working(with(functions(and(equations working with

Working with functions and equations 1

Copyright © 2016 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, Agile Mind, Inc.

WORKING WITH FUNCTIONS AND EQUATIONS Lesson 1: An introduction to functions

LESSON 1: OPENER

Casey and Stacey are playing a math game called “Guess my rule.”

Table 1

Input Output 2 5 5 26 7 50 10 101

Casey creates this table and shows it to Stacey. Stacey’s job is to find an algebraic rule that works for all of the input-‐output pairs.

“I found a rule that works for the table,” says Stacey.

1. What rule do you think she found? Write a rule that works for this table.

Table 2

Input Output -‐2 4 4 10 -‐2 -‐8 10 4

Now it is Stacey’s turn to create a table and Casey will find an algebraic rule. Stacey creates the table shown.

“Wait a second,” says Casey. “This is an illegal table. It’s impossible to find a rule for this table.”

2. Why does Casey say there is no rule for this table?

Table 3

Input Output -‐3 9 -‐1 1 0 0

1 1 3 9

Stacey creates the table shown. At first, Casey thinks this is another “illegal table.” Then Casey realizes that there is a rule that allows each output to be predicted from the input.

3. What rule do you think she found? Write a rule that works for this table.

2 Working with functions and equations


LESSON 1: CORE ACTIVITY

1. Which vending machine represents a function? Why?

2. Which vending machine does not represent a function? Why?

3. Examples of functions are listed here. Work with your partner. Take turns explaining why each example is a function, and

write your answers in the table.

Example Why this is an example of a function

If you save the same amount each week, the amount of money in your bank account depends on (is a function of) the number of weeks you have been saving.

The amount of time it takes to run a certain distance depends on (is a function of) the speed at which you run when you run at a steady speed.

If you are paid $7.00 an hour, the amount of money you earn at a summer job depends on (is a function of) the number of hours you work.



LESSON 1: CONSOLIDATION ACTIVITY

1. Look at the two snack machines labeled "Snacks" and "Healthy Snacks."

a. Describe how to fill the "Snacks" machine so that it represents a function.

b. Describe how to fill the “Healthy Snacks” machine so that it does not represent a function.

2. Complete the following math journal entry.

Topic Explanation

How I would explain the math concept of a function

3. With your partner, create:

• An example of a relationship that is a function

• An example of a relationship that is not a function

Write each of your examples on a separate whiteboard to share with your classmates.



LESSON 1: HOMEWORK

Notes or additional instructions based on whole-‐class discussion of homework assignment:

1. A stamp machine at the post office works very much like a vending machine: You press a button to select the type of stamp

you want to buy. The diagrams below show the buttons and types of stamp for one machine. All of the stamps cost the same amount of money.

a. Draw arrows from the buttons to the stamps in the diagram below to show a relationship that is a function.

b. Draw arrows from the buttons to the stamps in a different way than you did in part a to show a relationship that is a

function.

c. Draw arrows from the buttons to the stamps in a different way than you did in both part a and part b to show a relationship that is a function.

2. Create an example of a relationship that is a function. (Your example cannot be one of the examples discussed today in class.) Explain why your example is a function.

A B C D E

Flag President Elvis Flower Forever

A B C D E


A B C D E




3. Create an example of a relationship that is not a function. (Your example cannot be one of the examples discussed today in class.) Explain why your example is not a function.

4. Miriam is an aspiring meteorologist. She makes a table in which she numbers each day of the current calendar year—from 1 to 365—in the input column. She records the corresponding high temperature in her town for each of these days.

a. Does Miriam’s table represent a function? Answer “Yes,” “No,” or “Maybe,” and explain your answer.

b. Miriam wonders whether the table would be a function if she reversed the input and output columns. If the input column contained the daily high temperature and the output column contained the day of the year (from 1 to 365), would the relationship be a function? Answer “Yes,” “No,” or “Maybe,” and explain your answer.



LESSON 1: STAYING SHARP Practic

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1. Write a rule for the following graph:

Answer with supporting work:

2. Use the graph and your rule for question 1 to complete the table below.

x Y

-‐4

5

15

10

50

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3. Cordell writes the equation v = 0.10d + 0.05n to model the total value of the dimes and nickels in his pocket. (Dimes are worth ten cents and nickels are worth five cents.) What does each part of the equation represent?

Part Meaning

v

0.10

d

0.05

n

4. If Cordell has 6 nickels and 4 dimes, find the total value of his dimes and nickels by substituting and evaluating the expression 0.10d + 0.05n. Answer with supporting work:

Focus skill: Scalin

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5. State an appropriate scale to use to graph the data in the x-‐y table shown.

x 0 2 4 6 8 10 12 14 y 0 10 20 30 40 50 60 70

Minimum x-‐value: Maximum x-‐value: Increment for x-‐axis: Minimum y-‐value: Maximum y-‐value: Increment for y-‐axis:

6. Matt is trying to graph the data in this x-‐y table:

x 0 6 9 12 15 18 y 1 13 19 25 31 37

He chooses the scale:

Min. x: 0 Min. y: 0 Max. x: 20 Max. y: 20 x increment: 5 y increment: 5

Identify one problem with Matt’s scale, and suggest one improvement you would make.

-4 -2 2 4 6 8 10 12 14 16 18 20 22 24 26 28-1

1234567891011121314

x

y



Lesson 2: Mathematical functions

LESSON 2: OPENER

1. A television remote control can be thought of as a “function machine.” Fill in the blanks to complete the following statement.

For this situation, the input is ________________________

and the output is ___________________________.

2. Think of your own example of a “real-‐world” function machine. Describe your example, including a description of the input and output.


In this activity, you will use input-‐output machines to develop your understanding of the definition of a function. Then you will use what you've learned about functions to answer some questions.

