Alg2 Lesson 2.2.2: Function Transformations

Lesson: 2.2.2: Function

Transformations

Obj: I can describe transformations of functions
given graphs, formulas, and verbal descriptions.

EQ: How can I describe transformations in function
notation?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Poll

How do you plan to contribute during the lesson?

• Ask teacher/peers questions.

• Write notes in journal.

• Keep others on task.

• Complete the slides as many times as needed to demonstrate mastery.

• Look up additional info on sites such as YouTube.

Part 1: Characteristics of

Graphs

Parent Functions

Go to https://dashboard.blooket.com/set/6344b26040e455c01c1d85b1 and play the
Blooket when logged into an account.

The functions on this slide are the main functions we will go over in algebra 2, although the Blooket contains even more parent graphs (some are not functions, only equations).

Absolute Value Transformations (2.4.1)

Input
x

Process

Output
g(x)

Point

-2

-3|-2|+2=-3(2)+2=-6+2=-4

-4

(-2, -4)

-1

-3|-1|+2=-3(1)+2=-3+2=-1

-1

(-1, -1)

0

-3|0|+2= 0+2=2

2

(0, 2)

1

-3|1|+2=-3+2 = -1

-1

(1, -1)

2

-3|2|+2= -6+2 = -4

-4

(2, -4)

The function f(x) = |x| is transformed to g(x) = -3f(x) + 2. Graph the transformation
function, g(x), in the grid below. (Note: f(x) is the dashed line).
Make a table of values for the new function.

Graphing

The function f(x) = |x| is transformed to g(x) = -2f(x) + 3. Graph the integer points of the transformation function, g(x), in the grid.

Characteristics of Absolute Value (2.4.2)

X-Int(s): (-4, 0) and (2, 0)

Y-Int: (0, -2)

Vertex: (-1, -3)

Axis of Symmetry: x = -1

Translations from f(x) = |x|: left 1, down 3

Reflection from f(x) = |x|? No

Dropdown

From f(x) = |x|, describe the transformations.

Horizontal Translation:​

Select answer

Vertical Translation: ​

Select answer

Reflection: ​

Select answer

Dropdown

From f(x) = |x|, describe the transformations.

Horizontal Translation:​

Select answer

Vertical Translation: ​

Select answer

Reflection: ​

Select answer

Dropdown

From f(x) = |x|, describe the transformations.

Horizontal Translation:​

Select answer

Vertical Translation: ​

Select answer

Reflection: ​

Select answer

Dropdown

From f(x) = |x|, describe the transformations.

Horizontal Translation:​

Select answer

Vertical Translation: ​

Select answer

Reflection: ​

Select answer

Dropdown

From f(x) = |x|, describe the transformations.

Horizontal Translation:​

Select answer

Vertical Translation: ​

Select answer

Reflection: ​

Select answer

Characteristics of Quadratic Graphs (6.2.1)

X-Int(s): (-1, 0) and (3, 0)

Y-Int: (0, 3)

Vertex: (1, 4)

Axis of Symmetry: x = 1

Translations from f(x) = x²: right 1, up 4

Reflection from f(x) = x²? Yes

Multiple Choice

When finding transformations for f(x) = |x| and g(x) = x²,
The process is...

• the same

• different

Drag and Drop

When finding translations for f(x) = |x| and g(x) = x²,

find g(x)'s ​

Placeholder

and compare it the the origin ​

Placeholder

Reflections across the ​

Placeholder

will look like an ​

Placeholder

v or u.

Drag these tiles and drop them in the correct blank above

vertex

(0, 0)

slope

y-intercept

asymptote

upside down

x-axis

y-axis

Characteristics of Square Root Graphs (6.2.2)

Domain: (-∞, -1]

Range: [-2, ∞)

Reflection(s) from f(x) = √x:
Reflect across y-axis

Dilation from f(x) = √x: None

Translations from f(x) = √x:
Left 1, Down 2

sqrt is the abbreviation for for square root.

