Mastering End Behavior: Leading Coefficient Test for Polynomials

Algebra 2 Grades High School 3:26 Video
Print Lesson Plan Watch Video

Lesson Description

Explore the end behavior of polynomial functions using the leading coefficient test. Learn how the degree and leading coefficient of a polynomial determine its behavior as x approaches positive and negative infinity.

Video Resource

End Behavior of Polynomial Functions Using Leading Coefficient Test

Mario's Math Tutoring

Duration: 3:26
Watch on YouTube

Key Concepts

  • Leading Coefficient Test
  • End Behavior of Polynomial Functions
  • Degree of a Polynomial
  • Limit Notation

Learning Objectives

  • Students will be able to determine the end behavior of a polynomial function given its equation.
  • Students will be able to explain how the leading coefficient and degree influence the end behavior of a polynomial function.
  • Students will be able to describe end behavior using limit notation.

Educator Instructions

  • Introduction (5 mins)
    Begin by briefly reviewing the definition of a polynomial function and its key components (terms, coefficients, degree). Introduce the concept of 'end behavior' and its significance in understanding the overall shape and characteristics of polynomial functions.
  • Video Presentation (8 mins)
    Play the video 'End Behavior of Polynomial Functions Using Leading Coefficient Test' by Mario's Math Tutoring. Instruct students to take notes on the key concepts discussed, including the four scenarios for end behavior, the influence of the leading coefficient, and the effect of the degree of the polynomial.
  • Discussion and Explanation (10 mins)
    Facilitate a class discussion to reinforce the concepts presented in the video. Emphasize the relationship between the leading coefficient (positive or negative) and the right-end behavior (up or down). Explain how the degree (even or odd) determines whether the left-end behavior is the same or opposite to the right-end behavior. Use examples from the video to illustrate these principles.
  • Worked Examples (12 mins)
    Work through several examples, similar to those in the video (4:23 - 7:54), demonstrating how to determine the end behavior of polynomial functions. Encourage students to participate by asking them to identify the leading coefficient, degree, and resulting end behavior. Introduce and practice using limit notation to describe end behavior (4:53).
  • Practice Problems (10 mins)
    Assign students practice problems to work on independently or in small groups. These problems should involve various polynomial functions with different leading coefficients and degrees. Circulate to provide assistance and answer questions.

Interactive Exercises

  • Card Sort
    Create cards with polynomial functions on some and descriptions of end behavior on others (e.g., 'Up to the Right, Down to the Left'). Have students match the functions to their corresponding end behavior descriptions.

Discussion Questions

  • Why is the leading coefficient the primary factor determining the right-end behavior of a polynomial function?
  • How does the degree of a polynomial function affect its left-end behavior, and why does this relationship exist?
  • Explain, in your own words, how to use the leading coefficient test to determine the end behavior of any polynomial function.

Skills Developed

  • Analytical Skills
  • Problem-Solving
  • Critical Thinking
  • Mathematical Reasoning

Multiple Choice Questions

Question 1:

What determines the right-end behavior of a polynomial function?

Correct Answer: The leading coefficient

Question 2:

If a polynomial has a negative leading coefficient and an even degree, what is its end behavior?

Correct Answer: Down to the left, down to the right

Question 3:

If a polynomial has a positive leading coefficient and an odd degree, what is its end behavior?

Correct Answer: Up to the right, down to the left

Question 4:

Which of the following functions will go 'down to the right'?

Correct Answer: f(x) = -2x^3 + x

Question 5:

The function f(x) = -5x^6 + 2x^3 - 1 will approach what as x approaches infinity?

Correct Answer: Negative infinity

Question 6:

What does the degree of a polynomial tell you about the end behavior?

Correct Answer: Whether the left and right ends behave the same or opposite

Question 7:

Which notation correctly describes the end behavior of a function that goes 'up to the right'?

Correct Answer: lim x→∞ f(x) = ∞

Question 8:

What is the leading coefficient in the polynomial function f(x) = -7x^3 + 4x^2 - x + 2?

Correct Answer: -7

Question 9:

Which function has the same end behavior on both the left and the right?

Correct Answer: f(x) = 2x^2 - 3

Question 10:

What is the end behavior of f(x) = x?

Correct Answer: Up to the right, down to the left

Fill in the Blank Questions

Question 1:

The ________ coefficient determines the right end behavior of a polynomial function.

Correct Answer: leading

Question 2:

If the degree of a polynomial is _______, the left and right ends will behave oppositely.

Correct Answer: odd

Question 3:

The end behavior of f(x) = -x^4 is down to the left and down to the _______.

Correct Answer: right

Question 4:

The function f(x) = x^3 approaches _______ infinity as x approaches positive infinity.

Correct Answer: positive

Question 5:

When using limit notation, lim x→∞ f(x) = ∞ means the function goes up to the ________.

Correct Answer: right

Question 6:

A polynomial with an even degree will have _______ end behavior.

Correct Answer: the same

Question 7:

For f(x) = 2x^5, as x approaches negative infinity, f(x) approaches ________ infinity.

Correct Answer: negative

Question 8:

If a leading coefficient is positive and the degree is even, the function opens ________.

Correct Answer: up

Question 9:

The end behavior can be described using _______ notation.

Correct Answer: limit

Question 10:

If the leading coefficient is negative, the function will go _______ to the right.

Correct Answer: down

Educational Standards

A2.F.1.5:Analyze the graph of a polynomial function by identifying the domain, range, intercepts, zeros, relative maxima, relative minima, and intervals of increase and decrease. A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans