Secondary IIUtah Integrated Mathematics CoreStudent Edition - Honors

Unit 3: Functions

Cache County School District 2013-2014

Secondary II Unit 3 – Functions: Table of Contents

Homework Help (QR Codes and links to videos, tutorials, examples)……………………Section 3.1 – Exploring Functions with Sarah Task, Teacher Notes

Notes, Assignment Section 3.2 –Relations and Functions Teacher Notes

Notes, Task, Assignment Section 3.3 – Basic Function Shapes and Parent Graphs, Teacher Notes

Notes, Task, Assignment Section 3.4 – Domain and Range Task, Teacher Notes

Notes, Assignment Section 3.5a – More Domain and Range, Teacher Notes

Notes, AssignmentSection 3.5b – Appropriate Domain Task, Teacher Notes…………………………………

Notes, Assignment…………………………………………………………………..... Section 3.6 –Discovering Transformations Task, Teacher Notes

Notes, Assignment Extra Transformation Practice Worksheet …………….………………….……………Section 3.7 a,b – Piecewise Functions Task, Teacher Notes

Notes, Assignment Section 3.8 – Introduction to Step Functions, Teacher Notes

Notes, Assignment Section 3.9 – Greatest Integer/Step Function, Teacher Notes

Notes, Assignment Section 3.10 Even and Odd Function Task, Teacher Notes

Notes, AssignmentSection 3.11 Introduction to Inverse Functions Task…………………………………………..

Teacher Notes, Assignment

Domain and Range Matching Review Activity…………………………………...………Function Notation/ Domain&Range Review Task………………………………………. (not intended to be used as a test review)

Secondary II Unit 3 – Functions: Homework Help

Section 3.1

http://goo.gl/CDBNA

video

http://goo.gl/YafAuhttp://goo.gl/IJRJe

Section 3.2

http://goo.gl/QLvqhttp://goo.gl/lOAT0

video

http://goo.gl/Kav2l

Section 3.3

http://goo.gl/YjmjQhttp://goo.gl/faHXV

Section 3.4

http://goo.gl/3JLyr http://goo.gl/DwSFU

video

http://goo.gl/UH26V

Section 3.5Review of 3.4, see resources above

Section 3.6

http://goo.gl/nWVrohttp://goo.gl/5KQf4 http://goo.gl/JXxO9

Section 3.7

http://goo.gl/bDVR3 http://goo.gl/07lf4

video

http://goo.gl/PwUZP

Section 3.8

http://goo.gl/ptcE0http://goo.gl/q6Uwm http://goo.gl/o33B6

Section 3.9

http://goo.gl/OnAtvhttp://goo.gl/qqOMM

Section 3.10

http://goo.gl/RJEG6http://goo.gl/NU9yQ

Video

http://goo.gl/1QNVe

Unit 3 Lesson 1 – Exploring Functions with Sarah Task 3.1

Name________________________________

Date________ Hour________

Part 1

a. While visiting her grandmother, Sarah found markings on the inside of a closet door showing the heights of her mother, Tammy, and her mother’s brothers and sisters on their birthdays growing up. From the markings in the closet, Sarah wrote down her mother’s height each year from ages 2 to 16. Her grandmother found the measurements at birth and one year by looking in her mother’s baby book. The data is provided in the table below, with heights rounded to the nearest inch.

Age (yrs.) X 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Height (in.) Y 21 30 35 39 43 46 48 51 53 55 59 62 64 65 65 66 66

a. Which variable is the independent variable, and which is the dependent variable? Explain your choice.

b. Make a graph of the data.

c. Should you connect the dots on your graph? Explain.

d. Describe how Tammy’s height changed as she grew up.

e. How tall was Tammy on her 11th birthday?

f. What do you think happened to Tammy’s height after age 16? Explain. How could you show this on the graph?

Part 2

2. Function notation gives us another way to write about ideas that you began learning in middle school, as shown in the table below. In the case of the table above, h(2) mean the y-value when x is 2, which is Tammy’s height (in inches) at age 2, or 35. Thus, h(2) = 35.

StatementAt age 2, Tammy was 35 inches tall.When x is 2, y is 35.When the input is 2, the output is 35.h(2) = 35.

TypeNatural languageStatement about variablesInput-output statementFunction notation

a. What is h(11)? What does this mean?

b. When x is 3, what is y? Express this fact using function notation.

c. Find an x so that h(x) = 53. Explain your method. What does your answer mean?

d. From your graph or your table, estimate h(6.5). Explain your method. What does your answer mean?

e. Find an x so that h(x) = 60. Explain your method. What does your answer mean?

f. Describe what happens to h(x) as x increases from 0 to 16.

g. What can you say about h(x) for x greater than 16?

h. Describe the similarities and differences you see between these questions and the questions in part 1.

Additional Notes/Examples:

Unit 3 Lesson 1 – Exploring FunctionsReady, Set, Go! - Assignment 3.1

http://goo.gl/CDBNA

Name___________________________________

Date_________ Hour_______

Ready

1. What does it mean to evaluate a function?

x y Point (x, y)

-1012

2. Complete the following table using the equation: y=x−2

3. Evaluate the function f ( x )=−x2+1 at x=−1 , x=1 , x=2.

4. What “points” on the curve/graph of f ( x )=−x2+1 did you find?

5. , find 6. , find

7. , find

Set

8. Evaluate the following function at the following values:

a) f (1 )

b) f (−5 )

c) f (−4 )

d) f (6 )

e) f (0 )

f) f (−2 )

g) f (−7 )

Math Composer 1.1. 5ht tp: / / www.mathcomposer.com

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y 9. Evaluate the following function at:

a . f (0 ) =

b . f (2 ) =

c . f (−2 ) =

d . f (3 ) =

e . f (6 ) =

f . f ( 4.5 )=¿

Go!

