An odd function is symmetric with respect to which axis?

An odd function is symmetric with respect to which axis?

• A. The origin
• B. The x-axis
• C. The y-axis
• D. The line y = x

Answer: (B)

Odd functions have rotational symmetry about the origin, meaning if you rotate the graph 180 degrees around the origin, it looks the same. This is described by f(-x) = -f(x). So the symmetry is centered at the origin, not along the x-axis. If a graph were symmetric about the x-axis, reflecting across the x-axis would turn (x, y) into (x, -y). For an odd function, that would imply f(x) = -f(x) for all x, which forces f to be zero everywhere, not a general case. Similarly, symmetry about the y-axis would mean even behavior with f(-x) = f(x), which isn’t the defining property of odd functions. An example like f(x) = x shows the origin symmetry clearly: the point (1,1) pairs with (-1,-1) under a 180-degree rotation, and the graph remains the same.

Odd functions have rotational symmetry about the origin, meaning if you rotate the graph 180 degrees around the origin, it looks the same. This is described by f(-x) = -f(x). So the symmetry is centered at the origin, not along the x-axis.

If a graph were symmetric about the x-axis, reflecting across the x-axis would turn (x, y) into (x, -y). For an odd function, that would imply f(x) = -f(x) for all x, which forces f to be zero everywhere, not a general case. Similarly, symmetry about the y-axis would mean even behavior with f(-x) = f(x), which isn’t the defining property of odd functions. An example like f(x) = x shows the origin symmetry clearly: the point (1,1) pairs with (-1,-1) under a 180-degree rotation, and the graph remains the same.