Course Profile   Functions, Grade 11, University/College Preparation, Catholic and Public

 

Unit 2:  Function Notation, Inverses and Transformations

Time:  20 hours

 

Activity 2.1 | Activity 2.2 | Activity 2.3 | Activity 2.4 | Activity 2.5 | Activity 2.6 | Activity 2.7

 

Unit Description

Students, through authentic models, are introduced to the definition of a function and the notations associated with it. Students use graphing technology and paper-and-pencil tasks to investigate the properties of functions, their inverses and transformations of functions. The investigations are used to introduce and extend the use of function notation to inverses, and transformations. Students explore the domain and range of functions, inverses, and transformations.

 

Activity 2.1:  Wrap Around

Time:  150 minutes

Description

Students investigate the concept of functions and formalize the definition of a function. By extending and consolidating previously studied skills and concepts, students also investigate domain and range both graphically and algebraically.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2c - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE4f - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills;

CGE5e - a collaborative contributor who respects the rights, responsibilities and contributions of self and others;

CGE5g - a collaborative contributor who achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others.

Strand(s):  Tools for Operating and Communicating with Functions

Overall Expectations

OCV.02 - demonstrate an understanding of inverses and transformations of functions and facility in the use of function notation;

OCV.03 - communicate mathematical reasoning with precision and clarify throughout the course.

Specific Expectations

OC2.01 - define the term function;

OC2.02 - demonstrate facility in the use of function notation for substituting into and evaluating functions;

OC3.01 - explain mathematical processes, methods of solution, and concepts clearly to others;

OC3.02 - present problems and their solutions to a group, and answer questions about the
problems and the solutions;

OC3.04 - demonstrate the correct use of mathematical language, symbols, visuals (e.g., diagrams, graphs), and conventions.

Prior Knowledge & Skills

·         construct tables of values and graph linear and quadratic relationships

·         construct and manipulate relations

·         translate verbal relationships into algebraic terms (and vice versa)

Planning Notes

·         students are to be placed in groups of two or three

·         pipe cleaners of various lengths are needed (10 cm – 20 cm)

·         an overhead acetate for each group is needed (and water soluble markers)

·         graph paper is needed

Teaching/Learning Strategies

Teacher Facilitation

To help set a context and provide a motivation for functions the following statement can be provided to the class: “The average daily temperature and time of year are related to each other.”

This can prompt a class discussion as to how this can be represented and/or illustrated (i.e., graphically and/or algebraically). Students should be informed that this will be revisited later in this activity.

Even though the word “function” is not used in the first part of the student activity, it is designed for students to consolidate and extend their understanding of relationships between two variables. This will provide a platform for the teacher to define the term function and introduce the appropriate notations and definitions associated with it.

The teacher will give each group a pipe cleaner. The lengths can all be the same or different (suggested lengths are 10 cm, 12 cm, 14 cm, 16 cm, 18 cm, 20 cm).

The use of pipe cleaners is not essential but it does make the activity more tactile for the students.

Each group is to put their answers to questions 4, 7.c), 8.a), and 8.c) on their overhead acetates.

The teacher can choose selected groups (or all groups) to present their overhead solutions.

Student Activity

Every member of the group is to complete the following questions.

1.   What is the length of the pipe cleaner?

Bend 1 cm on each end of the pipe cleaner so that these ends will be parallel to each other and each perpendicular to the remaining piece of pipe cleaner.

 

 

2.   a)   The bottom part of the pipe cleaner is called the base. What is the length of the base in the           diagram?

b)   What is the area if the bent ends are joined to form a rectangle?

 

 

3.   Straighten out the pipe cleaner. Now do the same as above, bending each end 2 cm this time. Find the height, base, and area.

4.   Continue straightening the pipe cleaner and bending it to complete the following table.

Height (h)

Base (b)

Area (A)

1

 

 

2

 

 

3

 

 

4

 

 

 

 

 

5.   a)   Why is the height considered a variable in the activity?

b)   Are there any other variables? If so, what are they?

6.   Are there any constants in this activity? If so, what are they?

7.   a)   Is there a relationship between height (h) and base (b)?

b)   Describe this relationship?

c)   Write an algebraic equation for this relationship.

8.   a)   Describe the relationship between area (A) and height (h)

b)   Write an algebraic relationship.

c)   If the original length of the pipe cleaner was unknown (let it be k), write an algebraic relationship   between area (A) and height (h)

Teacher Facilitation

Teacher uses the results of the above activity to define function (special correspondence between the elements of two sets; value of one quantity depends on or is related to the value of another quantity), domain and range.

