https://file.aiquickdraw.com/mathbot/file/1728960659754bixug058.jpg

The problem asks you to determine the transformations needed to graph the function:

\[
h(x) = \frac{1}{2} \cdot \sqrt[3]{x + 7}
\]

This function is a transformation of the cube root function:

\[
f(x) = \sqrt[3]{x}
\]

Let's break down the transformations:

### 1. **Horizontal shift:**
The term \(x + 7\) inside the cube root indicates a horizontal shift. To shift a function horizontally, you replace \(x\) with \(x - h\), where \(h\) is the shift. Since it is \(x + 7\), the graph of \(f(x)\) is shifted **7 units to the left**.

### 2. **Vertical shrink:**
The factor \(\frac{1}{2}\) in front of the cube root function is a vertical transformation. Specifically, multiplying the function by \(\frac{1}{2}\) shrinks the graph vertically by a factor of \(\frac{1}{2}\).

### Conclusion:
The graph of \(f(x) = \sqrt[3]{x}\) is horizontally shifted 7 units to the left and vertically shrunk by a factor of \(\frac{1}{2}\).

### Correct Answer:
The correct answer is **Option F**: "The graph of \( f(x) = \sqrt[3]{x} \) should be horizontally shifted to the left by 7 units and shrunk vertically by a factor of \( \frac{1}{2} \)."

Do you need more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. What is the effect of multiplying a function by a factor greater than 1?
2. How would the transformation change if the function was \( h(x) = 2\sqrt[3]{x - 7} \)?
3. How do you determine whether a shift is to the left or right?
4. What does a vertical stretch of a function look like?
5. How would adding a constant outside the cube root (e.g., \( h(x) = \sqrt[3]{x} + 3 \)) affect the graph?

**Tip:** For horizontal shifts, always look inside the function, and for vertical transformations, check the factor outside the function.

What transformations are needed to graph the function h(x) = 1/2 * ∛(x + 7)? Choose the correct answer below.

Understand how to apply transformations such as horizontal shifts and vertical shrinkage to graph the function h(x) = 1/2 * ∛(x + 7). This problem covers key algebraic concepts for transforming functions and is suitable for students in Grades 10-12.

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