#4590. Parent Functions, Transformations, and Piecewise Functions

Mathematics, level: Senior
Posted Sun Apr 1 22:27:01 PDT 2012 by Heather Tyrrell (Heather Tyrrell).
University of Wyoming, Laramie, WY USA
Materials Required: Graphing Calculators and TI-84 Rangers
Activity Time: 7 days
Concepts Taught: Functions and Transformations

Parent Functions, Transformations, and Piecewise Functions
Lesson Title: Parent Functions Exploration (Day 1 of 7)
Class length: 42 minutes
Instructional Strategies: Cooperative learning and discovery learning

Objectives:
Students will investigate mathematical phenomena cooperatively to determine a pattern for parent function graphs.
Students will convert between the graphical, algebraic, and language representations of parent functions.
Students will identify parent functions by name, graph, and equation.

Materials:
Parent Functions Discovery Worksheet, 3 versions (graphics created with GeoGebra)
Parent Functions Graphic Organizer
Parent Functions Homework (created with Kuta Software)
Graphing calculators
Document camera
Smart Board

Procedure:
Teacher Activities: Display warm-up problems on the Smart Board as students enter the classroom. Circulate around the classroom and assist students with the warm-up problems by guiding students and posing questions. The warm-up problems deal with content the students learned in Algebra 1.
Warm-up:
What is a function?
Sketch a graph that is a function.
Sketch a graph that is NOT a function.
Student Activities: Students know to begin working on the warm-up problems when class begins. Students are able to work with a partner and reference their notes/textbook.

Teacher Activities: For question 1, ask students for their thoughts. Discuss the student answers and guide them towards the formal definition of a function and function notation. For questions 2 and 3, invite students to display their sketches on the document camera. Remind students of the vertical line test as a way to test whether a graph represents a function.
Student Activities: Students will participate in providing their answers to the warm-up problems. Students will be respectful of each other¡¯s ideas and will help a struggling peer. Students will correct and add to their warm-up problems if needed. If there is something that they do not understand, they will raise their hand and ask for clarification.

Teacher Activities: Tell students, ¡°We will be exploring parent functions today. Parent functions are basic functions that can be used as building blocks for more complicated functions.¡± Divide students into teams of three, and pass out a different version of the Parent Function Discovery Worksheet. Read over the directions with students, and answer any questions.
Student Activities: When divided into teams, students move their desks so they are able to work together. Students will listen to the directions and ask clarification questions.

Teacher Activities: Circulate around the classroom and assist students by guiding teams, troubleshooting technology, and posing questions. Guiding questions:
How many graphs do you need to find a pattern?
How can you test your hypothesized shape?
What is the relationship between the shape of the quadratic and square root functions?
What is the relationship between the shape of the cubic and cube root functions?
How can you remember the shape of each parent function?
Are all of these graphs functions? How do you know?
Student Activities: Students will graph the equations on their worksheet with their graphing calculator and copy down the graphs. Students will compare their graphs with their teammates to look for a pattern of the shape for each parent function. When working with their teammates, students will contribute and be respectful of each other¡¯s ideas. Students will ask their teammates questions before asking the teacher.

Teacher Activities: When all teams feel that they have identified the shape of every parent function, pass out the Parent Functions Graphic Organizer. Complete the Parent Functions Graphic Organizer as a class while debriefing the discovery activity. Go over the constant example from the graphic organizer first and then call on teams for the remaining parent functions. Guiding questions:
How did you find a pattern?
What is the relationship between the shape of the quadratic and square root functions?
What is the relationship between the shape of the cubic and cube root functions?
How can you remember the shape of each parent function?
Are all of these graphs functions? How do you know?
What did you learn about the investigating process?
Student Activities: Students will participate in providing their reasoning and findings to the Parent Functions Discovery Worksheet. They will fill in the graphic organizer with the relevant information and make any notes that will help them with their understanding of parent functions. Students will be respectful of each other¡¯s ideas and will help a struggling peer. If there is something that they do not understand, they will raise their hand and ask for clarification.

