Lesson 2.5

Symmetry: Even and Odd Functions

Some functions are perfectly balanced. Learn the algebraic tests that reveal hidden mirror images and rotational symmetries.

Introduction

Some functions are perfectly balanced. Learn the algebraic tests that reveal hidden mirror images and rotational symmetries.

Past Knowledge

In Lesson 2.2, we learned about reflections. Even functions are their own reflection over the Y-axis. Odd functions are their own reflection over the Origin.

Today's Goal

We are learning the Algebraic Test for Symmetry. You replace with and see if the function stays the same (Even) or flips completely (Odd).

Future Success

Knowing a function is Odd instantly tells you that its integral over a symmetric interval (like -5 to 5) is ZERO. This saves pages of unnecessary work.

Key Concepts

The Algebraic Test:

To test ANY function for symmetry, substitute into the function and simplify. Compare the result to the original .

Even Functions

The function "eats" the negative sign. It doesn't care if x is positive or negative.

Visual: Y-Axis Symmetry
Odd Functions

The function "spits out" the negative sign. The entire output flips sign.

Visual: Origin Symmetry
Neither: If is neither exactly the same nor exactly the opposite (e.g., some terms flipped, some didn't), the function has No Symmetry. Most functions are "Neither."

Worked Examples

Level: Basic

Example 1: The Classic Parabola

Test for symmetry.

Step 1: Plug in (-x)
Step 2: Simplify
is just . So, .
Conclusion
The result is identical to the original. is EVEN.
Level: Intermediate

Example 2: The Cubic

Test for symmetry.

Step 1: Plug in (-x)
Step 2: Simplify
Conclusion
Every sign flipped (pos became neg, neg became pos). It equals . Thus, is ODD.
Level: Advanced (Calculus Prep)

Example 3: The Mixture

Test for symmetry.

Check Graph
It looks symmetric... but the axis of symmetry is , not the y-axis ()!
Algebraic Proof
Is the same as ? NO.
Is the opposite of ? NO.
Conclusion
NEITHER even nor odd. (Even though it has symmetry, just not rotational around the origin or reflective over y-axis).

Common Pitfalls

  • Assuming exponents tell the whole story:

    Yes, is even and is odd. But is NEITHER. You must expand terms or use the test to be sure when shifts are involved.

  • Confusing Odd Function vs Odd Numbers:

    The function (odd degree) is geometric "Neither" because the shift breaks the origin symmetry. A polynomial is only ODD if ALL its terms have odd powers (and no constant!).

Real-Life Applications

Signal Processing: Fourier Series

Any signal (music, wifi, voice) can be decomposed into a sum of Even functions (Cosines) and Odd functions (Sines). This is the foundation of Fourier Analysis.

Engineers exploit symmetry to cut computing time in half. If a signal is known to be Even, they don't bother calculating the Sine coefficients (because they are mathematically guaranteed to be zero).

Practice Quiz

Loading...