Let's analyze each graph and identify the applicable symmetries.

Graph 1

This graph resembles a sinusoidal (wave) function, which often has symmetry:

• Y-axis symmetry: True, as the function appears to mirror across the y-axis.
• X-axis symmetry: False, as it does not mirror across the x-axis.
• Origin symmetry: False, as it is not symmetric about the origin.
• Correct Answer: Only y-axis symmetry applies.

Graph 2

This graph is a straight line passing through the origin with a positive slope, likely representing a linear function like $y = x$.

• Y-axis symmetry: False, as it does not mirror across the y-axis.
• X-axis symmetry: False, as it does not mirror across the x-axis.
• Origin symmetry: True, as it is symmetric about the origin (rotational symmetry).
• Correct Answer: Only origin symmetry applies.

Graph 3

This graph shows a square centered at the origin.

• Y-axis symmetry: True, as it mirrors across the y-axis.
• X-axis symmetry: True, as it mirrors across the x-axis.
• Origin symmetry: True, as it has rotational symmetry about the origin.
• Correct Answer: X-axis, Y-axis, and origin symmetry all apply.

Let me know if you want more details or have questions!

Here are some related questions for further exploration:

1. What types of functions typically have origin symmetry?
2. How can you determine x-axis symmetry in a function's graph?
3. What transformations preserve y-axis symmetry?
4. Are there functions with all three symmetries (x-axis, y-axis, and origin)?
5. How does symmetry affect the properties of a function?

Tip: Symmetries can often be verified algebraically by testing conditions like $f(-x) = f(x)$ for y-axis symmetry or $f(-x) = -f(x)$ for origin symmetry.