Teacher can use the input/output idea to illustrate results of Student Activity 1 in diagram form (mapping/arrow notation). e.g.,

 

 

In the assignment below, the teacher will give each group an opportunity to come up with three other examples of functions to present to the class.

The teacher must ensure that students are aware that functions can be described by algebraic formulas, written statements, as a table (or diagram) of input/output data or by a graph.

The teacher should also facilitate students’ understanding of the connection between the use of the “Vertical Line Test” for functions and the definition of a function. A variety of graphs could be given for the students to see the role of this excellent type of visual cue for functions.

The teacher may need to prompt some groups (or entire class) with examples (e.g., position of a speedometer needle depends on the value of the speed; square root key on a standard scientific calculator; linear and quadratic relationships; etc.).

It should be noted that task 5 (Question 5 on the Student Assignment – Part A) is the statement that was presented at the beginning of this activity. The teacher may need to prompt students to make the “proper” choice for the independent and dependent variables (and hence axes).

Student Assignment (Part A)

1.   State three examples of functions.

2.   Discuss why these examples are functions.

3.   What are the domain and range of each?

4.   Draw a diagram to illustrate each.

5.   “The average daily temperature and time of year are related to each other.”

a)   Sketch a reasonable graph that represents the above statement.

b)   Describe the graph/relationship.

Teacher Facilitation

Teacher uses the results from one group’s Activity 1 to introduce function notation.

Example: A = h(20 – 2h) is equivalent to f(h) = h(20 – 2h)

Teacher may want to illustrate and explain specific cases (e.g., f(1), f(2.5), f(0), f(-3), etc.). Particular attention must be given to the common misconception by students that the brackets in f(1) mean multiplication.

For the questions listed below, the students are to work in their groups, using the results from the pipe cleaner activity. Each student is to complete all questions.

Student Assignment (Part B)

(use the results from your pipe cleaner activity to answer the following)

1.   Evaluate the following:

a) f(1)                     b) f(2)               c) f(2.5)                        d) f(5)               e) f(10)             f) f(0)

g) f(-1)                   h) f(f(1))

2.   Give a practical interpretation (using the context of the pipe cleaner activity) to each answer in #1.

3.   Sketch a graph of the function.

4.   Pose 3 different questions about the pipe cleaner activity that can be answered by referencing to the graph.

5.   Explain how to answer your questions using your graph.

Assessment & Evaluation of Student Achievement

Learning skills, specifically teamwork, independence and initiative can be assessed in all the activities, especially in the group work components.

Conferencing can assess Knowledge/Understanding of groups and individuals during all activities.

Knowledge/Understanding can also be assessed by a quiz (or quizzes) that can be given after all activities or after any one of the activities. An objective marking scheme can be used.

Communication skills can be assessed using criteria such as; degree of clarity of explanations, appropriate use of mathematical language, use and clarity of mathematical diagrams or graphs, correct use of mathematical notations during any of the oral presentations recommended in the activities.

Individual student written reports (Student Assignment (Part A) question 5 and Student Assignment
(Part B) can be assessed using a rubric.

Category

Level 1

(50 – 59%)

Level 2

(60 – 69%)

Level 3

(70 – 79%)

Level 4

(80 – 100%)

Inquiry

- graph and describe a relationship

- limited understanding of a relationship

- some understanding of a relationship

- general understanding of a relationship

- thorough understanding of a relationship

Communication

- use of clear and concise mathematical language for explanations

- limited precision and clarity

- sometimes uses appropriate mathematical terminology clearly and precisely

- most often uses appropriate mathematical language clearly and precisely

- consistently uses appropriate mathematical language clearly, precisely

Application

- degree of independence for practical interpretation of function notation

- able to make connections with considerable assistance

- able to make connections with some assistance

- able to make connections with limited assistance

- able to make connections independently

Note: A student whose achievement is below level 1 (50%) has not met the expectations for this assignment or activity.

Follow-up Skills:  75 minutes

Teacher should supplement these activities with textbook exercises (include a wide variety of paper-and-pencil type questions):

·         evaluating functions expressed in function notation

·         find a given f(a)

·         finding domain and range (discrete and continuous types)

·         graphing functions expressed in function notation

·         interpreting function notation using the graph of a function (e.g., find f(1) from a given graph)

·         interpreting function notation from diagrams and other visual forms

·         use of “vertical line test” for functions (when is it a function? When is it a relation?)

Short quiz can be given after the follow-up skills to assess Knowledge and Application using an objective marking scheme.

 

Activity 2.2:  Home on the Range!

Time:  150 minutes

Description

Students graph various relations with and without technology on the Cartesian Plane. Using their results students investigate properties of functions (such as shape, domain, range, intercepts, asymptotes, etc.).

