Question

Use GeoGebra to graph f(x)=x−−√+9
and f(3x)=(3x)−−−−√+9
. Based on the graph, what is the value of k? Round your answer to the nearest whole number.(1 point)

Answer 1

INCORRECT

Answer 2

To analyze the functions \( f(x) = \sqrt{x} + 9 \) and \( f(3x) = \sqrt{3x} + 9 \) and find the value of \( k \), we can explore how the graphs of these functions behave.

### Step 1: Understanding the Functions

1. **Function \( f(x) \)**:
\[
f(x) = \sqrt{x} + 9
\]
This function starts from 9 when \( x = 0 \) and increases as \( x \) increases, reflecting the square root function shifted upwards by 9 units.

2. **Function \( f(3x) \)**:
\[
f(3x) = \sqrt{3x} + 9
\]
This function also starts from 9 when \( x = 0 \) and increases, but because of the \( 3x \) inside the square root, it grows faster than \( f(x) \).

### Step 2: Find Relationships and Behavior

To analyze their relationship, consider values of \( x \):
- For \( x = 1 \):
- \( f(1) = \sqrt{1} + 9 = 10 \)
- \( f(3 \cdot 1) = \sqrt{3} + 9 \approx 10.732 \)

- For \( x = 2 \):
- \( f(2) = \sqrt{2} + 9 \approx 10.414 \)
- \( f(3 \cdot 2) = \sqrt{6} + 9 \approx 10.449 \)

- For \( x = 3 \):
- \( f(3) = \sqrt{3} + 9 \approx 10.732 \)
- \( f(3 \cdot 3) = \sqrt{9} + 9 = 12 \)

### Step 3: Graphing

If you graph both functions in GeoGebra or any graphing tool, you would typically observe that:
- \( f(3x) \) starts and rises more steeply compared to \( f(x) \).

### Step 4: Finding \( k \)

Assuming the prompt is suggesting finding the intersection value where the heights of both functions equal each other, we might calculate the point where \( f(x) = f(3x) \):
\[
\sqrt{x} + 9 = \sqrt{3x} + 9
\]
Eliminating 9 gives:
\[
\sqrt{x} = \sqrt{3x}
\]

Squaring both sides leads to:
\[
x = 3x
\]
which reduces to \( 0 = 2x \), hence \( x = 0 \).

However, if the question refers to the height of the function or any integer \( k \), we'd consider the y-values they attain at specific x-values, or perhaps a specific coordinate reached within the graphing limits.

### Conclusion

Upon observing the graphs over a reasonable \( x \) range, you would find that at different \( x \) values, both functions attain a rounded \( y \) value that represents \( k \). If you can specify or simulate in GeoGebra, measure certain heights.

The best estimate from observations \( \sqrt{3} + 9 \approx 10.732 \) suggests \( k \) might be \( 11 \) depending on reference.

**Final round** answer: **\( k \approx 11 \)**.