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-7

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y

10. Evaluate:

a . f (−5 )=¿

b . f (−3 )=¿

c . f (−1 )=¿

What points on this graph did you find?

11. Translate the following statements into coordinate points:

a. f(–1) = 1

b. h(2) = 7

c. g(1) = –1

d. k(3) = 9

12. Given this graph of the function f(x):Find: a. f(–1)

b. f(0)

c. f(2)

d. f(3)

e. x when f(x) = 2

f. x when f(x) = 6

Unit 3 Lesson 2 – Relations and Functions Notes 3.2

Vocabulary

Relations

Examples

Function

Examples

Function Notation

Functions and relations can be written in many forms. Some of the possible forms include: a verbal description, a set of ordered pairs, a table, a correspondence or mapping, a graph, or an algebraic expression/equation. The following are a few examples.

Verbal description: Take a natural number, double it and add seven.

Algebra: f (x) = 2x + 7 for x ∈ N

x f(x)

1 9

2 11

3 13

4 15

... …

Table:

Domain (input) Range (output)

Mapping: 1 9

2 11

3 13

4 15

Examples:1. Write the following verbal description into an algebraic expression: Take a real number, triple it and subtract thirteen.

2. Draw a table showing at least 5 solutions for the function f(x) = 4x + 2.

3. Draw mapping diagram to represent the set of ordered pairs: {(5, -3), (6, -5), (2, 3), (3, 1)}.

Unit 3 Lesson 2 – Relations and Functions Task 3.2

Name____________________________________

Date_______________ Hour_______

To Function or Not to Function?

Determine if the following relationships are functions. Explain.

1. A person’s name versus their social security number.

2. A person’s social security number versus their name.

3. The cost of gas versus the amount of gas pumped.

4. { (3,6), (4,10), (8,12), (4,10) }

5. The time of day with respect to the temperature.

6.Distance Days6 210 46 59 8

7. The area of a circle as it relates to the radius.

8.

9. The radius of a cylinder is dependent on the volume.

10. The size of the radius of a circle dependent on the area.

11. Students letter grade and the percent. 12. The length of fence needed with respect to the amount of area to be enclosed.

3

5

7

5

5

7

13. The explicit formula for the recursive situation: f(1) = 3 and f(n+1) = f(n) + 4

14. If x is a rational number, then f(x) = 1 If x is an irrational number, then f(x)=0

Unit 3 Lesson 2 – Relations and Functions Ready, Set, Go! - Homework 3.2

http://goo.gl/QLvq

Name____________________________________

Date_________ Hour_______

ReadyDetermine whether each relation is function. If it is not a function, you must explain why.

1. 2. 3.

4. (1,3) (2,-1) (3,-1)(1,3)(-2,5) 5. (3,3) (2,3) (3,3)(1,3)(-2,3) 6. (-1,3) (6,4) (2,-1)(0,3)(-1,5)

7. (3,5) (3,1) (3,-2) (3,6) (3,0) 8. (2,5) (-3,4) (6,-7) (8,1) (-3,-3) 9. (3,5) (6,1) (-3,-2) (13,6) (-73,0)

x f(x)

-6

-3

0

1

-5

10. Given f ( x )=3−4 x . Fill in the table and then sketch a graph.

SetDetermine whether each relation is function. If it is not a function, you must explain why.

11. 12.

13. 14.

Create your own function in each space provided.

15. (Draw lines) (Points)

( , ) ( , )( , ) ( , )( , ) ( , )

Graph

Go!

16. Evaluate the following expressions given the functions below:

g(x) = -3x + 1 f(x) = x2 + 7 h( x )=12

x j( x )=2 x+9

a. g(10) = b. f(3) = c. h(–2) =

d. j(7) = e. h(a) f. g(b+c)

h. Find x if g(x) = 16 i. Find x if h(x) = –2 j. Find x if f(x) = 23

17. Swine flu is attacking Porkopolis. The function S below determines how many people have swine where t = time in days and S = the number of people in thousands. Sketch a graph to represent S(t).

S( t )=9 t−4a. Find S(4).

b. What does S(4) mean?

c. Find t when S(t) = 23.

d. What does S(t) = 23 mean?1 2 3 4 5 6

10

20

30

40

50

60

70

80

0 x

y

Unit 3 Lesson 3 – Basic Function Shapes and Parent Graphs Notes 3.3

Linear Greatest Integer/Step Function

Quadratic Exponential

Absolute Value Square Root

Piecewise-Functions Cube Root Functions

Extra Notes:

Unit 3 Lesson 3 – Basic Function Shapes and Parent Graphs TASK 3.3

Name_________________________________Date _________Hour __________

Based on what you learned in class, sketch the following graphs. They do not have to be perfect but they need to agree with the equation. Create tables, use calculators, and work as a group to be as accurate as possible.

1. 2. 3.

4. 5. 6.

Unit 3 Lesson 3 – Basic Function Shapes Ready, Set, Go! - Assignment 3.3

http://goo.gl/YjmjQ

Name__________________________________

Date_________ Hour_______

Ready Draw a sketch of the following graph descriptions.

1. 2. f ( x )=‖x‖ 3.

4.

5.

6. y=ax

Exponential

Set State which basic function shape corresponds to the following equation.

7. y=3 3√x−3 8. f ( x )=−5+x 9. y=−√x+4

10. f ( x )=7 x+5 11. y=−5|x+1|−4 12. g ( x )=4 x−3

GoMatch the name with the equation and the graph by connecting them with lines

Name Equation Graph

13. Quadratic

14. Linear

15. Square Root

16. Absolute Value

Unit 3 Lesson 4 – Domain and Range Notes 3.4

Domain – Range –

Relation Review– Function Review–

Examples 1-6: Find the domain and range of the following relations and determine if they are functions.

1.

Domain:

Range:

Function?

2.

Domain:

Range:

Function?

3. 4.

