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Algebra 1

Unit 2: Functions and Patterns

Note Packet

Date Topic/Assignment Page Due Date Score (For Teacher Use

Only) Warm-Ups

2-A Intro to Functions

2-B Tile Pattern Challenge

2-C Tile Pattern Challenge Continued

2-D Functions and Function Notation

2-E More on Functions

2-F Function Matrix Activity

2-G Types of Functions and Their Patterns

2-H Arithmetic Sequences

2-I Arithmetic Sequences Part 2

2-J Geometric Sequences (2 days of notes, 1 HW)

Check for Understanding

2-K Arithmetic an Geometric Sequences

2-L Review (2 Days)

Test

This packet will be turned in on the day of the test for 100 points. Whenever you’re absent, you can get these notes filled

out from a classmate or at my website www.skookummath.weebly.com. During the unit, I’ll check off homework each

day to keep track of who is doing their homework, but your homework assignments won’t be scored and entered into IC

until this packet is collected and graded at the end of the unit.

Name:

Period:

For Teacher Use

Packet Score:

2

Warm-Up Date:

Solve each equation below for the given variable. Record all work so that your teacher can follow your “legal

moves.”

a. −2x + 2 = − 8 b. 4x − 2 + x = 2x + 8 + 3x

c. 3y − 9 + y = 6 d. 9 − (2 − 3y) = 6 + 2y − (5 + y)

Warm-Up Date:

Use the pattern you discovered last unit to complete each diamond problem. Some of these may be

challenging!

a. b. c.

Warm-Up Date:

Use the Distributive Property to complete and simplify each expression below.

a. 2(x + 5) = b. 3(2x + 1) =

c. −2(x + 3) = d. −3(2x − 5) =

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Warm-Up Date:

Simplify the following expressions by combining like terms.

a. x + 3x − 3 + 2x2 + 8 − 5x b. 2x + 4y2 − 6y2 − 9 + 1 − x + 3x

c. 2x2 + 30y − 3y2 + 4xy − 14 – x d. 20 + 3xy − 3xy + y2 + 10 − y2

Warm-Up Date:

Solve each of the following equations for x. Show your steps.

a. 3x − 2(5x + 3) = 14 − 2x b. 3(x + 1) − 8 = 14 − 2(3x − 4)

c. 2(𝑥 − 3) + 5𝑥 = 3𝑥 + 14 d. 5𝑥 + 3(𝑥 − 4) = 24 − 2(𝑥 + 3)

Warm-Up Date:

Compute without using a calculator.

a.−15 + 7 b. 8 − (−21) c. 6 (−8)

d. −9 + (−13) e. −50 – 30 f. 3 − (−9)

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Warm-Up Date:

1. Joe’s age is three times Aaron’s age and Aaron is six years older than Christina. If the sum of their ages is

149, what is Christina’s age? Joe’s age? Aaron’s age?

2. A box of fruit has three times as many nectarines as grapefruit. Together there are 36 pieces of fruit. How

many pieces of each type of fruit are there?

Warm-Up Date:

1. Write an inequality that represents the solutions on each number line below.

a. b.

c. d.

2. Graph the following inequalities on the number lines below.

a. 𝑥 ≤ 5 b. 𝑥 < −3 𝑜𝑟 𝑥 ≥ 2

c. −4 ≤ 𝑥 < 7

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Warm-Up Date:

Solve the following equations below for x. Show all steps.

a. 3(2𝑥 − 5) − 2𝑥 = 2(𝑥 + 7) + 5 b. −2(𝑥 − 3) + 4 = 4𝑥 − 14

Warm-Up Date:

Solve each of the following inequalities for the given variable. Represent your solutions on a number line.

a. −3(2𝑝 − 1) > −15 b. −7 < 2𝑘 + 3 ≤ 19

Warm-Up Date:

Evaluate each expression if r = −4, s = 5 , and t = 6.

a. r2 + 2s b. 𝑡−𝑟

𝑠

Warm-Up Date:

1. Complete each of the Diamond Problems below.

Warm-Up Date:

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Evaluate the expression (𝑎+𝑐)2−𝑎

𝑏+𝑐 when 𝑎 = −2 , 𝑏 = 6 and c= 8. Show all work

below.

