Mastering Polynomial Graphs: End Behavior, Zeros, and Multiplicities

Algebra 2 Grades High School 12:08 Video
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Lesson Description

Learn to graph polynomial functions by analyzing end behavior, zeros, and multiplicities. This lesson provides a step-by-step approach with examples to help you understand these key concepts.

Video Resource

Graphing Polynomial Functions Using End Behavior, Zeros, and Multiplicities

Mario's Math Tutoring

Duration: 12:08
Watch on YouTube

Key Concepts

  • End Behavior: How the graph behaves as x approaches positive or negative infinity.
  • Zeros: The x-intercepts of the polynomial function's graph, found by setting the function equal to zero and solving for x.
  • Multiplicity: The number of times a zero occurs, affecting the graph's behavior at that x-intercept (e.g., bouncing or passing through).

Learning Objectives

  • Students will be able to determine the end behavior of a polynomial function using the leading coefficient test and the degree of the polynomial.
  • Students will be able to find the zeros of a polynomial function by factoring and applying the zero product property.
  • Students will be able to identify the multiplicity of each zero and understand its effect on the graph's behavior at that point.
  • Students will be able to sketch the graph of a polynomial function using end behavior, zeros, and multiplicities.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a polynomial function and its standard form. Briefly discuss the importance of graphing polynomial functions in real-world applications. Introduce the concepts of end behavior, zeros, and multiplicities as key components of graphing.
  • Video Presentation (15 mins)
    Play the YouTube video "Graphing Polynomial Functions Using End Behavior, Zeros, and Multiplicities" by Mario's Math Tutoring. Instruct students to take notes on the following: How to determine end behavior using the leading coefficient and degree. The process of finding zeros through factoring. The impact of multiplicity on the graph's shape at each zero.
  • Guided Practice (20 mins)
    Work through example problems similar to those in the video. Start with simple polynomials in factored form and gradually increase complexity. Guide students through the process of finding zeros, determining multiplicities, analyzing end behavior, and sketching the graph. Emphasize the connection between the algebraic representation and the graphical representation.
  • Independent Practice (15 mins)
    Assign practice problems for students to work on individually or in pairs. Circulate to provide assistance and answer questions. Encourage students to explain their reasoning and approach to each problem.
  • Wrap-up and Assessment (5 mins)
    Summarize the key concepts covered in the lesson. Administer the multiple-choice or fill-in-the-blank quiz to assess student understanding. Collect student work and provide feedback.

Interactive Exercises

  • Graphing Challenge
    Provide students with polynomial functions and challenge them to graph them using the concepts learned in the lesson. Students can compare their graphs and discuss any discrepancies or alternative approaches.
  • Desmos Exploration
    Use Desmos or another graphing calculator to explore the effects of changing the coefficients and exponents of polynomial functions on their graphs. Students can manipulate the equations and observe the resulting changes in end behavior, zeros, and multiplicities.

Discussion Questions

  • How does the leading coefficient of a polynomial function affect its end behavior?
  • Explain the difference between a zero with a multiplicity of 1 and a zero with a multiplicity of 2. How does each affect the graph?
  • What are some real-world applications where understanding polynomial functions and their graphs is important?

Skills Developed

  • Factoring Polynomials
  • Analyzing Graphs
  • Problem-Solving
  • Critical Thinking

Multiple Choice Questions

Question 1:

What does the leading coefficient test help determine about a polynomial function's graph?

Correct Answer: The end behavior

Question 2:

If a polynomial has a zero at x = 3 with a multiplicity of 2, what does the graph do at x = 3?

Correct Answer: Bounces off the x-axis

Question 3:

Which of the following is NOT a factor to consider when sketching a polynomial function?

Correct Answer: The color of the graph

Question 4:

A polynomial has a degree of 5. What could be the end behavior?

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

Question 5:

What does it mean for a zero to have a multiplicity of 3?

Correct Answer: The graph has a cubic shape as it crosses the x-axis

Question 6:

If a polynomial function 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 7:

Which method is primarily used to find the zeros of a polynomial function?

Correct Answer: Factoring

Question 8:

What is the y-intercept of a polynomial function?

Correct Answer: The point where the graph crosses the y-axis

Question 9:

What is the general shape of a graph if a zero has a multiplicty of 1?

Correct Answer: Line

Question 10:

How does multiplicity affect the shape of a polynomial function?

Correct Answer: It impacts the graphs shape at the x-intercept

Fill in the Blank Questions

Question 1:

The behavior of a graph as x approaches positive or negative infinity is called _____ behavior.

Correct Answer: end

Question 2:

The x-intercepts of a polynomial function are also known as its _____.

Correct Answer: zeros

Question 3:

The _____ of a zero indicates how many times that zero occurs as a root of the polynomial.

Correct Answer: multiplicity

Question 4:

If a polynomial function has an odd degree, its end behaviors will be _____.

Correct Answer: opposite

Question 5:

If a factor appears (x - a)^2, the zero x = a has a multiplicty of _____.

Correct Answer: 2

Question 6:

A positive leading coefficient and even degree mean the end behavior goes _____ to the left and _____ to the right.

Correct Answer: up, up

Question 7:

To find zeros of a polynomial, you must _____ it, if possible.

Correct Answer: factor

Question 8:

A zero with a multiplicity of 1 will pass _____ the graph.

Correct Answer: through

Question 9:

The y-intercept of a function is found by setting x = _____.

Correct Answer: 0

Question 10:

The shape of the graph when the multiplicity is 3 is referred to as a _____ shape.

Correct Answer: cubic

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). A2.A.2.1:Factor polynomial expressions including, but not limited to, trinomials, differences of squares, sum and difference of cubes, and factoring by grouping, using a variety of tools and strategies.

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