Teacher Activities: Pass out Parent Functions Homework to students and read over the directions with students. Circulate around the classroom and assist students by guiding teams and posing questions.
Student Activities: Students will listen to the directions and ask clarification questions. Students are able to work with a partner and reference their notes/textbook. Students will complete the homework assignment for the next class period. If students are having difficulty on the assignment, they will seek out the teacher for help during EHS time or afterschool.

Teacher Activities: Five minutes before the end of class, ask students to pack up and return the classroom to the normal set up. Once everyone is back in their seat, name the parent functions. Have students trace the shape of each parent function in the air using their finger. Use this opportunity as a quick informal formative assessment to gauge your students¡¯ knowledge of the parent function shapes. Then pose the following challenge problem to the students: Predict the shape of f(x)=2x^3-x^2+4x-2. Have students think about it and then ask them for their predictions. Write their predictions on the board with a question mark next to each. Ask students how they think we should test the predictions. Most likely, students will respond with ¡°graphing calculators¡±. Type the equation into the graphing calculator on the Smart Board, and determine the shape. Asks students to think about why the shape would resemble a cubic.
Student Activities: Students will return all materials that they used during class and move their desks back to their original location. Students will trace the shape of each parent function in the air as the teacher goes through them. The student will make note of any parent functions that they are struggling with. When the teacher poses the challenge problem, students will think carefully about it and make a prediction without typing it into their graphing calculator.

Assessments:
Informal
Warm-up
Questioning
Formal
Parent Functions Homework

Name: _______________________
Parent Functions Discovery WS#1
Directions: Use your graphing calculator to sketch a graph of each function. Compare with your teammates¡¯ graphs, and determine a pattern for the shape of every parent function.
1. f(x)=-2 2. f(x)=1/3 x+5

3. f(x)=-(x+4)^2+1 4. f(x)=(x-2)^3-5

5. f(x)=2¡Ì(x+1) 6. f(x)=∛x-3

7. f(x)=-2/3 |x|-3 8. f(x)=1/(x+2)

9. f(x)=sin⁡(2x) 10. f(x)=cos⁡x+4

Name: _______________________
Parent Functions Discovery WS #2
Directions: Use your graphing calculator to sketch a graph of each function. Compare with your teammates¡¯ graphs, and determine a pattern for the shape of every parent function.
1. f(x)=3 2. f(x)=-2x+1

3. f(x)=5/4 (x+1)^2 4. f(x)=¡¼-x¡½^3-2

5. f(x)=¡Ì(x+3)-5 6. f(x)=∛2x+2

7. f(x)=|x-4|-1 8. f(x)=1/(x-3)+1

9. f(x)=sin⁡(x)-3 10. f(x)=cos⁡(x-1)

Name: _______________________
Parent Functions Discovery WS #3
Directions: Use your graphing calculator to sketch a graph of each function. Compare with your teammates¡¯ graphs, and determine a pattern for the shape of every parent function.
1. f(x)=6 2. f(x)=x-5

3. f(x)=(x-6)^2-7 4. f(x)=¡¼(x+7)¡½^3-2

5. f(x)=-¡Ì(x+1)+2 6. f(x)=∛(x-2)

7. f(x)=-|x+2|+8 8. f(x)=-1/(x+1)

9. f(x)=sin⁡x+1 10. f(x)=cos⁡x-2

Name: _________________________
Period: ________
Parent Functions Graphic Organizer
Name Equation Sketch of graph

Constant

y=c

Linear

Quadratic

Cubic

Square root

Cube Root

Absolute Value

Reciprocal

Sine

Cosine

Parent Functions, Transformations, and Piecewise Functions
Lesson Title: Transformations (Day 2(instruction) & 3(practice) of 7)
Class length: 89 minutes
Instructional Strategies: Cooperative learning, discovery learning, and direct instruction

Objectives:
Students will investigate mathematical phenomena cooperatively to determine a pattern for transformations of parent functions.
Students will identify transformations in graphical and algebraic representations of parent functions.
Students will convert between the graphical, algebraic, and language representations of transformed parent functions.

