Analyzing Polynomial Functions

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SWBAT identify the degree of a Polynomial, the sign of the leading coefficient, the intercepts, the multiplicity of the zeros, and the end behavior to determine the shape of the graph.

Big Idea

To be able to sketch an approximate graph of a Polynomial function from key characteristics without a calculator.


10 minutes

The purpose of this lesson is for students to be able to determine the shape of the graph of a Polynomial Function without a calculator using key information from its equation. I start the lesson by projecting visuals of different Quadratic Functions. 

Teacher's Note: I have modified this lesson from the Mathematics Assessment Project lesson titled Representing Polynomials.  I use parts of the Introduction and the cards from this lesson. The complete lesson, Representing Polynomials, can be found at the following URL: (last accessed 4-25-15)

After displaying the graphs of four Parabolas on the board using 3 slides from the Representing Polynomials lesson, I ask students to write a possible equation for Quadratic Function A. Students are to write their response on their individual white boards.  Then I ask students to analyze Quadratic Function A for the following:

  1. Type of Function
  2. Degree
  3. Number of Turns
  4. Domain
  5. Range
  6. All intercepts
  7. Multiplicities of Zeros
  8. End Behavior

Our goal today is not necessarily to be precise about relative maximums and minimums, but rather to approximate their locations. Nonetheless, I will discuss with my students the idea that the intercepts and the shape should be described as precisely as possible.

My students are familiar with Quadratic Functions, so I expect this part of the lesson to move quickly.I will present a slide and students will provide quick responses. I have three other Parabolas posted on the slides to use if needed. The number of examples that we will create depends on the needs of the students. I will only have students analyze Quadratic Function B if I feel students need more practice before the lesson. Once I feel that they are ready, we will move on to the Partner Activity.


20 minutes

At the beginning of the Partner Activity, I post Slide 3 from the Introduction.  It has the Instructions for the Collaborative Partner Activity.  I instruct students to read the slide before we begin this activity.  I previously cut apart and put the Polynomial Cards into small envelopes for each set of Partners.  The cards are at each of their tables.  

After reading the instructions, I have students begin matching the Cubic Equations to their possible graphs. All of the equations are in Factored Form.  The students are to write all of the intercepts on each card as stated in the instructions.  Students should be able to match the equations to possible graphs based on an understanding of the relationship between Factors and Zeros. Students have previously learned the Zero Product Property. If the majority of students are struggling, I post Slide 2 from the Introduction to relate the characteristics of Polynomials to Cubic Functions.  Otherwise, I do not plan on using slide two in this lesson. 

Students should pay close attention to the sign of the leading coefficient and how it affects the shape of the graph. Students should also recognize that zeros with an even amount of multiplicities are tangent to the x-axis and bounce off.  All of these characteristics should help students determine which Polynomials match with which graph.  I also write the set of questions below on the board to prompt students while matching the Polynomial Cards.  These questions help struggling students to move forward.  I have posted a copy  of the questions in the resources of this lesson.

1.  What is the sign of the leading coefficient and how does that affect the shape of the graph?

2.  What is the End Behavior of the graph?

3.  How are factors of an equation and the zeros related?

4.  How do you find all of the intercepts of a graph from an equation?

5.  How does the multiplicities of the zeros change the behavior of the graph with the x-axis?

While students are working, I walk around to monitor and facilitate the learning.  I am looking for students that are able to explain certain matches to the class at the end of this activity.  I am also looking for incorrect answers or misconceptions to present to make students better understand the correct way to analyze Polynomials.  

After about 15 minutes of the students working on matching the cards, I call the students attention back to the front of the room.  I lead a short class discussion, calling on selected students about the certain matches that I chose when I was observing.  I purposely choose cards to move students toward meeting the objective. For example, if an y-intercept is incorrect, I may ask students if they agree with the y-intercept.  If a student does not agree, I have that student explain the reasoning and the correct way to find the y-intercept.  I may also call on random students during this discussion to continue to develop understanding to meet the objective.

Exit Slip

20 minutes

After the Partner Activity, I hand each student an Exit Ticket. I expect that we will reach this point with about 20 minutes left in the Period. The Exit Ticket is a quick formative assessment to check for student understanding.

I will give my students about ten minutes to complete the Exit Slip, monitoring to see when the majority of students complete it. I will instruct students to write their name on the back of the Exit Slip, and hand in to me.  On this Exit Slip students are to analyze the Polynomial Equation given, and sketch a possible graph.  After collecting the Exit Slips, I will have my students vote on which graphs are close to the actual graph on the calculator.  I post the graph on the calculator, as I show different responses on the Exit Slip.  At the end of the class, if time permits, as a class we calculate the percentage of the students that met the objective for the day.  Again, the objective is for students to be able to determine the shape of Polynomial Graphs from an Equation without a calculator.