1. Complete the questions below for these input-‐output tables.

a. Fill in the remaining output values for each x-‐y table.

x y -‐2 -‐4 3 6 5 10 10 20 30.5

b. Graph the coordinate pairs from each x-‐y table in part a. Write “function” on the graph that represents a functional relationship.

x y 9 3 9 -‐3 4 2 4 -‐2 1 1 0



2. For most of the input-‐output tables below, it is possible to find an algebraic rule. However, some of these tables do not

represent functions. For each table, circle "Function" or "Not a function." Then, either write the function rule, or explain why the relationship is not a function.

a. x y 2 -‐4 2 -‐2 2 0 2 2 2 4 2 6 2 82

b. x y -‐3 3 -‐2 7 -‐1 -‐1 0 5 1 8 2 0 3 -‐4

c. x y 0 0 1 1 4 2 9 3 16 4 25 5 36 6

d. x y -‐3 3 -‐2 2 -‐1 1 0 0 1 1 2 2 3 3

e. x y 3 4 2 8 1 8 1 4 2 4 3 8 3 4

f. x y -‐27 -‐3 -‐8 -‐2 -‐1 -‐1 0 0 1 1 8 2 27 3

g. x y -‐3 4 -‐2 4 -‐1 4 0 4 1 4 2 4 3 4

h. x y -‐6 3 -‐6 -‐4 -‐3 2 -‐3 -‐3 0 1 0 -‐2 3 -‐1

Table Is it a function? a.

Function Not a function

Function rule or explanation of why the relationship is not a function:

b.



c.



d.



e.



f.



g.



h.





For each table, make a graph of the relationship by plotting the ordered pairs on one of the grids provided. Make sure you graph each relationship you decided was a function on one of the grids labeled “Function” and each relationship you decided was not a function on one of the grids labeled “Not a Function.” Don’t forget to label each graph by indicating which table it represents.

Functio

n

Table _____

Table _____

Table _____

Table _____

Table _____

Table _____

Not a Fun

ction

Table _____

Table _____

Table _____

Table _____

Table _____

Table _____



3. How can you tell whether a relationship is a function by:

a. Looking at a table of values that represents the relationship? b. Looking at a graph that represents the relationship?

LESSON 2: CONSOLIDATION ACTIVITY 1. For each of the following graphs, indicate whether the graph represents a function. Circle "Function" or "Not a function."

Then provide a brief explanation to support your answer.

Graph Is it a function?

a.


Explanation:

b.


Explanation:

c.


Explanation:

d.


Explanation:

e.


Explanation:



2. The relationship shown in the table is not a function. What could be changed in the table so that it does represent a function?

3. Plot the points that are given in the table. From the graph, determine whether the relationship is a function. Explain your reasoning. If you are able, determine a function rule that represents the relationship.

x y

-‐3 9

-‐2 4

-‐1 1

0 0

1 1

2 4

3 9

-‐3 9

x Y

3 8

4 2

5 4

7 8

5 6



4. Now that you have some experience with functions and function vocabulary, complete this math journal entry. You may look back in your book and notes, but be sure to state the ideas in your own words and provide your own examples. For the term “function,” give both an example of a relationship that is a function and a relationship that is not a function. Then explain why the second relationship is not a function.

Vocabulary term

My understanding of what the term means

Examples that show the meaning of the term

Function

Example of a relationship that is a function: Example of a relationship that is not a function: Why is the second relationship not a function?

Write the classroom definition of “function” in the box below.

Function:

5. Compare the classroom definition of “function” to your definition/explanation. Underline the parts of your definition that

match the classroom definition. Then, add any information to your definition that you may be missing and delete any incorrect information in your definition.



LESSON 2: HOMEWORK Notes or additional instructions based on whole-‐class discussion of homework assignment:

1. For each table, determine whether the relationship is a function. Circle "Function" or "Not a function." Then, either write

the function rule, or explain why the relationship is not a function.

a. x y -‐2 8 0 0 2 -‐8 4 -‐16 6 -‐24 8 -‐32 10 -‐40

b. x y -‐3 0 -‐2 0 -‐1 0 0 0 1 0 2 0 3 0

c. x y

-‐100 10 -‐25 5 -‐4 2 0 0 -‐4 -‐2 -‐25 -‐5 -‐100 -‐10

d. x y -‐3 12 -‐2 12 -‐1 12 0 12 1 12 2 12 3 12

Table Is it a function?

a.


Function rule or explanation of why it is not possible to find a function rule:

b.



c.



d.





2. For each of the following graphs, determine whether the graph is a function. Circle "Function" or "Not a function." Provide a brief explanation to support your answer.

Graph Is it a function?

a.


Explanation:

b.


Explanation:

c.


Explanation:

3. Complete the following math journal entry.

Topic Explanation

a. How to determine whether a table of input-‐output data is a function

b. How to determine whether a graph of a mathematical relationship is a function




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2. Use the graph and/or your rule for question 1 to complete the table below.

x y

-‐2

10

60

100

100

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3. Caryssa makes bracelets to sell at craft fairs. The supplies she needed to start her business cost her $45. In addition, the supplies for each bracelet cost her $3. She sells her bracelets for $5. What is the minimum number of bracelets she must sell to make a profit? Answer with supporting work:

4. Use the information in question 3 to write an equation that Caryssa could use to determine her total cost, C, for making n bracelets.

Use the information in question 3 to write a separate equation that she could use to find the profit, P, she makes from selling n bracelets.

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5. Find an appropriate scale to use to graph the algebraic rule y = 3x – 11. Consider input values from -‐5 to 5. (Hint: Completing the input-‐output table may help.)


y = 3x – 11

x y -‐5 -‐4 -‐3 -‐2 -‐1 0 1 2 3 4 5

-4 -2 2 4 6 8 10 12 14 16 18 20 22 24 26 28

10

20

30

40

50

60

70

80

90

x

y



Lesson 3: Independent and dependent variables

LESSON 3: OPENER Allison’s class is investigating the relationship between height and arm span. The table shows some of the data the class has gathered so far.

Age (years) 5 8 10 12 15 16

Height (in.) 41 50 55 61 66 68

Arm span (in.) 39 48 53 60 66 67

1. If you were to make a graph to show the relationship between age and height, which variable would you plot on the x-‐axis and which variable would you plot on the y-‐axis? Why?

2. If you were to make a graph to show the relationship between arm span and height, which variable would you plot on the x-‐axis and which variable would you plot on the y-‐axis? Why?


Fill in the blanks to complete the statements in questions 1-‐3.

1. The type of drink you purchase depends on the button you push, so you would say that the buttons (A, B, C, D, E, and F) are

possible values of the _____________________________________. A(n) _____________________________________in a

function rule determines what happens to another variable in the rule.