Materials:
Translations and Reflections Discovery Worksheet
Transformations Homework (graphic created with KutaSoftware)
Graphing calculators
Smart Board
Document camera
Individual whiteboards

Procedure:
Teacher Activities: Display warm-up problems on the Smart Board as students enter the classroom. Circulate around the classroom and assist students with the warm-up problems by guiding students and posing questions. Collect homework from students.
Warm-up:
Name the parent function in the following graphs by name and equation.

Why is the shape of f(x)=2x^3-x^2+4x-2 resemble a cubic?
Sketch the reflection of a quadratic over the x-axis.
Student Activities: Students know to begin working on the warm-up problems when class begins. Students are able to work with a partner and reference their notes/textbook. Students will also place their homework assignment on their desk for the teacher to collect.

Teacher Activities: For question 1, ask students for their thoughts. Discuss the student answers and guide them towards square root, f(x)=¡Ìx and reciprocal, f(x)=1/x. For question 2, invite students to explain their reasoning and guide them towards thinking about the leading term in a polynomial determines the shape. To help students¡¯ understanding, create a table of values for the x^3 term, x^2 term, x term, and constant term. They will see that the x^3 term dominates as the x-values get really big (+¡Þ) and really small (-¡Þ). For question 3, invite a student to show his/her solution on the document camera.
Student Activities: Students will participate in providing their answers to the warm-up problems. Students will be respectful of each other¡¯s ideas and will help a struggling peer. Students will correct and add to their warm-up problems if needed. If there is something that they do not understand, they will raise their hand and ask for clarification.

Teacher Activities: Begin by having students name all of the parent functions. Have students trace the shape of each parent function in the air using their finger as the parent functions are named. The teacher will take note of any parent functions that the students are struggling with. Tell students, ¡°We will begin exploring transformations today. Transformations are what allow us to move the parent functions around the coordinate axes or stretch and compress the parent functions. The transformations that we will focus on today are translations and reflections.¡± The students have prior knowledge of translations and reflections from Geometry. Lead a discussion where students discuss what they know about translations and reflections. Guiding questions:
Where have you talked about translations and reflections?
What are translations?
What are reflections?
On your whiteboards, sketch the graph of a cubic translated right 3 units.
On your whiteboards, sketch the graph of a cube root reflected across the y-axis.
Student Activities: Students take turns naming the parent functions. Students will trace the shape of each parent function in the air as their peers name the parent functions. Students will participate in providing their prior knowledge of translations and reflections. Students will be respectful of each other¡¯s ideas and will help a struggling peer. If there is something that they do not understand, they will raise their hand and ask for clarification.

Teacher Activities: Divide students into teams of three, and pass out the Translations and Reflections Discovery Worksheet. Read over the directions with students, and answer any questions.
Student Activities: When divided into teams, students move their desks so they are able to work together. Students will listen to the directions and ask clarification questions.

Teacher Activities: Circulate around the classroom and assist students by guiding teams and posing questions.
Student Activities: Students will graph the equations on their worksheet with their graphing calculator and copy down the graphs. Students will work with their teammates to look for a pattern between the equation and the transformation observed in the graph. When working with their teammates, students will contribute and be respectful of each other¡¯s ideas. Students will ask their teammates questions before asking the teacher.

Teacher Activities: When all teams feel that they have identified the transformations, begin a debriefing discussion. Ask students what they discovered for each transformation. During the debriefing discussion, introduce the function notation used to represent each transformation. The f(x) represents a general parent functions and k is any real number. Stress that it is mathematical shorthand used to indicate which type of transformation is occurring. Instruct students to write down each function notation on the corresponding page of the worksheet.
Vertical shift f(x)+k
Horizontal shift f(x+k)
x-axis reflection -f(x)
y-axis reflection f(-x)
Vertical compression kf(x)
Horizontal compression f(kx)
If students are uncomfortable using the function notation, work through a couple of examples.
Student Activities: Students will participate in providing their reasoning and findings to the Translations Reflections Discovery Worksheet. They will make any notes that will help them with their understanding of translations and reflections on their worksheet. Students will be respectful of each other¡¯s ideas and will help a struggling peer. If there is something that they do not understand, they will raise their hand and ask for clarification.