Domain:

Range:

Function?

Domain:

Range:

Function?

5. Math Composer 1. 1. 5ht tp: / /www.mathcomposer. com

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Domain:

Range:

Function?

6.

Domain:

Range:

Function?

-5 -4 -3 -2 -1 1 2 3 4 5

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12345

x

y

Additional Notes:

Unit 3 Lesson 4 – Domain and Range TASK 3.4

Name___________________________________Date_________Hour __________

With a partner – or in a group, decide if the following graphs are functions. Then, determine the domain and range of each. a) Domain b) Range c) Function? Write your answers in interval notation.

1.

a)

b)

c)

2.

a)

b)

c)

3.

a)

b)

c)

4.

a)

b)

5.

a)

b)

c)

6.

a)

b)

c)

c)

Unit 3 Lesson 4 – Domain and Range Ready, Set, Go! - Assignment 3.4

http://goo.gl/3JLyr

Name_____________________________________

Date_________ Hour_______

Ready State the domain and range:

1. {(0 , 3 ) , (2 , 4 ) , (6 ,−2 ) , (2 ,−3 ) } 2. {(−1 , 1 ) , (−2 , 2 ) , (−3 ,3 ) , (1 ,−1 ) }

Domain: Domain:

Range: Range:

State the domain and range, then determine if the following relations are functions.

x y0 -

52 -

42 0-8

-1

3. 4. 5.

State the domain and range: 6. 7.

Set

8.

Domain:

Range:

Function?

9.

Domain:

Range:

Function?

10. 11.

Domain:

Range:

Function?

Domain:

Range:

Function?

12.

Domain:

Range:

Function?

13.

Domain:

Range:

Function?

14. 15.

Math Composer 1. 1. 5http: / / www. mathcomposer. com

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Domain:

Range:

Function?

Domain:

Range:

Function?

Unit 3 Lesson 5a – More Domain and Range TASK 3.5a

\Name_________________________________Date_________Hour __________

With your partner/group complete the following task. Take your time and be sure to answer each part of the question completely.

Describe a context where the domain of a function would be:

All real numbers Whole Numbers Rational Numbers Integers Even Natural Numbers 2 to 10.

Label each part of your answer and use the back of this paper if necessary. Be prepared to present your results with the class.

Unit 3 Lesson 5a – More Domain and Range Ready, Set, Go! - Assignment 3.5a

Name___________________________________

Date_________ Hour_______

ReadyFind the domain and range of each graph. Note: When the graph “runs off/goes off” the coordinate plane, that means the graph has “arrows” and therefore goes on forever.

1. 2. 3.

4. 5. 6.

Set 7. Given f ( x )=√ x+1 . Fill in the table and then sketch a graph.

x f(x)30

-102

6

Find the domain and range of each. Then, determine if each graph represents a function.

8. Domain:

Range:

Function?

9.Domain:

Range:

Function?

10. Domain:

Range:

Function?

11. Domain:

Range:

Function?

12. Domain:

Range:

Function?

13. Domain:

Range:

Function?

-5 -4 -3 -2 -1 1 2 3 4 5

-5-4-3-2-1

12345

x

y

Go!

14. Translate the following statements into coordinate points, and then plot them.

a. f(–1) = 1

b. f(2) = 7

c. f(1) = –1

d. f(3) = 0

15. Given this graph of the function f(x):

Find:

a. f(–4) = b. f(0) = c. f(3) = d. f(-5) =

e. x when f(x) = 2 f. x when f(x) = 0

-5

-5

5

5x

yf(x)

Unit 3 Lesson 5b – Appropriate Domain TASK 3.5 b

Name____________________________________

Date_______ Hour_____

Directions: Determine a reasonable domain for the model each situation. Write your responses using interval notation and include units of measure. Answers will vary.

1. Years of life for a human. 2. Hours worked in a day.

3. Gallons of gas in a car. 4. Calories in a candy bar.

5. Amount earned per hour tending 2 children.

6. The amount of fencing necessary to enclose a garden with a perimeter of 50 feet.

7. Speed a car is moving on the freeway.

8. Write your own situation and its domain.

Unit 3 Lesson 5b – Appropriate Domain Ready, Set, Go! - Assignment 3.5b

Name___________________________________

Date_________ Hour_______

ReadyDetermine a reasonable domain and range for each model of the given situation. Write your answer using interval notation and include appropriate units of measure.

1. A pencil thrown up in the air. Domain:

Range:

2. A firework launched up in the air. Domain:

Range:

3. Write your own situation and its domain and range.

Domain:

Range:

4. Create a problem situation.

a) Draw the graph of the situation

b) What is the domain of the function? c) What is the range of the function?

5. Describe a context where the domain of the function would be [ 0,2 ] .

SetDetermine a reasonable domain for each situation. Write your answers using interval notation and include units of measure.

6. Hours of sleep in a day. 7. Years of attending school over a lifetime.

8. Height of a statue in a hotel lobby with a ceiling height of 20 feet.

9. Someone holding his/her breath. 10. Temperature during a year in Utah.

Go!Draw a graph in the coordinate planes below and state the domain and range for each.

11. Domain:

Range:

12. Domain:

Range:

Unit 3 Lesson 6 – Discovering Transformations TASK 3.6

Name_________________________________Date_________Hour __________

Discovering Transformations: TaskUse the following functions to complete each table. Then graph each function on the indicated coordinate

plane. Use the parent function of

x f(x)

-2

-1

0

1

2

3

x f(x-3)

-2

-1

0

1

2

3

6

4

2

-2

-5 5

6

4

2

-2

-5 5

What is the difference between the two functions? How do the two graphs compare?

Make a conjecture about how n changes the graph of f(x) in f(x - n).