2. The different types of drinks (orange juice, apple juice, skim milk, chocolate milk, and water) form the set of possible values

of the _____________________________________. A(n) _____________________________________in a function rule

changes its value when a value of the independent variable is changed.

3. A pot of water is placed on a stove to heat. The water begins with a temperature of 70° F. As the water heats, the temperature of the water rises 20° F every 30 seconds.

a. What are the variables in this situation?

b. Which variable is the dependent variable? Which is the independent variable? Provide a convincing explanation.

c. Create an input-‐output table and a graph for this situation. Add data to the table that make sense for this situation.

d. On a whiteboard, plot the data on the graph. Add the appropriate labels on the graph axes.

Input Output



4. The fee to rent a van from Ralph’s Rent-‐a-‐Car is $0.30 per mile plus a one-‐time charge of $40. Answer the following questions about this relationship to demonstrate that you understand important ideas about functions. Refer to the vocabulary listed.

a. The amount of money charged is a ______________________________because it can change.

b. The number of miles you drive is a ______________________________because it can change.

c. The amount of money charged is the ____________________________________, because it is calculated based on the number of miles driven.

d. The number of miles you drive is the _____________________________________.

e. To make a graph of this relationship, you would plot ______________________________ on the x-‐axis and _______________________ on the y-‐axis.

f. Is this relationship a function? Explain.


1. Consider the following situation: A restaurant is preparing for a banquet. The banquet tables are square and one person can be seated on each side, as shown in the diagram. The manager sets up the restaurant’s private room based on the size of the group that is renting it. The square tables are always arranged in a single row. The variables in this situation are as follows:

• Number of people to be seated • Number of tables

a. Which variable is the dependent variable? Which is the independent variable? Provide a convincing explanation.

b. Create an input-‐output table and a graph for this situation. Add data to the table that make sense for this situation. Add the appropriate labels (Number of People or Number of Tables) to the table heading and the graph axes. Then plot the data on the graph.

Input INDEPENDENT VARIABLE ( )

Output DEPENDENT VARIABLE ( )

Outpu

t De

pend

ent V

ariable

Input

Independent Variable

variable amount of money charged

number of miles driven

independent variable

dependent variable



2. Consider the following situation: The height of six buildings is measured in both feet and yards. Analyze the relationship between these measurements. The variables in this situation are as follows:

• Height in feet • Height in yards


b. Create an input-‐output table and a graph for this situation. Add data to the table that make sense for this situation. (Make up reasonable heights for six buildings.) Add the appropriate labels (Height in Feet or Height in Yards) to the table heading and the graph axes. Then plot the data on the graph.



Outpu

t De

pend

ent V

ariable

Input


3. Now make up your own situation and list the two variables. Fill in only the “Situation” and “Variables” information. Your partner will then try to solve the problem.

Situation:

Variables:

To be answered by your partner: • What is the independent variable? Why? • What is the dependent variable? Why?

Create an input-‐output table and a graph for this situation. Add the appropriate labels to the table heading and the graph axes. Then plot the data on the graph.



Outpu

t De

pend

ent V

ariable

Input




LESSON 3: HOMEWORK


1. Consider the following situation: A high school assistant principal is making the school’s lunch period schedule. As shown in the diagram, the school has only one size of table for the lunch room and each table can seat 8 students. The assistant principal has decided that he will not push any tables together. When more tables are placed in the room, more students can be seated and fewer lunch periods are needed. The variables in this situation are as follows:

• Number of people to be seated

• Number of tables


b. Create an input-‐output table and a graph for this situation. Add data to the table that make sense for this situation. Add the appropriate labels (Number of People or Number of Tables) to the table heading and the graph axes. Then plot the data on the graph.

Input

INDEPENDENT VARIABLE ( )

Process


Outpu

t De

pend

ent V

ariable

Input


c. Define the variables and write a function rule to describe the relationship between the number of people to be seated and the number of tables.

d. Explain why your function rule makes sense in the context of the problem situation.

e. Use your function rule to determine how many tables will be needed for 500 students.



2. Consider the following situation: The number of bacteria in a petri dish on the first day is 2, and the number doubles every day. The variables in this situation are as follows:

• Number of bacteria (measured in number) • Time (measured in days)


b. Create an input-‐output table and graph for this situation. Add data to the table that make sense for this situation. Add the appropriate labels (Number of Bacteria or Time) to the table heading and graph axes. Then plot the data on the graph.

Input

INDEPENDENT VARIABLE ( )

Process


Outpu

t De

pend

ent V

ariable

Input


c. Define the variables and write a function rule to describe the relationship between the number of bacteria and the number of days.

d. Use your function rule to determine how many bacteria will be present after 20 days.

3. Now that you have some understanding of independent and dependent variables, complete this math journal entry. You may look back in your book and notes, but be sure to state the ideas in your own words and provide your own examples.

Vocabulary term My understanding of what the term means

An example that shows the meaning of the term

Independent variable

Dependent variable




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1. What is an algebraic rule for the following table?

x y

3 0

5 2

10 7

13 10


2. Which graph could represent the relationship shown in question 1?

Graph A Graph B

Answer with supporting explanation:

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3. Rewrite each of the following expressions using exponents.

a. 4⋅4⋅4⋅4⋅4 = b. 2⋅2⋅2⋅2 = c. –3⋅–3⋅–3 =

4. Translate the following statements in to an expression using exponents.

a. The number 5 is multiplied to itself x times. b. The number 3 is multiplied to itself n times.

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x 1 4 6 7 12 14 21 y -‐4 5 11 14 29 35 56

Minimum x-‐value:

Maximum x-‐value:

Increment for x-‐axis:

Minimum y-‐value:

Maximum y-‐value:

Increment for y-‐axis:

6. Arturo is trying to graph the data in this x-‐y table:

x -‐40 -‐20 0 20 40 60 y 45 25 5 -‐15 -‐35 -‐55

He chooses the scale:

Min. x: -‐40 Min. y: -‐40 Max. x: 60 Max. y: 60 x increment: 10 y increment: 10

Identify one problem with Arturo’s scale and suggest one improvement you would make.



Lesson 4: Equations and functions

LESSON 4: OPENER

Suppose a medium pizza with no toppings costs $7.00. Each topping costs an additional $0.50.

