Mastering End Behavior of Polynomial Functions

Algebra 2 Grades High School 5:10 Video
Print Lesson Plan Watch Video

Lesson Description

Explore the end behavior of polynomial functions, learning to predict graph direction based on leading terms and exponents.

Video Resource

End Behavior Algebra

Kevinmathscience

Duration: 5:10
Watch on YouTube

Key Concepts

  • End behavior of polynomial functions
  • Leading term and its influence on end behavior
  • Effect of even and odd exponents on end behavior
  • Positive and negative coefficients and their impact

Learning Objectives

  • Define end behavior of a polynomial function.
  • Determine the end behavior of a polynomial function based on its leading term.
  • Predict the direction of the graph (up or down) on the left and right sides of the coordinate plane.
  • Apply the concepts to solve related problems.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of a function and its graphical representation. Introduce the term 'end behavior' as what happens to the y-values as x approaches positive and negative infinity. Briefly explain its importance in understanding polynomial functions.
  • Video Viewing and Note-Taking (10 mins)
    Play the Kevinmathscience video 'End Behavior Algebra'. Instruct students to take notes on the key concepts, paying close attention to the method of substituting large positive and negative values for x in the leading term.
  • Guided Practice (15 mins)
    Work through example problems similar to those in the video, focusing on identifying the leading term and substituting infinity and negative infinity. Emphasize the importance of the exponent's parity (even or odd) and the coefficient's sign. Provide clear step-by-step explanations.
  • Independent Practice (15 mins)
    Provide students with a set of polynomial functions to analyze. Have them determine the end behavior for each function and sketch a possible graph representing the end behavior. Circulate to provide assistance and answer questions.
  • Wrap-up and Discussion (5 mins)
    Review the main points of the lesson. Address any remaining questions and preview upcoming topics. Assign homework for further practice.

Interactive Exercises

  • Matching Game
    Create cards with polynomial functions and corresponding end behavior descriptions (e.g., 'up on the left, down on the right'). Students match the functions to their correct end behavior.
  • Graph Sketching
    Provide students with functions and their end behaviors. Have them sketch possible graphs of the functions, ensuring the ends of the graphs match the specified behavior. Discuss the multiple possible graphs that can satisfy the end behavior conditions.

Discussion Questions

  • How does the degree of the polynomial affect its end behavior?
  • How does the sign of the leading coefficient affect the end behavior?
  • Can a function have different end behavior on the left and right sides?
  • How does end behavior relate to the range of a polynomial function?

Skills Developed

  • Analyzing polynomial functions
  • Predicting graphical behavior
  • Abstract reasoning
  • Problem-solving

Multiple Choice Questions

Question 1:

What does the end behavior of a polynomial function describe?

Correct Answer: The behavior of the function as x approaches positive or negative infinity

Question 2:

Which part of a polynomial function primarily determines its end behavior?

Correct Answer: The leading term

Question 3:

If a polynomial function has a leading term with an even exponent and a positive coefficient, what is its end behavior?

Correct Answer: Up on both sides

Question 4:

If a polynomial function has a leading term with an odd exponent and a negative coefficient, what is its end behavior?

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

Question 5:

What is the end behavior of the function f(x) = x^3 + 2x - 1 as x approaches positive infinity?

Correct Answer: y approaches positive infinity

Question 6:

What is the end behavior of the function f(x) = -x^2 + 5x + 3 as x approaches negative infinity?

Correct Answer: y approaches negative infinity

Question 7:

Which of the following functions has the end behavior of going down on both sides?

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

Question 8:

The end behavior of a polynomial function is up on the left and up on the right. Which of the following could be the leading term of the function?

Correct Answer: x^2

Question 9:

How does the constant term of a polynomial function affect its end behavior?

Correct Answer: It does not affect the end behavior

Question 10:

The graph of a polynomial function shows that as x approaches both positive and negative infinity, y approaches negative infinity. What can you conclude about the leading term?

Correct Answer: It has an even exponent and a negative coefficient

Fill in the Blank Questions

Question 1:

The behavior of a polynomial function as x approaches positive or negative infinity is called its __________ ___________.

Correct Answer: end behavior

Question 2:

The __________ __________ of a polynomial function is the term with the highest power of x.

Correct Answer: leading term

Question 3:

If the leading term of a polynomial function has an odd exponent and a positive coefficient, the graph goes _______ on the right.

Correct Answer: up

Question 4:

If the leading term of a polynomial function has an even exponent and a negative coefficient, the graph goes _______ on both sides.

Correct Answer: down

Question 5:

For the function f(x) = -2x^5 + 3x^2 - 1, the leading coefficient is _________.

Correct Answer: -2

Question 6:

As x approaches positive infinity, the y-value of the function f(x) = x^4 + 2x - 5 approaches _________ ___________.

Correct Answer: positive infinity

Question 7:

As x approaches negative infinity, the y-value of the function f(x) = -x^3 + x - 2 approaches _________ ___________.

Correct Answer: positive infinity

Question 8:

A polynomial function with the end behavior going up on the left and down on the right has a leading term with an _______ exponent and a _________ coefficient.

Correct Answer: odd, negative

Question 9:

The end behavior of a polynomial function is not affected by its __________ term.

Correct Answer: constant

Question 10:

The direction of the graph on either side depends on the sign of the coefficient and whether the exponent of the variable is ___________ or __________.

Correct Answer: even, odd

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.

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