8.2: Short-Run Behavior

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Objectives

• Given the factored form of a polynomial, find the zeros and their multiplicities
• Sketch the graph of a given polynomial
• Find the formula for a polynomial based off a graph

Notes:

In class, we will often be speaking of zeros and $x$-intercepts. We should use them correctly, but the distinction between the two doesn't need to be emphasized to the students. Either answer will be accepted on an exam.

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Lesson Guide

Warm-Up

Have students do Problems 1 and 2.

Given the factored form of a polynomial, find the zeros and their multiplicities

Give a few examples of polynomials in factored form and use them to talk about zeros and multiplicities. In anticipation of the following discussion, include polynomials with a mixture of even and odd multiplicities.

Students may have trouble understanding that having a factor of $x$ implies having 0 as a zero. To make this clearer, include one example that has a factor of $x$ and rewrite it as $(x-0)$.

• Example:

• Example:

Have students do Problem 3. Assign each group one or two functions to work on. Either project or redraw the table on the whiteboard and have each group fill in the row they worked on.

Once the table is filled in, ask students to see if they can find relationships between the number of zeros, the multiplicities, the degree, and the way the graph "acts" around zeros.

* You can add the multiplicities together to get the degree.
* The number of zeros is less than or equal to the degree.
* If a zero has even multiplicity, then the graph ``bounces" off of the $x$-axis at the $x$-intercept.
* If a zero has odd multiplicity, then the graph ``crosses" the $x$-axis at the $x$-intercept.

Have students do Problem 4.

Sketch the graph of a given polynomial

Do several examples to demonstrate this objective, using these observations to demonstrate how to sketch the graph of a polynomial without using a calculator. Combine all material seen thus far in Chapter 8 to show students how one can sketch a decent graph of a polynomial function.

• Example:

Have students do Problem 5 and 8.

Find the formula for a polynomial based off a graph

Do an example similar to Problem 6.

• Example:

Have students do Problems 6-7 and 9. As students finish these problems, have them work on the rest of the worksheet.

Comments

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• Problem 4 no longer makes sense since students aren't allowed to have graphing calculators. One could instead, however, plot the function using DESMOS.
• We never actually give a definition of multiplicity in this lesson. Since the lesson plan encourages that we do Problem 3 before formally discussing the implications of the multiplicities of zeros, I felt it wasn't clear what the problem means when it says to determine the multiplicities of the zeros. Hence, I gave my students the following definition: "For a polynomial, the multiplicity of a zero (a,0) is the number of times the factor $(x-a)$ occurs in the factored form of the polynomial."