1. How much would it cost to order a medium pizza with 3 toppings?

2. How much would it cost to order 3 medium pizzas with no toppings?

3. How many medium pizzas could you have ordered if your bill was $15.00?


1. Use function notation to represent the relationship between each button

and the type of beverage the machine delivers when that button is pushed.

2. Consider the function f(x) = 3x – 9.

a. Evaluate the function at x = 3

b. Solve the equation 3x – 9 = 48.

3. Use the function rule for the Banquet Table Problem, p(n) = 4n + 2, to calculate

a. how many people can be seated around 12 hexagonal tables.

b. the number of tables needed to seat 100 people.

4. A rule that relates the number of frames in the DVD to the number of seconds the DVD is played is y = 30x. Write this rule using function notation.

5. Use your function rule for the DVD data to calculate

a. how many frames will be shown if the DVD plays for 1 minute.

b. the number of seconds the DVD is played if 1350 frames are shown.




1. Recall Martina’s sequence: 2, 6, 10, 14, 18, 22… The 1st term is 2, the 2nd term is 6, the 3rd term is 10, and so on.

Term number Term

1 2

2 6

3 10

4 14

5 18

6 22

a. Suppose n and f(n) are chosen as variables. Which quantity does n represent? Which quantity does f(n) represent?

b. You can think of this sequence as a function. What is the domain?

c. You can think of this sequence as a function. What is the range?

d. Using function notation, write each term in terms of its term number. For example, the first term is 2, so f(1) = 2. The 2nd term is 6, so f(2) = 6. Continue this process for the first 6 terms of the sequence.

f(1) = 2

f(2) = 6

=

=

=

=

e. How do you get from one term to the next in Martina’s sequence?



f. There are several ways to represent how to get from one term to the next using function notation. Study the examples in the table. Then complete the table using function notation to represent the relationship between one term and the next.

Term number, n Process Term, f(n)

1 2

2 f(2) = 2 + 4 = f(1) + 4 6

3 f(3) = 6 + 4 = f(2) + 4 10

4 f(4) = 10 + 4 = f(3) + 4 14

5 f(5) = 14 + 4 = f(4) + 4 18

6 f(6) = 18 + 4 = f(5) + 4 22

7 f(____) = ____ + 4 = f(____) + 4

8 f(____) = ____ + 4 = f(____) + 4

9 f(____) = ____ + 4 = f(____) + 4

2. Now think of another sequence you saw. In a previous lesson, you found the perimeters of shapes when you attached equilateral triangles to each other.

Figure number, n 1 2 3 4

Visual

Perimeter, f(n)

3 units 4 units 5 units 6 units

a. What is the domain of the function that represents this sequence?

b. What is the range of the function that represents this sequence?

c. What is the relationship between a term and the one after it? .

d. Complete the table using function notation to show the relationship between one term and the next.

Figure number, n Process Perimeter, f(n)

1 3

2 f(2) = 3 + = f( ) +

3 f(3) = + = f( ) +

4 f(4) = + = f( ) +

5 f(5) = + = f( ) +



3. So far, you have considered the relationship between a term and the term before it. Now look for a direct relationship between the figure number and the perimeter in the sequence in question 2.

a. Write a rule, using function notation, that relates the perimeter directly to the figure number.

b. Using this rule, what is the perimeter of the 23rd figure?

c. If one of the terms of the perimeter sequence is 23, what is the term number?

4. Karina’s sequence was a different type of sequence. Her sequence was 2, 6, 18, 54…

a. How do you get from one term to the next?

b. Complete the table to develop a function rule to represent this sequence.

Term number, n

Process Term, f(n)

1 f(1) = 2 2

2 f(2) = 3( ) = 3⋅f( )

3 f(3) =3( ) = 3⋅ f( )

4 f(4) = 3( ) = 3⋅f( )

5 f(5) = 3( ) = 3⋅f( )

n f(n) = 3⋅f( ) 3⋅f( )



LESSON 4: HOMEWORK


Use the function rule from the Banquet Table Problem to answer the following questions. In the rule, n represents the number of hexagonal tables used and p(n) represents the number of people seated. Show your work for each question on your own paper.

Banquet Table

p(n) = 4n + 2

1. If 10 hexagonal tables are used, then people can be seated.

2. If 65 people need to be seated, then tables should be used.

Apply what you have learned about function notation to answer these questions. Show your work on your own paper.

3. f(x) = 2x

a. f(6) = b. f(-‐11) = c. f(2.75) =

d. If f(x) = 20, x = e. If f(x) = -‐14, x = f. If f(x) = 29, x =

4. f(x) = 5x + 50

a. f(7) = b. f(-‐12) = c. f 2

45

⎛⎝⎜

⎞⎠⎟=

d. If f(x) = 100, x = e. If f(x) = 20, x = f. If f(x) = -‐50, x =

5. h(x) = x2 + 9

a. h(2) = b. h(–3) =

c. If h(x) = 9, x = d. If h(x) = 13, x =

(Careful—there are two answers!)



6. Consider the sequence –3, –7, –11, –15…

a. Is this sequence a function? Explain.

b. How do you get from one term to the next in this sequence?

c. Create a function rule for this sequence.

d. Use your rule to find the value of the 8th term.

e. Which term number has a value of –47? How did you find your answer?




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1. What is an algebraic rule for the table shown below?

x y

0 10

10 10

30 10

50 10


2. Graph the rule in question 1 on the grid below. Label and scale your axes.

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3. Recall that in football, touchdowns are worth 7 points each (including the extra point), and field goals are worth 3 points each. List all of the different ways a football team could score 63 points, if they make at least one touchdown and at least one field goal. Answer with supporting work:

4. Recall that in basketball, a team can score 1-‐point baskets (free throws), 2-‐point baskets, or 3-‐point baskets. List all of the different ways a basketball team could score 24 points, without making any free throws. Answer with supporting work:

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5. For science class, Joan watches a bird as it moves up and down in a tree over time. You could make a graph to represent the relationship between the bird’s height off the ground and time. If you made a graph, what variable would you put on each axis? Explain your answers. Independent (x-‐axis) variable:

Dependent (y-‐axis) variable:

6. As Joan watches, the bird starts 10 feet above the ground and stays there for 5 seconds. Then it takes 10 seconds to fly up to 30 feet. After staying there for 15 seconds, it flies down to 20 feet. State an appropriate scale to use to graph the height of the bird during this period.




Lesson 5: Sequences as functions

LESSON 5: OPENER Leonardo Fibonacci was an Italian mathematician who described a sequence of numbers. The sequence he created was

1, 1, 2, 3, 5, 8, 13, 21…

1. Is the Fibonacci sequence a function? If so, what is the domain and range of the function?

2. Explain how you get from one term to the next in this sequence. How did you discover this pattern?

3. Is this sequence an example of an arithmetic or geometric sequence? Explain.


1. Show how the Fibonacci pattern holds to find f(3), f(4), and f(5). What do you need to know in order to find f(13)?

2. What do you need to know in order to find f(n)?

3. Create a recursive rule that defines the Fibonacci sequence. Complete the blanks with the correct information given in the tiles.

1 2 n

n – 1 n – 2

f(____) = _____ f(____) = ______

f(__ __) = f(_____) + f(______), where n is an integer and n ≥ 3.



4. Consider the following recursive definitions. Find the first 5 terms of the sequence that the function defines. Then classify the sequence as arithmetic, geometric, or neither.

a. f(1) = 12

f(n) = f(n – 1) – 3, where n is an integer and n ≥ 2.

b. f(1) = 2

f(n) = 4 ⋅ f(n – 1), where n is an integer and n ≥ 2.

5. Karina’s sequence is 2, 6, 18, 54….

a. Use a table to create an explicit rule for the sequence.

Term number, n

Process Term, f(n)

1 2

2

3

4

5

n

b. Use the explicit rule to find the value of the 9th term.



6. Think back to the mosaic Reuben created. You explored this figure in an earlier topic.

The sequence that is formed by the number of tiles in the first five mosaics is 5, 8, 11, 14, 17…

a. Can this sequence be represented by a function? If so, what is the domain and range of the function?

b. Create an explicit rule and a recursive rule to represent this sequence.

.

c. Each tile used in this mosaic costs $0.20. How much will it cost Reuben to create the 11th mosaic?

7. Joyce created the images below by taking a red square, dividing it into 9 smaller squares, and removing the center square.

The sequence formed by the number of red squares in the first five images is, 1, 8, 64, 512, 4096…

a. Can this sequence be represented by a function? If so, what are the domain and range?

b. Create an explicit rule and a recursive rule to represent this sequence.



8. Use the information below to either find terms of a sequence or create a rule for the nth term of a sequence.

a. A sequence is defined recursively as

f(1) = –2

f(n) = –2⋅f(n – 1), where n is an integer and n ≥ 2.

Find f(2), f(3), and f(4).

b. Create a function rule for the nth term of the sequence 1, 2, 4, 8…

LESSON 5: ONLINE ASSESSMENT

Today you will take an online assessment.



LESSON 5: HOMEWORK


Write a recursive rule and an explicit rule for each of the following sequences.

1. 7, 5, 3, 1, –1…

2. 42, 53, 64, 75, 86…

3. 100, 50, 25, 12.5…

4. In the design for an auditorium the first row of will have 15 seats. Each row after that will have 5 more seats than the row in front of it.

a. Find a formula for the number of seats in any row.

b. How many seats are in the 18th row?



5. For each rule, generate the first 5 terms of the sequence that is defined. In each case, n is an integer and n ≥ 2.

a. f(1) = 4

f(n) = 5 + f(n – 1)

b. f(1) = –3

f(n) = –2 ⋅ f(n – 1)

c. f(1) = –4

f(n) = 3 ⋅ f(n – 1) + 4

6. A sequence is formed by the perimeters of adjacent congruent pentagons. Each pentagon has a side length of 1 unit.

a. Create a table showing the relationship between the figure number, n, and the perimeter, f(n) for the first five figures.

b. Write the sequence formed by the first five perimeters.

c. Create a recursive rule that represents the sequence of perimeter values, where n is the figure number and f(n) is the perimeter. Don’t forget to include the restriction on n.

d. Create an explicit rule that relates n and f(n) directly.

e. What is the perimeter of the 20th figure? Which rule did you use, and why?




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1. Study this graph. What conclusions can you draw from the graph? (Look for patterns in the data. State your conclusions in the context of the specific situation shown in the graph.)

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2. The rule 2d + 3.5m = c can be used to find the total cost in dollars, c, of buying d donuts, and m muffins at a bakery. What does the rule tell you about the cost of a donut and the cost of a muffin at this bakery? Answer with supporting work:

3. Trish is selling lemonade at a school fundraiser. She bought 2 bags of cups. Each bag contained 90 cups and cost $3.60. The total cost of the ingredients to make the lemonade was $80. She plans to sell the lemonade for 50¢ a cup. Assuming she has enough lemonade, did she buy enough cups to make a profit?

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4. You want to graph the algebraic rule y = 7x – 20. Suppose you want to consider only positive values for the input. State an appropriate scale for the axes for the graph. (Hint: Consider what an input-‐output table for this rule might look like.)


5. The amount of time it takes to repaint a school decreases as the number of helpers increases. You could make a graph to represent this relationship. If you made a graph, what variable would you put on each axis? Explain your answer. Independent (x-‐axis) variable: Dependent (y-‐axis) variable:



Lesson 6: Functions and proportionality

LESSON 6: OPENER

The primary colors are red, yellow, and blue. You probably know it is possible to create new colors from primary colors. For example, a shade of purple paint can be made from mixing 1 drop of blue paint with 2 drops of red paint.

1. List five more different combinations of blue and red paint that could be used to create the same shade of purple paint. Organize your information into a table and then plot the six combinations on the graph. Label and scale your graph axes.

Number of drops of blue paint

Number of drops of red paint

1 2

2. What patterns do you see in the graph?



LESSON 6: CORE ACTIVITY 1. A cheese pizza costs $7.00, and increasing the number of cheese pizzas increases the total cost of your order. Answer these

questions about this situation.

a. Complete the table and graph for this pizza relationship.

Number of pizzas Process

Cost in dollars

0

1

2

3

4

b. Find a function rule that represents the relationship between the number of pizzas, n, and the total cost, C.

c. Write your function rule using function notation.

d. Using your function rule, find out how much it would cost to buy 8 pizzas.

e. If you have exactly $84 to spend in this situation, how many pizzas can you order?



