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We are tasked with analyzing the graph of the function \( g(x) = \frac{1}{3} \sqrt{x - 5} \) using transformations of the basic square root function \( f(x) = \sqrt{x} \).

Let's break down the transformations in \( g(x) \):

1. **Horizontal Shift**: The term \( x - 5 \) indicates a horizontal shift. The graph of \( f(x) = \sqrt{x} \) will shift **to the right by 5 units** because the general form of a horizontal shift is \( f(x - h) \), which shifts the graph right by \( h \) units.

2. **Vertical Stretch/Shrink**: The coefficient \( \frac{1}{3} \) outside the square root indicates a **vertical shrink**. A number less than 1 (in absolute value) multiplies the output of the function, thus compressing the graph vertically.

### Combining both transformations:
- A horizontal shift to the **right by 5 units**.
- A **vertical shrink** by a factor of \( \frac{1}{3} \).

Now, let's review the multiple-choice answers:
- **Option A**: Horizontal shift left by 5 units and horizontal stretch by \( \frac{1}{3} \) – Incorrect.
- **Option B**: Horizontal shift right by 5 units and vertical shrink by \( \frac{1}{3} \) – Correct.
- **Option C**: Horizontal shift left by 5 units and vertical shrink by \( \frac{1}{3} \) – Incorrect.
- **Option D**: Horizontal shift right by 5 units and horizontal stretch by \( \frac{1}{3} \) – Incorrect.

### Correct Answer: **B**.

Would you like further details on how these transformations affect the graph? Here are some related questions to consider:

1. How do horizontal shifts impact the graph of any function?
2. What is the effect of vertical stretches and shrinks on a graph?
3. How would the graph change if the \( \frac{1}{3} \) coefficient was negative?
4. Can you explain the difference between horizontal and vertical transformations?
5. How do transformations differ for other types of functions, like quadratics?

**Tip**: When analyzing transformations, always address horizontal and vertical changes separately for clarity.

What transformations are needed to graph the function g(x) = 1/3 √(x - 5)? Choose the correct answer from the multiple-choice options.

Explore the transformations needed to graph the function g(x) = 1/3 √(x - 5). This problem involves horizontal shifts and vertical shrinking of the basic square root function. Suitable for students in Grades 9-11.

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