Functions & Their Graphs

Even & Odd Functions

Identifying a function's symmetry algebraically and graphically.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Graphing Transformations

Functions with Built-In Symmetry

Before reading on, picture the graph of y = x² reflected over the y-axis. Does the reflected graph look any different from the original? Now picture y = x³ rotated 180° around the origin — does that graph look any different either?

Definition — Even and Odd Functions

A function is even if f(−x) = f(x) for every x — its graph is symmetric across the y-axis. A function is odd if f(−x) = −f(x) for every x — its graph has 180° rotational symmetry about the origin. Most functions are neither.

These algebraic tests aren't arbitrary — they directly encode the symmetries they name. Substituting −x into a function evaluates the mirror-image point across the y-axis. If that mirror-image point lands at the exact same height as the original (f(−x) = f(x)), the graph truly is symmetric across the y-axis — that's what 'even' checks. If it lands at the exact opposite height (f(−x) = −f(x)), the point directly opposite through the origin matches, which is precisely 180° rotational symmetry — that's what 'odd' checks.

Worked Example — Testing a Function for Even or Odd Symmetry

Test f(x) = x⁴ − 3x² + 5 for even/odd symmetry. Substitute −x: f(−x) = (−x)⁴ − 3(−x)² + 5 = x⁴ − 3x² + 5. Since f(−x) = f(x) exactly, this function is even.

Worked Example — Testing a Function That's Neither Even Nor Odd

Test f(x) = x² + x for even/odd symmetry. Substitute −x: f(−x) = (−x)² + (−x) = x² − x. Compare to f(x) = x² + x (not a match) and to −f(x) = −x² − x (also not a match). Since f(−x) equals neither f(x) nor −f(x), this function is neither even nor odd.

This connects directly to polynomial structure from Algebra II: a polynomial built entirely from even-degree terms (like x⁴ − 3x² + 5, degrees 4, 2, and 0) is always even, and one built entirely from odd-degree terms (like x³ − 4x) is always odd — the pattern in the name isn't a coincidence, since each individual even-degree term already satisfies (−x)ⁿ = xⁿ on its own, and each odd-degree term satisfies (−x)ⁿ = −xⁿ on its own.

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

For a polynomial, check the exponents directly: if every term's exponent is even, the whole polynomial is even; if every term's exponent is odd, the whole polynomial is odd; if the exponents are mixed, it's neither — no substitution needed for that quick check.

Common Mistakes

Concluding a function is odd just because it contains a mix of positive and negative terms, without actually testing f(−x).
Even and odd depend specifically on the exponents' parity (or on directly substituting −x and comparing), not on whether the coefficients happen to be positive or negative.
Assuming every function must be either even or odd, since those are the two names given.
Most functions are neither — even and odd describe two specific kinds of symmetry that many functions simply don't have.

Key Takeaways

An even function is symmetric across the y-axis (f(−x) = f(x)); an odd function has 180° rotational symmetry about the origin (f(−x) = −f(x)).
Substituting −x directly tests these symmetries algebraically.
A polynomial built entirely from even-degree terms is even; one built entirely from odd-degree terms is odd.

Summary

Even and odd symmetry reveal a function's built-in structure without needing to graph it. The next unit applies everything learned about function behavior to polynomials and rational functions of higher degree.

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