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!