Lesson 2.2

End Behavior (Even Degrees)

What happens to the graph when gets really, really big? For even degree polynomials, the ends act just like a parabola.

Introduction

"End Behavior" describes what the arrows on the far left and far right of the graph are doing. We ignore the twists and turns in the middle.

Past Knowledge

goes Up/Up. goes Down/Down.

Today's Goal

Extend this rule to ALL even degrees ().

Future Success

This allows you to sketch rough graphs of complex functions in seconds.

Key Concepts

1. The Even Degree Rule

If the highest exponent is EVEN, the ends must point in the SAME direction.

Positive a

UP / UP

Like a Happy Face

Negative a

DOWN / DOWN

Like a Sad Face

2. Why?

When you square a huge negative number, it becomes positive.

Even powers "eat" the negative sign, forcing the left side to match the right side.

Interactive Graph

Zoom out to see that and eventually look just like .

Worked Examples

Example 1: Analyzing End Behavior

Basic

Describe the end behavior of .

Identify Degree and Sign

Degree: (Even)
Leading Coefficient: (Negative)

Conclusion

Down / Down

(As and as )

Example 2: Hidden Terms

Concept

Describe .

Find the Highest Degree

It's not written first! The highest degree is .

Conclusion

Degree 8 (Even)
Coefficient +1 (Positive)

Up / Up

Example 3: Factored Form

Advanced

Find the end behavior of .

Sum the Exponents

Add the exponents of all terms to find total degree.
.

Identify Sign of LC

Multiply the coefficients of each term.
(Negative).

Even () + Negative () = Down / Down

Common Pitfalls

Ignoring Standard Form

If the polynomial is , students often look at the 4. You MUST find the term with the big exponent ().

Looking at the End

Usually "end behavior" matches the "Leading Term". Don't look at the Constant term at the end of the equation!

Real-Life Applications

Even degree polynomials are used in bridge design (cables hang in shapes resembling parabolas or catenaries). Knowing that the ends inevitably go UP means the structure must be supported from above or anchored deeply into the ground.