6

4

2

-2

-5 5

6

4

2

-2

-5 5

Fill out the table for each of the following versions of f(x), then graph the functions.

x f(x)-4

-2

-1

0

1

2

3

x f(2x)

-2

-1

0

1

2

3

6

4

2

-2

-4

-5 5

6

4

2

-2

-4

-5 5

Compare each graph to the “parent function” f(x).

Make a conjecture about how n changes the graph of f(x) for each variation.

Fill out the table for each of the following versions of f(x), then graph the functions.

x 2f(x)

-2

-1

0

1

2

3

x -f(x)

-2

-1

0

1

2

3

6

4

2

-2

-4

-5 5

6

4

2

-2

-4

-5 5

Compare each graph to the “parent function” f(x).

Make a conjecture about how n changes the graph of f(x) for each variation.

Let’s compare as a class and summarize what we have investigated and learned.

f (−x )

Unit 3 Lesson 6 – Discovering Transformations Ready, Set, Go! - Assignment 3.6

http://goo.gl/JXxO9

Name___________________________________

Date_________ Hour_______

Ready

For each function below, (A) identify the parent function, then (B) Describe in words the transformations

made to the parent function.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

Set

Sketch a graph of the function with the indicated transformations. (No Calculator)

13. 14. 15.

Write the function for with the indicated transformations.

16. Vertical stretch by a factor of 3, horizontal shift left 5

17. Moved 4 units right and 5 units down.

18. moved 6 units left and 2 units up.

GoUse the graphs below to identify each function. Write the function that corresponds to each graph.19. 20.

6

4

2

-2

-4

-5 5

6

4

2

-2

-4

-5 5

6

4

2

-2

-4

-5 5

Extra Transformation Practice

Name___________________________Hour_____

Match each core graph with its equation, from the options given below a thru f.

a. b. c.

d. e. f ( x )= ⟦ x ⟧ f.

1. _______ 2. _______ 3. _________ 4. _______

Match each graph with the appropriate equations, a thru h.

a. b. c. d.

e. f. g. h.

5. ______ 6. _______ 7. _________ 8. _________

Graph each function without the aid of a graphing calculator.

9. 10. 11.

12. 13. 14.

15. 16. 17. f ( x )=|x−2|+4

18. f ( x )=−( x−2 )2 19. f ( x )=x3−4 20. f ( x )=−√x+2−2

Unit 3 Lesson 7a – Pool Party TASK 3.7a

Name_________________________________Date_______Hour __________

Pool Party: Task

Marie has a small pool full of water that needs to be emptied and cleaned, then refilled for a pool party. During the process of getting the pool ready, Marie did the following activities. Each activity took place during a different time interval.

Removed water with a single bucket Filled the pool with a hose (same rate as emptying the pool)

Drained water with a hose (same rate as filling the pool)

Cleaned the empty pool

Marie and her two friends removed the water with three buckets

Took a break

1. Sketch a possible graph showing the height of the water level in the pool over time. Be sure to include all of the activities Marie did to prepare the pool for the party. Remember that only one activity happened at a time. Think carefully about how each section of your graph should look. Label all sections indicating where each activity begins and ends.

2. Create a story connecting Marie’s process for emptying, cleaning, and then filling the pool to the graph you have created. Do your best to use appropriate math vocabulary.

3. Does your graph represent a function? Why or why not? Given your reason, would all graphs of this task have the same result? Explain.

Unit 3 Lesson 7b – Exploring Piecewise Functions TASK 3.7b

Name___________________________________Hour __________

Exploring Piecewise Functions: Task

1. Look for patterns in the function below. Would it be possible to mistakenly judge the nature of this function if you based your analysis on just one part? _________ Explain. _________________

____________________________________________________________________________________

_____________________________________________________________________________________

2.

a) Let’s assume that someone described the function above as linear. Explain how is their statement is partially correct.

b) Explain how that same statement is also false.

c) Explain how you could accurately identify the parts of the function that are linear.

3. Let’s look at the individual pieces of the previous piecewise function. Identify each type of function and provide its equation.

a.

f(x) =

equation

function type

2018161412

10

8

6

4

2

108642

b.

f(x) =

equation

function type

2018161412

10

8

6

4

2

108642

c.

f(x) =

equation

function type

2018161412

10

8

6

4

2

108642

4. Only a piece of each of the functions above was used to form the piecewise function in question 1. Using numbers and symbols, identify the piece of each function that is used.

a. ____________________ b. ____________________ c. ____________________

Definition –

The following graph is called a piecewise function because the function is defined by two or more different equations applied to different parts of the function’s domain.

Notice that it appears to be composed of three segments, each a different linear function over a particular domain. Please note a filled circle includes that point, while an open circle does not include that point.

1. What is the domain for the first (left) segment? The Range?

2. What is the domain for the second (middle) segment? The Range?

3. What is the domain for the third (right) segment? The Range?

4. How many equations do you think you would have to use to write a rule for the above piecewise function? Explain.

Example 1:

f(x) = {2x

5

,−5≤x<2,2≤x≤6

5. Complete the following table of values for the piecewise function over the given domain. Then graph.

x -5 -3 0 1 1.7 1.9 2 2.2 4 6

f(x)

Math Composer 1. 1. 5http: / / www. mathcomposer. com

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6. How many pieces does your graph have? Why?

7. Are the pieces rays or segments? Why?

8. Are all the endpoints solid dots or open dots or some of each? Why?

Example 2

Graph this piecewise function: f(x) = { x+310−2 x

,−8≤x<1,1≤x≤7

Math Composer 1.1.5http: / /www.mathcomposer. com

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Example 3

Graph the piecewise function f ( x )=¿ {x+1¿ {5 ¿¿¿¿,,,

−6≤x<−2−2≤x<1

x≥3

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Unit 3 Lesson 7b – Piecewise Functions Ready, Set, Go! - Assignment 3.7b

http://goo.gl/bDVR3

Name___________________________________

Date_________ Hour_______

ReadyPart I. Find the domain and range for each piecewise function. Then, evaluate the graph at the specified domain value.

