Sketching Polynomial Functions

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SWBAT sketch the graph of a cubic function based on the zeros of the functions as well as the leading coefficient test.

Big Idea

Both the leading coefficient test and the roots can give students a fairly accurate idea for the shape of a polynomial function.

Warm Up

3 minutes

Today's warm up provides an opportunity for a quick review of what the class accomplished during yesterday's lesson (see cubic_functions_sketch_warmup). I have students work on the warm up individually and use their calculator to check the reasonableness of the answer that they obtain algebraically (MP6).  While students are finishing up, I ask one student put the solution on the board so that others can check their work.


7 minutes

After completing the Warmup, I  transition to Slide 3 of cubic_functions_sketch. At first, only the two graphs appear.  I ask students to compare the two graphs. At first, I want them to work by themselves, so I ask them to work independently and describe the difference between the two graphs. We will complete this activity as a Think-Pair-Share exercise. 

After students have had a minute to think about the two graphs and jot down one or two ideas, I will ask them to share their initial thoughts with a partner.  Have several students share their ideas with the class.  Students may say things like:

  • The graph on the left goes up while the one on the right goes down.
  • The graph on the right starts on the bottom and ends at the top while the one on the right starts at the top and ends at the bottom.

On the board, I record all of the students' ideas. Once we have a useful list of thoughts to consider, I tell students, "We are going to investigate how the equation of a function determines the shape of its graph."  

For this exploration, I ask students to use either a graphing calculator or graphing calculator to investigate how the value of the parameter "a" changes the shape of the graph of a function.  I write the equation on the board:

f(x)= ax^3 - 2x^2 + 1

If students type in the function from into, the graphing tool will ask them to make a slider that can be used to represent the values for a in their function.  (This is a great way to perform this type of investigation!)

Once students have a few minutes to investigate. I bring the class back together to share results.  Students should have come to the understanding that when a > 0 the graph looks similar to the one on the left in the slide.  If a < 0 the graph will appear like the one on the right (MP2). After yesterday's investigation, I expect my students will arrive at this conclusion relatively quickly. 

Graphing Functions

25 minutes

From my perspective, the two keys for today are to have students graph cubic functions using:

  1. The Leading Coefficient Test
  2. The Zeros of the Function

I developed this resource (cubic_functions_sketch_practice) to reach these objectives. Before students start to find the zeros of each of these functions, I have them go through each and predict the general shape of the graph based on the leading coefficient (MP7).  I plan to call on several students and ask them to share their ideas with respect to the shape of the graph of each function. I will also have them justify their thinking to the rest of the class (MP3).  

Next, I have students work with their partner to find the zeros of each function. Once they establish locations for the roots, I ask students to use the shape of the graph and the zeros to make a sketch.

Teaching Notes:

  1. Students are "sketching" the graph which is why the axis is given with no grid lines or numbers.  Students do not need to find the value of any other coordinates except the zeros.
  2. I expect some of my students to struggle with Question #4 because the function has only two zeros.  Have students graph the function using their graphing calculator to see that the function touches the x-axis at x=-1 and passes through the x-axis at x=0.


5 minutes

The Cubic_Functions_Sketch_Close may seem fairly obvious at first.  Students will recognize that they can determine the zeros of the function based off of the work that was "completed so far." However, once students have the zeros, they still need to determine the shape of the graph. Students will need to correctly identify the leading coefficient of the function. Since the trick here is subtle, I plan to let students work on this Exit Ticket individually.