Unlocking End Behavior: Mastering Polynomial Graphs

Algebra 2 Grades High School 10:24 Video
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Lesson Description

Learn to predict the end behavior of polynomial functions using the leading coefficient test. This lesson covers identifying leading coefficients, understanding even/odd degrees, and using proper notation.

Video Resource

End Behavior of Polynomials Using Leading Coefficient Test

Mario's Math Tutoring

Duration: 10:24
Watch on YouTube

Key Concepts

  • Leading Coefficient Test
  • Even and Odd Degree Polynomials
  • End Behavior Notation (Limit and Arrow Notation)

Learning Objectives

  • Students will be able to determine the right and left end behavior of a polynomial function using the leading coefficient test.
  • Students will be able to write the end behavior of a polynomial function using proper notation (arrow and limit notation).

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the general form of a polynomial function and the definition of the leading coefficient and degree. Briefly discuss the concept of 'end behavior' as the behavior of the graph as x approaches positive or negative infinity.
  • Video Presentation (10 mins)
    Play the YouTube video 'End Behavior of Polynomials Using Leading Coefficient Test' by Mario's Math Tutoring. Encourage students to take notes on the key concepts and examples presented.
  • Guided Practice (15 mins)
    Work through several examples together, similar to those in the video. Emphasize identifying the leading coefficient and degree, and then applying the leading coefficient test. Write out the end behavior using both arrow notation and limit notation for each example.
  • Independent Practice (15 mins)
    Provide students with a worksheet or online assignment containing polynomial functions. Have them determine the end behavior of each function and write it in both arrow and limit notation. Circulate to provide assistance as needed.
  • Wrap-up and Assessment (10 mins)
    Review the key concepts and address any remaining questions. Administer a short quiz (multiple choice and fill-in-the-blank) to assess student understanding.

Interactive Exercises

  • Polynomial End Behavior Matching
    Create a matching activity where students match polynomial functions with their corresponding end behavior descriptions in arrow and limit notation.
  • Graphing Calculator Exploration
    Have students graph polynomial functions using a graphing calculator or online tool. Observe the end behavior of the graphs and relate it to the leading coefficient and degree.

Discussion Questions

  • How does the leading coefficient affect the right-hand behavior of a polynomial?
  • How does the degree of the polynomial affect the left-hand behavior?
  • Explain the difference between arrow notation and limit notation for describing end behavior.
  • Can you think of real-world scenarios that can be modeled by polynomial functions and where end behavior might be important?

Skills Developed

  • Identifying key features of polynomial functions
  • Applying mathematical rules to predict outcomes
  • Using mathematical notation to communicate results

Multiple Choice Questions

Question 1:

What determines the right-hand 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 left-hand behavior?

Correct Answer: Goes down to negative infinity

Question 3:

Which of the following correctly describes the arrow notation for a function going to positive infinity as x goes to infinity?

Correct Answer: As x -> ∞, f(x) -> ∞

Question 4:

What does 'lim x-> -∞ f(x) = ∞' mean?

Correct Answer: As x approaches negative infinity, f(x) approaches infinity

Question 5:

A polynomial function has a positive leading coefficient and an odd degree. What is its right-hand behavior?

Correct Answer: Goes up to positive infinity

Question 6:

For the polynomial f(x) = -3x^4 + 2x^2 - 1, what is the leading coefficient?

Correct Answer: -3

Question 7:

Which degree will have the opposite end behaviors?

Correct Answer: Odd

Question 8:

What does the degree of a polynomial influence regarding end behavior?

Correct Answer: Left-hand behavior based on right-hand behavior

Question 9:

The leading coefficient is positive, what will the right end behavior do?

Correct Answer: Increase

Question 10:

For the polynomial f(x) = 5x^3 - 2x + 1, what is the degree of the polynomial?

Correct Answer: 3

Fill in the Blank Questions

Question 1:

The number in front of the term with the highest power of x is called the ________ coefficient.

Correct Answer: leading

Question 2:

If a polynomial has an even degree, its left and right end behaviors are the ________.

Correct Answer: same

Question 3:

As x approaches infinity, we are examining the graph's ________ end behavior.

Correct Answer: right

Question 4:

The notation 'f(x) -> -∞' means the y-values are approaching ________ ________.

Correct Answer: negative infinity

Question 5:

If the leading coefficient is negative, the right end behavior will go ________.

Correct Answer: down

Question 6:

If a polynomial has an odd degree, its left and right end behaviors are ________.

Correct Answer: opposite

Question 7:

When examining the left end behavior, the x values approach ________ ________.

Correct Answer: negative infinity

Question 8:

If the exponent on the leading coefficient is odd, the left and right end behavior are ________.

Correct Answer: opposite

Question 9:

Using limit notation, end behavior can be described with ________.

Correct Answer: limits

Question 10:

The leading coefficient tests, tells about the ________ and ________ end behavior.

Correct Answer: right, left

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

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