2. Cheese pizzas cost $7.00 each, and there is a one-‐time charge of $5.00 to have your pizzas delivered. Answer the following questions about this situation.

a. Complete the table and graph for this pizza and delivery charge relationship.

Number of pizzas

Process Cost in dollars

0

1

2

3

4

b. Find a function rule that represents the relationship between the number of pizzas, n, and the total cost, C.

c. Write your function rule using function notation.

d. How much would it cost to buy 8 pizzas and have the pizzas delivered?

e. If you have exactly $75 to spend in this situation, how many pizzas can you order?



3. Compare the two pizza problems. a. How are the pizza situations similar and how are they different?

Similar Different

b. How are the tables of data for the pizza problems similar and how are they different?

Similar Different

c. How are the graphs of the data similar and how are they different?

Similar Different

d. How are the function rules similar and how are they different?

Similar Different

e. What is the domain for each situation? In other words, what kind of numbers make sense as inputs for the function rules if you allow only numbers that make sense in the context of the situation?

Pizzas without delivery Pizzas with delivery charge



4. For which situation can you just multiply the input value by a constant to get an output value?

5. Answer these questions about pizzas without delivery.

a. When you double the number of pizzas you order, what happens to the cost of the order? Give specific examples.

b. When you triple the number of pizzas you order, what happens to the cost of the order? Give specific examples.

c. How does the number of pizzas you can order for $42 compare to the number of pizzas you can order for $14?

6. Answer these questions about pizzas with delivery.

a. When you double the number of pizzas you order, does the cost of the order double? Give specific examples.

b. When the cost of the order is cut in half, is the number of pizzas you order also cut in half? Give specific examples.

7. Which pizza scenario represents a proportional linear relationship? Explain.

8. What are some characteristics of the graphs of proportional linear relationships? Of non-‐proportional linear relationships?

9. Use your rule to answer the following questions.

a. If you have a pizza delivered, how much will it cost to order 5 toppings?

b. If your pizza is delivered, how many toppings can you order for $60?



LESSON 6: REVIEW ONLINE ASSESSMENT

Today you will review the online assessment.

Problems we did well on: Skills and/or concepts that are addressed in these problems:

Problems we did not do well on: Skills and/or concepts that are addressed in these problems:

Addressing areas of incomplete understanding

Use this page and notebook paper to take notes and re-‐work particular online assessment problems that your class identifies.

Problem #_____ Work for problem:





LESSON 6: HOMEWORK Notes or additional instructions based on whole-‐class discussion of homework assignment:

Next class period, you will take an end-‐of-‐unit assessment. One good study skill to prepare for tests is to review the important skills and ideas you have learned. Use this list to help you review these skills and concepts, especially by reviewing related course materials.

Important skills and ideas you have learned so far in the unit Introduction to functions and equations:

• Identify, describe, extend, analyze, represent, and generalize patterns and mathematical relationships in words, and with tables, graphs, and algebraic rules

• Solve problems and model real-‐world situations using patterns and mathematical relationships

• Make connections among different representations of mathematical relationships, using verbal descriptions, tables, graphs, and algebraic rules

• Analyze and create equivalent algebraic expressions and rules

• Distinguish between discrete and continuous data

• Scale graph axes

• Recognize functional relationships

• Identify independent and dependent variables in functional relationships

• Represent functions using words, tables, graphs, and symbols, including function notation

• Formulate equations that arise from functions and solve these equations by inspection or “undoing”

• Recognize proportional and non-‐proportional situations represented by linear functions

Homework Assignment

Part I: Study for the end-‐of-‐unit assessment by reviewing the key ideas listed above.

Part II: Complete the online More practice for this topic. Note the skills and ideas for which you need more review, and refer back to related activities and animations to help you study.

Part III: Complete Lesson 6: Staying Sharp




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1. Valerie is planning a birthday party for her best friend. She is trying to figure out how much food to buy for the party. Which statement makes more sense?

a. The number of people she invites to the party depends on the amount of food she buys.

b. The amount of food she buys depends on the number of people she invites to the party.

Answer with supporting explanation:

2. If you made a graph to represent the relationship you identified in question 1, what variable would you put on each axis? Explain your answer. Independent (x-‐axis) variable:

Dependent (y-‐axis) variable:

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3. Using n to represent “a number,” write an algebraic equation to represent the statement “19 more than 5 times a number is 244.”

4. Sofía believes that 7 is a value of x that makes the equation 3x + 4 = 25 true. Is she correct? Answer with justification:

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x 2 5 7 8 10 13 19 25 y 26 20 16 14 10 4 -‐8 -‐20

Minimum x-‐value:

Maximum x-‐value:


Minimum y-‐value:

Maximum y-‐value:


6. Amy is trying to graph the algebraic rule y = 10x + 50 with input values from 0 to 20. She chooses this scale:

Min. x: 0 Min. y: 0 Max. x: 20 Max. y: 250

x increment: 5 y increment: 1

Identify one problem with Amy’s scale and suggest one improvement you would make. (Hint: Consider what an input-‐output table for this rule might look like.)



Lesson 7: Checking for understanding

LESSON 7: OPENER Giant Julius Pizza sells medium pizzas for $9.50 and $1.25 per topping.

1. Fill in the table. Then graph the points.

2. Write a function rule relating the price of the pizza to the number of toppings.

3. Use your function rule to find

a. the cost for a pizza with 10 toppings b. the number of toppings on a pizza that costs $17

LESSON 7: END-OF-UNIT ASSESSMENT

Today you will take the end-‐of-‐unit assessment.

Number of toppings Price in dollars

0

1



LESSON 7: CONSOLIDATION ACTIVITY Mariah is trying different sidewalk designs for her garden. She wants one walkway with “wings” that run in opposite directions.

Mariah realizes that there may be a relationship between the number of tiles in a “wing” (the darker tiles) and the number of tiles in the entire sidewalk.

1. Draw the next two models in Mariah's concrete representation next to the models shown.

2. Make a table representing the relationship between the number of tiles in a “wing” and the number of tiles in the entire sidewalk. Explain how the pattern in your tabular representation relates to the concrete models.

3. Write an algebraic representation for the relationship between the number of tiles in a “wing” and the number of tiles in the entire sidewalk. Explain how your rule relates to the concrete models.