A.

7

5 C.

19

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B. 5

7 D.

17

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Warm-Up Date:

1. Christian has twice as much money as Vanessa? Gonzo has ten more dollars than Vanessa. How much

money does each person have it the sum of their money is $90?

2. A board that is 100 centimeters long is cut into three pieces? The first piece and the second piece have

the same length. The third piece is four more than twice the length of the first piece. Find the length of

each piece.

Warm-Up Date:

Simplify the following expressions by combining like terms.

a. -15x + 3x + 9 - 2x2 +10 + 7x2 b. -4x +3y2 + 7y2 + 9x - 10 + 5x - 3

Warm-Up Date:

Solve the following equations for x. Show all steps that lead to your solution.

a. 3

8𝑥 −

1

12= −

3

4 b.

5

9−

5𝑥

6= −

2

3

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2-A: Intro to Functions

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2-B: Tile Pattern Challenge

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2-D: Functions and Function Notation

What is a function?

Opening Task: Think about it…

Is there another way you could represent these functions?

(Eureka Math Grade 9 – Functions)

Part 1: How can you determine whether a relation is a function?

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a) Table

x f(x)

-2 11

-1 7

0 3

1 -1

2 -5

3 -9

b) Set of Points

{(2

5, −1) , (−3.5, 6), (3, −10), (−2.1, 6), (−1,

1

5)}

c) Mapping

d) Graph

Give an example of a function and not a function for the following Choose: Choose: Table or Set of Coordinates Graph or Map

Part 3: The variables of a function

Domain: Range:

Determine the domain and range for the functions in part 1 a)-c) a) D: b) D: c) D: R: R: R:

Part 3: Function Notation

Ex 1. Given the coordinate: (-2, 5) Ex 3. 𝑓(𝑥) = −3𝑥2 + 1 𝑓(𝑥) = 𝑦 Write in function notation. 𝑓(3)= input output Ex 2. Given: 𝑔(5) = −2 𝑓(−.5) = Write as a coordinate.

Does this represent a function? Why or why not?

Does this represent a

function? Why or why

not?

Does this represent a

function? Why or why not?

Does this represent a

function? Why or why

not?

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2-E: More on Functions

Opening Task: Think about it… Swine flu is attacking Porkopolis. The function 𝑆(𝑡) = 9𝑡 − 4 determines how many people have swine where t = time in days and S = the number of people in thousands. a) What would reasonable values for t (input values)? b) Graph the function Make a table of 4 different values. t 𝑆(𝑡) c) Find 𝑆(4). d) Find t when S(t) = 23. What does 𝑆(4) mean? What does S(t) = 23 mean?

Part 4: Discrete versus continuous functions.

Discrete Functions: Domain and ranges of discrete functions:

Continuous Functions: Domain and ranges of continuous functions:

Ex1: Graphs

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Ex2: Real World Situations – For each of the “context” situations 1-3 described in the opening task from last class: 1) Tell whether it describes a discrete or continuous situation 2) Tell which variable represents the domain and which variable represents the range. 1) Discrete or Continuous?

Why? Description of the domain: Description of the range:

2) Discrete or Continuous? Why?

Description of the domain: Description of the range:

3) Discrete or Continuous? Why?

Description of the domain: Description of the range:

Bringing it all together

Domain: Range:

Domain: Range:

Function?

Discrete/Continuous?

f(6) = __________

If f(x) = 12, then x = ____

Function?

Discrete/Continuous?

f(0) = __________

If f(x) = 1, then x =

_________

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Give a real world example of a situation modeled by a continuous function.

If 𝑣(𝑡) = 3𝑥2 − 2𝑥 + 1, find the value of 𝑣(5).

If ℎ(𝑥) =𝑥2−9

𝑥−5, find the value of ℎ(5).

Function?

Discrete/Continuous?

f(-2) = __________

If f(x) = -2, then x = ____

Domain:

Range:

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Equation Word Rule Chart of Values Ordered Pairs Graph

f(x) = x2 +3

Domain:

Range:

To find the

output, you

square the

input and add 3.

