Which graph represents an odd function?
There are 6 points on a coordinate plane. The points are (negative 5, 3), (negative 3, 1), (negative 1, 1), (1, 3), (3, negative 5), (5, negative 1).
There are 8 points on a coordinate plane. The points are (negative 4, 2), (negative 3, negative 1), (negative 1, negative 3), (0, 1), (1, 0), (2, negative 4), (4, 5), (5, 4).
There are 8 points on a coordinate plane. The points are (negative 4, 0), (negative 3, 1), (negative 2, negative 3), (negative 1, negative 2), (1, 2), (2, 3), (3, negative 1), (4, 0)
There are 11 points on a coordinate plane. The points are (negative 4, 3), (negative 4, 4), (negative 3, 5), (negative 2, 1), (negative 1, negative 1), (0, negative 3), (1, negative 1), (2, 1), (3, 5), (4, 3), (4, 4).

To determine which graph represents an odd function, we need to recall the definition of an odd function:

- **A function \( f(x) \) is odd if it satisfies \( f(-x) = -f(x) \) for all \( x \) in the domain.**
- Graphically, an odd function is symmetric about the origin. That means if a point \( (x, y) \) is on the graph, then \( (-x, -y) \) must also be on the graph.

### Evaluation of Each Option:

#### 1. Six points:
\[
(-5, 3), (-3, 1), (-1, 1), (1, 3), (3, -5), (5, -1)
\]
- Symmetry test:
  - \( (1, 3) \neq (-1, -3) \): fails symmetry.
  - \( (3, -5) \neq (-3, 5) \): fails symmetry.
- **This is NOT an odd function.**

#### 2. Eight points:
\[
(-4, 2), (-3, -1), (-1, -3), (0, 1), (1, 0), (2, -4), (4, 5), (5, 4)
\]
- Symmetry test:
  - \( (4, 5) \neq (-4, -5) \): fails symmetry.
  - \( (1, 0) \neq (-1, 0) \): fails symmetry.
- **This is NOT an odd function.**

#### 3. Eight points:
\[
(-4, 0), (-3, 1), (-2, -3), (-1, -2), (1, 2), (2, 3), (3, -1), (4, 0)
\]
- Symmetry test:
  - \( (1, 2) = (-1, -2) \): passes symmetry.
  - \( (2, 3) = (-2, -3) \): passes symmetry.
  - \( (3, -1) = (-3, 1) \): passes symmetry.
  - \( (4, 0) = (-4, 0) \): passes symmetry.
- **This IS an odd function.**

#### 4. Eleven points:
\[
(-4, 3), (-4, 4), (-3, 5), (-2, 1), (-1, -1), (0, -3), (1, -1), (2, 1), (3, 5), (4, 3), (4, 4)
\]
- Symmetry test:
  - \( (3, 5) \neq (-3, -5) \): fails symmetry.
  - \( (4, 3) \neq (-4, -3) \): fails symmetry.
- **This is NOT an odd function.**

### Conclusion:
The graph with **eight points: (-4, 0), (-3, 1), (-2, -3), (-1, -2), (1, 2), (2, 3), (3, -1), (4, 0)** represents an odd function.

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Do you have any further questions or want additional details?

### Follow-Up Questions:
1. What are the characteristics of even functions, and how do they differ from odd functions?
2. Can a function be both odd and even? Explain why or why not.
3. How can you algebraically verify if a function is odd using its formula?
4. What is an example of a real-world scenario where an odd function applies?
5. How do transformations (shifts and stretches) affect the symmetry of odd functions?

**Tip:** Remember, an odd function always has origin symmetry, so it helps to visually check the points for this property!

Which graph represents an odd function? There are 6 points on a coordinate plane. The points are (negative 5, 3), (negative 3, 1), (negative 1, 1), (1, 3), (3, negative 5), (5, negative 1). There are 8 points on a coordinate plane. The points are (negative 4, 2), (negative 3, negative 1), (negative 1, negative 3), (0, 1), (1, 0), (2, negative 4), (4, 5), (5, 4). There are 8 points on a coordinate plane. The points are (negative 4, 0), (negative 3, 1), (negative 2, negative 3), (negative 1, negative 2), (1, 2), (2, 3), (3, -1), (4, 0). There are 11 points on a coordinate plane. The points are (negative 4, 3), (negative 4, 4), (negative 3, 5), (negative 2, 1), (negative 1, -1), (0, -3), (1, -1), (2, 1), (3, 5), (4, 3), (4, 4).

Explore how to identify odd functions using symmetry tests on given sets of coordinate points. Learn the mathematical properties of odd functions, including f(-x) = -f(x), and determine which graph satisfies the symmetry about the origin. Suitable for high school students.

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