conceptual understanding: program of study outcomes

Learning Intentions Conceptual Understanding: Transformations of functions can be used to create representations of objects and natural phenomena. Program of Study Outcomes:

§ RF2/3 – Demonstrate an understanding of the effects of horizontal and vertical translations and stretches on the graphs of functions and their related equations [C, CN, R, V]

§ RF4 – Apply translations and stretches to the graphs and equations of functions. [C, CN, R, V] § RF5 – Demonstrate an understanding of the effects of reflections on the graphs of functions and

their related equations, including reflections through the: x-axis, y-axis, line y = x. [C, CN, R, V] Task

§ Explore examples of graphic designs/objects/phenomena and how they can be represented with function transformations

§ Create a visual representation using transformations of functions Assessment Plan

§ Cycles of Formative Assessment § Peer feedback on Desmos Function Transformation Project (conversation) § Exit slip check-ins for individual outcomes (product)

§ Body of Summative Evidence

§ Desmos Function Transformation Project § Final Product including image and written analysis and description (product) § Interview response (conversation)

§ Transformations Outcome Check (product)

RF2____ RF3_____ Exit Slip #1 Name:______________Class:____

Draw the graph resulting from each transformation. Write the transformation as both an equation and a mapping.

RF2____ RF3_____ Exit Slip #1 Name:______________Class:____


Transformation Equation: y=af(bx-h)+k

Co-ordinate

Mapping: (x,y)-> (1/b x-h, ay+k)

A) the graph of f(x) is horizontally translated 6 units left.

B) the graph of f(x) is vertically translated 4 units down.

Write a sentence describing the transformation, then write the transformation equation and mapping.

Description:

Transformation Equation: y=af(bx-h)+k

Co-ordinate

Mapping: (x,y)-> (1/b x-h, ay+k)

A) the graph of f(x) is horizontally translated 6 units left.

B) the graph of f(x) is vertically translated 4 units down.


Description:



Transformation Equation

Transformation Mapping

A) the graph of f(x) is reflected in the x-axis

B) the graph of f(x) is reflected in the y-axis.


Description:



A) the graph of f(x) is reflected in the x-axis

B) the graph of f(x) is reflected in the y-axis.


Description:

RF4____ Exit Slip #2 Name:______________Class:____


RF4____ Exit Slip #2 Name:______________Class:____




A) the graph of f(x) is horizontally stretched by a factor of ½.

B) the graph of f(x) is horizontally stretched by a factor of 4.


Description:



A) the graph of f(x) is horizontally stretched by a factor of ½.

B) the graph of f(x) is horizontally stretched by a factor of 4.


Description:





A) the graph of f(x) is vertically stretched by a factor of ½.

B) the graph of f(x) is vertically stretched by a factor of 3.


f(x) is narrow, Image is wide

Description:



A) the graph of f(x) is vertically stretched by a factor of ½.

B) the graph of f(x) is vertically stretched by a factor of 3.


f(x) is narrow, Image is wide

Description:

RF2/3 RF4 RF5 Desmos Transformation Project

Name: Due Date:

Task: You have been asked by Desmos Graphing to create a picture using a combination of the following functions. This needs to be completed no later than ___________, because they would like to post them on their website as advertisement of the ease and flexibility of their graphing product. The functions are detailed below (other functions may be used in addition to a combination of the required ones): Required functions: radical, rational, polynomials (linear, quadratic, cubic, quartic, quantic) Optional functions: absolute value, sine, cosine, tangent, exponents, logs etc. Please list, on a separate sheet of paper, details of the transformations on 5 of your functions (See exemplars). You may pick any of the functions used and identify the original function, and any/all transformations that were applied.

Learning Outcomes:

RF2/3 Demonstrate an understanding of the effects of horizontal and vertical translations and stretches on

the graphs of functions and their related equations [C, CN, R, V]

RF4 Apply translations and stretches to the graphs and equations of functions. [C, CN, R, V]

RF5 Demonstrate an understanding of the effects of reflections on the graphs of functions and their related

equations, including reflections through the: x-axis, y-axis, line y = x. [C, CN, R, V]

Create your picture using: https://www.desmos.com – Create an account and make sure to save your work

To submit your assignment you will be required to submit a link of your picture on the following Google form.

Suggestions:

Domain and Range restrictions will be necessary, use { }

brackets to enter these on the site. You can adjust the colour

of functions drawn by following directions on the site.

https://www.desmos.com/

http://goo.gl/forms/4oa7faKjVp

Outcome Excellent2 Excellent1 Good2 Good1 Basic2 Basic1

RF2/3

Mathematical analysis, transformation and stretches are always accurately expressed using appropriate mathematical language, units and symbols in written form.

