Activity 1: Explore Transformationsmathlandwithmszimmer.weebly.com/uploads/2/2/6/4/22649728/... · Activity, students practice a more visual approach. Many students will “see”

I. Foundations 1.1 Transformations: Activity 1

TEXTEAMS Part 1: Algebra II and Precalculus Institute 17DRAFT 3-29-03

Activity 1: Explore Transformations

Given: Graph of f Table for ff �2( ) = 1f �1( ) = �1f 0( ) = 2f 1( ) = 1f 2( ) = 3f 3( ) = 0

Complete tables and sketch graphs of:

1. y = f x + 2( )

x y-4-3-2-101



2. y = f x � 2( )

x y012345

3. y = f x( ) + 2x y-2-10123

4. y = f x( ) � 2x y-2-10123



5. y = f 2x( )

x y-1

-1/20

1/21

3/2

6. y = f12

x� �

� �

x y-4-20246

7. y = 2 f x( )

x y-2-10123



8. y =12

f x( )

x y-2-10123

9. y = f � x( )

x y-3-2-1012

10. y = � f x( )

x y-2-10123



11. y = f x( )

x y-2-10123

12. y = f x( )x y-3-2-10123



Activity 2: Domains

1. Look back at Activity 1. Notice that input values, the x-values,change for some of the transformations. The function is defined forthe x-values: –2, –1, 0, 1, 2, 3. But, for example, Exercise 1,

y = f x + 2( ) , provides the x-values of –4, –3, –2, –1, 0, 1 andExercise 2, y = f x � 2( ) , provides the x-values of 0, 1, 2, 3, 4, 5.Why? Use an example in your explanation.

2. Design a simple function g that is similar to f. Show a table andgraph that would allow your students to fill in the table and graph thetransformation g x + 3( ) .

Given: Graph of g Table for gg( ) =g( ) =g( ) =g( ) =g( ) =g( ) =

Complete the table and sketch the graph of:

y = g x +3( )

x y



Activity 3: Transformations with Technology

I. Explore Vertical Translations

1. Enter into y2 : y2 = y1 + 1Enter the following functions, one at a time, into y1 .Use a friendly window.Sketch the graph of y1 and y2 .

a. y1 = x b. y1 = ln x c. y1 = cos x d. y1 = �x + 2

2. Now enter y2 = y1 � 2

a. y1 = 0.5x b. y1 = sin x c. y1 = 2 x d. y1 = � x

3. Generalize: What happens to the graph of a function when youadd a constant to the function rule?



II. Explore Horizontal Shifts

1. Enter into y2 : y2 = y1 x � 1( )

Enter the following functions, one at a time, into y1 .Use a friendly window.Sketch the graphs of y1 and y2 .

a. y1 = x b. y1 = x2

c. y1 =

1x

d. y1 = x +1

2. Now enter y2 = y1 x + 2( )

a. y1 = 9 � x2 b. y1 = sin x c. y1 = x3 d. y1 = x

3. Generalize: What happens to the graph of a function when youreplace x with x � a ?



III. Explore Vertical Stretches and Compressions

1. Enter into y2 : y2 = 3y1

Enter the following functions, one at a time, into y1 .Use a friendly window.Sketch the graphs of y1 and y2 .

a. y1 = e x b. y1 = x 3 c. y1 = cos x d. y1 = x

2. Now enter y2 =13

y1 .

a. y1 = x b. y1 = sin x c. y1 = e x d. y1 = x

3. Generalize: What happens to the graph of a function when youmultiply the function rule by a?

I. Foundations 1.1 Transformations: Student Activity 1

TEXTEAMS Part I: Algebra II and Precalculus Institute 31DRAFT 3-29-03

Student Activity 1: Move the Monster Algebra II

Overview: Students sketch the graphs of transformations of a piece-wise definedfunction.

Objective: Algebra II TEKS(b.1.A) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations.(b.1.B) In solving problems, the student collects data and records results,organizes the data, makes scatter plots, fits the curves to the appropriate parentfunction, interprets the results, and proceeds to model, predict, and makedecisions and critical judgments.(c.1.B) The student extends parent functions with parameters such as m in

y = mx and describes parameter changes on the graph of parent functions.