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE4f - a self-directed, responsible, life long learner who applies effective communication decision-making, problem-solving, time and resource management skills;

CGE5a - a collaborative contributor who works effectively as an interdependent team member.

Strand(s):  Functions and Relations

Overall Expectations

OCV.02 - demonstrate an understanding of inverses and transformation of functions and facility in the use of function notation;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC2.02 - demonstrate facility in the use of function notation for substituting into and evaluating functions;

OC2.03 - demonstrate, through investigation, the properties of functions defined by f(x) = ,
[e.g., domain, range, relationship to f(x) = x2] and f(x) = 1/x [e.g., domain, range,
relationship to f(x) = x];

OC3.03 - communicate solutions to problems and to findings of investigations clearly and concisely, orally and in writing, using an effective integration of essay and mathematical forms;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

·         use a graphing calculator or graphing software

·         construct table of values and graph functions

Planning Notes

·         The teacher may want to have the students use graphing calculators to verify their manually graphed functions on their graph paper. (Dynamic software may be used as an alternative to graphing calculators.)

·         The teacher may need to book computer time, if dynamic software is to be used.

·         Use an overhead projection tablet, if available.

Teaching/Learning Strategies

Teacher Facilitation

Divide the class into five groups. Every group is to do all graphs in question 1. If there is a concern that graphs could become messy doing all five graphs on the same axes, the teacher can have students do them on separate axes or group them in whatever fashion may be appropriate.

Graphs could be verified using technology. The teacher must ensure that students are aware the “f(x) =” is the same as “y =”, since graphing technology has defining equations in the letter. Each group will be designated a specific function to present to the class using technology. Group presentations will be used to address question 1 and 2 in the assignment below.

The teacher should use the results from question (1.iv)) and question (1.v)) to introduce and define the concept of asymptotes.

Students are to do questions 3, 4, and 5 individually.

In 3.ii), students should be analysing the nature of the asymptotes and the conditions for increasing and decreasing functions.

In question 5, students use f(0) to analyse conditions for y-intercepts vs. asymptotes

Student Activity

Graphing Assignment: (verify your graphs with graphing technology)

1.   Construct a table of values and graph each function on the same axes. (use different colours)

i)    f(x) = x2                        ii) f(x) = x3        iii) f(x) =     iv) f(x) = 1/x     v) f(x) = 2x

2.   Find the domain, range, and any other special characteristics of each function.

3.   Compare/contrast each pair of graphs (analyse shape, domain, range, asymptotes, etc.)

i)    y =  and y = x2

ii)   y = x and y = 1/x

iii)   y = x2 and y = x3

4.   For parts 3. i), ii) and iii) find f(0), f(1), f(2), f(3), f(4) and note similarities or differences.

5.   Are there any pattern(s) for functions that have similar f(0) and different f(0) values?

Assessment & Evaluation Student Achievement

By ensuring that all members of each group are involved in the presentation, the teacher can assess Knowledge (accuracy of graphing, properties of functions and their graphs as criteria) and Communication (clarity of presentation, use of mathematical language and notations as criteria)

The teacher should observe and conference with individual students while they are doing questions 3, 4, and 5. A class discussion can be used to consolidate concepts.

Upon completion of the assignment, a quiz with an objective marking scheme can be used to assess Knowledge and Application.

Accommodations

Partner students who may be experiencing difficulties with the use of graphing technology with a team/group that has at least one member who is proficient in the use of graphing technology.

Follow-up Skills:  75 minutes

Teacher should supplement this activity with textbook exercises that should include finding domain, range, asymptotes and other characteristics of a wide variety of functions given algebraic equations or other forms of functions (e.g., written form, table of values, diagrams, graphs, etc.).

An open ended task in which a domain and/or range is given, then require students to give a function that would fit the domain and/or range would serve as an excellent assessment task.

 

Activity 2.3:  Follow the Bouncing Ball

Time:  105 minutes

Description

Students use a graphing calculator and motion sensor (e.g., Calculator-Based Ranger; CBR) to gather data from a bouncing ball. The students fit an equation to the data (and graph) that is generated. Students investigate the properties of the function that best fits the data.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2b - an effective communicator who reads, understands and uses written materials effectively;

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE5a - a collaborative contributor who works effectively as an interdependent team member.

Strand(s):  Functions and Relations

Overall Expectations

OCV.02 - demonstrate an understanding of inverses and transformation of functions and facility in the use of function notation;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC2.02 - demonstrate facility in the use of function notation for substituting into and evaluating functions;

OC3.03 - communicate solutions to problems and to findings of investigations clearly and concisely, orally and in writing, using an effective integration of essay and mathematical forms;

OC3.04 - demonstrate the correct use of mathematical language, symbols, visuals (e.g., diagrams, graphs), and conventions;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

·         Use a graphing calculator and a device to measure motion.

