1. Chapter 1:Foundations for Functions
Algebra II

2. 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

3. 1.1 - Sets of Numbers
Algebra II

4. A set is a collection of items called elements.
1-1
Algebra 2 (Bell work)
Copy the following definitions down
A set is a collection of items called elements.
A subset is a set whose elements belong to another set.
The empty set, denoted , is a set containing no elements.

5. 1-1

6. Order the numbers from least to greatest.
1-1
Example 1
Ordering and Classifying Numbers
Consider the numbers
Order the numbers from least to greatest.
Write each number as a decimal to make it easier to compare them.
 ≈ 3.14
The numbers in order from least to greatest are

7. 1-1
Consider the numbers
Classify each number by the subsets of the real numbers to which it belongs.
Numbers
Real
Rational
Integer
Whole
Natural
Irrational
2.3 
2.7652          

8. Math Humor
Q: Why do the other numbers refuse to take √2, √3, √5 seriously?
A: They are completely irrational

9. Consider the numbers –2, , –0.321, and .
1-1
Consider the numbers –2, , –0.321, and
Classify each number by the subsets of the real numbers to which it belongs.
Numbers
Real
Rational
Integer
Whole
Natural
Irrational –2 
–0.321           

10. 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.

11. 1-1
In interval notation, use [ ] to include an endpoint. Use ( ) to exclude an endpoint
Pg. 8
Do Not Copy

12. Use interval notation to represent the set of numbers.
1-1
Example 2
Interval Notation
Use interval notation to represent the set of numbers.
7 < x ≤ 12
(7, 12]
Use interval notation to represent the set of numbers.
– – –
There are two intervals graphed on the number line.
[–6, –4] or (5, ∞)

13. Use interval notation to represent each set of numbers.
1-1
Use interval notation to represent each set of numbers. a.
(–∞, –1]
b. x ≤ 2 or 3 < x ≤ 11
(–∞, 2] or (3, 11]

14. {x | 8 < x ≤ 15 and x  N} Read the above as
1-1
Algebra 2 (bell work)
Day 2
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

15. Some representations of the same sets of real numbers are shown.
1-1
Some representations of the same sets of real numbers are shown.

16. Rewrite each set in the indicated notation.
1-1
Example 3
Translating Between Methods of Set Notation
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, …} C.
; set-builder notation
{x | x ≤ –2}

17. Rewrite each set in the indicated notation.
1-1
Rewrite each set in the indicated notation.
a. {2, 4, 6, 8}; words
even numbers between 1 and 9
b. {x | 2 < x < 8 and x  N}; roster notation
{3, 4, 5, 6, 7}
The order of the elements is not important.
c. [99, ∞}; set-builder notation
{x | x ≥ 99}

18. 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

19. 1.2 - Properties of Real Numbers
Algebra II

20. 1-2
Bell work (Algebra II)
Write down the following properties and leave two lines below each for notes
Additive Identity Property
Multiplicative Identity Property
Additive Inverse Property
Multiplicative Inverse Property
Closure Property
Commutative Property
Associative Property
Distributive Property

21. 3 + 0 = 3 n + 0 = 0 + n = n For all real numbers n, WORDS
1-2
Properties Real Numbers
Identities and Inverses
For all real numbers n,
WORDS
Additive Identity Property
The sum of a number and 0, the additive identity, is the original number.
NUMBERS
3 + 0 = 3
ALGEBRA
n + 0 = 0 + n = n

22. n  1 = 1  n = n For all real numbers n, WORDS
1-2
Properties Real Numbers
Identities and Inverses
For all real numbers n,
WORDS
Multiplicative Identity Property
The product of a number and 1, the multiplicative identity, is the original number.
NUMBERS
ALGEBRA
n  1 = 1  n = n

23. 5 + (–5) = 0 n + (–n) = 0 For all real numbers n, WORDS
1-2
Properties Real Numbers
Identities and Inverses
For all real numbers n,
WORDS
Additive Inverse Property
The sum of a number and its opposite, or additive inverse, is 0.
NUMBERS
5 + (–5) = 0
ALGEBRA
n + (–n) = 0

24. For all real numbers n, WORDS Multiplicative Inverse Property
1-2
Properties Real Numbers
Identities and Inverses
For all real numbers n,
WORDS
Multiplicative Inverse Property
The product of a nonzero number and its reciprocal, or multiplicative inverse, is 1.
NUMBERS
ALGEBRA