Terms: translation, reflection, dilation

Materials:

Procedures: This activity is intended to reinforce students’ understanding after studentshave completed an activity similar to 1.1.1 Introducing Transformations. In1.1.1, students use a table to graph the transformations of a function. In thisActivity, students practice a more visual approach. Many students will “see”a table of values in their minds to help graph the transformations. The ideahere is to apply the “rules” they wrote in their summaries of transformations.Have students complete the activity and share graphing strategies.

• What is the domain of h ? .• What is the range of h ?• Compare the domain and range of f x( ) with the domains and ranges

of af x( ) , f x( ) + b , f ax( ) , f �x( ) , � f x( ) , f x + h( ) .

Summary: The big idea here is that students can use the general summary statements theymade about transformations to graph the transformations of a function withoutdoing it numerically first.



Activity: Move the Monster

Given h x( ):

Sketch the graph of:

1. �h x( ) 2. h �x( )



3. h x( )+ 2 4. h x( )� 2

5. h x + 2( ) 6. h x � 2( )

I. Foundations 1.1 Introducing Transformations: Student Activity Precalculus


7. 2h x( )8.

12

h x( )

9. h 2x( )10. h

12

x� �

� �



Student Activity 2: Move the Monster Precalculus

Overview: Students sketch the graphs of transformations of a piece-wise definedfunction.

Objective: Precalculus TEKS (c.2.A) The student is expected to apply basic transformations, includinga • f x( ), f x( ) + d, f x � c( ), f b • x( ), f x( ) , f x( ) , to the parent functions.(c.2.B) The student is expected to perform operations including compositionson functions, find inverses, and describe these procedures and results verbally,numerically, symbolically, and graphically.(c.3.c) The student is expected to use properties of functions to analyze andsolve problems and make predictions.


Materials:

Procedures: This activity is intended to reinforce students’ understanding after studentshave completed an activity similar to 1.1.1 Introducing Transformations. In1.1.1, students use a table to graph the transformations of a function. In thisActivity, students practice a more visual approach. Many students will “see”a table of values in their minds to help graph the transformations. The ideahere is to apply the “rules” they wrote in their summaries of transformations.Have students complete the activity and share graphing strategies.

• What is the domain of h ?• What is the range of h ?• Compare the domain and range of f x( ) with the domains and ranges

of af x( ) , f x( ) + b , f ax( ) , f �x( ) , � f x( ) , f x + h( ) .

Summary: The big idea here is that students can use the general summary statements theymade about transformations to graph the transformations of a function withoutdoing it numerically first.



Activity: Move the Monster

Given h x( ):

Sketch the graph of:

1. �h x( ) 2. h �x( )



3. h x( )+ 2 4. h x( )� 2

5. h x + 2( ) 6. h x � 2( )



7. 2h x( )8.

12

h x( )

9. h 2x( )10. h

12

x� �

� �



11. h x( ) 12. h x( )

13. Design a function, f , so that

f x( ) = f � x( )14. Design a function, g , so that

g �x( ) = �g x( )



Student Activity 3: Combinations of Transformations

Overview: Students sketch the graphs of function built from transformations of a parentfunction.

Objective: Algebra II TEKS(b.1.A) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations. (c.1.B) The student extends parent functions with parameters such as m in



Materials: Graphing calculators

Procedures:Activity: Combinations of Transformations

A hint about “order of transformations”—there are rules similar to “order ofoperations”. When you are trying to sketch the graph of a new function, it isgenerally helpful to do any stretches, shrinks, or reflections before you do thehorizontal or vertical shifts. In the same sense that multiplication takesprecedence over addition, choose to do the stretches, shrinks, or flips first.

Solution: 1. Begin with the parent function y = x2. The transformations are:

Flip or reflect across the x-axis to get y = -x2.Vertical stretch by a factor of 2 (y-values are doubled) to get y = -2x2.Horizontal slide left 3 units to get y = -2(x + 3)2.Vertical slide down 1 unit to get y = -2(x + 3)2 – 1.The result will be a parabola that opens downward. It will be “narrower”than normal, since y-coordinates are doubled.

2. All points, including the vertex, will then be moved 3 units to the left anddown 1 unit. The vertex of this parabola will be (-3, -1). The domain ofthe function will still be all real numbers; the range becomes y � -1.

