Unveiling the Secrets of Multiplicity in Polynomial Functions

Algebra 2 Grades High School 11:35 Video
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Lesson Description

Explore the concept of multiplicity in polynomial functions and how it influences the behavior of their graphs, including identifying roots, end behavior, and turning points.

Video Resource

Multiplicity Functions

Kevinmathscience

Duration: 11:35
Watch on YouTube

Key Concepts

  • Roots/Zeros/x-intercepts of polynomial functions
  • End behavior of polynomial functions
  • Multiplicity of roots and its impact on graph behavior (touching vs. crossing the x-axis)

Learning Objectives

  • Students will be able to determine the roots (zeros, x-intercepts) of a polynomial function from its factored form.
  • Students will be able to determine the end behavior of a polynomial function based on its leading term.
  • Students will be able to identify the multiplicity of a root and explain its effect on the graph of the polynomial function at that root (i.e., whether the graph crosses or touches the x-axis).

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concepts of roots, zeros, and x-intercepts of polynomial functions. Briefly discuss end behavior of polynomial functions. Ask students to recall how to find these features.
  • Video Presentation (10 mins)
    Play the YouTube video 'Multiplicity Functions' by Kevinmathscience. Instruct students to take notes on key concepts like multiplicity, even/odd powers, and how they relate to graph behavior.
  • Guided Practice (15 mins)
    Work through example problems similar to those in the video, demonstrating how to determine the multiplicity of a root and how it affects the shape of the graph at that x-intercept. Emphasize the difference between roots with odd multiplicity (graph crosses the x-axis) and even multiplicity (graph touches the x-axis).
  • Independent Practice (15 mins)
    Provide students with practice problems where they need to factor polynomials, identify roots and their multiplicities, determine end behavior, and sketch the graph of the polynomial function. Circulate to provide assistance and answer questions.
  • Wrap-up and Assessment (5 mins)
    Review the main points of the lesson. Assign a short quiz or exit ticket to assess student understanding.

Interactive Exercises

  • Graphing Activity
    Provide students with a set of polynomial functions in factored form. Have them determine the roots and their multiplicities, predict the end behavior, and then sketch the graph. Use graphing software or calculators to verify their sketches.

Discussion Questions

  • How does the multiplicity of a root affect the behavior of the graph at that point?
  • Can you give an example of a polynomial function where all the roots have even multiplicity? What does its graph look like?
  • How can knowing the end behavior and the roots of a polynomial function help you sketch its graph?

Skills Developed

  • Factoring polynomials
  • Graphing polynomial functions
  • Interpreting the relationship between algebraic representation and graphical representation

Multiple Choice Questions

Question 1:

What does the multiplicity of a root tell you about the graph of a polynomial function at that point?

Correct Answer: Whether the graph crosses or touches the x-axis

Question 2:

If a root has an even multiplicity, what does the graph do at that point?

Correct Answer: Touches and turns around at the x-axis

Question 3:

If a root has an odd multiplicity, what does the graph do at that point?

Correct Answer: Crosses the x-axis

Question 4:

What is the multiplicity of the root x = 3 in the function f(x) = (x - 3)^2(x + 1)?

Correct Answer: 2

Question 5:

The graph of a polynomial function touches the x-axis at x = -2 and then turns around. What can you conclude about the multiplicity of the root x = -2?

Correct Answer: It must be even

Question 6:

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

Correct Answer: Goes up

Question 7:

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

Correct Answer: Goes down

Question 8:

Which of the following factored forms indicates that the graph has a root at x = 5 with a multiplicity of 3?

Correct Answer: (x - 5)^3

Question 9:

What is the sum of all multiplicities for function f(x)=(x-2)(x+1)^2(x-3)^3

Correct Answer: 6

Question 10:

What happens to the end behavior when the function is multiplied by -1?

Correct Answer: The end behavior flips.

Fill in the Blank Questions

Question 1:

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

Correct Answer: roots

Question 2:

If a factor (x - a) appears with an exponent of 4 in a polynomial, then 'a' is a root with multiplicity ____.

Correct Answer: 4

Question 3:

A root with an ______ multiplicity will cause the graph to cross the x-axis at that point.

Correct Answer: odd

Question 4:

A root with an ______ multiplicity will cause the graph to touch the x-axis and turn around at that point.

Correct Answer: even

Question 5:

The ______ of a polynomial function describes what happens to the y-values as x approaches positive or negative infinity.

Correct Answer: end behavior

Question 6:

The highest power of x determines the _______ of a polynomial function.

Correct Answer: end behavior

Question 7:

The other names for roots are x-intercepts, and ______

Correct Answer: zeros

Question 8:

If a graph touches and turns around on the x-axis, the multiplicity of that point is _______

Correct Answer: even

Question 9:

If a root is odd, then it will _____ the x axis.

Correct Answer: cross

Question 10:

If a graph has an even number of negatives, the answer will be _____.

Correct Answer: positive

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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