25. 1-2
Example 1
Finding Inverses
Find the additive and multiplicative inverse of each number. 12
additive inverse: –12
additive inverse:
Check – = 0 
multiplicative inverse:
multiplicative inverse:
Check 

26. 1-2
500
–0.01
additive inverse: –500
additive inverse: 0.01
Check (–500) = 0 
multiplicative inverse: –100
multiplicative inverse:
Check 

27. 2 + 3 = 5 2(3) = 6 a + b   ab   For all real numbers a and b,
1-2
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
Closure Property
The sum or product of any two real numbers is a real number
NUMBERS
2 + 3 = 5
2(3) = 6
ALGEBRA
a + b  
ab  

28. 7 + 11 = 11 + 7 7(11) = 11(7) a + b = b + a ab = ba
1-2
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
Commutative Property
You can add or multiply real numbers in any order without changing the result.
NUMBERS =
7(11) = 11(7)
ALGEBRA
a + b = b + a
ab = ba

29. 1-2
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
Associative Property
The 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)

30. 5(2 + 8) = 5(2) + 5(8) (2 + 8)5 = (2)5 + (8)5 a(b + c) = ab + ac
1-2
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
Distributive Property
When 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.
NUMBERS
5(2 + 8) = 5(2) + 5(8)
(2 + 8)5 = (2)5 + (8)5
ALGEBRA
a(b + c) = ab + ac
(b + c)a = ba + ca

31. 1-2
Example 2
Identifying Properties of Real Numbers
Identify the property demonstrated by each question.
A. 2  3.9 = 3.9  2
Commutative Property of Multiplication
Associative Property of Addition

32. 1-2
Example 4
Classifying Statements as Sometimes, Always or Never True
Classifying each statement as sometimes, always, or never true. Give examples or properties to support your answers.
a  b = a, where b = 3
sometimes true
true example: 0  3 = 0
false example: 1  3 ≠ 1
3(a + 1) = 3a + 3
a + (–a) = b + (–b)
always true
Always true by the Distributive Property.
Always true by the Additive Inverse Property.

33. 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

34. 1.3 - Square Roots
Algebra II

35. Put the following definitions in your notes
1-3
Bell work (Algebra II)
Put the following definitions in your notes
= radical symbol.
The number or expression under the radical symbol is called the radicand.
The radical symbol indicates only the positive square root of a number, called the principal root.

36. or –5 The side length of a square is the square root of its area.
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

37. 1-3
Pg. 22

38. Simplify each expression.
1-3
Example 2
Estimating Square Roots
Simplify each expression. A. C. D. B.

39. Simplify each expression. A. C.
1-3
Simplify each expression. A. C. B. D.

40. Simplify by rationalizing the denominator.
1-3
Example 3
Rationalizing the Denominator
Day 2
Simplify by rationalizing the denominator.

41. Simplify by rationalizing the denominator.
1-3
Simplify by rationalizing the denominator.

42. 1-3
Square roots that have the same radicand are called like radical terms.
To add or subtract square roots, first simplify each radical term and then combine like radical terms by adding or subtracting their coefficients.

43. Math Joke Teacher: Lets find the square root of 1 million
1-3
Math Joke
Teacher: Lets find the square root of 1 million
Student: Don’t you think that’s a bit too radical?

44. 1-3
Example 4
Adding and Subtracting Square Roots

45. 1-3

46. HW pg.24
1.3
Day 1: 6-9, 22-29, (Odd), 78-81
Day 2: 10-17, (Odd), 42, 46, 57, 62-65
Ch: 67
HW Guidelines or ½ off
Always staple Day 1&2 Together
Put assignment in planner

47. 1.4 - Simplifying Algebraic Expressions
Algebra II

48. Just Read 1-4 Algebra II (Bell work)
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.

49. Possible Context Clues
1-4
Don’t Copy
Action
Operation
Possible Context Clues
Combine
Add
How many total?
Combine equal groups
Multiply
How many altogether?
Separate
Subtract
How many more? How many remaining?
Separate into equal groups
Divide
How many in each group?

50. Write an algebraic expression to represent each situation.
1-4
Example 1
Translating Words into Algebraic Expressions
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
12 + n
Add n to 12.
Divide 100 by M.

51. Write an algebraic expression to represent each situation.
1-4
Write an algebraic expression to represent each situation.
a. Lucy’s age y years after her 18th birthday
18 + y
Add y to 18.
b. the number of seconds in h hours
3600h
Multiply h by 3600.

