https://file.aiquickdraw.com/mathbot/file/1727640574281moukx4uu.jpg

Let's analyze the problem presented in the image:

We are given two functions:
1. \( f(x) = \sqrt{x} \)
2. \( g(x) = \sqrt{x} + 5 \)

You are asked to:
- Use integer values of \(x\) from the given set \( \{0, 1, 4, 9\} \) to find ordered pairs for both functions.
- Graph the functions in the same rectangular coordinate system.
- Describe how the graph of \(g(x)\) is related to the graph of \(f(x)\).

### Step 1: Calculate the ordered pairs for \( f(x) \) and \( g(x) \)

For \( f(x) = \sqrt{x} \):
- \( f(0) = \sqrt{0} = 0 \)
- \( f(1) = \sqrt{1} = 1 \)
- \( f(4) = \sqrt{4} = 2 \)
- \( f(9) = \sqrt{9} = 3 \)

So, the ordered pairs for \( f(x) \) are:
- \( (0, 0), (1, 1), (4, 2), (9, 3) \)

For \( g(x) = \sqrt{x} + 5 \):
- \( g(0) = \sqrt{0} + 5 = 0 + 5 = 5 \)
- \( g(1) = \sqrt{1} + 5 = 1 + 5 = 6 \)
- \( g(4) = \sqrt{4} + 5 = 2 + 5 = 7 \)
- \( g(9) = \sqrt{9} + 5 = 3 + 5 = 8 \)

So, the ordered pairs for \( g(x) \) are:
- \( (0, 5), (1, 6), (4, 7), (9, 8) \)

### Step 2: Graphing the functions

The graph of \( g(x) = \sqrt{x} + 5 \) is a vertical translation of the graph of \( f(x) = \sqrt{x} \). Specifically, the graph of \( g(x) \) is shifted 5 units upward from the graph of \( f(x) \). This is because adding a constant (in this case, 5) to the function shifts it vertically.

### Conclusion:
The graph of \( g(x) \) is the same shape as the graph of \( f(x) \), but every point on the graph of \( g(x) \) is shifted 5 units higher than the corresponding point on the graph of \( f(x) \).

Would you like me to graph these functions or provide further details? Here are some additional related questions:

1. What is the domain of both \( f(x) \) and \( g(x) \)?
2. How would the graph change if \( g(x) = \sqrt{x} - 5 \)?
3. Can the functions \( f(x) \) and \( g(x) \) ever intersect?
4. How would you modify the equation to shift the graph of \( f(x) \) 5 units to the right instead?
5. What are some real-world applications of square root functions?

**Tip**: When graphing, shifts like \( +5 \) or \( -5 \) are vertical translations; similarly, terms added or subtracted inside the square root (e.g., \( \sqrt{x - h} \)) result in horizontal shifts.

Graph the given functions, f(x) = √x and g(x) = √x + 5, in the same rectangular coordinate system. Use the integer values of x (0, 1, 4, 9) to obtain ordered pairs. Describe how the graph of g is related to the graph of f.

Learn how to graph the square root functions f(x) = √x and g(x) = √x + 5 in the same coordinate system. Explore how the graph of g(x) is a vertical translation of f(x), with detailed steps and ordered pairs for x values 0, 1, 4, and 9. Suitable for students in Grades 8-10.

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