1.

Domain:________________

Range:_________________

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Domain:________________

Range:_________________

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Domain:________________

Range:_________________

4.

Domain:________________

Range:_________________

5.

Domain:________________

Range:_________________

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Write a piecewise function for each graph. Also, give the domain and range. 6.

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8. (adapted from Math Visions Project Secondary math 2 Book)

Isaac lives 3 miles away from his school. School ended at 3 pm and Isaac began his walk home with his friend Kaili who lives 1 mile away from the school, in the direction of Isaac’s house. Isaac stayed at Kaili’s house for a while and then started home. On the way he stopped at the library, then he hurried home. The graph at the right is a piece-wise defined function that shows Isaac’s distance from home during the t time it took him to arrive home.

a. How much time passed between school ending and Isaac’s arrival home?

b. How long did Isaac stay at Kaili’s house?

c. How far is the library from Isaac’s house?

d. Where was Isaac 3 hours after school ended?

e. Use function notation to write a mathematical sentence that says the same thing as question (d).

9. (adapted from MVP)Create a story that would match the graph below. Use a different example than the one given in question 8. Be specific about what is happening for each part of your story. Include what you know about linear equations, domain, and rates of change.

9. If you were to write equations to match each piece of your story (or section of the graph), how many would you write? Explain.

10. Write each of these equations. Explain how the equations connect to your story and to the graph.

Unit 3 Lesson 8 – Absolute Value Graphs and Piecewise Functions – TASK 3.8

Name_________________________________Date________Hour __________

Part A1. Explain your knowledge about absolute value using words.

2. Using past knowledge to create new knowledge, try graphing the following function:

y=|x|

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3a. Explain your reasoning for the graph you created.

3b. Please justify this method (using another method).

Part B

Now try graphing the following absolute value equations. Create your own table to justify values.

x f ( x )

4. f ( x )=|x−3|

x f ( x )

5. g ( x )=|x+3|

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Use a table to create the following graph.

x f ( x )

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Now try graphing the following absolute value equations. Create a table to justify values.

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10. Make a conjecture about functions that come in the form: y=|x|+k. Is that the result you expected? Why or why not?

Part C

Absolute value functions can be written without absolute value bars if they are separated into two equal parts.

Example:

y=|x| can be written as two different linear functions.

x f ( x )

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11. Sketch the graph.

12. Where in the domain do you think the graph will change from one function to the next?

13. If you answered x=0 for number 12, thank you. Now, let’s break it apart and write the equation:

f ( x )=¿

Unit 3 Lesson 8 - Absolute Value and Piecewise Functions Ready, Set, Go! - Assignment 3.8

http://goo.gl/ptcE0

Name__________________________________

Date_________ Hour_______

Ready

1) What will be the equation of the resulting graph if the graph of y = ∣x∣ is shifted 4 units down?A) y=∣x+4∣ C) y=∣x∣+4B) y=∣x - 4∣ D) y=∣x∣ - 42) What will be the equation of the resulting graph if the graph of y=∣x∣ is shifted 3 units up?A) y= ∣x - 3∣ C) y=∣x∣+3B) y=∣x+3∣ D) y=∣x∣ - 33) What will be the equation of the resulting graph if the graph of y=∣x∣ is shifted 3 units to the left?A) y=∣x∣-3 C) y=∣x+3∣B) y=∣x∣+3 D) y=∣x-3∣4) What will be the equation of the resulting graph if the graph of y=∣x∣ is shifted 4 units to the right?A) y=∣x∣+4 C) y=∣x+4∣B) y=∣x∣-4 D) y=∣x-4∣5) What will be the equation of the resulting graph if the graph of y=∣x∣ is reflected in the x-axis?A) y=∣x∣-1 C) y=-∣x∣B) y= ∣ x ∣2 D) y=2∣x∣6) When compared to the graph of y=∣x∣, the graph of y=∣x∣ - 8 isA) Shifted to the left 8 units C) shifted down 8 unitsB) Shifted up 8 units D) shifted to the right 8 units7) When compared to the graph of y=∣x∣, the graph of y=∣x+5∣ isA) Shifted up 5 units C) shifted down 5 unitsB) Shifted to the right 5 units D) shifted to the left 5 units

SetGraph each absolute value equation. State the domain and range of each:

8. y=|−x−3| 9.

D:

R:D:

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10. f ( x )=−|x+3| 11. y=12|x−4|

D:

D:

R:

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12. 13.

D:

R:

D:

R:

Graph each of the following absolute value equations on the graph provided.

14. y=4|2−x| 15.

16. 17.

18. 19.

Go!20. (adapted from MVP)Michelle and Heidi love going on long bike rides. Every Saturday, they have a particular route they bike together that takes four hours. Below is a piecewise function that estimates the distance they travel for each hour of their bike ride.

f ( x )=¿ 16 x ,0< x≤110 ( x−1 )+16 ,1< x≤ 214 ( x−2 )+26 ,2<x ≤312 ( x−3 )+40 ,3<x ≤ 4

a. What part of the bike ride do they go the fastest?

b. What is the domain of the function?

c. Find f (2 ). Explain what this means in terms of context.

d. How far have they traveled in 3 hours? Write the answer using function notation.

e. What is the total distance they travel on this bike ride?

f. Sketch a graph of the bike ride as a function of distance traveled over time.

Unit 3 Lesson 9– Parking Deck Pandemonium TASK 3.9

Name_________________________________Hour __________

Part 1

Directions: The fee schedule at parking decks is often modeled with a step function. Let’s look at a few different parking deck rates to see the step functions in action. (Most parking decks have a maximum daily fee. However, for our exploration, we will assume that this maximum does not exist.)

