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To determine the symmetry of the graph of the function \( y = \sqrt{x^2 + 9} \), we can analyze it with respect to the origin, the x-axis, and the y-axis.

### Step-by-Step Analysis:

1. **Symmetry with Respect to the y-axis:**
   To check for y-axis symmetry, replace \( x \) with \( -x \) in the equation:
   \[
   y = \sqrt{(-x)^2 + 9} = \sqrt{x^2 + 9}.
   \]
   Since we get the original equation back, the function is symmetric with respect to the y-axis.

2. **Symmetry with Respect to the x-axis:**
   To check for x-axis symmetry, replace \( y \) with \( -y \):
   \[
   -y = \sqrt{x^2 + 9}.
   \]
   Squaring both sides gives:
   \[
   y^2 = x^2 + 9.
   \]
   This equation is not the same as the original because we started with \( y = \sqrt{x^2 + 9} \), which only has non-negative values of \( y \). Thus, the function is **not** symmetric with respect to the x-axis.

3. **Symmetry with Respect to the Origin:**
   To check for origin symmetry, replace \( x \) with \( -x \) and \( y \) with \( -y \):
   \[
   -y = \sqrt{(-x)^2 + 9} = \sqrt{x^2 + 9}.
   \]
   As seen above, this is not the original equation, so the function is not symmetric with respect to the origin.

### Conclusion:

The function \( y = \sqrt{x^2 + 9} \) is symmetric with respect to the **y-axis only**.

**Correct Answer: A. Symmetric with respect to the y-axis only.**

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Would you like further details, or do you have any questions?

### Follow-Up Questions:
1. How can we determine if a function is symmetric with respect to a particular axis or the origin in general?
2. What are other common types of symmetry in mathematical functions?
3. How does squaring both sides of an equation affect the analysis of symmetry?
4. Can you give an example of a function that is symmetric with respect to the x-axis?
5. What is the geometric interpretation of a function symmetric about the y-axis?

**Tip:** When checking for symmetry, always substitute the appropriate negative variables and see if the original equation is preserved.

Determine whether the graph of y = √(x^2 + 9) is symmetric with respect to the origin, the x-axis, or the y-axis.

Determine the symmetry properties of the graph of y = √(x^2 + 9). This solution covers step-by-step analysis of symmetry with respect to the x-axis, y-axis, and the origin, exploring key concepts in graphing and functions. Suitable for high school students in Grades 9-12.

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