·         Retrieve data from the graphing calculator and plot the data.

·         Understanding and use of function notation (y = f(x)).

Planning Notes

·         The students are placed in groups of two or three to do the activity

·         Basketball (or similar size bouncing ball), graphing calculator and motion sensor supplied to every group

·         The teacher will review the skills needed to use the graphing calculators and motion sensor

Teaching/Learning Strategies

Teacher Facilitation

Convey to the students that they will be gathering data of a uniformly accelerated basketball (gravity ensures uniform acceleration). Prior to this activity students have been introduced to the notation y = f(x). The teacher should take the opportunity to discuss the context for this activity since this data represents a function that can be written in the form d = f(t) (i.e., distance is a function of time).

Groups should gather their data by dropping the ball from an original height > 3 m (e.g., top of a staircase, bleachers, etc.).

The teacher may do a short demonstration of the calculator and the motion sensor set up, if necessary.

All students are to complete the student assignment component individually.

Student Activity

One member of the group holds the motion sensor at the top of an open staircase (height > 3 m) and another member holds the ball approximately 0.5 m below the motion sensor.

Set the measuring time of the motion sensor to the approximate time it takes the basketball to drop. (This may require a few test runs.)

When the CBR is ready, release the ball.

Observe the corresponding distance-time graph for the ball.

Repeat the activity at least once to verify the results.

Retrieve the data from the graphing calculator, construct a table of values and plot it on graph paper (to be done by every student).

Only the data gathered until the ball touches the floor (or ground) is to be plotted. Do not include any data after the first bounce.

Teacher Facilitation

When the activity is done, students can see the actual results on the calculator  (it should resemble a half parabola opening upward) Distance on the graph represents the distance the basketball is away from the motion sensor at any point in time.

Student Assignment

1.   State the dependent and independent variables.

2.   a)   Is distance a function of time or time a function of distance? Explain your answer.

b)   Using a graphing calculator, find the equation that corresponds to the function.
(quadratic regression)

3.   Find the domain and range.

4.   What does the domain represent in this activity?

5.   What does the range represent in this activity?

6.   Find f(1), f(0.5), f(0). What do they represent in this activity?

9.   Find f(-1). What does it represent in this activity? (time cannot be negative)

10.  List as many properties as possible of the graph.

Assessment & Evaluation of Student Achievement

The student assignment can be collected as a written report.

·         Knowledge can be assessed by asking students to find domain and range and to use function notation in context.

·         Application can be assessed by asking students to interpret f(1), f(0.5), f(0) on the d = f(t) graph.

·         Communication skills can be assessed by using criteria: the degree of clarity of explanations and the use of appropriate mathematical language.

Accommodations

Partner students in teams/groups to provide support for students who require peer assistance.

 

Activity 2.4:  Let’s Switch Seats

Time:  105 minutes

Description

Through the investigation of the graphs of various relations students discover properties of inverse functions. Students also discover the algebraic approach to finding the inverse of a function.

Strand(s) & Learning Expectations

Ontario Catholic Graduate Expectations

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE4f - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills.

Strand(s):  Functions and Relations

Overall Expectations

OCV.02 - demonstrate an understanding of inverses and transformations of functions and facility in the use of function notation;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC2.04 - explain the relationship between a function and its inverse (i.e., symmetry of their graphs in the line y = x; the interchange of x and y in the equation of the function; the interchanges of the domain and range), using examples drawn from linear and quadratic functions, and form the functions f(x) =  and f(x) = 1/x);

OC3.03 - communicates solutions to problems into findings of investigations clearly and concisely, orally and in writing, using an effective integration of essay and mathematical forms;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

Graphing Relations

·         graph a relation using a table of values

·         operate a graphing calculator or graphing software

Planning Notes

The use of a graphing calculator or graphing software may be used to allow the students to verify the graphs of the relations and the accompanying tables of values.

Teaching/Learning Strategies

Teacher Facilitation

The activity is to be done as an individual assignment with the teacher assisting with any difficulties. The teacher can introduce inverse function by having the students observe “inverses” on their scientific calculators (i.e., what is on top of the x2 key? and consolidating previous concepts and characteristics of
y = x2 and y = ,). As part of the class discussion the teacher can emphasize inverses in other aspects of mathematics and life.