Mathematical analysis, transformation and stretches are mostly accurately expressed using mathematical language, units and symbols.

Mathematical analysis, transformation and stretches are sometimes accurately expressed using mathematical language, units and/or symbols.

RF4Transformations and stretches on the function expression and graph are always accurately applied.

Transformations and stretches on the function expression and graph are mostly accurately applied.

Transformations and stretches on the function expression and graph are sometimes accurately applied.

RF5

The effects of reflections are always accurately expressed using appropriate mathematical language, units and symbols in written form.

The effects of reflections are mostly accurately expressed using mathematical language, units and symbols.

The effects of reflections are sometimes accurately expressed using mathematical language, units and/or symbols.

Check-in Interview Student Selected

Outcome: RF2/3 , RF4, RF5

Howdidyouknowfromyourexpressionthatyouwereabletorepresentacombinationoftranslations,stretchesandreflections?Whenexplainingatransformation,whatisuniquewhenlookingathowthe‘x’iseffectedinthealgebraicfunctionversusthemappingnotation?

Canyoubesurethatyouhadallfunctiontypesrepresent?How?Howisthedomainandrangealteredwhenusingatranslation/stretch/reflection?

Whichgraphsusedonlyonetranslation/stretch/reflection?Howdidthisimpactyourimage?Howdidyouuseavertical/horizontaltranslationinyourimage?Whenusingaparentgraph,howdidithelptopredictthenewtransformedgraph?Whydidyouuse_____________toalterthegraphandnota__________?

DesmosTransformationProjectRubric

RF2/3 RF4 RF5 Outcome Check Transformations

Name:

1

RF2/3 – Horizontal and Vertical Translations & Stretches

Part 1: Multiple Choice - Record your answer in the multiple choice section on the answer sheet.

1. (Basic) The function 𝑦 = 𝑓(𝑥) = 𝑥3 containing the ordered pair (3, 7) is transformed to become

𝑦 = 𝑓(𝑥 + 1) + 2. Which of the ordered pairs can we be certain are on the transformed graph?

A. (4, 5) B. (2, 9) C. (4, 9) D. (2,5)

2. (Basic) The function 𝑦 = 𝑓(𝑥) is translated 3 units right and 2 units down. The translated image could be

represented by:

A. 𝑦 + 2 = 𝑓(𝑥 + 3) B. 𝑦 − 2 = 𝑓(𝑥 + 3) C. 𝑦 + 2 = 𝑓(𝑥 − 3) D. 𝑦 − 2 = 𝑓(𝑥 − 3)

3. (Good) The range of the function ( )y f x is 3,y y R . The range of the function 2 ( 4)y f x must

be:

A. 7,y y R B. 1,y y R C. 5,y y R D. 1,y y R

4. (Basic) The graph of 𝑓(𝑥)𝑎𝑛𝑑 𝑔(𝑥) are shown

to the right. Based on the graph, g(x) must

equal:

A. 𝑔(𝑥) =1

3𝑓(x)

B. 𝑔(𝑥) = 2𝑓(x)

C. 𝑔(𝑥) = 𝑓(2x)

D. 𝑔(𝑥) = 𝑓( 𝑥

3 )

5. (Good) The graph of the function 𝑦 = 𝑓(𝑥) has zeros at x = -2 and x = 0. The graph of 𝑦 = 𝑓(2𝑥) will

have zeros at:

A. x = -4 and x = 0 B. x = -2 and x = 0 C. x = -1 and x = 0 D. x= -4 and x = -2

6. (Good) The function, ( )y f x is shown in the graph on

the right. It has x-intercepts at (-2, 0) and (4, 0). The

intercepts of the function 2

xf

must be:

A. (-1, 0) and (2, 0)

B. (-4, 0) and (2, 0)

C. (-4, 0) and (8, 0)

D. (0, 0) and (6, 0)

Part 2: Numerical Response - Record your answer in the numerical-response section on the answer sheet.

NR1. (Excellent) The graph of 𝑦 = 𝑓(𝑥) is shown to the

right and contains asymptotes at x = -2, 3. The

transformation of 𝑦 = 2 𝑓 ( 𝑥

3 ) has a positive

asymptote at x = _____.

NR2. (Good) The graph of 𝑦 = 𝑓(𝑥) is a polynomial

function with x – intercepts at -2 and 4. The

transformed graph 𝑦 = 𝑓(𝑘𝑥) is shown to the

right. To one decimal place, the value of k =

___.

Part 3: Written Response – Communicate and show all reasoning in the space provided below the question.

