chapter 1:foundations for functions algebra ii. table of contents 1.1 – sets of numbers 1.1 1.2...

Chapter 1:Foundations for Functions

Algebra II

Table of Contents

• 1.1 – Sets of Numbers• 1.2 – Properties of Real Numbers• 1.3 – Square Roots• 1.4 - Simplifying Algebraic Expressions• 1.5 - Properties of Exponents

1.1 - Sets of Numbers

Algebra II

Algebra 2 (Bell work)

1. A set is a collection of items called elements.

2. A subset is a set whose elements belong to another set.

3. The empty set, denoted , is a set containing no elements.

1-1

Copy the following definitions down

1-1

Order the numbers from least to greatest.

Write each number as a decimal to make it easier to compare them.

≈ 3.14

Consider the numbers

The numbers in order from least to greatest are

1-1 Example 1 Ordering and Classifying Numbers

Numbers Real Rational Integer Whole Natural Irrational

2.3

2.7652

Consider the numbers

Classify each number by the subsets of the real numbers to which it belongs.

1-1

Math Humor

• Q: Why do the other numbers refuse to take √2, √3, √5 seriously?

• A: They are completely irrational

Classify each number by the subsets of the real numbers to which it belongs.

Consider the numbers –2, , –0.321, and .

Numbers Real Rational Integer Whole Natural Irrational

–2

–0.321

1-1

You can also use roster notation, in which the elements in a set are listed between braces, { }.

Words Roster Notation

The set of billiard balls is numbered 1 through 15.

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

A finite set has a definite, or finite, number of elements.

An infinite set has an unlimited, or infinite number of elements.

1-1

1-1In interval notation, use [ ] to include an

endpoint. Use ( ) to exclude an endpoint

Pg. 8 Do Not Copy

Use interval notation to represent the set of numbers.

7 < x ≤ 12

(7, 12]

There are two intervals graphed on the number line.

[–6, –4] or (5, ∞)

–6 –4 –2 0 2 4 6

Use interval notation to represent the set of numbers.

1-1 Example 2 Interval Notation

Use interval notation to represent each set of numbers.

a.

(–∞, –1]

b. x ≤ 2 or 3 < x ≤ 11

(–∞, 2] or (3, 11]

-4 -3 -2 -1 0 1 2 3 4

1-1

The set of all numbers x such that x has the given properties

{x | 8 < x ≤ 15 and x N}

Read the above as “the set of all numbers x such that x is greater than 8 and less than or equal to 15 and x is a natural number.”

The symbol means “is an element of.”

So x N is read “x is an element of the set of natural numbers,” or “x is a natural number.”

Helpful Hint

1-1 Day 2Algebra 2 (bell work)

Some representations of the same sets of real numbers are shown.

1-1

Rewrite each set in the indicated notation.

A. {x | x > –5.5, x Z }; words

integers greater than 5.5

B. positive multiples of 10; roster notation

{10, 20, 30, …}

{x | x ≤ –2}

-4 -3 -2 -1 0 1 2 3 4; set-builder notationC.

1-1 Example 3 Translating Between Methods of Set Notation

Rewrite each set in the indicated notation.

a. {2, 4, 6, 8}; words

b. {x | 2 < x < 8 and x N}; roster notation

c. [99, ∞}; set-builder notation

even numbers between 1 and 9

{3, 4, 5, 6, 7}

{x | x ≥ 99}

The order of the elements is not important.

1-1

HW pg. 10• 1.1

– Day 1: 3, 5-7, 15-17, 46, 47, 53-56, 75– Day 2: 8-11, 18-21, 31-35, 44

– HW Guidelines or ½ off– Always staple Day 1&2 Together– Put assignment in planner

1.2 - Properties of Real Numbers

Algebra II

Bell work (Algebra II)

Write down the following properties and leave two lines below each for notes

1. Additive Identity Property2. Multiplicative Identity Property3. Additive Inverse Property4. Multiplicative Inverse Property5. Closure Property6. Commutative Property7. Associative Property8. Distributive Property

1-2

For all real numbers n,

WORDSAdditive Identity PropertyThe sum of a number and 0, the additive identity, is the original number.

NUMBERS 3 + 0 = 3

ALGEBRA n + 0 = 0 + n = n

Properties Real Numbers Identities and Inverses

1-2

For all real numbers n,

WORDS

Multiplicative Identity PropertyThe product of a number and 1, the multiplicative identity, is the original number.

NUMBERS

ALGEBRA n 1 = 1 n = n

Properties Real Numbers Identities and Inverses

1-2

For all real numbers n,

WORDSAdditive Inverse PropertyThe sum of a number and its opposite, or additive inverse, is 0.

NUMBERS 5 + (–5) = 0

ALGEBRA n + (–n) = 0

Properties Real Numbers Identities and Inverses

1-2

For all real numbers n,

WORDSMultiplicative Inverse PropertyThe product of a nonzero number and its reciprocal, or multiplicative inverse, is 1.

NUMBERS

ALGEBRA

Properties Real Numbers Identities and Inverses

1-2

Find the additive and multiplicative inverse of each number.

