AP PRECALCULUS — UNIT 1A · POLYNOMIAL & RATIONAL FUNCTIONS
1.5B — Even and Odd Polynomials
Notes — Symmetry of Functions
💡 Learning Objectives (1.5.B.1 and 1.5.B.2) By the end of this lesson you will be able to:
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1. The Two Symmetries
Many functions have one of two symmetric shapes. Recognizing the symmetry up front simplifies graphing, integration, equation-solving, and reasoning.
1.1 Even functions
💡 Even function: definition A function f is called even if for every x in its domain, f(−x) = f(x). Geometrically, the graph of f is symmetric across the y-axis: reflect the right half across the y-axis to recover the left half (and vice versa). |
1.2 Odd functions
💡 Odd function: definition A function f is called odd if for every x in its domain, f(−x) = −f(x). Geometrically, the graph is symmetric about the origin (a 180° rotation about the origin maps the graph to itself). |
Even functions are mirror images across the y-axis. Odd functions look the same after a 180° spin around the origin.
2. Power Functions Are the Building Blocks
💡 Pure power functions Consider f(x) = aₙxⁿ where aₙ ≠ 0.
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Even powers all share the same y-axis symmetry; odd powers all share the same origin symmetry. The names ‘even/odd polynomial’ come directly from this.
3. Polynomials with Mixed Terms
A general polynomial like p(x) = 2x⁴ + 3x² + 1 has only even-power terms (treating the constant 1 as 1 · x⁰, an even power). Substituting −x for x leaves it unchanged, so it is even.
Likewise, p(x) = x⁵ − 4x has only odd-power terms (treating x as x¹, odd). Substituting −x gives −p(x), so it is odd.
But a polynomial like p(x) = x³ + x² has one odd-power term and one even-power term — it is neither even nor odd.
💡 Quick test for polynomials
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📘 Example — Classifying polynomials • p(x) = 4x⁶ − x⁴ + 7 → all exponents even (6, 4, 0) → even function. • p(x) = 5x⁷ + 2x³ − x → all exponents odd → odd function. • p(x) = x⁴ − x³ + 2 → mix of even and odd exponents → neither. • p(x) = 0 → trivially both even and odd. (The unique exception.) |
4. Verifying with the Algebraic Definition
📘 Example — Confirming evenness algebraically Verify that p(x) = 3x⁴ − 2x² + 7 is even. p(−x) = 3(−x)⁴ − 2(−x)² + 7 = 3x⁴ − 2x² + 7 = p(x). ✓ |
📘 Example — Confirming oddness algebraically Verify that p(x) = x⁵ − 6x is odd. p(−x) = (−x)⁵ − 6(−x) = −x⁵ + 6x = −(x⁵ − 6x) = −p(x). ✓ |
📘 Example — Showing a polynomial is neither Show that p(x) = x³ − x² + 1 is neither even nor odd. p(−x) = −x³ − x² + 1, which is neither equal to p(x) (even check) nor equal to −p(x) (odd check). So p is neither. |
5. Why Symmetry Helps
Recognizing symmetry pays off in several ways:
- Halve the graphing work — sketch one side of the y-axis, then mirror or rotate to get the other
- Quickly check answers — a value computed at x = 5 should match (or negate) the value at x = −5
- Spot zeros faster — for an odd function, x = 0 is automatically a zero
- Predict end behavior — both ends of an even-degree polynomial behave the same way; the two ends of an odd-degree polynomial behave oppositely
⚠️ Even/Odd ≠ Even-degree/Odd-degree Don't confuse the parity of a function (even or odd, defined by the symmetry rules) with the parity of a polynomial's degree (even or odd integer). • An even-degree polynomial may not be an even function. Example: p(x) = x² + x has degree 2 (even) but is not an even function. • An odd-degree polynomial may not be an odd function. Example: p(x) = x³ + 1 has degree 3 (odd) but is not an odd function. |
6. Summary
- Even function: f(−x) = f(x); graph is symmetric across the y-axis
- Odd function: f(−x) = −f(x); graph is symmetric about the origin
- A pure power function f(x) = aₙxⁿ is even when n is even and odd when n is odd
- A polynomial is even ⇔ all of its terms have even exponents; odd ⇔ all odd
- If terms mix odd and even exponents, the polynomial is neither even nor odd