52. 1. Parentheses and grouping symbols. 2. Exponents.
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.
PEMDAS
Please Excuse My Dear Aunt Sally
Evaluate the expression for the given values of the variables.
2x – xy + 4y for x = 5 and y = 2
Example 2
2(5) – (5)(2) + 4(2)
10 –
0 + 8 8

53. 1-4
Math Joke
Surgeon: Nurse! I have so many patients! Who do I work on first?
Nurse: Simple, use order of operations

54. Evaluate x2y – xy2 + 3y for x = 2 and y = 5.
1-4
q2 + 4qr – r2 for r = 3 and q = 7
(7)2 + 4(7)(3) – (3)2
49 + 4(7)(3) – 9
– 9
124
Evaluate x2y – xy2 + 3y for x = 2 and y = 5.
(2)2(5) – (2)(5)2 + 3(5)
4(5) – 2(25) + 3(5)
20 –
–15

55. Simplify the expression.
1-4
Example 3
Simplifying Expressions
Simplify the expression.
3x2 + 2x – 3y + 4x2
3x2 + 2x – 3y + 4x2
7x2 + 2x – 3y

56. Simplify the expression.
1-4
Simplify the expression.
j(6k2 + 7k) + 9jk2 – 7jk
6jk2 + 7jk + 9jk2 – 7jk
15jk2
–3(2x – xy + 3y) – 11xy.
–6x + 3xy – 9y – 11xy
–6x – 8xy – 9y

57. Apples cost $2 per pound, and grapes cost $3 per pound.
1-4
Example 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.
Let A be the number of pounds of apples.
Then 10 – A is the number of pounds of grapes.
2A + 3(10 – A)
= 2A + 30 – 3A
= 30 – A
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.

58. 1-4
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.
Then 100 – h is the remaining Cancun packages.
50h + 80(100 –h)
= 50h –80h
= 8000 – 30h
How much will he make if he sells 28 Hawaii packages?
Evaluate 8000 –30h for h = 28.
8000–30(28) = 8000–840
He will make $7160.
= 7160

59. 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

60. 1.5- Properties of Exponents
Algebra II

61. Bell work (Algebra II) Copy the information below
1-5
Bell work (Algebra II)
Copy the information below
Squared means to the 2nd power x2
Cubed means to the third power, x3
In an expression of the form an, a is the base, n is the exponent, and the quantity an is called a power.

62. 1-5
Example 1
Write the expression in expanded form.
(5z)2
(5z)2
(5z)(5z)

63. –s4 3h3(k + 3)2 3b4 3b4 1-5 Write the expression in expanded form. –s4
–(s  s  s  s) = –s  s  s  s
3(h)(h)(h) (k + 3)(k + 3)
3b4
(2a)5
3b4
(2a)5
3  b  b  b  b
(2a)(2a)(2a)(2a)(2a)

64. 1-5
Write the expression in expanded form.
–(2x – 1)3y2
–(2x – 1)3y2
–(2x – 1)(2x – 1)(2x – 1)  y  y

65. 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

66. 1 æ - ç è ö ÷ ø 1-5 Example 2 Simplify the expression. 3–2 (–5)–5 32
3  3 = 9

67. 1-5

68. 1-5
Simplify the expression. Assume all variables are nonzero.
3z7(–4z2)
3  (–4)  z7  z2
(yz3 – 5)3 = (yz–2)3
–12z7 + 2
y3(z–2)3
–12z9
y3z(–2)(3)

69. 1-5
Simplify the expression. Assume all variables are nonzero.
(–2a3b)–3
(5x6)3
53(x6)3
125x(6)(3)
125x18

70. 1-5
Day 2
Example 4
Simplify the expression. Write the answer in scientific notation.
3.0  10–11

71. 1-5
Simplify the expression. Write the answer in scientific notation.
(2.6  104)(8.5  107)
(2.6)(8.5)  (104)(107)
22.1  1011
0.25  10–3
2.5  10–4
2.21  1012

72. 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?
Example 5
Skip

73. 1-5
First, convert the speed of light from

74. 1-5
It takes light approximately 328 minutes to travel from the Sun to Pluto.

75. HW pg. 38
1.5-
3-9 (Odd), 10-19, 21, 37, 43, 44, 74,
Challenge: 55,