More Cubic Function Graphs

I prepared this lesson to focus more explicitly on the behavior of graphs of cubic functions, even though during this unit I would not expect all students to master this content. The goal of students being able to sketch graphs of polynomials and write function rules that could fit given polynomial graphs will be more prominent in the next unit, but for today’s lesson I want all students to start thinking about how to use their understanding of cubic functions to develop these skills.

The warm-up to this lesson is really quick for many students—I use it as a quick informal assessment. It is important not to have computers available initially, to ensure that students are actually thinking about their match-ups.

There are many important questions to discuss with students, either individually or as a class, related to these match-ups:

• How can the number of x-intercepts help you determine which function matches the graph?

• How can two different graphs have the same x-intercepts but different function rules?

• How can you use points other than the x-intercepts to help you match the graphs to the functions?

Once students have shown that they understand these ideas, they can transition to the next investigation.

If students struggle with these ideas, feel free to have those students hold off on the new investigation completely. For some of my struggling students, they had enough of a challenge today just finishing the investigations from the previous days, so there was no need to push them to complete this next investigation, because the same skills will be coming up in the next unit.

The new challenge of the day is for students to write functions to match graphs. I encourage students to avoid using computers until they come up with their initial ideas about the functions. Write Functions to Match Cubic Graphs is pretty self-explanatory—find the functions to match these graphs. Some students may struggle with different aspects of this:

• How can you make sure that the function has the right shape once you have identified the roots and factors?

• Do you need to use the additional points provided to ensure you have the correct function?

• Is it possible to find the coefficient without using guess-and-check?

When it comes to the writing component of this investigation, it is helpful to give students a list of key terms to help them describe the big ideas:

• Coefficient
• Roots
• X-intercepts
• Factors
• Key points
• Input/output

The Cubic Polynomial Graph Sketches includes some problems that ask students to sketch graphs. I decided not to emphasize this skill yet, and made sure to emphasize it in the upcoming unit. For students who were ready to try this, I asked them to give it a shot without the computer, and then to check their sketches on the computer.