Teacher Activities: Ask students to get out their notebook and get ready to take a couple of notes. Work through the identifying transformations notes with the students that focus on identifying transformations in equations, graphing equations by hand, and writing an equation from a graph. Make sure that students are following along and answering questions. When done with the notes, give students whiteboard problems one by one to gauge student understanding.
Whiteboard problems:
Identify the parent function and transformations for f(x)=(x+2)^2.
Identify the parent function and transformations for g(x)=¡¼-|x-3|¡½^2-1.
Graph h(x)=(x+2)^2 without a graphing calculator.
Graph f(x)=-¡Ì(x-1)+5 without a graphing calculator.
Write an equation for the following graph.

Write an equation for the following graph.

Identify the parent function and transformations for f(x)=¡¼5/4 (x-6)¡½^3.
Identify the parent function and transformations for g(x)=-1/3 |x|-1.
Graph h(x)=¡¼3(x+1)¡½^2 without a graphing calculator.
Graph f(x)=-1/2 ¡Ì(x-2)+5 without a graphing calculator.
Write an equation for the following graph.

Write an equation for the following graph.

Student Activities: Students will have their notebook with them and open it when instructed. They will follow along and take notes. If they do not understand, they will raise their hand and ask for clarification. Students will work the whiteboard problems on their whiteboards. They may work with a partner or reference their notes.

Teacher Activities: Pass out Transformations Homework to students and read over the directions with students. Circulate around the classroom and assist students by guiding teams and posing questions.
Student Activities: Students will listen to the directions and ask clarification questions. Students are able to work with a partner and reference their notes/textbook. Students will complete the homework assignment for the next class period. If students are having difficulty on the assignment, they will seek out the teacher for help during EHS time or afterschool.

Teacher Activities: Five minutes before the end of class, ask students to pack up and return the classroom to the normal set up. Ask students to write a quick write about horizontal shifts.
Student Activities: Students will return all materials that they used during class and move their desks back to their original location. Students will think carefully about the quick write.

Assessment:
Informal:
Warm-up
Questioning
Whiteboards
Quick write
Formal:
Transformations HW

Parent Functions, Transformations, and Piecewise Functions
Lesson Title: Bouncy Ball Activity (Day 4 of 7)
Class length: 42 minutes
Instructional Strategies: Discovery learning and cooperative learning

Objectives:
Students will investigate mathematical phenomena cooperatively.
Students will identify parent functions by name, equation, and graph.
Students will identify transformations in graphical and algebraic representations of parent functions.
Students will convert between the graphical, algebraic, and language representations of transformed parent functions.

Materials:
Graphing calculators
TI-84 Ranger
Bouncy Ball
Ball Bounce Worksheets

Procedure:
Teacher Activities: Display warm-up problems on the Smart Board as students enter the classroom. Circulate around the classroom and assist students with the warm-up problems by guiding students and posing questions. Collect homework from students.
Warm-up:
Identify the transformations in the following equation f(x)=-1/2 (x-1)^3+5
Student Activities: Students know to begin working on the warm-up problems when class begins. Students are able to work with a partner and reference their notes/textbook. Students will also place their homework assignment on their desk for the teacher to collect.

Teacher Activities: Discuss the answer for question 1 with students.
Student Activities: Students will participate in providing their answers to the warm-up problems. Students will be respectful of each other¡¯s ideas and will help a struggling peer. Students will correct and add to their warm-up problems if needed. If there is something that they do not understand, they will raise their hand and ask for clarification.

Teacher Activities: Tell students, ¡°We will be exploring parent functions today in a real world context.¡± Divide students into teams of three, and pass out the Ball Bounce Worksheet and the TI-84 Ranger. Read over the directions with students, and answer any questions. Demonstrate how to use the TI-84 Ranger for students. Circulate around the classroom and assist students by guiding teams, troubleshooting technology, and posing questions.
Student Activities: When divided into teams, students move their desks so they are able to work together. Students will listen to the directions and ask clarification questions. Students will complete the Ball Bounce Worksheet. When working with their teammates, students will contribute and be respectful of each other¡¯s ideas. Students will ask their teammates questions before asking the teacher.