1. As you drive through town, Pete’s Parking Deck advertises free parking up to the first hour (i.e., the first 59 minutes). Then, the cost is $1 each additional hour or part of an hour. (If you park for 1 ½ hours, you owe $1; 2 hours costs $2.)

a. Make a table listing the fees for parking at Pete’s for up to 5 hours. Be sure to include some non-integer values. Then draw the graph that illustrates the fee schedule at Pete’s.

b. Use your graph to determine the fee if you park for 3 ½ hours. What about 3 hours, 55 minutes? 4 hours, 5 minutes?

c. What are the x- and y-intercepts of this graph? What do they represent?

d. What do you notice about the time and the corresponding fees? Make a conjecture about the fee if you were to park at Pete’s Parking Deck for 10 ½ hours (assuming no maximum fee). Similar to the other situations explored in this unit, this graph could be represented by a piecewise function.

e. Write a piecewise function to model the fee schedule at Pete’s Parking Deck.

Part 2

2. Paula’s Parking Deck is down the street from Pete’s. Paula recently renovated her deck to make the parking spaces larger, so she charges more per hour than Pete. Paula’s Parking Deck offers free parking up to the first hour (i.e., the first 59 minutes). Then, the cost is $2 each additional hour or part of an hour. (If you park for 1 ½ hours, you owe $2.)

a. Graph the fee schedule for Paula’s Parking Deck for up to the first 5 hours.

b. How does this graph compare with the graph of Pete’s Parking Deck? To what graphical transformation does this change correspond?

c. If you were to connect the left endpoints of the steps in 1a, what would be the equation of the resulting function? If you connected the left endpoints of the steps in 2a, what would be the equation of the resulting function? How do these answers relate to your answer in 2b?

Part 3

3. Pablo’s Parking Deck is across the street from Paula’s deck. Pablo decided to make his fee schedule even more straightforward than Pete’s and Paula’s. Rather than provide any free parking, Pablo charges $1 for each 0 – 59 minutes. (If you park for 59 minutes, you owe $1; if you park for 1 hour, you owe $2; etc.)

a. Graph the fee schedule for Pablo’s Parking Deck for up to the first 5 hours.

b. How does this graph compare with the graph of Pete’s Parking Deck? To what two different graphical transformations does this change correspond?

c. Write the function, h, in terms of the greatest integer function, that represents this graph. (What are the two different forms that this function could take?)

Part 4

4. Padma’s Parking Deck is the last deck on the street. To be a bit more competitive, Padma decided to offer parking for each full hour at $1/hour. (If you park for 59 minutes or exactly one hour, you owe $1; if you park for up to and including 2 hours, you owe $2.)

a. Graph the fee schedule for Padma’s Parking Deck for up to the first 5 hours.

b. To which of the graphs of the other parking deck rates is the most similar? How are the graphs similar? How are they different?

Unit 3 Lesson 9 - Step Functions Ready, Set, Go! - Assignment 3.9

http://goo.gl/OnAtv

Name__________________________________

Date_________ Hour_______

ReadyEvaluate the following greatest integer expressions. 1. ⟦7.1 ⟧ 2. ⟦1.8 ⟧ 3. ⟦ π ⟧

4. ⟦−6.8 ⟧ 5. ⟦−2.1 ⟧ 6. ⟦ 0 ⟧

Solve the following equations for x and write the answers in interval notation.

7. ⟦2 x7 ⟧=1 8. ⟦3 x ⟧=12

Set

Using what you learned about translations/transformations of the form: y=a|x−h|+k∧ y=a ( x−h )2+k , graph the following by hand and check your answer on your calculator.

9. f ( x )= ⟦ x ⟧+2 10. g ( x )=⟦ x+2 ⟧Math Composer 1.1.5http: / /www.mathcomposer.com

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Explain the shift in each graph and how they differ.

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Explain the relection in these graphs and how they differ.

Go!

Prior to September 2000, taxi fares from Washington DC to Maryland were described as follows: $2.00 up to and including ½ mile, $.70 for each additional ½ mile incremement.

15. Desribe the independent and dependent variables and explain your choices.

16. Graph the fares for the first 2 miles. Make sure to label the axes.

17. Write the piecewise function for 0 to 2 miles.

18. Discuss why this step function and it is different from the greatest integer parent function f ( x )= ⟦ x ⟧

Unit 3 Lesson 10 –Even and Odd Functions Notes 3.10

An even function has symmetry about _____________________________________________. (like a reflection)

Algebraically, a function is even when for all x.

** This means when you replace x with –x in an equation, you will get the original function if that function is even.

EX: Show f ( x )=x4+3 x2−5 is an even function, algebraically.

This is the curve f ( x )=x2+1

This curve is an even function. This function is symmetrical about the y-axis.

However, all parabolas are not even functions. An even exponent does not always make an even function. For example, f ( x )= (x+1 )2 is not an even function.

Some examples of even functions are:

a) ________________________________

b) ________________________________

Even Functions

f ( x )=¿______________

An odd function has symmetry with ________________________________.

Algebraically, a function is odd when for all x.

This means when you replace x with –x in an equation, you get the opposite of the original function. (i.e., all signs change)

EX: Show that f ( x )=2x3−4 x5 is an odd function.

An odd exponent does not always make an odd function. For example, f ( x )=x3− x+1 is NOT an odd function although it has an odd exponent.

Some examples of odd functions are:

a) ________________________________

b) ________________________________

Odd Functions

This curve is f ( x )=x3− x

f ( x )=¿______________

This curve is NOT odd.

Don’t be misled by the names “odd” and “even” … they are just names. A function does not have to be even or odd. In fact most functions are neither odd nor even. We classify these as “neither.”

EX: Show that f ( x )=7 x3+4 x−2 is neither even nor odd.

Additional Examples

Determine whether the function is even, odd, or neither.

