Odd Functions

Odd functions are symmetric about the origin, not an axis — rotate the graph 180° and it lands on itself. The two-step reflection to draw the missing half, the algebra test f(−x) = −f(x), and the classic examples: y = x, y = x³, and sine.

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An odd function is symmetric about the origin, not an axis. The graph on the right is the opposite of the graph on the left — rotate it around the origin and it lands right back on itself.

Symmetric About the Origin Opposite inputs, opposite outputs

The value at is the negative of the value at . The value at is the negative of the value at . That pattern — every point paired with its exact opposite across the origin — is what makes a function odd.

Drawing The Other Half Two reflections instead of one
visible part step 1: reflect in y-axis step 2: reflect in x-axis
Seeing only the right side is enough. First reflect it across the y-axis, as if the function were even. Then reflect that new part across the x-axis. The result is the same as rotating the original about the origin.
The Algebra Test

A function is odd when for every in its domain.

and

is exactly — the negative of the value at .

Common Examples , , and sine

Both pass straight through the origin with the same point symmetry — and so does the sine function.

Summary
  1. Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.
  2. A rotation about the origin leaves an odd function’s graph unchanged.
  3. To complete an odd graph from one side: reflect across the y-axis, then across the x-axis.
  4. Algebraically, . Common examples: , , and .
287. Function Symmetry289. Functions Are Everywhere
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