Download A2 Ch 1.1-1.5

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
1-1
Algebra 2 (Bell work)
Copy the following definitions down
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
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
1-1
Consider the numbers
Classify each number by the subsets of the real numbers
to which it belongs.
Numbers
Real
Rational
2.3




Whole
Natural
Irrational



2.7652
Integer




Math Humor
• Q: Why do the other numbers refuse to take
√2, √3, √5 seriously?
• A: They are completely irrational
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
–2





–0.321





Whole
Natural
Irrational


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
In interval notation, use [ ] to include an
endpoint. Use ( ) to exclude an endpoint
Pg. 8
Do Not Copy
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.
–6
–4
–2
0
2
4
6
There are two intervals graphed on the number line.
[–6, –4] or (5, ∞)
1-1
Use interval notation to represent each set of numbers.
a.
-4
-3
-2
-1
0
1
2
(–∞, –1]
b. x ≤ 2 or 3 < x ≤ 11
(–∞, 2] or (3, 11]
3
4
1-1
Algebra 2 (bell work)
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.”
Helpful Hint
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.”
Day 2
1-1
Some representations of the same sets of real
numbers are shown.
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, …}
; set-builder notation
C.
-4 -3 -2 -1
{x | x ≤ –2}
0 1 2 3 4
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}
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
1-2
Bell work (Algebra II)
Write down the following properties and leave
two lines below each for notes
1.
2.
3.
4.
5.
6.
7.
8.
Additive Identity Property
Multiplicative Identity Property
Additive Inverse Property
Multiplicative Inverse Property
Closure Property
Commutative Property
Associative Property
Distributive Property
1-2
Identities and Inverses
Properties Real Numbers
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
1-2
Identities and Inverses
Properties Real Numbers
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
1-2
Identities and Inverses
Properties Real Numbers
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
1-2
Properties Real Numbers
Identities and Inverses
For all real numbers n,
WORDS
NUMBERS
ALGEBRA
Multiplicative Inverse Property
The product of a nonzero number and its
reciprocal, or multiplicative inverse, is 1.
1-2
Example 1
Finding Inverses
Find the additive and multiplicative inverse of each number.
12
additive inverse: –12
additive inverse:
Check –12 + 12 = 0 
multiplicative inverse:
Check

multiplicative inverse:
1-2
500
–0.01
additive inverse: –500
Check 500 + (–500) = 0 
multiplicative inverse:
Check