1. f ( x )=−3x3+2 x 2. f ( x )=5 x2+2 x4−1

3. f ( x )=x+ 1x 4. f ( x )=17

Neither Odd nor Even

Unit 3 Lesson 10 – Even and Odd Functions Task 3.10

Name______________________________

Date_________ Hour_______

Directions:Part 1:Sort the function into the following categories: even, odd, and neither. Be sure to justify your work.

f ( x )=x+3 g ( x )=5 x h ( x )=( x−4 )2

j ( x )=2|x|+1 m (x )=−7 x2 p ( x )=2 x

Part 2:

For any function in the “neither” category, describe how you could transform it into an even or odd function.

Unit 3 Lesson 10 – Even and Odd Functions Ready, Set, Go! - Assignment 3.10

http://goo.gl/RJEG6

Name___________________________________

Date_________ Hour_______

Ready1. The graphs of various functions are shown. Determine if the functions are odd, even, or neither.

2. Each of the graphs below shows a portion of a graph of an EVEN function over the interval [-5, 5]. Complete these graphs.

3. Table A shows a portion of a table of an even function and Table B shows a portion of a table of an odd function. Complete these tables.

Table A -5 8 -4 6-3-2-1 -2012 03 445

Table B-5 -6-4-3 -2-2-101 32 534 -35

Set

Algebraically, determine whether each function is odd, even, or neither.

4. f ( x )=3 x4−5 x2+17 5. f ( x )=|x|

6. f ( x )=12 x7+6 x3−2x 7. f ( x )=4 x3−7

8. f ( x )=x2+2 x+2 9. f ( x )= x2−52 x3+x

10. The graphs of an odd function are symmetric about the origin. What geometric property characterizes even functions?

11. Suppose f is an odd function whose domain includes zero. Explain why f(0)=0 must be true.

12. Can a function be both even and odd? (Hint: consider constant functions of the form f(x)=c).

Go!

Determine whether the following functions are even, odd, or neither.

13. f(x)= 4x – 3 14. f(x)=|x|+1 15. f(x)= -x2 – 4

16. f(x)= 13 x3

17. f(x)= 7x 18. f(x)= √ x+5

19. f(x)= 3x2 20. f(x) = x3 – 2 21. f(x)= 3x + 4

22. f(x)= x2 – 5 23. f(x)= 10x + 5 24. f(x)= 2(x+1)3

Unit 3 Lesson 11 – Introduction to Inverse FunctionsTask 3.11

Name________________________________________

Date_____________ Hour_________

1. Reflect ∆ ABC crosses the line y=x . Label the new image as ∆ A ' B ' C ' . Connect the segments A A' ,B B ' , C C ' . Describe how these segments are related to each other and to the line y=x .

2. On the graph provided to the right, draw a 5-sided figure in the 4th quadrant. Label the vertices of the pre-image. Include the coordinates of the vertices. Reflect the pre-image across the line y=x. Label the image, including the coordinates of the vertices.

3. A table of values for a four-sided figure is given in the first two columns. Reflect the image across the line y = x, and write the coordinates of the reflected image in the space provided.

A (−6 , 2 ) A’

B (−4 , 5 ) B’

C (−2,3 ) C’

D (−3 ,−1 ) D’

PART 2

Chandler and Isaac both like to ride bikes for exercise. They were discussing whether or not they have a similar pace so they could plan a bike ride together. Chandler said he bikes about 12 miles per hour (12 miles in 60 mins). Isaac looked confused and said he doesn’t know how many miles he bikes in an hour because he calculates his pace differently.

1. Using Chandler’s information, determine the independent and dependent variables.

2. Since Chandler uses time to determine the distance he travels, determine how far she will go in 1 minute? 5 minutes? 10 minutes? 20 minutes? 30 minutes? t minutes? Fill in the following chart.

t 1 5 10 20 30 60 t

d (t )

3. Write the equation for Chandler’s pace using time (in minutes) as the independent variable and distance (in miles) as the dependent variable.

d ( t )=¿

4. Sketch the graph for this situation whose domain goes from [ 0 ,120 ] .

Isaac says he calculates his pace differently. He explains that he bikes a five minute mile, meaning that for every five minutes he bikes, he travels one mile.

5. How is this different than how Chandler described his rate?

6. Who goes a faster rate? Explain.

7. Since Isaac uses distance to determine how long he has ridden, determine how long it will take him to travel 1 mile? 2 miles? d miles? (complete the table)

d 1 2 4 5 6 d

t (d )

8. Write the equation for Isaac’s pace using miles as the independent variable and minutes as the dependent variable.

t ( d )=¿

9. Sketch a graph of Isaac’s function. As always, be sure to label.

Notes and Observations:

10. Using the equations, tables, and graphs, make a list of observations of what happens what you have to functions that are inverses of each other.

11. Why do you think inverse function have these characteristics?

Unit 3 Lesson 11 – Introduction to Inverse FunctionsNotes 3.11

Inverse Functions:

Find the inverse of each function, if it exists.

1. f ( x )=2 x−35 2. g ( x )=3 x2

3. h ( x )=x−3 4. j ( x )=2 x3

5. Give an example of a function that does not have an inverse function and explain how you know it does not.

6. Prove that the inverse of a non-horizontal linear function is also linear and that the slopes are reciprocals.

Graph the given function and its inverse.

7. f ( x )=2 x+3 8. g ( x )=√ x+2

9. Why would creating an inverse of a quadratic function require a restricted domain? Explain.

Unit 3 Lesson 11 – Introduction to Inverse Functions Ready, Set, Go! - Assignment 3.11

Name___________________________________

Date_________ Hour_______

Ready 1. The function f(x) is shown on the graph. Graph f−1 (x ) on the same set of axes.

Determine the inverse for each function, then sketch the graphs and state the domain and range for both the original function and its inverse.