Use the following graph for the WR1

WR1. The function f(x) is the image of the function g(x) in the graph above.

a) (Basic) Describe, in words, the translations that were applied to g(x) to generate the graph of f(x).

b) (Excellent) Determine the equation of f(x) in terms of g(x) and the mapping notation for the transformation of g(x)

to f(x).

WR2. The graph of 𝑦 = 𝑓(𝑥) is shown to the

right.

a) (Excellent) Describe the transformations that

have been applied to 𝑓(𝑥) to generate the mapping

notation of (𝑥, 𝑦) → (𝑥 + 3, 𝑦 − 4) and Sketch the

image graph.

b) (Good) Determine the equation of the image

graph, g(x) in terms of f(x) .

WR3. The function 𝑦 = 𝑓(𝑥) is transformed to 𝑦 = 𝑎𝑓(𝑏𝑥). Determine the equation for the following

transformations:

a) (Basic) A horizontal stretch by a factor of 4

3 about the 𝑦-axis

b) (Basic) A vertical stretch about the 𝑥-axis by a factor of 3

c) (Good) A vertical stretch about the 𝑥-axis by a factor of 1

3 and a horizontal stretch about the 𝑦-axis by a factor

of 1

10

WR4. The graph of 𝑦 = 𝑓(𝑥) is shown to the below:

a) (Excellent) Sketch the graph of 𝑦 = 𝑓(2𝑥) and label the invariant points on the graph.

b) (Excellent) Describe the transformation in reference to the x or y axis.

c) (Basic) State the domain and range of the image graph.

RF5 - Reflections

Part 1: Multiple Choice - Record your answer in the multiple choice section on the answer sheet.

1. (Basic) The function 𝑦 = 𝑓(𝑥) is shown to

the right. Which of the following

transformations would result in an identical

graph to 𝑦 = 𝑓(𝑥)?

A. 𝑦 = 𝑓(−𝑥)

B. 𝑦 = −𝑓(𝑥)

C. 𝑦 = − 𝑓(−𝑥)

D. – 𝑦 = 𝑓(𝑥)

2. (Basic) A Math 30-1 student graphs the function 3( ) 3 2f x x x in a window :[ 5,5,1]x , :[ 5,5,1]y .

Under the transformation ( ) ( )g x f x the graph of ( )y g x and ( )y f x share

A. No invariant points B. One invariant point C. Two invariant points D.Three invariant points

Use the following graph for the next question

3. (Good) Given the function 𝑦 = 𝑓(𝑥) shown in the graph above, which of the following functions, 𝑦 =

𝑔(𝑥) will share the same domain and range with 𝑓(𝑥).

A. 𝑔(𝑥) = 𝑓(−𝑥) B. 𝑔(𝑥) = −𝑓(𝑥) C. 𝑔(𝑥) = − 𝑓(−𝑥) D. 𝑔(𝑥) = 𝑓(−𝑥2)

Part 2: Written Response – Communicate and show all reasoning in the space provided below the question.

WR1. The graph of 𝑓(𝑥) = 𝑥2 − 4𝑥 + 3

shown to the right.

a) (Basic) Sketch the graphs of 𝑔(𝑥) =

− 𝑓(𝑥) and ℎ(𝑥) = 𝑓(−𝑥). Then state the

invariant point(s) for each transformation.

b) (Good) Determine the simplified equation of

the image graph, (𝑥) = − 𝑓(𝑥) .

c) (Excellent) Determine the simplified equation of the image graph, ℎ(𝑥) = 𝑓(−𝑥).

WR2. (Excellent) Determine a simplified equation, 𝑔(𝑥), which represents the graph of 𝑓(𝑥) = 2𝑥2 − 4𝑥 + 5 after

the following transformation: 𝑔(𝑥) = −𝑓(−𝑥)

RF4 - Combinations of Transformations

WR1. (Basic-Excellent) Describe all the transformations, and identify the mapping notation, for the following

equation: 𝑦 = −3𝑓(2𝑥 + 8) − 1

WR2. (Basic-Excellent) The graph of 𝑦 = 𝑔(𝑥) is the image of the graph of 𝑦 = 𝑓(𝑥) after a combination of

transformation. Describe the transformations that have occurred. Write the mapping notation and identify

an equation of the image graph in terms of the function 𝑓(𝑥). State the domain and range of both functions.

f(x)

g(x)

RF2/3 RF4 RF5

Answer Sheet (Version #2)

Name:

Section:

Multiple-Choice Questions

Record your answers below

RF5

1.

2.

3.

Numerical Response

Enter your answer, beginning in the left-hand box and leaving any unused boxes blank.

RF2/3

1.

2.

3.

4.

5.

6.

RF4

1.

2.

3.

RF2/3

1. 2.

conceptual understanding: program of study outcomes

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