4. A possible solution: Let f x( ) = x 2 , g x( ) = x + 3 , h x( ) = �2x , andj x( ) = x �1. Describe y = �2 x + 3( )

2�1 as a composition of functions.

Solution: y = �2 x + 3( )2

�1 can be expressed as y = j h f g( )( )( ) x( ) .



To understand better the “order of transformations,” use the graphingcalculator to do the steps listed in 1. A non-example and an example follow.Note that in the non-example, the vertical shift is done first and then thereflection across the x-axis. In the example the reflection is done first,followed by the vertical shift. If possible, use two overhead calculators toshow both the non-example and the example step by step at the same time.

Non-example (vertical shift before reflection) Example (reflection before vertical shift)



• Why does the order of transformations matter in this example? Show algebraically.Non-example example

y1 = x2

y2 = y1 x + 3( ) = x +3( )2

y3 = 2y2 = 2 x + 3( )2

y4 = y3 +1 = 2 x + 3( )2

+1 y4 = �y3 = �2 x + 3( )2

y5 = �y4 = � 2 x + 3( )2+1( )

= �2 x + 3( )2�1

y5 = y4 +1 = �2 x + 3( )2

+1

Note that in the last step in the non-example, the –1 is multiplied throughthe entire expression producing a shift down instead of the correct shift up.

Summary: The big idea here is that students can combine transformations to graph thetransformations of a parent function but they must be done in a certain order.



Activity: Combinations of Transformations

1. List the transformations applied to y = x2 to get y = �2 x +3( )2�1.

2. Describe the changes to domain and range.

3. Sketch the graphs of y = x2 and y = �2 x +3( )2�1.

4. Describe y = �2 x +3( )2�1 as a composition of functions.

5. Write your own function with at least 3 transformations. Sketch thegraph. Verify with a grapher.



Student Activity 4: Transformations on Generic Graphs

Overview: Students sketch the graphs of transformations of a generic function.

Objective: Algebra II TEKS(b.1.A) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations. (c.1.B) The student extends parent functions with parameters such as m in



Materials:

Procedures:Activity: Transformations on Generic GraphsA good assessment of whether someone understands the notation and thedescriptions of these transformations is determined by using a “generic”graph. This can be any function, such as the one graphed below, where only agraph is given, with no analytical formula. Now the graphing calculatorcannot provide the answers for the student. The calculator has been used asan exploration tool; now the student is applying mathematical concepts.Our “generic” graph is defined as follows:



1. y = � f x( ) 2. y = f x + 2( )

3. y =1

2f x( )�

� � �

� � � 34. y = f �x( )

5. y = f x( ) 6. y = 2 f x( )

Summary: The big idea here is that after students have learned the affects oftransformations on parent functions, they can apply this knowledge to findtransformations of a generic graph.



Activity: Transformations on Generic Graphs

Given: the generic graph shown below.

1. Sketch the graphs of the following transformations and describe thetransformations in words.

1. y = � f x( ) 2. y = f x + 2( )



3. y =12

f x( )�

� � �

� � � 3 4. y = f �x( )

5. y = f x( ) 6. y = 2 f x( )


TEXTEAMS Part 1: Algebra II/Precalculus Institute 48DRAFT 3-29-03

Student Activity 5: Transformation Practice Algebra II

Overview: Students graph the effects of transformations of parent functions, using agraphing calculator to check.

Objective: Algebra II TEKS(b.1.A) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range values forgiven situations.(c.1.B) The student extends parent functions with parameters such as m in y=mxand describes parameter changes on the graph of parent functions.(d.2.B) The student uses the parent function to investigate, describe, and predictthe effects of changes in a, h, and k on the graphs of y=a(x-h)2+k form of afunction in applied and purely mathematical situations.

Terms: Parent functions, transformation, domain, range, slide, shift, stretch,reflection, symmetry

Materials: Graphing calculator

Procedures:Students are to:(a) predict what the graph of each of the following transformations will look

like and(b) identify the effects of the transformation on the domain and range of the

parent function.Then they are to use their graphing calculators to verify their predictions.