Domain:

Range:

Domain:

Range:

x f(x)

7 3

-2 -6

1 -3

3 -1

-4 -8

0 -4

page 1

f(x)

x

f(x)

x

f(x)

x

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Equation Word Rule Chart of Values Ordered Pairs Graph

Domain:

Range:

(-1,1/4) (2,16)

(1,4)

(-2, 1/16)

(3,64)

Domain:

Range:

Domain:

Range:

To find the

output, square

the input and

subtract 1

page 2

f(x)

x

f(x)

x

f(x)

x

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Graphically

2-G: Types of Functions and Their Patterns

Linear Functions (arithmetic pattern)

Algebraically Numerically

I noticed… I wonder…

Summary of important things I heard from my classmates…

Exponential Functions (geometric pattern)

Algebraically Numerically

I noticed… I wonder…

Summary of important things I heard from my classmates…

𝑦 = 𝑓(𝑥) =1

2𝑥 − 3

x f(x)

-2 -4

-1 -3.5

0 -3

1 -2.5

2 -2

3 -1.5

Graphically

𝑦 = 𝑓(𝑥) = 1 ∗ 2𝑥 x f(x)

-2 ¼

-1 ½

0 1

1 2

2 4

3 8

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Quadratic Functions

Algebraically Numerically

I noticed… I wonder…

Summary of important things I heard from my classmates…

Identifying types of functions

Type/Eqaution Pattern “Look fors” Vocabulary

Linear –

_____________________

x 1 2 3 4 5

y -5 -1 3 7 11

Constant 1st Difference

Exponential –

_____________________

x -1 0 1 2 3

y 2/5 2 10 50 250

Common Ratio

Quadratic –

______________________

x 0 1 2 3 4

y -3 -1 3 9 17

Constant 2nd Difference

𝑦 = 𝑓(𝑥) = 𝑥2 + 1 x f(x)

-2 5

-1 2

0 1

1 2

2 5

3 10

Graphically

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Practice: Identify the type function from the pattern given or described. Justify your answer using correct

mathematical language.

1. 2. 3.

4. Bacteria can multiply at an alarming rate when each bacteria splits into two new cells, thus doubling. For

example, if we start with only one bacteria which can double every hour, by the end of one day we will have

over 16 million bacteria.

5. I get a $100 iTunes gift card for my birthday and then start buying $1 songs. What type of function describes the

amount of money, m, I have left on the card after buying s songs?

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2-H: Arithmetic Sequences

Today’s Goal: Identify arithmetic sequences and write recursive and explicit formulas for arithmetic

sequences.

Warm-up: Find the next 3 terms in the patterns below

So what is a sequence?!?

Try this… Find the missing terms in the sequences using correct notation.

a. a1, a2, a3,…, a____, a56, a____,…, an-1, a____, an+1, …

b. f(___), f(2), f(3), … , f(98), f(___), f(___), … , f(_____), f(_____), f(n+1), …

What is an arithmetic sequence?

Describe how you know if a sequence is arithmetic in your own words:

Let’s look back at the warm-up… Which of the following are arithmetic sequences? How do you know?

A sequence is a list or an ordered arrangement of numbers, figures, or objects. The members, which are also

elements, are called the terms of sequences. A general sequence can be written using the following 2 different

notations:

Subscript Notation: a1, a2, a3, a4, a5, a6, a7, a8,…

Function Notation: f(1), f(2), f(3), f(4), f(5), …

where a1 or f(1) is the first term, a2 or f(2) is the second term, and so on. The nth term is denoted with an or f(n)

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How do you write formulas of arithmetic sequences?

Recursive Formulas: use ___________________ terms to find the _________________ term

Helpful hint:

Ex1: Write the recursive formula for the following sequence; 14, 17, 20, 23, …

Verbal: 1) __________=______ 2) ____________ = ______________ + ____________

Subscript: 1) ______= ______, 2) _______ = _______ + _______

Functions: 1) ______= ______, 2) _______= _______ + _______

Explicit

Formulas:

Do not

rely on previous terms to find the nth term.

