Polynomials in the Short Run (Lesson 2.2 Day 1)
Unit 2  Day 3
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Learning Objectives

Determine a polynomial’s x and yintercepts from its equation

Use a root’s multiplicity to describe the graph’s behavior at an xintercept

Determine the maximum number of turning points and roots of a polynomial using the Fundamental Theorem of Algebra
Quick Lesson Plan
Experience First
Students will use Desmos to explore the relationship between the roots, factors, yintercepts, and turning points of a polynomial. Have them graph each function and answer the corresponding questions. Make sure they really pay attention to the behavior around the xintercepts. Does the curve go straight through? Does it “bounce” and change directions? How do the factors play into this? We want them to see the repeated factors have different behavior at the xintercepts to introduce the concept of multiplicity.
Formalize Later
When you formalize, make sure you use the proper language to define x and yintercepts. The xintercept isn’t just the “opposite sign of the factor” like we anticipate many students will put in 1d, but the solution when you set function equal to 0. The yintercept isn’t just where the graph crosses the yaxis, but the value of y when you evaluate f(0).
Show on the graph where the multiplicity changes the behavior by outlining the “bounce” versus the “goes through.” Connect the word xintercept to root and zero since we will be using these interchangeably moving forward.
Next lesson we will explore polynomials in the long run to create a complete picture of how polynomial graphs are related to their equations.