Lesson 10

Multiplicity

Lesson Narrative

This lesson is meant to be a culmination of what students have learned in the unit so far. In order to sketch a graph of a polynomial function from a factored equation, students must understand the relationship between the factors, zeros, and horizontal intercepts. They need to calculate the leading term of the equation in order to use the degree and the sign of the leading coefficient to identify the end behavior of the function.

New in this lesson is the relationship between the multiplicity of a factor, which is the power to which the factor occurs in the factored form of a polynomial, and the shape of the graph at the intercept associated with that factor. While students did preview this idea in their study of quadratics in previous courses when they looked at the graph of equations such as \(y=(x-5)^2\), here we name the effect and study the differences between factors with a multiplicity of 1, 2, and 3.

Students will have more opportunities to sketch polynomials from factored equations in future lessons, but this lesson is the main opportunity to make solid connections between the shape of a graph and the structure of the factored equation for polynomial functions (MP7).

Learning Goals

Teacher Facing

  • Comprehend the effect that the multiplicity of factors has on the shape of the graph of a polynomial function.
  • Use zeros and multiplicities to create a rough graph of a polynomial function given in factored form.

Student Facing

  • Let’s sketch some polynomial functions.

Required Materials

Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

Learning Targets

Student Facing

  • I can use zeros and multiplicities to sketch a graph of a polynomial.

CCSS Standards

Building On

Addressing

Building Towards

Glossary Entries

  • end behavior

    How the outputs of a function change as we look at input values further and further from 0.

    This function shows different end behavior in the positive and negative directions. In the positive direction the values get larger and larger. In the negative direction the values get closer and closer to -3.

  • multiplicity

    The power to which a factor occurs in the factored form of a polynomial. For example, in the polynomial \((x-1)^2(x+3)\), the factor \(x-1\) has multiplicity 2 and the factor \(x+3\) has multiplicity 1.