additive inverse: 0.01
multiplicative inverse: –100
1-2
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
NUMBERS
ALGEBRA
Closure Property
The sum or product of any two real
numbers is a real number
2+3=5
2(3) = 6
a+b
ab  
1-2
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
NUMBERS
ALGEBRA
Commutative Property
You can add or multiply real numbers in
any order without changing the result.
7 + 11
7(11)
a+b
ab
=
=
=
=
11 + 7
11(7)
b+a
ba
1-2
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
NUMBERS
ALGEBRA
Associative Property
The sum or product of three or more real
numbers is the same regardless of the
way the numbers are grouped.
(5 + 3) + 7
(5  3)7
(a + b) + c
(ab)c
=
=
=
=
5 + (3 + 7)
5(3  7)
a + (b + c)
a(bc)
1-2
Properties Real Numbers
Addition and Multiplication
For all real numbers a and b,
Distributive Property
WORDS
NUMBERS
ALGEBRA
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.
5(2 + 8) = 5(2)
(2 + 8)5 = (2)5
a(b + c) = ab
(b + c)a = ba
+
+
+
+
5(8)
(8)5
ac
ca
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
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.
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
1-3
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
Example 2
Estimating Square Roots
Simplify each expression.
A.
B.
C.
D.
1-3
A.
B.
Simplify each expression.
C.
D.
1-3
Example 3
Rationalizing the Denominator
Simplify by rationalizing the denominator.
Day 2
1-3
Simplify by rationalizing the denominator.
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.
1-3
Math Joke
• Teacher: Lets find the square root of 1 million
• Student: Don’t you think that’s a bit too
radical?
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
1-4
Algebra II (Bell work)
Just Read
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
Don’t Copy
Action
Combine
Combine equal
groups
Separate
Separate into
equal groups
Operation
Add
Possible Context Clues
How many total?
Multiply
How many altogether?
Subtract
How many more? How
many remaining?
Divide
How many in each group?
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
12 + n
Add n to 12.
B. the number of days it will take to walk 100
miles if you walk M miles per day
Divide 100 by M.
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.
1-4
Order of Operations
1.
2.
3.
4.
Parentheses and grouping symbols.
Exponents.
Multiply and Divide from left to right.
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 – 10 + 8
0+8
8
1-4
Math Joke
• Surgeon: Nurse! I have so many patients! Who
do I work on first?
• Nurse: Simple, use order of operations
1-4
Example 2
Evaluating Algebraic Expressions
q2 + 4qr – r2 for r = 3 and q = 7
(7)2 + 4(7)(3) – (3)2
49 + 4(7)(3) – 9
49 + 84 – 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 – 50 + 15
–15
1-4
Example 3
Simplifying Expressions
Simplify the expression.
3x2 + 2x – 3y + 4x2
3x2 + 2x – 3y + 4x2
7x2 + 2x – 3y
1-4
Simplify the expression.
j(6k2 + 7k) + 9jk2 – 7jk
–3(2x – xy + 3y) – 11xy.
6jk2 + 7jk + 9jk2 – 7jk
–6x + 3xy – 9y – 11xy
15jk2
–6x – 8xy – 9y
1-4
Example 4
Application: Writing in terms of 1 variable
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.
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 + 8000 –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
HW pg. 31
• 1.4
– 9-21 (Odd), 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
1-5
Algebra 2 (bell work)
1. Copy the information below
In an expression of the form an, a is the base, n is the exponent,
and the quantity an is called a power.
Squared means to the 2nd power x2
Cubed means to the third power, x3
1-5
Example 1
Writing Exponential Expressions in Expanded Form
(5z)2
(5z)2
(5z)(5z)
1-5
Write the expression in expanded form.
–s4
3h3(k + 3)2
–s4
3h3(k + 3)2
–(s  s  s  s) = –s  s  s  s
3(h)(h)(h) (k + 3)(k + 3)
(2a)5
(2a)5
(2a)(2a)(2a)(2a)(2a)
3b4
3b4
3bbbb
–(2x – 1)3y2
–(2x – 1)3y2
–(2x – 1)(2x – 1)(2x – 1)  y  y
1-5
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
Example 2
Simplifying Expressions with Negative Exponents
3–2
(–5)–5
32
33=9
 1
 
 5
5
1-5
1-5
Example 4
Using Properties of Exponents to Simplify Expressions
Simplify the expression. Assume all variables are nonzero.
3z7(–4z2)
3  (–4)  z7  z2
–12z7 + 2
(yz3 – 5)3 = (yz–2)3
y3(z–2)3
y3z(–2)(3)
–12z9
1-5
Simplify the expression. Assume all variables are nonzero.
(5x6)3
53(x6)3
125x(6)(3)
125x18
(–2a3b)–3
1-5
Day 2
Example 4
Simplifying Expressions involving Scientific Notation
Simplify the expression. Write the answer in scientific notation.
3.0  10–11
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
2.21  1012
0.25  10–3
2.5  10–4
1-5
Example 5
Application
Skip (15_16)
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
First, convert the speed of light from
1-5
It takes light approximately 328 minutes to travel from the Sun to Pluto.
HW pg. 38
• 1.5– Day 1: 3-17 (Odd), 27-31 (Odd)
Day 2: 18-21, 35, 37 43, 44, 74
– Ch: 38, 55
– HW Guidelines or ½ off
– Always staple Day 1&2 Together
– Put assignment in planner