Graphing Polynomials: Zeros and End Behavior

Lesson 1 of 2

In this lesson:

• Find zeros from factored form
• Determine end behavior from degree and leading coefficient
• Use technology when factoring is not available

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Identify zeros from a factored polynomial
2. Determine end behavior from degree and leading coefficient
3. Use technology when factoring is not possible
4. Sketch a polynomial using zeros and end behavior

Extending What You Know About Quadratics

You already know quadratics — degree 2 with parabolas:

• , zeros at
• End behavior: rises on both ends (positive leading coefficient)

Today: degree 3, 4, 5, and beyond — same ideas, more structure

Zeros Are the Skeleton of the Graph

Zeros (or roots) of : x-values where

• Zeros correspond to x-intercepts on the graph
• A degree- polynomial has at most real zeros
• In factored form, zeros are read directly from the factors

Key strategy: plot zeros first — they anchor everything else

Reading Zeros from Factored Form

• Factor
• Factor
• Factor

X-intercepts: , ,

Factor First, Then Read the Zeros

Step 1: GCF →
Step 2: Difference of squares →
Step 3: Zeros: , ,

When Factoring Is Not Available

Use technology: graph in Desmos, then use the "zero" feature

• Approximate zeros: , ,

Exact zeros need factoring; technology gives approximate zeros

Find the Zeros: Your Turn

What are the zeros of ?

Set each factor equal to zero — try it before the next slide.

Reading Zeros: Checking Your Answer

X-intercepts at , ,

From Zeros to the Full Picture

Zeros tell you where the graph hits the x-axis.

End behavior tells you what happens at the edges:

• Does the graph rise or fall as ?
• Does the graph rise or fall as ?

End Behavior: Only Two Things Matter

End behavior of as :

1. Degree: even or odd?
2. Leading coefficient: positive or negative?

All other terms are irrelevant for extreme x-values

Why? At : . The leading term dominates.

All Four End-Behavior Cases at Once

Two questions: Even or odd? Positive or negative leading coefficient?

Odd Degree: Tails Go Opposite Directions

Odd degree, positive leading coefficient:

Odd degree, negative leading coefficient: both directions flip

Key: odd degree → one tail rises, the other falls

Even Degree: Both Tails Same Direction

Even degree, positive leading coefficient:

Even degree, negative leading coefficient:

Key: even degree → both tails go the same direction

End Behavior Example: Even Degree

• Degree: 4 (even)
• Leading coefficient: (negative)
• End behavior: falls on both ends

"No matter how complicated the middle, both tails point downward."

Example: Odd Degree in Action

• Degree: 5 (odd)
• Leading coefficient: (positive)
• End behavior: falls left, rises right

What would change if the leading term were ?

State the End Behavior: Your Turn

What is the end behavior of ?

Two questions: even or odd degree? Positive or negative leading coefficient?

End Behavior: Checking Your Answer

• Degree: 6 (even)
• Leading coefficient: (negative)
• End behavior: falls on both ends

Find Zeros and End Behavior: Practice

Find zeros and state end behavior for each:

3. (end behavior only)

Work through each, then check the next slide.

Zeros and End Behavior: Full Answers

1. Zeros at ; degree 3, positive → falls left, rises right

2. Zeros at ; degree 3, negative → rises left, falls right

3. Degree 4, positive → rises on both ends

Lesson 1 Summary: Zeros and End Behavior

✓ Zeros come from factored form — set each factor to zero

✓ Only degree and leading coefficient determine end behavior

Watch out: middle terms do NOT affect end behavior

Watch out: a degree- polynomial has AT MOST zeros

Coming Up: Multiplicity and Polynomial Sketching

Lesson 2 of 2:

• Why graphs cross at some zeros and bounce at others
• Three-step sketch: zeros → end behavior → smooth curve
• Maximum turning points: at most

Combine everything into a complete polynomial sketch.