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.

1

Introduction

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

Why This Matters for Calculus: 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.

2

Explanation of 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."
3

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).
4

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!).

5

Real-World Application

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).

6

Practice Quiz

Loading...