Symmetry about the origin

Definition

A function is symmetric about the origin if rotating its graph 180 degrees around the origin does not change the graph. Mathematically, this means $f(-x) = -f(x)$ for all $x$ in the domain of $f$.

5 Must Know Facts For Your Next Test

1. Symmetry about the origin implies that the function is an odd function.
2. If a function is symmetric about the origin, then its graph will always pass through the origin.
3. Checking for symmetry about the origin involves substituting $-x$ into the function and verifying if it equals $-f(x)$.
4. Common examples of functions with symmetry about the origin include $y = x^3$ and $y = \sin(x)$.
5. Not all functions have symmetry properties; some may be neither even nor odd.

Review Questions

• What does it mean for a function to be symmetric about the origin?
• How can you test whether a given function is symmetric about the origin?
• Provide an example of a function that is symmetric about the origin.

Related terms

Odd Function: A function $f$ is called odd if $f(-x) = -f(x)$ for all $x$ in its domain.

Even Function: $A function f is called even if f(-x)=f(x)$ for all x in its domain.

Symmetry About The Y-Axis: $A graph exhibits symmetry about the y-axis if replacing x with -x yields an equivalent equation, which means f(-x)=f(x).