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To determine the symmetry of the function represented by the given table of values, we need to analyze the \( (x, y) \) pairs and see if any symmetrical properties can be identified. Here are the values:

\[ 
\begin{array}{cc}
x & y \\
\hline
-3 & -6 \\
-2 & 5 \\
-1 & 4 \\
0 & -3 \\
1 & 4 \\
2 & 5 \\
3 & -6 \\
\end{array}
\]

### Symmetry Analysis

1. **Symmetric about the Origin (Odd Function):**
   For a function \( f(x) \) to be symmetric about the origin, \( f(-x) = -f(x) \) for all \( x \).

   - \( f(-3) = -6 \) and \( f(3) = -6 \), so \( -f(3) = 6 \neq f(-3) \)
   - \( f(-2) = 5 \) and \( f(2) = 5 \), so \( -f(2) = -5 \neq f(-2) \)
   - \( f(-1) = 4 \) and \( f(1) = 4 \), so \( -f(1) = -4 \neq f(-1) \)
   - \( f(0) = -3 \) and \( f(0) = -3 \), so \( -f(0) = 3 \neq f(0) \)

   The function is not symmetric about the origin.

2. **Symmetric about the Y-axis (Even Function):**
   For a function \( f(x) \) to be symmetric about the y-axis, \( f(-x) = f(x) \) for all \( x \).

   - \( f(-3) = -6 \) and \( f(3) = -6 \), so \( f(3) = f(-3) \)
   - \( f(-2) = 5 \) and \( f(2) = 5 \), so \( f(2) = f(-2) \)
   - \( f(-1) = 4 \) and \( f(1) = 4 \), so \( f(1) = f(-1) \)
   - \( f(0) = -3 \), which does not contradict anything

   The function is symmetric about the y-axis.

3. **Symmetric about the X-axis:**
   For a function \( f(x) \) to be symmetric about the x-axis, both \( f(x) \) and \( -f(x) \) would need to be valid outputs for the same \( x \). This is typically not a property of functions, as each input should have only one output.

   The function is not symmetric about the x-axis.

4. **Not Symmetric:**
   If the function does not exhibit any of the above symmetries.

Given the values, the function shows symmetry about the y-axis.

**Answer:**
c) symmetric about the y-axis

Would you like any further details or have any questions?

### Additional Related Questions

1. What is an example of a polynomial that is symmetric about the origin?
2. How can you test for symmetry about the y-axis in a general polynomial function?
3. What are the implications of a function being symmetric about the origin?
4. Can a function be symmetric about both the x-axis and the y-axis? Explain.
5. How would you determine the degree of the polynomial from a table of values?
6. What are the key characteristics of even and odd functions?
7. How can you graphically determine if a function is symmetric about the y-axis?
8. What are some real-world examples of functions that are symmetric about the y-axis?

### Tip:
To test for symmetry in a function, substituting \(-x\) for \(x\) and comparing the results can be very useful. For even functions, \(f(-x) = f(x)\), and for odd functions, \(f(-x) = -f(x)\).

Explore the concept of symmetry in functions with a detailed analysis of odd and even functions. Understand how to determine symmetry about the y-axis using specific examples and mathematical analysis. Suitable for students in Grades 9-12.

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