Graphing Piecewise Defined Functions

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SWBAT graph piecewise defined functions.

Big Idea

Through the use of technology students see how to graph functions whose domains are restricted and then develop a process to graph piecewise-defined functions by hand.

Bell work

5 minutes

Today's Bell Work shows students how a piecewise-defined function may be undefined. Students are given different domain values to evaluate. The first value, f(2), is similar to the problems they worked in previous lessons. The second value, f(-1), gives students a value that is undefined. This is also the point where the function changes from one definition to another.

I anticipate that there will be some student confusion about what the answer should be. Some students will choose a definition to evaluate. Others will say there is no answer.

After a minute or 2, I will ask student volunteers for answers. I will have students defend their answers. I'll ask"

  • Does -1 agree with the first definition? How about the second part of the graph?
  • Can a function have a point where it is not defined?
  • So what should the answer be? 

As we finish the discussion I will ask if there is a bigger idea behind this: "What is important when you evaluate a function?" I am hoping students will comment on the importance of analyzing the restrictions on the domain, to make clear where some functions have places where they are undefined.

Using "Desmos Calculator" to explore graphing piecewise-defined functions

25 minutes

Prior to this lesson, my students have evaluated and written piecewise-defined functions. Today we will use the computer application Desmos Graphing Calculator to see how to graph a function with restrictions on the domain.

Students are given Writing Piecewise Activity to read over. I make sure students understand the goal is for them to find a process for graphing piecewise-defined functions by hand. As students work on this activity I move around the room. I answer questions and make sure that students see know the value of the left endpoint for Questions 2 and 3 is undefined. If a student has not noticed that the graph for questions 2 has no endpoint I make sure return to the student and see how they have completed question 4. To help students focus on the left endpoint, I ask students about the point and ask if there was a point there when you graphed question 2. 

I sometimes need to help students determine how to input the inequality for question 4 Students need to use < then = immediately after the < for the application to use the correct notation.


Sharing graphing procedures

10 minutes

After students have explored using Desmos for 20-25 minutes, I bring the class back together to see if students have developed a method to graph piecewise-defined functions by hand.

I put up a problem on the board for students to try. I tell them to use the process they wrote on their worksheet. I give students 2-3 minutes to find the graph and then ask a student to share the graph on the board. Some questions I use are:

  • Do you graph the function over all real numbers? Why or Why not?
  • How will you decide what part of the function to graph?
  • How do you determine the type of endpoint to use (closed circle or open circle)?
  • Should the graphs overlap?

Students also ask each other questions about their procedures. When the students ask questions I am able to assess student understanding of the process. I am also able to assess how well students are explaining their process.


5 minutes

Once we have discussed how to graph students need to try more problems.

Students are given Graphing Piecewise Worksheet. The worksheet was designed to make sure students are understanding how to graph by hand. The student of course can use the technology we used today but they will need to be able to produce a sketch of a piecewise-defined function on their assessment.

The worksheets begins with students determining the restrictions on the domain. The worksheet then has problems where students graph a single function on a restricted domain. The last two problems give students a chance to graph a piecewise-defined function. By scaffolding the homework I can see what is confusing students.

Each student must answer the exit problem before they can leave class. This problem allows me to assess if the students understand how to identify if the endpoint of each piece is included or not included.  If a student is incorrect I ask the student to recheck the response and then explain what the original answer was wrong.