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Introductory terms and symbols:
• Algebraic expression– One or more numbers or
variables along with one or more arithmetic operations
– You may evaluate and simplify expressions, but you cannot solve expressions…you solve equations!
• Variable– A letter or symbol to
represent an unknown
• Term- A term may be a
number, variable, or product or quotient of numbers and variables
Identify the variable and term in each expression(What could each represent?)
• .10d• 2x - 4• 3 + z/3• Pq• 2(x + 5)
• 3x²• 5x³ + 16• 16u² - 3u + 4• ½a - 6b/7
Translate verbal expressions to algebraic expressions
7 less than the product of 3 and a number
• The product of 7 and a number divided by the product of 8 and a number
• 5 more than half a number
• The quotient of 3 and the square of a number
• Twice the sum of 15 and a number
Real Life Connection
• Mr. Martinez orders 250 key chains printed with his athletic teams logo and 500 pencils printed with their web address. Write an expression to represent the cost of each order
• Katie bakes 40 pastries and makes coffee for 200 people. Write and expression to represent the situation
Order of Operations
• Evaluate Numerical Expressions
• How????• PEMDAS
• 16 – 8/2^2 + 14• 3 + 42 * 2 – 5• 4/2 + 5(10 – 6)• 6[32 – ( 2 + 3)^2]
• 2^5 – 6*2 3^3 – 5*3 - 2
Evaluate Algebraic Expressions
• 3x^2 + (2y + z^3) if x=4, y=5, z=3
• A^2(3b + 5) /C IF A=2, B= 6, C=4
• Real Life Connection• Find the volume of a 3
foot radius sphere
Algebraic Properties
• Reflexive• Symmetric• Transitive• Substitution
• Additive Identity• Additive Inverse• Multiplicative
Identity• Multiplicative
Inverse• Multiplicative
Property of Zero
These properties say:
• Reflexive– Any quantity is equal
to itself– For any number a, a=a
• Symmetric– If one quantity equals
a second, then the second equals the first
– For any numbers a and b, if a=b, then b=a.
• Transitive– If one quantity equals a
second and the second equals a third, then the first equals the third.
– For any numbers a and b, and c, If a = b, and b=c, then a=c
• Substitution– A quantity may be substituted
for its equal expression– If a =b, the a may be replaced
with b in any expressions
More Algebraic Properties
• Additive Identity– For any number
a , a + 0 = 0 + a = a
Additive Inverse a + (-a) = 0
Multiplicative Identity– For any number a, (a)(1) = 1a = a
• Multiplicative Inverse (reciprocal)
For every number a/b where a,b = 0, (a/b)(b/a) = 1
Multiplicative Property of zeroFor any number a, a(0)=0 0(a) = 0
Algebraic Properties You Already Know
• Distributive Property– For any numbers a, b, and c, a(b + c) = ab + ac and (b + c)a = ba + ca a(b - c) = ab - ac and (b - c)a = ba - ca
• Associative Property– For any numbers a and b, a + b = b + a and ab = ba
• Commutative Property– For any numbers a, b, c, ( a + b ) + c = a + ( b + c ) and (ab)c = a(bc)
ExpressionsVocabulary
• Equivalent expression– denote the same number
• Simplify expressions– Write an expression with the least
amount of symbols, numbers, and variables
Termsvocabulary
• Term– a number or variable or the product of a number
and variable• Like terms– Terms that contain the same variable– Like terms can be grouped (combined)
• Constant– A numerical term containing NO variables
• Coefficient– The numerical factor of a term
Terms
Like Terms• 8m and m• 4g and 7g• 9b and ¼ b• 5x and x/8• 6y and –y• 6a³ and -9a³
Non Like Terms• a and 9• -4a and 8• 2x and 3xy• 5j and -7j²• 2d and 2cd
Equivalent Expressions
Expression• 8m - m• 4g + 7g• 9b + ¼ b• 5x + x/8• 6y + (–y)• 6a³ - 9a³
Simplified expression• 7m• 11g• 9 1/4b• 5 1/8x• 5y• -3a3
Open Sentences
Vocabulary• Set• Element• Replacement set• Solution set• Solution• Equation• inequality
Examples• {-2,-1, 0, 1, 2, 3}• -2,-1, 0, 1, 2, 3• {1, 0, 1}• {0,1}• 1
Find the solution (set). The replacement set is {0,1,2,3,4,5}
• 6b + 7= 37• y + 5 < 7• 8 – x > 7• t + 3 = 3 4
Symbols
• =• =• <• >• <• >• 0
• Equal to• Not equal to• Less than• Greater than• Less than or equal to• Greater than or equal to• no solution
Relation~ A set of Ordered Pairs
Input• Independent
variable• X - coordinate• domain
Output• Dependent
variable• Y-coordinate• range
A Preview to Functions
• A function is a relationship between input and output values (a relation)
• With a function, there is exactly one output for each input!
• A function (relation) can be expressed as ordered pairs
Discrete and Continuous Functions
• Non-continuous data• Points not connected• Sometimes points are
connected to show trends
• Examples:• number of items
• Points connected by curves or lines
• Step functions too!
Discrete Continuous