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Polynomial Functions & End Behavior

Eleventh graders learn to predict the end behavior of polynomial functions — which way the ends of the graph point — using just the degree (even or odd) and the sign of the leading coefficient.

Grade 11Polynomials55 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Identify the degree.
  • ✓Identify the leading coefficient.
  • ✓Predict end behavior.
  • ✓Use even vs. odd degree.
Essential Question

Without graphing a single point, you can predict which way a polynomial’s graph shoots off at the far ends. The secret is just two things: the degree and the leading coefficient.

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Polynomial Explorer · Interactive

The Polynomial Explorer

Project this and tap each idea. Have students learn how degree and leading coefficient control a polynomial’s end behavior!

📈 Explore polynomial end behavior — tap eachTry it

The Lesson · Gradual Release (I Do · We Do · You Do)

55 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — Predict Without Graphing

5 min

Can you tell where a graph’s ends point — without graphing it?

👩‍🏫 Teacher Moves

  • Show a polynomial.
  • Ask about its ends.
  • Set the goal: end behavior.

🎒 Student Actions

  • See the polynomial.
  • Wonder the ends.
  • Predict the idea.
2

I Do — Degree & Leading Coefficient

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Degree sets the shape.
  • Leading coefficient sets direction.
  • Even vs. odd degree.

🎒 Student Actions

  • Learn degree.
  • Learn coefficient.
  • Compare even/odd.
3

We Do — Explore Together

13 min

Class uses the Polynomial Explorer.

👩‍🏫 Teacher Moves

  • Tap each idea.
  • Predict end behavior.
  • Explain the rule.

🎒 Student Actions

  • Tap each.
  • Predict it.
  • Explain it.
4

You Do — On Your Own

12 min

Students predict.

👩‍🏫 Teacher Moves

  • Find degree and leading coefficient.
  • Predict end behavior.
  • Sketch the ends.

🎒 Student Actions

  • Find them.
  • Predict it.
  • Sketch it.
5

Close — Close

5 min

One polynomial.

👩‍🏫 Teacher Moves

  • Give a polynomial.
  • Predict its end behavior.
  • Hand out the exit ticket.

🎒 Student Actions

  • Predict it.
  • Explain it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School Algebra II).

CCSS
F-IF.7c

Graph polynomial functions, identifying zeros and end behavior.

CCSS
A-APR.3

Identify zeros of polynomials and use them to sketch a graph.

CCSS
F-IF.4

Interpret key features of graphs.

CCSS
F-BF.3

Identify the effect of transformations on graphs.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • End-behavior diagram.
  • Sentence frame: “Odd degree → ends point ___.”
  • Use graph sketches.

Support & Access

IEP / 504
  • Start with even vs. odd.
  • Use visual sketches.
  • Focus on the ends only.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Find zeros and multiplicity.
  • Sketch a full polynomial graph.
  • Match graphs to equations.
Materials

What to gather

  • 📽️Projector / board
  • 📈Graphing calculators
  • 💻The Polynomial Explorer
  • 📄Graph sketches
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

polynomiala sum of terms with whole-number exponentsdegreethe highest exponentleading coefficientthe coefficient of the highest-degree termend behaviorwhat the graph does at the far endseven degreea highest exponent that is evenodd degreea highest exponent that is oddzeroan x-value where the graph crossesturning pointwhere the graph changes direction
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
What two things determine a polynomial’s end behavior?
The degree (even or odd) and the sign of the leading coefficient.
QUESTION 2
An even-degree polynomial with a positive leading coefficient — which way do both ends point?
Both up.
QUESTION 3
Does x³ (odd degree) have ends pointing the same way or opposite ways?
Opposite ways (one up, one down).

Going deeper? Zeros and turns.

Have students combine end behavior with zeros to sketch a full polynomial graph. A printable polynomial sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Polynomial End BehaviorFlip
Card 1
Term
Tap to flip →
Meaning
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Nice work!

Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Polynomial End BehaviorQuiz
Score: 0
1 / 6
Question 1
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Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Polynomial End BehaviorPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: degree, end behavior, even degree, leading coefficient, odd degree, polynomial, turning point, zero
  1. the highest exponent
  2. a highest exponent that is even
  3. an x-value where the graph crosses
  4. where the graph changes direction
  5. the coefficient of the highest-degree term
  6. what the graph does at the far ends
  7. a highest exponent that is odd
  8. a sum of terms with whole-number exponents

Part B · Show what you learned

  1. What two things determine a polynomial’s end behavior?
  2. An even-degree polynomial with a positive leading coefficient — which way do both ends point?
  3. Does x³ (odd degree) have ends pointing the same way or opposite ways?
Answer key — Part A: 1) degree · 2) even degree · 3) zero · 4) turning point · 5) leading coefficient · 6) end behavior · 7) odd degree · 8) polynomial
Part B: 1) The degree (even or odd) and the sign of the leading coefficient. 2) Both up. 3) Opposite ways (one up, one down).