For Example: What is the inverse of:       i) multiplication              ii) squared         iii) up

iv) backwards               v) right              vi) subtraction

While working through the activity students may need some direction to realize the following about inverses:

1.   they can be found by interchanging x and y in the table of values (i.e., in the ordered pairs);

2.   they can be found algebraically by interchanging x and y in the defining equation;

3.   they can be found graphically by reflection in the line y = x (i.e., .symmetry in the line y = x);

Students should be informed, before starting the assignment, that the given pair of graphs in Part 1 are inverses and they will be looking for some special properties of inverses by using them.

Particular attention must be given to the introduction of the notation f-1(x) in Part 2 and that the students understand this notation.

Student Assignment

Part 1

Graph each relation using a table of values.

Graph 1            f(x) = 3x + 2                 Graph 2            f = {(x,y)/ y = x2, x>0}

g(x) =                                          g = {(x,y)/ y = , x>0}100

 

Part A

For Graph 1: (graph y = x, using a different colour, on your axes)

1.   Find the domain and range of the functions f and g.

2.   Compare the table of values for f and g, listing all similarities and differences.

3.   Compare the graphs of f and g, listing all similarities and differences.

4.   How are the graphs of f and g related with respect to the line y = x?

5.   By putting the equations for f and g in the form “y = …” and “x = …,” explain how the equation of one function can be used to find the other?

6.   Determine the intercepts of f and g. Describe any special characteristics.

7.   Determine the slopes of f and g. Describe any unique characteristics.

Part B

Repeat questions 1-6 for Graph 2.

Part 2

Given: f(x) = 2x + 4.

Interchange x and y in the defining equation of f(x).

Solve for y in this new equation. The result will be called f -1(x).

1.   Graph y = f(x), y = f -1(x) and y = x on the same Cartesian Plane.

2.   Compare the slopes of y = f(x) and y = f -1(x).

3.   Compare the x and y intercepts of y = f(x) and y = f -1(x).

4.   Find any point [P(x,y)] on y = f(x) and find the distance from point P to the line y = x.

5.   Interchange the x and y coordinates of P (let this new point be Q). Find the distance from point Q to the line y = x.

6.   How do the results in question 4 and 5 compare? Prove this result.

7.   Find the slope of line y = x and the slope of line PQ. How do they compare? Prove this result.

Assessment & Evaluation of Student Achievement

The student assignment can be used to assess Application (appropriate algebraic manipulations and calculations; validity of interpretations; quality of connections that are made between a function and its inverse).

The teacher should use a class discussion to consolidate all the algebraic and graphical relationships of a function and its inverse and well as the appropriate notations used for inverses.

Follow-up Skills:  75 minutes

The teacher should supplement this activity with textbook exercise questions (include a wide variety of pencil-and-paper type questions) that should include the following:

·         given defining equation of a function, find its inverse algebraically;

·         given a written definition of a function, describe its inverse;

·         given a graphical model of a function, find its inverse graphically given a table of values (or parts of one), find the inverse;

·         multi-step questions: e.g., given f -1(1) = 3 and f -1(4) = 6 and f is a linear function, find its equation. Find the inverse;

·         find f -1(f -1(x)) for a variety of functions. What do you notice?

Activity 2.5:  On the Move

Time:  150 minutes

Description

Students investigate the effect of transformations on the mathematical functions f(x) = x, f(x) = x2, f(x) = x3, f(x) = , f(x) = , 1/x using graphing technology. The student also formalize the algebraic approach to the transformation of functions using appropriate function notation.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2c - be an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

CGE5a - be a collaborative contributor who respects the rights, responsibilities and contributions of self and others.

Strand(s):  Functions and Relations

Overall Expectations

OCV.02 - demonstrate an understanding of inverses and transformations of functions and facility in the use of function notation;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC2.06 - represent transformations (e.g., translations, reflections, stretches) of the functions defined by f(x) = x, f(x) = x2, f(x) = , f(x) = sin x, f(x) = cos x, using function notation;

OC2.07 - describe, by interpreting function notation, the relationship between the graph of a function and its image under one or more transformations;

OC2.08 - state the domain and range of transformations of the functions defined by f(x) = x, f(x) = x2, f(x) = , f(x) = sin(x), and f(x) = cos(x);

OC3.02 - present problems and their solutions to a group, and answer questions about the problems and the solutions;

OC3.04 - demonstrate the correct use of mathematical language, symbols, visuals (e.g., diagrams, graphs), and conventions.

Prior Knowledge & Skills

Graphing Relations

·         graph a relation using a table of values and operate a graphing calculator or graphing software

·         understanding of the shape the basic functions used in the activity

Planning Notes

·         Provide a graphing calculator available for every student (or at least one for each pair) or book computer time for the use of appropriate graphing software

·         The teacher can use an overhead projection tablet for demonstration purposes

·         The activity can be done without the use of graphing technology but the students will spend a significantly longer period of time graphing using a table of values.

