Lesson 2.3

End Behavior (Odd Degrees)

Odd degree polynomials () are "opposites." If one end goes up, the other MUST go down. They act just like lines.

Introduction

Unlike even powers, odd powers preserve the sign. A positive number stays positive, and a negative number stays negative. This creates ends that point in opposite directions.

Past Knowledge

goes Down/Up. goes Up/Down.

Today's Goal

Extend this rule to ALL odd degrees ().

Future Success

This confirms that every odd-degree polynomial MUST cross the x-axis at least once (Intermediate Value Theorem).

Key Concepts

1. The Odd Degree Rule

If the highest exponent is ODD, the ends must point in OPPOSITE directions.

Positive a

DOWN / UP

Like a Line with +Slope

Negative a

UP / DOWN

Like a Line with -Slope

2. Why?

Odd powers keep the sign of the input.

The left side (negative x) stays huge and negative. The right side (positive x) stays huge and positive.

Interactive Graph

Compare with and . They all start low and end high.

Worked Examples

Example 1: Positive Leading Coefficient

Basic

Describe .

Identify Degree and Sign

Degree: (Odd)
Leading Coefficient: (Positive)

Conclusion

Down / Up

(As and as )

Example 2: Negative Leading Coefficient

Concept

Describe .

Identify Degree and Sign

Degree: (Odd)
Leading Coefficient: (Negative)

Conclusion

Up / Down

Example 3: Tricky Factors

Advanced

Analyze .

Find Degree

(Odd).

Watch Calculation!

The term has a negative !
(Negative LC).

Odd + Negative = Up / Down

Common Pitfalls

Confusing Positive Slope

Students see "Positive" and think "Up/Up". For odd degrees, "Positive" means "Increasing" (starts low, ends high).

Real-Life Applications

Odd degree polynomials are used to model things that must change direction completely, like population growth that eventually stabilizes or price fluctuations. The fact that they go from negative to positive ensures there is always a "break-even point" (a real zero).