Table

____________ ____________

4. Make a graphical representation of the number of tiles in a “wing” and the number of tiles in the entire sidewalk.

5. Explain how the graphical representation relates to the concrete representation, the tabular representation, and the algebraic representation.

6. Mariah claims that the relationship between the number of tiles in a “wing” and the number of tiles in the entire sidewalk is

a function. She is correct.

a. What is the domain of this function in the sidewalk context?

b. What is the domain of the mathematical function without the real-‐world context?



LESSON 7: HOMEWORK


The Flower Bed problem

Anthony Chen's family operates a full-‐service pool and landscaping business. Sometimes their clients want low decorative fences around their flower beds.

Anthony's father wants a quick way to estimate the amount of fencing needed. Mr. Chen asked Anthony to help him. Anthony realizes that this is just a perimeter question.

He starts his task by analyzing the relationship between the length of the side of a square flower bed and the perimeter of the flower bed. This will tell him the amount of fence needed to enclose the flower bed. Anthony realizes that lengths of sides of flower beds are not always whole numbers, but he decides to use square tiles to build models of flower beds of various sizes to help him find a pattern. In his models, 1 tile represents 1 square foot.

1. Draw the next two models in Anthony's concrete representation next to the models shown.

2. Make a table representing the relationship between the length of the side of a square flower bed and the perimeter of the flower bed. Explain how the pattern in your tabular representation relates to the concrete models.

Table

__________ ____________



3. Write an algebraic representation for the relationship between the length of the side of a square flower bed and the perimeter of the flower bed. Explain how your rule relates to the concrete models.

4. Make a graphical representation of the relationship between the length of the side of a square flower bed and the perimeter of the bed.

5. Explain how the graphical representation relates to the concrete representation, the tabular representation, and the algebraic representation.

6. Anthony claims that the relationship between the length of the side of a square flower bed and the perimeter of the flower

bed is a function. He is correct.

a. What is the domain of this function in the garden context?

b. What is the domain of the mathematical function without the real-‐world context?

c. If the domain of the mathematical function is the same as for the garden situation, explain why. If it is different than the garden situation, explain why.




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1. Use the rule y = 10x + 30 to complete the table below.

x y

2

80

11

140

31

2. In question 1, if y represents the total cost of a nature tour and x represents the number of people attending the nature tour, write a verbal description that could represent the relationship described by the algebraic rule

10 30y x= + .

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3. Complete the table for this magic number puzzle.

Specific number

Any number

Step 1. Pick a number. 11 n

Step 2. Multiply by 4.

Step 3. Add 9.

4. Joaquin goes through the steps of the magic number puzzle in question 3 and gets an ending number of 37. What was his starting number? Answer with supporting work:

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5. Henry collects rainwater for his garden. His tank begins with 30 gallons of water. Over 5 days, the tank collects 12 more gallons of water from the rain. Then in 1 day, he uses 18 gallons to water his garden. Then over 4 days, 7 gallons evaporate. Then over 3 days, the tank collects 11 more gallons from the rain.

State an appropriate scale to use to graph the volume of water in the tank over time.

Minimum x-‐value:

Maximum x-‐value:


Minimum y-‐value:

Maximum y-‐value:



x -‐10 -‐8 -‐6 -‐4 -‐2 0 2 4 6 8 10 y 95 59 31 11 -‐1 -‐5 -‐1 11 31 59 95

Minimum x-‐value:

Maximum x-‐value:


Minimum y-‐value:

Maximum y-‐value:




Lesson 8*: Using functions to solve problems

LESSON 8: OPENER

Consider the following statement: “The amount of money you earn at a summer job depends on (is a function of) the number of hours you work.” Using the terms provided, complete the statements in questions 1-‐4 to describe this situation.

variable first variable second variable independent variable

dependent variable

1. The amount of money you earn is a ______________________________because it can change.

2. The number of hours you work is a _______________________________because it can change.

3. The number of hours you work is the ________________________________________________.

4. The amount of money you earn is the ________________________________________________ because it is

calculated based on the number of hours you work.


A soccer team plans a special fund-‐raising project: Taking orders for roses and delivering them on Valentine’s Day. The team finds two flower shops willing to work with it: Roses-‐R-‐Red and Flower Power.

• Roses-‐R-‐Red sells roses for a service charge of $20 plus 75¢ per rose.

• Flower Power sells its roses for a service charge of $60 plus 50¢ per rose.

1. Analyze each flower shop’s offer.

Roses-‐R-‐Red Flower Power

a. What is the service charge for each company?

b. How much does each company charge per rose?

c. How much would each company charge for 3 roses?

2. Complete the following sentences describing the relationship between the variables to consider in the flower shop offers.

cost number unique

a. The of the roses depends on the of roses in the order. Each number of roses

gives a cost because each different number of roses gives a different cost. In other words,

the of the roses is a function of the of roses.

b. The of roses is the independent variable.

c. The of the roses is the dependent variable.



To complete the Roses Project, you will work with your partner to analyze the two offers and determine which flower shop offers the best deal for the soccer team. Your analysis must include representations of each shop’s offers algebraically, numerically, and graphically, as well as comparisons of the prices of the two companies, specifying the advantages and disadvantages of choosing either company. You and your partner will then make a final recommendation based on your analysis.

Now, work with your partner to complete the Roses Project Worksheet to aid in your analysis.

Roses Project Worksheet

3. Complete tables and graphs of the data for the costs of ordering flowers from each of the flower shops.

a. Complete the table to calculate the costs for different numbers of roses from Roses-‐R-‐Red, and develop an expression to show the general process for finding the cost.

Cost of Roses from Roses-‐R-‐Red

Number ordered Process Cost in dollars

12

24

36

x y

b. Why can we say that the cost of the roses from Roses-‐R-‐Red is a function of the number of roses ordered? In other words, why is the relationship between number and cost a function?

c. Complete the table to calculate the costs for different numbers of roses from Flower Power, and develop an expression to show the general process for finding the cost.

Cost of Roses from Flower Power

Number ordered Process Cost in dollars

12

24

36

x y



d. Why can we say that the cost of roses from Flower Power is a function of the number of roses ordered? Why is the relationship between number and cost a function?

e. Make graphs to show the relationship between the number of roses ordered and the cost of the roses for each flower shop.