Teaching/Learning Strategies

Teacher Facilitation

Students are to be placed into six groups. Each group presents to the class the result of its transformation(s) using (and answering) the questions assigned to each group incorporating the graphing calculator and the overhead tablet

The teacher should take the opportunity for class discussion to formalize the appropriate function notations associated with the various transformations (e.g., for a vertical stretch the defining equation for the function y = f(x) becomes y = af(x).

Student Activity

Part 1:  Group Presentations

For the functions:

i) f(x) = x          ii) f(x) = x2          iii) f(x) =     iv) f(x) = 1/x     v) f(x) = x3

Groups 1 to 4:

1.   Find and simplify the defining equation for all the graphs assigned to your group (look below).

{Example: for the function f(x) = x3, f(x – 2) = (x – 2)3 = x3 – 6x2 + 12x – 8}

2.   Graph each of the five functions listed above along with the group of transformations defined for your group on a separate Cartesian Plane.

Group 1      y = f(x), y = f(x) + 5, y = f(x) – 6

Group 2      y = f(x), y = f(x – 4), y = f(x + 2)

Group 3      y = f(x), y = 2f(x), y = ˝ f(x), y = -3f(x)

Group 4      y = f(x), y = f(2x), y = f, y =

3.   How is the original function affected? (shape, orientation, key points, position, etc.)
Describe the transformation that is applied.

Groups 5 and 6:

1.   Find and simplify the defining equation for all the functions assigned to your group (look below)

{Example: for the function f(x) = x2, 3f(x + 1) = 3(x + 1)2 then expand and simplify)

2.   Graph the functions listed above according to the transformations defined for your group on a separate Cartesian Plane.

Group 5      y = f(x), y = f(x + 4), y = f(x) + 4, y = f(x – 3), y = f(x) – 3

Group 6      y = f(x), y = 2f(x), y = f(2x), y = -0.5f(x), y = f(-0.5x)

3.   Describe how each transformation affects the function. Describe all similarities and differences.

Part 2:  Individual Assignment

Teacher Facilitation

Each student is to be assigned two of the five functions listed at the beginning of Part 1. This part is to be completed and submitted by each student individually.

1.   Find the defining equation for and graph f(x) according to the following transformations.

y = f(x)                   y = f(2x) – 6                 y = 1/2f(x + 4)               y = f(x – 4) + 3

y = 3f(2x)               y = – 0.5f(x – 7) + 2      y = 2f(0.5x)                   y = -2f(3x – 1) – 5

2.   Describe how each transformation affects the function. Describe all similarities and differences.

Assessment & Evaluation of Student Achievement

By conferencing with the students during their group work, Knowledge can be assessed using the following criteria: i) accuracy of content, ii) correct algebraic manipulation, iii) appropriate use of function notation for transformations.

Communication can be assessed using the following criteria: i) clarity of presentation, ii) proper use of mathematical language, iii) proper use of technology.

The teacher can use Part 2 to assess Application (algebraic manipulation, use of function notation, interpretation of appropriate transformations) using an objective marking scheme.

Accommodations

Partner students in teams/groups to provide support for students who require peer assistance.

Follow-up Activities

1.   An interesting and fun activity is to have the students draw a simple picture or shape (a box, face, house, logo, etc.) on a Cartesian Plane and state the key coordinates of that shape. Students can then perform one of each type of transformation (and/or multiple transformations) to their picture and show the results (which are often quite funny and interesting) to the class.

2.   Draw a cartoon character on a Cartesian Plane. Students must apply (and describe) a series of transformations that will allow the character to be “animated.”

Follow-up Skills:  75 minutes

The teacher should supplement these activities with textbook exercises (include a wide variety of paper-and-pencil type questions):

·         Find the defining equation of a function with a given equation under the transformations.

·         Find the defining equation of a function with a given equation under more than one transformation.

·         Find the transformation(s) that would be necessary to transform the equation of a given function to another given function.

·         Find the transformation(s) that would be necessary to transform the graph of a given function to another given function.

 

Activity 2.6:  Be My Valentine

Time:  105 minutes

Description

Students, with the aid of graphing technology, construct a greeting card using appropriate functions and transformations. Students also analyse the functions and transformations both algebraically and graphically.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2b - an effective communicator who reads, understands and uses written materials effectively;

CGE4f - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills.