12

additive inverse: –12

Check –12 + 12 = 0

multiplicative inverse:

Check

additive inverse:

multiplicative inverse:

1-2 Example 1 Finding Inverses

500

Check 500 + (–500) = 0

additive inverse: –500

multiplicative inverse:

Check

–0.01

additive inverse: 0.01

multiplicative inverse: –100

1-2

For all real numbers a and b,

WORDSClosure PropertyThe sum or product of any two real numbers is a real number

NUMBERS2 + 3 = 52(3) = 6

ALGEBRAa + b

ab

Properties Real Numbers Addition and Multiplication

1-2

For all real numbers a and b,

WORDSCommutative PropertyYou can add or multiply real numbers in any order without changing the result.

NUMBERS7 + 11 = 11 + 7

7(11) = 11(7)

ALGEBRAa + b = b + a

ab = ba

Properties Real Numbers Addition and Multiplication

1-2

For all real numbers a and b,

WORDS

Associative PropertyThe sum or product of three or more real numbers is the same regardless of the way the numbers are grouped.

NUMBERS(5 + 3) + 7 = 5 + (3 + 7)

(5 3)7 = 5(3 7)

ALGEBRA(a + b) + c = a + (b + c)

(ab)c = a(bc)

Properties Real Numbers Addition and Multiplication

1-2

For all real numbers a and b,

WORDS

Distributive PropertyWhen you multiply a sum by a number, the result is the same whether you add and then multiply or whether you multiply each term by the number and add the products.

NUMBERS5(2 + 8) = 5(2) + 5(8)(2 + 8)5 = (2)5 + (8)5

ALGEBRAa(b + c) = ab + ac (b + c)a = ba + ca

Properties Real Numbers Addition and Multiplication

1-2

Identify the property demonstrated by each question.

A. 2 3.9 = 3.9 2

Commutative Property of Multiplication Associative Property of Addition

1-2 Example 2 Identifying Properties of Real Numbers

3(a + 1) = 3a + 3

Always true by the Distributive Property.

always true

a + (–a) = b + (–b)

Always true by the Additive Inverse Property.

1-2

a b = a, where b = 3

Classifying each statement as sometimes, always, or never true. Give examples or properties to support your answers.

sometimes true

true example: 0 3 = 0

false example: 1 3 ≠ 1

Example 4 Classifying Statements as Sometimes, Always or Never True

HW pg. 17

• 1.2– 15-19 (Odd), 21-23, 26-34, 51, 52, 63-65

– HW Guidelines or ½ off– Always staple Day 1&2 Together– Put assignment in planner

1.3 - Square Roots

Algebra II

Bell work (Algebra II)

1. Put the following definitions in your notes

1. = radical symbol.

2. The number or expression under the radical symbol is called the radicand.

3. The radical symbol indicates only the positive square root of a number, called the principal root.

1-3

The side length of a square is the square root of its area.

To indicate both the positive and negative square roots of a number, use the plus or minus sign (±).

or –5

1-3

Pg. 22

1-3

A.

B.

C.

D.

1-3

Simplify each expression.

Example 2 Estimating Square Roots

A.

B.

C.

D.

1-3Simplify each expression.

Simplify by rationalizing the denominator.

1-3 Example 3 Rationalizing the Denominator Day 2

1-3 Simplify by rationalizing the denominator.

Square roots that have the same radicand are called like radical terms.1-3

To add or subtract square roots, first simplify each radical term and then combine like radical terms by adding or subtracting their coefficients.

Math Joke

• Teacher: Lets find the square root of 1 million

• Student: Don’t you think that’s a bit too radical?

1-3

1-3 Example 4 Adding and Subtracting Square Roots

1-3

HW pg.24

• 1.3– Day 1: 6-9, 22-29, 49-53 (Odd), 78-81– Day 2: 10-17, 31-41 (Odd), 42, 46, 57, 62-65– Ch: 67

– HW Guidelines or ½ off– Always staple Day 1&2 Together– Put assignment in planner

1.4 - Simplifying Algebraic Expressions

Algebra II

There are three different ways in which a basketball player can score points during a game.

There are 1-point free throws, 2-point field goals, and 3-point field goals.

An algebraic expression can represent the total points scored during a game.

1-4 Algebra II (Bell work)

Just Read

Action Operation Possible Context Clues

Combine Add How many total?

Combine equal groups

Multiply How many altogether?

Separate SubtractHow many more? How many remaining?

Separate into equal groups Divide How many in each group?

1-4 Don’t Copy

Write an algebraic expression to represent each situation.

A. the number of apples in a basket of 12 after n more are added

B. the number of days it will take to walk 100 miles if you walk M miles per day

Add n to 12.

Divide 100 by M.

12 + n

1-4 Example 1 Translating Words into Algebraic Expressions

a. Lucy’s age y years after her 18th birthday

Write an algebraic expression to represent each situation.

18 + y

3600h

Add y to 18.

Multiply h by 3600.

b. the number of seconds in h hours

1-4

Order of Operations

1. Parentheses and grouping symbols.2. Exponents.3. Multiply and Divide from left to right.4. Add and Subtract from left to right.