Cost of Roses from Roses-‐R-‐Red

Cost of Roses from Flower Power

4. Complete the following sentences about the graphs you created in question 3. Refer to the vocabulary list to fill in parts d-‐g

rising falling lower higher increases decreases

a. The Roses-‐R-‐Red line intersects the y-‐axis at the point _____________.

b. The Flower Power line intersects the y-‐axis at the point __________________________.

c. These data points on the y-‐axis represent the cost of the __________________________ for each flower company.

d. For both graphs, the plotted points fall in a line that is __________________________ from left to right.

e. The shapes of both graphs show a steady increase in the cost of the roses, as the number of roses ordered

__________________________.

f. The Roses-‐R-‐Red line starts__________________________ but is steeper.

g. Compared with the Flower Power graph, this shape means that the service charge for the Roses-‐R-‐Red flowers is

__________________________ but the charge per rose is __________________________.



5. Create a single table comparing the Roses-‐R-‐Red and Flower Power data. Be sure to use a range of values so that you can compare the costs for ordering small and large numbers of roses.

Number ordered Roses-‐R-‐Red cost ($) Flower Power cost ($)

Now use your table to create a graph that shows the costs for each company. Plot both sets of points and draw a line through each set of plotted points on your graph. Be sure to label the lines for each company.

6. At this point, you have enough information to choose between the flower shops. Which flower company should the soccer team use? Why?




Complete the Rose Project and be prepared to share your solutions with the class.

1. Use the Roses-‐R-‐Red data to answer the following questions.

a. What is the function rule that expresses the total cost of the flowers as a function of the number of flowers ordered?

b. How much will 30 roses from Roses-‐R-‐Red cost?

c. How many roses can the soccer team order from Roses-‐R-‐Red for $100?

2. Use the Flower Power data to answer the following questions.

a. What is the function rule that expresses the total cost of the flowers as a function of the number of flowers ordered?

b. How much will 30 roses from Flower Power cost?

c. How many roses can the soccer team order from Flower Power for $100?



3. Consider the graphs of the cost of roses at the two companies.

a. Complete the following statements to compare the graphs. Use the answer choices provided.

The Roses-‐R-‐Red line intersects the y-‐axis at _________.

The Flower Power line intersects the y-‐axis at _____________.

These data points on the y-‐axis represent the cost of ______________________ for each flower company.

The characteristics of these two graphs tell you that these relationships _______________ proportional.

b. Complete the following statements to further compare the graphs. Use the answer choices provided.

higher lower increases decreases rising falling

On both graphs, the plotted point falls in a line that is ______________ from left to right.

The shapes of both graphs show a steady increase in the cost of the roses, as the number of roses ordered

_____________.

The Roses-‐R-‐Red line starts ______________ but is steeper.

Compared with the Flower Power graph, this shape means the service charge for the Roses-‐R-‐Red flowers is

____________ but the charge per rose is ______________.

(0,60) (0,20) a rose the service charge are are not



LESSON 8: HOMEWORK


1. For each of the problems below, identify the independent and dependent variable and write an algebraic rule to represent the relationship between the variables.

a.

Number of CDs, n

Cost (in dollars), c

1 15 2 30 3 45 4 60 5 75

Independent variable:

Dependent variable:

Algebraic rule:

Function rule:

b. Chris works at Fran’s Frozen Yogurt Parlor. He earns $7.75 per hour. The amount of money that Chris earns each week depends on the number of hours that he works.

Independent variable

Dependent variable:

Algebraic rule:

Function rule:

c.

2. On the next page, functions are represented in different ways (A -‐ L). Compare and contrast the function rules, tables, graphs, and situations. Sort the representations into six pairs that show representations of the same functional relationship.

Write your answers for matching pairs of functional relationships in the first column in the table below. In the second column of the table, provide a convincing explanation for why the two representations constitute a match.

Matching Pairs Explanation

Independent variable:

Dependent variable:

Algebraic rule:

Function rule:



A. y = 2x – 1

G. y = 1.5x

B.

H.

C. The plant was growing at a rate of 1.5 inches per week.

I.

x y

-‐2 -‐5

-‐1 -‐3

0 -‐1

1 1

2 3

3 5

D.

x y

-‐2 1.5

-‐1 1.5

0 1.5

1 1.5

2 1.5

3 1.5

J. The Math Club found a company that will sell it t-‐shirts for $5.00 each, but there is a set-‐up fee of $50.

E.

x y

0 50

1 55

2 60

3 65

4 70

5 75

K.

x y

0 0

1 20

2 40

3 60

4 80

5 100

F.

y = 2x + 1 L.

y = 1.5



Use the function rules from the Roses Project to answer the following questions. In each rule, x represents the number of roses ordered and f(x) represents the cost of the order. Show your work for each question on your own paper.

Roses-‐R-‐Red Flower Power

f(x) = 20 + 0.75x f(x) = 60 + 0.50x

3. If the soccer team orders 120 roses from Roses-‐R-‐Red, they will pay $ .

4. If the soccer team orders 120 roses from Flower Power, they will pay $ .

5. From Roses-‐R-‐Red, the soccer team can get roses for $95.

6. From Flower Power, the soccer team can get roses for $95.




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2. Use the graph and/or your rule for question 1 to complete the table below.

x y

-‐4

0

16

5

60

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3. To calculate a basketball team’s score in a game, you can use the equation 3x + 2y + z = p, where x represents the number of three-‐point baskets, y represents the number of two-‐point baskets, z represents the number of free throws (one-‐point baskets) and p represents the team’s total points. How many points did the team score if x = 8, y = 27, and z = 17?


4. Use the equation in question 3 to figure out the number of two-‐point baskets a team scored if the team made 7 three-‐point baskets and 11 free throws, and scored a total of 84 points.


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5. Find an appropriate scale to use to graph the algebraic rule y = 3x + 11. Consider input values from -‐5 to 5.

(Hint: Completing the input-‐output table may help.)

Minimum x-‐value:

Maximum x-‐value:


Minimum y-‐value:

Maximum y-‐value:


y = 3x + 11 x y

-‐5

-‐4

-‐3

-‐2

-‐1

0

1

2

3

4

5

-4 -2 2 4 6 8 10 12 14 16 18 20 22 24 26 28-1

1

2

3

4

5

6

7

8

9

x

y

working(with(functions(and(equations working with

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