Labelling

Match each graph to its transformations from f(x) = √x

Drag labels to their correct position on the image

Reflect across x-axis & y-axis

Reflect across y-axis

No Reflection

Reflect across x-axis

Characteristics of Cubic Graphs (7.2.1)

Reflection from f(x) = x³: Yes

Dilation from from f(x) = x³: V Compress by ⅕

Translations from from f(x) = x³: left 2, down 1

(The Dashed line is f(x) = x³)

Drag and Drop

Reflection from f(x) = x³: ​

Placeholder

Dilation from from f(x) = x³: ​

Placeholder

by 5

Translations from from f(x) = x³: ​

Placeholder

1, ​

Placeholder

2(The Dashed line is f(x) = x³)

Drag these tiles and drop them in the correct blank above

No

Yes

V Stretch

H Stretch

right

left

up

down

Characteristics of Cube Root Graphs (7.2.2)

cbrt is the abbreviation for for
cube root.

Reflection from f(x) = ∛x: No

Dilation from from f(x) = ∛x: None

Translations from from f(x) = ∛x: up 2

(The Dashed line is f(x) = ∛x)

Domain and Range of Cube & Cube Root Functions

Domain and Range will always be all
real numbers for cubic and cube root functions.

Drag and Drop

Reflection from f(x) = ∛x: ​

Placeholder

Dilation from from f(x) = ∛x: ​

Placeholder

by 2

Translations from from f(x) = ∛x: ​

Placeholder

1(The Dashed line is f(x) = ∛x)

Drag these tiles and drop them in the correct blank above

Yes

No

V Stretch

H Stretch

right

left

Characteristics of Rational Graphs (8.1.1)

Reflection from f(x) = 1/x: No

Vertical Translation from f(x) = 1/x: down 3

Horizontal Translation from f(x) = 1/x: left 2

Drag and Drop

Reflection from f(x) = 1/x: ​

Placeholder

Vertical Translation from f(x) = 1/x: ​

Placeholder

Horizontal Translation from f(x) = 1/x: ​

Placeholder

Drag these tiles and drop them in the correct blank above

Yes

No

up 2

down 2

none

left 2

right 2

Characteristics of Rational Graphs (8.1.1)

Reflection from f(x) = 1/x^2: Yes

Vertical Translation from f(x) = 1/x^2: up 2

Horizontal Translation from f(x) = 1/x^2: right 1

Drag and Drop

Reflection from f(x) = 1/x^2: ​

Placeholder

Vertical Translation from f(x) = 1/x^2: ​

Placeholder

Horizontal Translation from f(x) = 1/x^2: ​

Placeholder

Drag these tiles and drop them in the correct blank above

No

Yes

none

left 2

right 2

up 2

down 2

Part 2: Characteristics of

Formulas

Characteristics of Rational Equations (8.1.2)

Hole: (-3, -27/4); Set canceled out factor = 0 (x=-3). Evaluate with solution f(-3)=9(-3-3)/(-3-1)(-3+1).

Vertical Asymptote (VA): x = -1 and x = 1; set simplified denominator = 0 (x-1)(x+1)=0

Domain Verbal Notation: All Real Numbers except x ≠ -3, -1, 1

Domain Interval Notation: (-∞, -3) ∪ (-3, -1) ∪ (-1, 1) ∪ (1, ∞)

Everything is in domain besides the excluded values (Holes and VAs).

Horizontal Asymptote (HA): y = 0; look at the leading coefficients of each part; it is 9x/1x^2 in this case.
Change it so that both leading terms have the same degree: 0x^2/1x^2. Divide the leading coefficients. 0/1=0

X-Intercept: (3, 0); set the simplified numerator = 0 9(x-3)=0

Y-Intercept: (0, 27); Evaluate f(0): 9(0-3)/(0-1)(0+1) = 9(-3)/(-1)(1) = -27/-1 = 27

Simplify Fractions &
Find Holes before other
characteristics.

Multiple Choice

What is the hole of $$f(x)=\frac{8\left(x+5\right)}{\left(x+5\right)\left(x+1\right)}$$ ?

• (-5, -2)

• (-1, 8)

• (5, 4/3)

• (1, 4)

Multiple Choice

What is the domain of $$f(x)=\frac{2}{x-7}$$ ?