Strand(s): Tools for Operating and Communicating with Functions

Overall Expectations

OCV.02 - demonstrate an understanding of inverses and transformations of functions and facility in the use of function notation;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC2.06 - represent transformations (e.g., translations, reflections, stretches) of the functions defined by f(x) = x, f(x) = x2, f(x) = , f(x) = sin x,  f(x) = cos x, using function notation;

OC2.07 - describe, by interpreting function notation, the relationship between the graph of a function and its image under one or more transformations;

OC2.08 - state the domain and range of transformations of the functions defined by f(x) = x, f(x) = x2, f(x) = , f(x) = sin(x), and f(x) = cos(x);

OC3.01 - explain mathematical processes, methods of solution, and concepts clearly to others;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

·         use a graphing calculator (and a graph link cable for computer interfacing)

·         analyse and graph algebraic relationships

Planning Notes

·         need a graphing calculator for every student

·         need an overhead projection tablet for demonstration purposes

·         do the class demonstration in advance

·         provide or ask students to bring graph paper

Teacher/Learning Strategies

Teacher Facilitation

Teacher will demonstrate a “Have A Happy Valentines Day” greeting card using a graphing calculator (and overhead projection tablet).

Discuss with the class the instructions necessary to get the picture.

Discuss with the class the analysis (appropriate functions and transformations) necessary to obtain the defining equations used to generate the picture.

If the teacher wants students to print out their cards, a demonstration must be given on the process of downloading calculator images to a computer.

 

 

Instructions for generating a Valentine’s Day card:

(These generic calculator instructions; depending on the model adjustments may be necessary)

[WINDOW]                 [MODE]

Xmin=0             Dot

Xmax=12                      [Y=]setting

Xscl=1                          y1=(-x+8)(x$2)(x#6)

Ymin=0                        y2=(x-4)(x$6)(x#10)

Ymax=10                      y3=((-0.5)(x-4)2+8)(x$2)(x#6)

Yscl=1                          y4=((-0.5)(x-8)2+8)(x$6)(x#10)

**the $ and # are under [2nd][TEST]

 

[2nd][QUIT]Home Screen

Text (5,10,”HAVE A HAPPY”)

Text (50,30,”VALENTINES DAY”)

**Text is under [2nd][DRAW]

**to type letters, LOCK the ALPHA key with [2nd][A-LOCK]

*to change position of the words recall that Text (row#,column#, …………)

where the corners of the screen are:       (0,0)     (0,94)

(57,0)    (57,94)

Student Activity

Construct a Greeting Card that includes an appropriate picture and saying, using a graphing calculator. As part of a journal assignment include a discussion and analysis on the process that was used to arrive at the picture. Be sure to include both a graphical and algebraic analysis, along with appropriate transformations and any adjustments that you made throughout the process.

Assessment & Evaluation of Student Achievement

If students are able to download their greeting card and instructions to a computer, a formal written report (including computer downloads and journal) can be submitted and assessed by the teacher. The graphical and algebraic analysis of the functions can be used as a basis for assessing Knowledge and Application using an objective marking scheme. If computer downloads are not used, students can submit those parts of the written report without technology (i.e., paper-and-pencil).

Upon completion of the activity, each student can be assigned a partner that they would have to present their greeting card and journal. Communication and Inquiry can be assessed using a self-and peer assessment checklist. This should be developed prior to the activity as part of a class discussion. Criteria that can be used include: the depth and clarity of the written and oral explanations, appropriate use of notations, symbols and diagrams, incorporation of the appropriate inquiry actions. Students can use column headings: Needs Improvement, Satisfactory, Good and Excellent.

Accommodations

Since student skill in using the graphing calculator may vary, it may be appropriate to allow students who need assistance to seek some help from peers who are proficient in the use of graphing technology.

 

Activity 2.7:  Consolidating and Connecting

Time:  105 minutes

Description

In this activity students will consolidate and connect their understanding of functions with emphasis on the quadratic function. The activity can be used as either a teaching/learning opportunity or an assessment opportunity, depending on the needs and readiness of the students.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE3c - thinks reflectively and creatively to evaluate situations and solve problems;

CGE4f - applies effective communication, decision-making, problem-solving, time and resource management skills;

CGE5a - works effectively as an interdependent team member.

Strand(s):  Functions and Relations

Overall Expectations

OCV.01 - demonstrate facility in manipulating polynomials, rational expressions, and exponential expressions;

OCV.02 - demonstrate an understanding of inverses and transformations of functions and facility in the use of function notation;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC1.03 - determine the maximum or minimum value of a quadratic function whose equation is given in the form y = ax2 + bx + c, using the algebraic method of completing the square;

OC2.02 - demonstrate facility in the use of function notation for substituting into and evaluating functions;

OC2.06 - represent transformations (e.g., translations, reflections, stretches) of the functions defined by f(x) = x, f(x) = x2, f(x) = , f(x) = sin x,  f(x) = cos x, using function notation;

OC2.07 - describe, by interpreting function notation, the relationship between the graph of a function and its image under one or more transformations;

OC3.02 - present problems and their solutions to a group, and answer questions about the problems and the solutions;

OC3.03 - communicate solutions to problems and to findings of investigations clearly and concisely, orally and in writing, using an effective integration of essay and mathematical forms.

