   - **Interpretation**: The constant $$ +5 $$ adds to the output of the cube root function.
   - **Effect**: This translates the graph **5 units upward**.

**Summary of Transformations**:
- **Translated 3 units to the right**.
- **Translated 5 units upward**.
- **Undergoes a vertical stretch**.

**Therefore, the correct transformations are:**

1. **The function is translated 3 units right.**
2. **The function is translated 5 units up.**
3. **The function undergoes a vertical stretch.**

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**Correct Answers:**
- The function is translated 3 units right.
- The function is translated 5 units up.
- The function undergoes a vertical stretch.
The function is translated 3 units to the right, 5 units up, and undergoes a vertical stretch.

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