Function: y1 = x2� 4x + 3

Viewing window: (–4.7, 4.7, 1, –2, 6, 1)Domain: all real numbersRange: y � �1 or [�1, �)Vertex: (2, –1)


Viewing window: (–3, 6.4, 1, –4, 6, 1)Domain: all real numbersRange: y � �2 or [�2, �)Vertex: (3, –2)

Function: y = x + 4Viewing window: (–4.7, 4.7, 1, –2, 5, 1)Domain: x� –4 or [–4, �)Range: y�0 or [0, �)



To check using a graphing calculator, you can use the following:

y = f x( ) � 3 Use y2 = y1 � 3y = f x � 3( ) Use y2 = y1 x � 3( )y = 2 f x( ) Use y2 = 2y1

y = � f x( ) Use y2 = �y1

Answers for transformations to y1 = x2� 4x + 3.

Function Transformation Domain/Rangey=f(x)+2 graph is shifted up 2;

vertex becomes (2,1)range is affected;range becomes [1, �)

y=f(x+2) graph is shifted left 2;vertex becomes (0, -1)

neither is affected

y = 2 f x( ) vertex becomes (2,-2);graph is steeper

range is affected;range becomes [-2, �)

y = � f x( ) graph is reflected aboutthe x axis;vertex becomes (2, 1)

range is affected;range becomes (-�, 1]


Function Transformation Domain/Rangey = f x( ) � 3 graph is shifted down 3;

vertex becomes (3, -5)range is affected;range becomes [-5, �)

y = f x � 3( ) graph is shifted right 3;vertex becomes (6, -2)

neither is affected



y = � f x( ) graph is reflected aboutthe x-axis;vertex becomes (3, 2)




Answers for transformations to y = x + 4 .

Function Transformation Domain/Rangey = f x( ) � 3 graph is shifted down 3 domain is not affected;


y = f x � 3( ) graph is shifted right 3 domain is affected;domain becomes [-1, �)

y = 2 f x( ) graph is vertically stretched neither is affected;

y = � f x( ) graph is reflected aboutthe x-axis

domain is not affected;range is affected;range becomes (-�, 0]

Summary: Students graph the effects of transformations of parent functions, using agraphing calculator to check.



Transformation Practice 1

Function: y = x2� 4x + 3

Window:Domain:

Range:

Vertex:

a. Predict the graph of each transformation.b. Describe the transformation in words.c. Identify effects on the domain and range.

1. y = f x( ) + 2 2. y = f x + 2( )

a.

b.

c.

a.

b.

c.

3. y = 2 f x( ) 4. y = � f x( )

a.

b.

c.

a.

b.

c.





Window:Domain:

Range:

Vertex:


1. y = f x( ) � 3 2. y = f x � 3( )

a.

b.

c.

a.

b.

c.

3. y = 2 f x( ) 4. y = � f x( )

a.

b.

c.

a.

b.

c.




Function: y = x + 4Window:Domain:

Range:


1. y = f x( ) � 3 2. y = f x � 3( )

a.

b.

c.

a.

b.

c.

3. y = 2 f x( ) 4. y = � f x( )

a.

b.

c.

a.

b.

c.


TEXTEAMS Part 1: Algebra II/Precalculus Institute Draft 54DRAFT 3-29-03

Student Activity 6: Transformation Practice Precalculus

Overview: Students graph the effects of transformations of parent functions, using agraphing calculator to check.

Objective: Precalculus TEKS(c.1) The student defines functions, describes characteristics of functions, andtranslates among verbal, numerical, graphical, and symbolic representationsof functions, including polynomial, rational, radical, exponential, logarithmic,trigonometric, piecewise-defined functions.(c.2.A) The student is expected to apply basic transformations, includinga•f(x), f(x)+d, f(x-c), f(b•x), |f(x)|, f(|x|), to the parent functions.

Terms: Parent functions, transformation, domain, range, slide, shift, stretch,reflection, symmetry

Materials: Graphing calculator

Procedures:Students are to:(a) predict what the graph of each of the following transformations will look

like and(b) identify the effects of the transformation on the domain and range of the

parent function.Then they are to use their graphing calculators to verify their predictions.