Prior Knowledge & Skills

Quadratic Equation

·         the properties of quadratic functions represented in algebraic, graphical, numerical, or contextual form.

·         function notation

·         domain and range

·         transformations of functions

·         difference tables

Planning Notes

·         If this activity is used as a teaching/learning activity, form students placed into heterogeneous groups of 2 or 3.

·         If the activity is used as an assessment activity, organize the students into small groups to discuss the problem set for approximately 10 minutes, without taking notes. Then require students to complete the activity independently

·         Provide toothpicks

·         graph paper and graphing calculators, if requested by the students

Teaching/Learning Strategies

Teacher Facilitation

Circulate around the classroom as students clarify their understanding of the problems. If any group is in need of extra support, provide toothpicks for them to create concrete models before moving to pictorial form. Students experiencing difficulties may need assistance in constructing appropriate categories (i.e., vertical toothpicks, horizontal toothpicks, inside (interior) toothpicks, outside (exterior) toothpick.

If this activity is being used as a teaching/learning activity, ensure that students are aware of the inquiry/problem solving stages they are engaged in for each part of question 1. The intent of question 2 is to help students connect their understanding of discrete vs. continuous functions, introduced in Grade 9, to the quadratic functions studied in Grade 10 and Units 1 and 2 of this course, and to set the stage for sequences to be introduced later in this course.

Ensure that students understand how they could be expected to deduce that the required equation in question 3 is 2(x2 – 2) despite the fact that they cannot read off the exact x-intercepts.

The intent of question 4 is to help students connect their understanding of difference tables to the coefficients of the corresponding equation. It also sets the stage for derivatives, by leading students to say that the “rate of change is increasing at a constant rate of 8”.

More questions of a skill-based nature can be added at the discretion of the teacher.

Student Activity

 

               

 

1.   Use the toothpick diagrams above to answer the following questions:

a)   Predict the number of toothpicks needed to complete the 100th such toothpick diagram, providing detailed evidence of your analysis.

b)   Using function notation, write a formula for the number of toothpicks needed to complete the kth toothpick diagram. Name the type of function and define its domain and range.

c)   Describe a function that, with appropriate domain restrictions, could be modeled by f(k) = 4k, where k is the number of toothpicks along one outside edge of the square.

d)   Describe a function that, with appropriate domain restrictions, could be modeled by f(k) = k2, where k is the number of toothpicks along one outside edge of the square.

e)   Form a hypothesis about the number of interior toothpicks (not including those toothpicks on the outside perimeter). Confirm or refute your hypothesis.

2.   The formula from question 1b) should be related to the function y = 2x2 + 2x

a)   Describe the transformations that relate y = 2x2 + 2x to y = x2

b)   Illustrate, on a graph, differences between the entire function y = 2x2 + 2x and the part of the function used to model the number of toothpicks in the kth diagram. Explain your reasoning.

3.   Find the equation of the function shown above, given that it is a polynomial of degree 4 or less and has integral numerical coefficients between –5 and 5.

 

 

4.   a)   Construct difference tables for the functions y = x2 , y = 2x2 , y = 2x2 + 2x, y = 2x2 – 4x

b)   What do all the difference tables have in common?

c)   Describe how the values in the different difference tables relate to the coefficients in the equation of the function.

d)   If a function has constant 2nd differences of 8, what would the equation of the function look like?

e)   Describe what 8 measures in question 4d), using the phrase “rate of change.”

Extension

1.   Construct rectangles of the form k ´ (k + 1) and find the corresponding f(x)

2.   Allow students to manipulate dimensions and find corresponding relationships

Assessment & Evaluation of Student Achievement

By conferencing with students during the teaching/learning activity the teacher can assess Knowledge (use of function notation, transformations and understanding of domain and range) and oral Communication (correct use of mathematical language and notations).

If the activity is being used to gather assessment data, question 1 could be assessed for Thinking/Inquiry/Problem Solving and Communication, question 2 for Application and Communication, question 3 for Thinking/Inquiry/Problem Solving and question 4 for Knowledge/Understanding. It would be appropriate to add more questions to gather further Knowledge/Understanding and Application assessment data.

Accommodations

Partner students in teams/groups to provide support for students who require peer assistance.

Allow extra time for those students who have it identified on their IEP.

 

 

 

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