• All Real Numbers except x ≠ 7

• All Real Numbers

• All Real Numbers except x ≠ -7

• All Real Numbers except x ≠ 2

Characteristics of Exponential Equations (9.1.3)

Horizontal Asymptote (HA): y = -1 (look at the # add/sub to the exponent.
If there is no #, y = 0)

Range: y > -1

Y-Intercept: (0, 31) (evaluate h(0))

Translations from f(x) = 2^x: left 5 and down 1

Multiple Choice

What are the translations from $$f(x)=2^x$$ to $$g(x)=2^{\left(x+8\right)}+10$$ ?

• Left 8 and Down 10

• Left 8 and Up 10

• Right 8 and Up 10

• Right 8 and Down 10

Characteristics of Logarithmic Equations (9.1.4)

Vertical Asymptote (VA): x = -7

Domain: x > -7

X-Intercept: (-6 ,0) (log(1) = 0, so set x+7=1)

Translation from f(x) = log(x): left 7

Multiple Choice

What is the translation from $$f(x)=\log(x)$$ to $$g(x)=\log(x-12)$$ ?

• left 12

• right 12

• up 12

• down 12

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the wheel as a class and spend a minute or so
discussing our answers.

Part 3: Transformations of

Verbal Descriptions

Monster Transformations

https://tinyurl.com/f3jc4923

Click the link to go to
Desmos Monster graph.
Use the sliders for a, b,
h, and k to make these
functions for 1 through
9. Match each
transformation function
to its transformed
monster.

Labelling

Match each transformed monster function to the image.

Drag labels to their correct position on the image

2f(-x)

-f(x+5)

2f(x) - 5

f(x) - 5

f(2x)

f(-x) + 5

-f(0.5x)

f(x-5)

0.5f(x)

Transformations in Function Notation (1.1.7)
Reflections from f(x) to g(x)

Transformation

Other Names

New Function

Reflects Vertically

Reflect across x-axis

g(x) = -f(x)

Reflects Horizontally

Reflect across y-axis

g(x) = f(-x)

Match

What is the new function g(x) if f(x) was reflected...

• Reflect f(x) vertically and horizontally

• Reflect f(x) vertically

• Reflect f(x) horizontally

g(x) = -f(-x)

g(x) = -f(x)

g(x) = f(-x)

• g(x) = -f(-x)

• g(x) = -f(x)

• g(x) = f(-x)

Transformations in Function Notation (1.1.7)
Dilations from f(x) to g(x)

Transformation

Other Names

New Function

Dilates Vertically by a factor of 2

Vertical Stretch by 2

g(x) = 2f(x)

Dilates Vertically by a factor of ½

Vertical Compression by 1/2

g(x) = ½f(x)

Dilates Horizontally by a factor of 2

Horizontal Compression by 2

g(x) = f(½x)

Dilates Horizontally by a factor of ½

Horizontal Stretch by ½

g(x) = f(2x)

Drag and Drop

What is the relationship between vertical stretches and horizontal stretches? ​

Placeholder

What is the relationship between vertical stretches and horizontal compressions? ​

Placeholder

Drag these tiles and drop them in the correct blank above

They are expanding in different directions.

They would look the same on a graph.

Transformations in Function Notation (1.1.7)
Translations from f(x) to g(x)

Transformation

New Function

Translates Vertically Up 2 Units

g(x) = f(x) + 2

Translates Vertically Down 2 Units

g(x) = f(x) - 2

Translates Horizontally Left 2 Units

g(x) = f(x + 2)

Translates Horizontally Right 2 Units

g(x) = f(x -2)

Multiple Choice

Which of the following is the transforms from f(x) to g(x) when g(x) is translated 3
units right and 9 units up?

• g(x) = f(x - 3) + 9

• g(x) = f(x + 3) + 9

• g(x) = f(x - 3) - 9

• g(x) = f(x + 3) - 9

Write the new transformation from f(x)
that reflects vertically across the x-axis,
vertically stretches (dilates) by a factor of 3, translates left 5 units, and translates down 7 units.

Exit Ticket Algebra 2 Lesson 2.2.2