Viewing window: (–4.7, 4.7, 1, –2, 6, 1)Domain: all real numbersRange: y � �1 or [�1, �)Vertex: (2, –1)


Viewing window: (–3, 6.4, 1, –4, 6, 1)Domain: all real numbersRange: y � �2 or [�2, �)Vertex: (3, –2)

Function: y1 = x3� 4x

Viewing window: (–4.7, 4.7, 1, –6, 6, 1)Domain: all real numbersRange: all real numbersTurning points: approx. (–1.15, 3.08) (1.15, –3.08)



To check using a graphing calculator, you can use the following:

y = f x( ) � 3 Use y2 = y1 � 3y = f x � 3( ) Use y2 = y1 x � 3( )y = 2 f x( ) Use y2 = 2y1

y = f 2x( ) Use y2 = y1 2x( )y = � f x( ) Use y2 = �y1

y = f �x( ) Use y2 = y1 �x( )

y = f x( ) Use y2 = abs y1( )y = f x( ) Use y2 = y1 abs x( )( )


Function Transformation Domain/Rangey=f(x)+2 graph is shifted up 2;

vertex becomes (2,1)range is affected;range becomes [1, �)

y=f(x+2) graph is shifted left 2;vertex becomes (0, -1)

neither is affected



y = f 2x( ) Vertex becomes (1, -1);graph is steeper

Neither is affected

y = � f x( ) graph is reflected aboutthe x axis;vertex becomes (2, 1)


y = f �x( ) graph is reflected aboutthe y axis;vertex becomes (-2, -1)

neither is affected

y = f x( ) the section of the originalgraph that was below thex axis is reflected aboutthe x axis

range is affected;range becomes [0, �)

y = f x( ) the section of the originalgraph that was to the rightof the y axis is kept andreflected about the y axis;the graph looks like an “w”

neither is affected





vertex becomes (3, -5)range is affected;range becomes [-5, �)

y = f (x � 3) graph is shifted right 3;vertex becomes (6, -2)

neither is affected

y = 2 f (x) vertex becomes (3,-4);graph is steeper


y = f (2x) vertex becomes (1.5, -2);graph is steeper

neither is affected

y = � f x( ) graph is reflected aboutthe x-axis;vertex becomes (3, 2)


y = f �x( ) graph is reflected aboutthe y-axis;vertex becomes (-3, -2)

neither is affected

y = f x( ) the section of the originalgraph that was below thex-axis is reflected aboutthe x-axis


y = f x( ) the section of the originalgraph that was to the rightof the y-axis is kept andreflected about the y-axis;the graph looks like an “w”

neither is affected



Answers for transformations to y1 = x3� 4x .


turning points (-1.15, 0.08)and (1.15, -6.08)

neither is affected

y = f x � 3( ) graph is shifted right 3;turning points (1.85, 3.08)and (4.15, -3.08)

neither is affected

y = 2 f x( ) graph is vertically stretched;turning points (-1.15, 6.16)and (1.15, -6.16)

neither is affected;

y = f 2x( ) graph is horizontally shrunk;turning points (-0.575, 3.08)and (0.575, -3.08)

neither is affected

y = � f x( ) graph is reflected aboutthe x-axis;turning points (-1.15, -3.08)and (1.15, 3.08)

neither is affected

y = f �x( ) graph is reflected aboutthe y-axis; turning points(-1.15, -3.08) and (1.15, 3.08)

neither is affected

y = f x( ) the section of the originalgraph that was below thex-axis is reflected aboutthe x-axis


y = f x( ) the section of the originalgraph that was to the rightof the y-axis is kept andreflected about the y-axis

range is affected;range becomesapproximately [-3.08, �)

Summary: Students graph the effects of transformations of parent functions, using agraphing calculator to check.





Window:Domain:

Range:

Vertex:


1. y = f x( ) + 2 2. y = � f x( )

a.

b.

c.

a.

b.

c.

3. y = f x + 2( ) 4. y = f � x( )

a.

b.

c.

a.

b.

c.



5. y = 2 f x( ) 6. y = f x( )

a.

b.

c.

a.

b.

c.

7. y = f 2x( ) 8. y = f x( )

a.

b.

c.

a.

b.

c.





Window:Domain:

Range:

Vertex:


1. y = f x( ) � 3 2. y = f x � 3( )

a.

b.

c.

a.

b.

c.

3. y = 2 f x( ) 4. y = f 2x( )

a.

b.

c.

a.

b.

c.

Documents

Activity 1: Explore Transformationsmathlandwithmszimmer.weebly.com/uploads/2/2/6/4/22649728/... · Activity, students practice a more visual approach. Many students will “see”