Arithmetic sequences have a specific

structure so writing a formula is easy!

Slope – Intercept Form y = b

+ m x

Explicit Forms of Arithmetic Seq. an = a1 + d (n – 1)

f(n) = f(1) + d (n – 1)

Ex2: Write an explicit formula in both subscript and function notation for the following arithmetic sequence

76, 72, 68, 64, ….

Subscript Notation:

Function Notation:

Verbal Subscript Function Notation

Previous term ________ f(n-1)

______________ an __________

Next term ________ __________

IMPORTANT IDEA: A recursive formula always consists of 2 parts.

1) the value of the 1st term

2) the formula to find the current term based on the previous term.

Connection Time! This structure is just like …

Output Input 0-term vs. 1st term Slope vs. common dif.

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2-I: Arithmetic Sequences Part 2

Warm-up:

1. Write a recursive formula in subscript notation for the following sequence.

2

5 ,

4

5, 1

1

5, 1

3

5, …

2. Write an explicit formula in function notation for the following sequence.

76.3, 75.5, 74.7, 73.9, …

Lesson Activity: Card Sort

With a partner sort the cards into matching pairs.

You will have 20 minutes to complete this task – Good Luck!

While you are working on the card sort, think about the class discussion questions.

Class Discussion Questions:

1. Which pair of cards were easiest to match and why?

2. Which pair of cards were hardest to match and why?

3. How are arithmetic sequences similar to linear functions

4. How are arithmetic sequences different to linear functions

5. Is an arithmetic sequence a function?

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2-J: Geometric Sequences (2 days of notes, 1 HW)

Doctors need to know approximately how long medications stay in a person’s body. A half-life is the approximate time it takes for the body to remove 1/2 of the active ingredient in a medicine. Caffeine is in medications, foods, and energy supplements. The half-life of caffeine is about 4 hours. A student drinks an energy drink that contains 16 mg of caffeine.

a. Complete the table showing the amount of caffeine in the body over time

Time (hr.) 0 4 8

Caffeine in body (mg)

b. Create a graph showing the relationship between the time in hours and the amount of caffeine in the

body

What are geometric Sequences? Find the next 3 terms in the sequences below. Explain how you found these terms.

a. 48, 24, 12, 6, ______, ______, ______, … b. 7, 10, 15, 22, ______. ______, ______, …

c. 18, 14, 10, 6, ______, ______, ______, … d. 2, 6, 18, 54, ______, ______, ______, …

Which two sequences in number 2 are geometric sequences? How do you know?

c. Describe the domain and range of the

relationship.

d. Does this represent a continuous or

discrete situation? Explain.

e. What type of graph describes this

relationship?

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A geometric sequence is a sequence of numbers where each successive number is determined by multiplying by a constant value, called the common ratio.

Consider the sequence 2, 6, 18, 54, ______, ______, _______, …. a. Write the recursive form of the sequence: now = previous ______, starting at ____ b. Write the recursive form in sequence notation: an = an-1 _______, ______ = 2 c. Write the recursive form in function notation: f(n) = _________ _______, _______ = 2 Practice: Find the first four terms of this sequence: 𝑓(𝑛 + 1) = 𝑓(𝑛) · 2, 𝑓(1) = 9

Explict Form of Geometric Sequences. How would you find the explicit form? 2, 6, 18, 54, …. x 3 x 3 x 3 Term 1 = 2 Term 2 = 2 x 3 Term 2 is multiplied by one 3 Term 3 = 2 x 3 x 3 Term 3 is multiplied by two 3s Term 4 = 2 x 3 x 3 x 3 Term 4 is multiplied by three 3s … … Term n is multiplied by _________ 3s So, f(n) = 2•(3) _________ Does it work? Is f(1) = 2? f(2) = 6? f(3) = 18?

More Practice: Write each sequence below in recursive and explicit form. For each sequence, find the given term. a. 48, 24, 12, 6, … Find a15 b. -90, -30, -10, …. Find f(6) c. 6, -18, 54, -162, ….. Find f(10)

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2-K: Arithmetic and Geometric Sequences

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2-L: Review

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