Evaluate the expression for the given values of the variables.

2(5) – (5)(2) + 4(2)

10 – 10 + 80 + 8

8

2x – xy + 4y for x = 5 and y = 2

1-4

PEMDAS Please Excuse My Dear Aunt Sally

Example 2

Math Joke

• Surgeon: Nurse! I have so many patients! Who do I work on first?

• Nurse: Simple, use order of operations

1-4

q2 + 4qr – r2 for r = 3 and q = 7

49 + 4(7)(3) – 9

49 + 84 – 9

124

(7)2 + 4(7)(3) – (3)2

4(5) – 2(25) + 3(5)

20 – 50 + 15

–15

Evaluate x2y – xy2 + 3y for x = 2 and y = 5.

(2)2(5) – (2)(5)2 + 3(5)

1-4

Simplify the expression.

3x2 + 2x – 3y + 4x2

3x2 + 2x – 3y + 4x2

7x2 + 2x – 3y

1-4

Example 3 Simplifying Expressions

Simplify the expression.

j(6k2 + 7k) + 9jk2 – 7jk

6jk2 + 7jk + 9jk2 – 7jk

15jk2

–6x + 3xy – 9y – 11xy

–3(2x – xy + 3y) – 11xy.

–6x – 8xy – 9y

1-4

Apples cost $2 per pound, and grapes cost $3 per pound.

Write and simplify an expression for the total cost if you buy 10 lb of apples and grapes combined.

2A + 3(10 – A)

Let A be the number of pounds of apples.

= 30 – A

Then 10 – A is the number of pounds of grapes.

= 2A + 30 – 3A

What is the total cost if 2 lb of the 10 lb are apples?

Evaluate 30 – A for A = 2.30 – (2) = 28

The total cost is $28 if 2 lb are apples.

1-4Example 4

A travel agent is selling 100 discount packages. He makes $50 for each Hawaii package and $80 for each Cancún package.

Write an expression to represent the total the agent will make selling a combination of the two packages.

Let h be the number of Hawaii packages.

50h + 80(100 –h)

= 8000 – 30h

Then 100 – h is the remaining Cancun packages.

= 50h + 8000 –80h

How much will he make if he sells 28 Hawaii packages?

8000–30(28) = 8000–840

Evaluate 8000 –30h for h = 28.

He will make $7160. = 7160

1-4

HW pg. 31

• 1.4– 9-21, 27, 47-53 (Odd)– Challenge: 26, 30

– HW Guidelines or ½ off– Always staple Day 1&2 Together– Put assignment in planner

1.5- Properties of Exponents

Algebra II

Bell work (Algebra II)1. Copy the information below

Squared means to the 2nd power x2

Cubed means to the third power, x3

1-5

In an expression of the form an, a is the base, n is the exponent, and the quantity an is called a power.

Write the expression in expanded form.

(5z)2

(5z)2

(5z)(5z)

1-5

Example 1

Write the expression in expanded form.

–s4

–s4

–(s s s s) = –s s s s 3h3(k + 3)2

3h3(k + 3)2

3(h)(h)(h) (k + 3)(k + 3)

(2a)5

(2a)5

(2a)(2a)(2a)(2a)(2a)

3b4

3 b b b b

3b4

1-5

Write the expression in expanded form.–(2x – 1)3y2

–(2x – 1)3y2

–(2x – 1)(2x – 1)(2x – 1) y y

1-5

Math Joke

• Q: Why won’t Goldilocks drink a glass of water with 8 pieces of ice in it?

• A: Its’ too cubed

1-5

Simplify the expression.3–2

32

3 3 = 9

(–5)–5

1 1 1 1 1 1

5 5 5 5 5 3125

51

5

æ-çè

ö÷ø

1-5Example 2

1-5

Simplify the expression. Assume all variables are nonzero.

3z7(–4z2)

3 (–4) z7 z2

–12z7 + 2

–12z9

(yz3 – 5)3 = (yz–2)3

y3(z–2)3

y3z(–2)(3)

1-5

(5x6)3

53(x6)3

125x(6)(3)

125x18

Simplify the expression. Assume all variables are nonzero.

(–2a3b)–3

1-5

Simplify the expression. Write the answer in scientific notation.

3.0 10–11

1-5 Day 2

Example 4

Simplify the expression. Write the answer in scientific notation.

22.1 1011

(2.6)(8.5) (104)(107)

(2.6 104)(8.5 107)

2.21 1012

0.25 10–3

2.5 10–4

1-5

Light travels through space at a speed of about 3 105 kilometers per second.

Pluto is approximately 5.9 1012 m from the Sun.

How many minutes, on average, does it take light to travel from the Sun to Pluto?

1-5

Example 5

Skip 2014-2015

First, convert the speed of light from

1-5

It takes light approximately 328 minutes to travel from the Sun to Pluto.

1-5

HW pg. 38

• 1.5-– 3-9 (Odd), 10-19, 21, 37, 43, 44, 74, – Challenge: 55,