Graphing Polynomials: Zeros, Multiplicities, and End Behavior

Algebra 2 Grades High School 8:01 Video
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Lesson Description

Learn how to sketch polynomial graphs by identifying zeros, understanding multiplicities, and analyzing end behavior. This lesson uses examples to guide you through the process.

Video Resource

How to Graph Polynomials Using Zeros, Multiplicities, and End Behavior (2 Examples)

Mario's Math Tutoring

Duration: 8:01
Watch on YouTube

Key Concepts

  • Zeros of a polynomial
  • Multiplicity of zeros and its effect on the graph
  • End behavior of polynomials and its relation to the leading coefficient and degree

Learning Objectives

  • Identify the zeros of a polynomial function in factored form.
  • Determine the multiplicity of each zero and explain its impact on the graph at that point.
  • Analyze the leading coefficient and degree of a polynomial to determine its end behavior.
  • Sketch a graph of a polynomial function using zeros, multiplicities, and end behavior.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a polynomial and its factored form. Introduce the concepts of zeros, multiplicity, and end behavior as key elements in sketching polynomial graphs. Briefly mention the connection to solving polynomial equations.
  • Video Presentation (15 mins)
    Play the YouTube video 'How to Graph Polynomials Using Zeros, Multiplicities, and End Behavior (2 Examples)'. Encourage students to take notes on the examples provided. Pause at key points to emphasize important concepts.
  • Guided Practice (15 mins)
    Work through a similar example as a class. Guide students through the steps of finding the zeros, determining their multiplicities, analyzing the end behavior, and sketching the graph. Emphasize the connection between the algebraic representation and the graphical representation.
  • Independent Practice (10 mins)
    Provide students with a polynomial in factored form and ask them to sketch its graph individually. Circulate to provide assistance and answer questions.
  • Wrap-up and Discussion (5 mins)
    Review the key concepts and answer any remaining questions. Preview the next lesson, which could involve factoring polynomials to find the zeros.

Interactive Exercises

  • Zero and Multiplicity Matching
    Provide a list of polynomials and a list of zeros with multiplicities. Students must match each polynomial with its corresponding zeros and multiplicities.
  • End Behavior Prediction
    Present various polynomials with different leading coefficients and degrees. Students must predict the end behavior of each polynomial.

Discussion Questions

  • How does the multiplicity of a zero affect the graph's behavior at that point?
  • Explain how the leading coefficient and degree of a polynomial determine its end behavior.
  • Why is it important to factor a polynomial before graphing it using this method?
  • How can we find the y-intercept of the polynomial?

Skills Developed

  • Graphing Polynomial Functions
  • Analyzing Polynomial Functions
  • Problem Solving

Multiple Choice Questions

Question 1:

What does the multiplicity of a zero tell you about the graph at that point?

Correct Answer: Whether the graph crosses or bounces off the x-axis.

Question 2:

If a polynomial has a leading coefficient of -2 and a degree of 3, what is its end behavior?

Correct Answer: Down to the left, up to the right.

Question 3:

Which of the following is the correct way to find the zeros of a polynomial in factored form?

Correct Answer: Set each factor equal to zero and solve for x.

Question 4:

A zero with an even multiplicity will cause the graph to:

Correct Answer: Bounce off the x-axis

Question 5:

A polynomial with an odd degree and a positive leading coefficient will have which end behavior?

Correct Answer: Goes down to the left and up to the right

Question 6:

What is the y-intercept of the function f(x) = (x+2)(x-1)(x-3)?

Correct Answer: (0, 6)

Question 7:

What is the zero of the factor (x + 5)?

Correct Answer: x = -5

Question 8:

The degree of a polynomial tells us about its:

Correct Answer: End Behavior

Question 9:

The leading coefficient of a polynomial helps determine its:

Correct Answer: End behavior

Question 10:

If a polynomial has a factor of (x-2)^3, what is the multiplicity of the zero x=2?

Correct Answer: 3

Fill in the Blank Questions

Question 1:

The points where a polynomial crosses or touches the x-axis are called the ______.

Correct Answer: zeros

Question 2:

The ______ of a zero tells us how many times that zero occurs in the factored form of the polynomial.

Correct Answer: multiplicity

Question 3:

The behavior of the graph as x approaches positive or negative infinity is called the ______ ______.

Correct Answer: end behavior

Question 4:

If the leading coefficient is positive and the degree is even, the graph goes ______ on both ends.

Correct Answer: up

Question 5:

A zero with a multiplicity of 1 will cause the graph to ______ the x-axis.

Correct Answer: cross

Question 6:

The leading coefficient affects the _______ end behavior of a polynomial.

Correct Answer: right

Question 7:

When a zero has an even multiplicity, the graph will _______ off the x-axis at that point.

Correct Answer: bounce

Question 8:

The _______ of a polynomial is the highest exponent of the variable.

Correct Answer: degree

Question 9:

The y-intercept of a polynomial can be found by setting x = _____ in the equation.

Correct Answer: 0

Question 10:

Before graphing, it's helpful to write the polynomial in _______ form to easily identify zeros.

Correct Answer: factored

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.1.4:Solve polynomial equations with real roots using various methods (e.g., polynomial division, synthetic division, using graphing calculators or other appropriate technology).

Teaching Materials

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