2. 𝑓(𝑥) = 𝑥2 − 1 ; 𝑓−1(𝑥) Domain:

Range:

Domain:

Range:

3. 𝑔(𝑥) = 3𝑥+ 2 ; 𝑔−1(𝑥) Domain: Domain:

Range: Range:

4. Find the inverse for each relation.

a. {(1 ,−3 ) , (−2 , 3 ) , (5 , 1 ) , (6 , 4 ) } b. { (−5 ,7 ) , (−6 ,−8 ) , (1,−2 ) , (10 , 3 ) }

Set Find an equation for the inverse of each of the following relations if it exists.

5. y=−3x+2 6. f ( x )=12 x−3

7. g ( x )=23

x−5 8. y=−58

x+10

9. y=x2+5 10. h ( x )=( x+3 )2

11. f ( x )= (x−6 )2 12. y=√x−2 , y ≥0

13. g ( x )=√ x+5 , y ≥ 0 14. y=√x+8 , y≥ 8

Go! 15. Solve the following equations (find the value of x):

a) 5(x + 2) = 25 b) 2(2x + 10) = 40

c) 3(2x – 5) = 21 d) 4(5x –3) = 7(2x + 3)

e) 3(4 + x) = 5(10 + x) f) 2(3x – 4) = 4x + 3

Match each domain and range given in this table with a graph labeled from A t o L on the attached page. Only use Graphs A – L for this page. Write the letter of your answer in the blank provided for each problem.

1.

Domain: {-4 ≤ x ≤ 4}

Range: {-4 ≤ y ≤

4} Function: NO

2.

Domain: {-3 < x ≤

5} Range:{y = -1}

Function: YES

3.

Domain: {-4 ≤ x ≤ 2}

Range: {-2 ≤ y ≤

4} Function: YES

4.

Domain: {x > 0}

Range: {y =

4} Function:

YES

5.

Domain: {-6 ≤ x ≤ 6}

Range: {0 ≤ y ≤

6} Function:

YES

6.

Domain: {x = -

5}

Range: {-2 < y <

6} Function: NO 7.

Domain: {x ≥

0}

Range: {all real

numbers} Function: NO

8.

Domain: {-3 ≤ x ≤ 4}

Range: {-2 ≤ y ≤

4} Function: NO

9.

Domain: {all real numbers}

Range: {all real numbers}

Function: YES

10.

Domain: {-7 ≤ x < 5}

Range: {-3 ≤ y <

1} Function: YES

11.

Domain: {all real

numbers} Range: {y ≥ 0}

Function: YES

12.

Domain: {-3 < x < 4}

Range: {0 ≤ y ≤

5} Function: YES

Match each domain and range given in this table with a graph labeled from M t o X on the attached page. Only use Graphs M to X for this page. Write the letter of your answer in the blank provided for each problem.

13.

Domain: {-6 ≤ x ≤ 3}

Range: {-6 ≤ y ≤ -

1} Function: YES

14.

Domain: {0 ≤ x < 5}

Range: {0 ≤ y <

7} Function: YES

15.

Domain: {-5 ≤ x < 0}

Range: {-5 < y ≤ -

1} Function: YES

16.

Domain: {-6 ≤ x ≤ 3}

Range: {-5 ≤ y ≤ -

1} Function:

YES

17.

Domain: {0 ≤ x ≤ 6}

Range: {0 ≤ y ≤

7} Function:

YES

18.

Domain: {-4 ≤ x ≤ 7}

Range: {-7 ≤ y ≤ -

2} Function: NO

19.

Domain: {x ≤ 0}

Range: {y ≥ 0}

Function: YES

20.

Domain: {2 ≤ x ≤ 7}

Range: {1 ≤ x ≤

6} Function: NO

21.

Domain: {0 ≤ x ≤ 4}

Range: {0 ≤ y ≤

6} Function: YES

22.

Domain: {-4 < x < 5}

Range: {-2 ≤ y <

5} Function: YES

23.

Domain: {x ≤ 5}

Range: {y = 0}

Function: YES

24.

Domain: {-7 < x < 0}

Range: {-3 < y <

4} Function: YES

Use these graphs to answer questions 1-12.

A B C

D E F

G H I

J K L

Use these graphs to answer questions 13-24

M N O

P Q R

S T U

V W X

Function Notation/Domain&Range Review Task

Name____________________________________

Date________ Hour_______

#1-6. Work with a partner or on your own to investigate the equations listed in the chart using technology, a t-chart, your notes, etc. Every equation in the table is a function. Determine the domain and range for each function from the possibilities listed below the chart. Select the appropriate domain and range and complete your chart.

Function Domain (write the #)

Range (write the #)

1 y=−3 x+4

2 y=x2−6 x+5

3 y=9 x−x2

4 y=|x+1|

5 y=3+√x

6 y= 4x

Possible Domains1) all real numbers2) all real x, such that x≠−2.3) all real x, such that x≠ 0.4) all real x such that x≠ 2.5) all real x, such that x≤ 0.6) all real x, such that x≥ 0.

Possible Rangesa) all real numbersb) all real y, such that y ≠ 0.c) all real y, such that y ≥−4.d) all real y, such that y ≥0.e) all real y, such that y ≥1.f) all real y, such that y ≥3.

For each pair of variables below, a) tell which is the independent variable, andb) write a sentence giving the relationship between the two variables using either “depends on” or “is a function of.”

7. spring rainfall and wildflowers

8. number of wild rabbits and available food supply.

9. amount of money spent and articles of clothing bought.

10. amount of stagnant water and number of mosquito larva.

For each of the following write a sentence giving the relationship between the two variables using either “depends on” or “is a function of”. Vary your responses.

11. W(c), where c = calories and W = weight

12. T(t), where T = temperature and t = time of day

13. A(d), where A = area and d = diameter

14. W(e), where e = exercise and W = weight

15. G(t), where G = grade and t = hours of TV watched per day

16. R (r ) ,where r = amount of rainfall and R = river level