Teacher Activities: Five minutes before the end of class, ask students to pack up and return the classroom to the normal set up. Once everyone is back in their seats, conduct a quick debrief of the Bouncy Ball Activity.
Student Activities: Students will return all materials that they used during class and move their desks back to their original location.

Assessment:
Informal
Warm-up
Formal
Bouncy Ball Worksheet

Name: ____________________
Period: _______
Ball Bounce Worksheet

Sketch a graph of what you think happens when a ball is dropped in the space below. The x-axis is time and the y-axis is height.

Next you will observe the demonstration of a ball bouncing using a TI-84 Ranger. It may take a couple of trials before you are satisfied with your data.

When satisfied with your data, sketch a graph in the space below.

What type of relationship is there between time and height?

Write an equation for one of the parabolas on your calculator.

Parent Functions, Transformations, Piecewise Functions
Lesson Title: Piecewise Functions (Day 5 of 7)
Class length: 89 minutes
Instructional Strategies: Direct instruction

Students will identify transformations in graphical and algebraic representations of parent functions.
Students will convert between the graphical and algebraic representations of piecewise functions.

Materials:
Piecewise Functions HW
Graphing calculators
Smart Board
Document camera
Individual whiteboards

Procedure:
Teacher Activities: Display warm-up problems on the Smart Board as students enter the classroom. Circulate around the classroom and assist students with the warm-up problems by guiding students and posing questions.
Warm-up:
Graph f(x)=-2∛(x+3)-5.
Graph x<2.
Student Activities: Students know to begin working on the warm-up problems when class begins. Students are able to work with a partner and reference their notes/textbook.

Teacher Activities: Invite students to show his/her solution on the document camera. Remind students about the open and closed dot rules for inequalities.
Student Activities: Students will participate in providing their answers to the warm-up problems. Students will be respectful of each other¡¯s ideas and will help a struggling peer. Students will correct and add to their warm-up problems if needed. If there is something that they do not understand, they will raise their hand and ask for clarification.

Teacher Activities: Ask students to get out their notebook and get ready to take a couple of notes. Work through the piecewise function notes with the students that focus on graphing piecewise functions by hand and with a calculator. Make sure that students are following along and answering questions. When done with the notes, give students whiteboard problems one by one to gauge student understanding.
Whiteboard problems:
Graph by hand: f(x)={¨€(2 if x>0@5x if x¡Ü0)©È.
Graph with your graphing calculator: f(x)={¨€(¡Ì(x-1) if x>1@-¡Ì(x-1) if x¡Ü1)©È.
Graph by hand: f(x)={¨€(4 if x<-3@-2/3 x-1 if x¡Ý-3)©È.
Graph with your graphing calculator: f(x)={¨€(x^3 if x¡Ý-1@-¡Ìx+2 if x<-1)©È.
Student Activities: Students will have their notebook with them and open it when instructed. They will follow along and take notes. If they do not understand, they will raise their hand and ask for clarification. Students will work the whiteboard problems on their whiteboards. They may work with a partner or reference their notes.

Teacher Activities: Pass out Piecewise Homework to students and read over the directions with students. Circulate around the classroom and assist students by guiding teams and posing questions.
Student Activities: Students will listen to the directions and ask clarification questions. Students are able to work with a partner and reference their notes/textbook. Students will complete the homework assignment for the next class period. If students are having difficulty on the assignment, they will seek out the teacher for help during EHS time or afterschool.

Assessment:
Informal
Warm-up
Questioning
Whiteboards
Formal
Piecewise Homework

Parent Functions, Transformations, and Piecewise Functions
Lesson Title: Unit Project (Day 6 of 7)
Class length: 89 minutes

Objectives:
Students will investigate mathematical phenomena.
Students will convert between the graphical, algebraic, and language representations of transformed parent functions.

