To check for symmetry of the equation $y = \frac{x}{x^2 + 6}$ with respect to the y-axis, x-axis, and the origin, we use algebraic tests.

1. Symmetry with respect to the y-axis:

For y-axis symmetry, the function should satisfy: $f(-x) = f(x)$ Substitute $-x$ into the equation: $y = \frac{-x}{(-x)^2 + 6} = \frac{-x}{x^2 + 6}$ This is not equal to the original function $\frac{x}{x^2 + 6}$, so the function is not symmetric with respect to the y-axis.

2. Symmetry with respect to the x-axis:

For x-axis symmetry, the function should satisfy: $-y = f(x)$ Substitute $-y$ for $y$: $-y = \frac{x}{x^2 + 6}$ This gives: $y = -\frac{x}{x^2 + 6}$ Again, this is not the original equation, so the function is not symmetric with respect to the x-axis.

3. Symmetry with respect to the origin:

For origin symmetry, the function should satisfy: $f(-x) = -f(x)$ Substitute $-x$ into the equation: $f(-x) = \frac{-x}{(-x)^2 + 6} = \frac{-x}{x^2 + 6}$ Now check if this is equal to $-f(x)$: $-f(x) = -\frac{x}{x^2 + 6}$ Since $f(-x) = -f(x)$, the function is symmetric with respect to the origin.

Conclusion:

• Not symmetric with respect to the y-axis.
• Not symmetric with respect to the x-axis.
• Symmetric with respect to the origin.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How can a function be symmetric about the x-axis in general?
2. Can you provide another example of a function that is symmetric with respect to the origin?
3. How does the symmetry property affect the graph of the function?
4. How would you test for symmetry with respect to other types of lines or curves?
5. Can a function be symmetric with respect to both axes and the origin at the same time?

Tip: Origin symmetry implies that the graph of the function reflects through the point (0,0). This means rotating the graph 180 degrees around the origin will result in the same graph.