Materials:
Smiley Face Project Worksheet
Graphing calculators
Graphing calculator transfer cable
Smart Board
Document camera

Procedure:
Teacher Activities: Display warm-up problems on the Smart Board as students enter the classroom. Circulate around the classroom and assist students with the warm-up problems by guiding students and posing questions.
Warm-up:
Graph by hand: f(x)={¨€(x if x>0@-4 if x¡Ü0)©È.
Graph with your graphing calculator: f(x)={¨€(x^2 if x>1@-x if x¡Ü1)©È.
Student Activities: Students know to begin working on the warm-up problems when class begins. Students are able to work with a partner and reference their notes/textbook.

Teacher Activities: For question 1 and 2, invite students to display their solutions on the document camera. Discuss any incorrect answers.
Student Activities: Students will participate in providing their answers to the warm-up problems. Students will be respectful of each other¡¯s ideas and will help a struggling peer. Students will correct and add to their warm-up problems if needed. If there is something that they do not understand, they will raise their hand and ask for clarification.

Teacher Activities: Pass out the Smiley Face Project Worksheet, and explain to students that they will be working on this project individually. Read over the directions with students, and answer any questions.
Student Activities: Students will listen to the directions and ask clarification questions.

Teacher Activities: Circulate around the classroom and assist students by guiding, troubleshooting technology problems, and posing questions. Prepare SET on the Smart Board for students who finish early. Challenge students who finish early to create a picture of their choice.
Student Activities: Students will create a smiley face by experimenting with parent functions and transformations. Students can use their notes/textbook to help them with the graphing or identifying of transformations. When they are finished, they will turn in their worksheets and upload their graphs to Mrs. Tyrrell¡¯s computer. Students who finish early are allowed to play SET on the Smart Board.

Teacher Activities: Print out smiley faces and display around the room.

Assessment:
Informal
Warm-up
Formal
Smiley Face Project

Smiley Face Project

Name: _____________________
Period: ______

Directions: Create a smiley face on your graphing calculator using transformations of the parent functions and piecewise functions.

Criteria:
Each equation used must be a transformation of a parent function. No other equations or plots may be used.
Your calculator window must be limited to the first quadrant.

Each equation can include multiple transformations and multiple parent functions.
For each equation, you must give an accurate description of all of the transformations.
You must transfer your graph from your calculator to Mrs. Tyrrell¡¯s computer and turn in a written list of your equations.

Rubric:
8 6 4 2 0
Descriptions of transformations All descriptions of transformations are accurate 1 to 2 descriptions of transformations have minor errors More than 2 descriptions of transformations have minor errors Descriptions of transformations contained major errors No descriptions of transformations
Equations Students used 8 or more equations to create a smiley face Students used 7 equations correctly to create a smiley face Students used 6 equations correctly to create a smiley face Students used fewer than 6 equations correctly to create a smiley face Student used zero equations or student did not create a smiley face

Total: _________

Equation Part of Face Description of transformation(s)

y_1=

y_2=

y_3=

y_4=

y_5=

y_6=

y_7=

y_8=

y_9=

y_10=
Example:
y_1=-8¡Ì(1-(1/6 x)^2 )
Bottom of Face Vertical reflection over the x-axis
g(x) = -f(x)
Vertical stretch by a factor of 8
g(x) = 8f(x)
Horizontal stretch by a factor of 6
g(x) = f(1/6 x)

Parent Functions, Transformations, and Piecewise Functions

Lesson Title: Unit Assessment (Day 7 of 7)
Class length: 42 minutes

Objectives:
Students will identify transformations in graphical and algebraic representations of parent functions.
Students will convert between the graphical, algebraic, and language representations of transformed parent functions.
Students will convert between the graphical and algebraic representations of piecewise functions.

Materials:
Unit Assessment

Procedure:
Teacher Activities: Ask students if they have any questions before you pass out the unit assessment.
Student Activities:

Assessment:
Formal
Unit Assessment

References
Common Core Standards Initiative (2011). Common core mathematics standards. Retrieved from http://www.corestandards.org/the-standards/mathematics
GeoGebra (2012). GeoGebra (version 4) [computer software]. Available from http://www.geogebra.org/cms/
Kuta Software LLC (2010). Infinite Algebra 2 [computer software]. Available from http://www.kutasoftware.com/
Tyler, R.W. (1949). Basics principles of curriculum and instruction. Chicago, IL: The University of Chicago Press.