Download Expressions, Equations, and Functions

Expressions, Equations,
and Functions
Chapter 1
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
Verbal Translations
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
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•
•
•
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
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•
•
•
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
zero
For 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)
These properties
allow algebra to
work!
Expressions
Vocabulary
• Equivalent expression
– denote the same number
• Simplify expressions
– Write an expression with the least
amount of symbols, numbers, and
variables
Terms
vocabulary
• 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
8m
a
-7j²
2cd
¼b
5x
4g
m
6a³
9
-4a
x/8
3xy
j
–y
2d
8
7g
9b
6y
-9a³
Coefficients
Term
Coefficient
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•
•
•
•
•
•
•
•
•
•
•
2b
1/8c²
K
-5t³
2x
3
• 9
• -c
2
1/8
1
-5
2/3
9
-1
Terms
Like Terms
Non Like Terms
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•
•
•
•
•
•
•
•
•
•
8m and m
4g and 7g
9b and ¼ b
5x and x/8
6y and –y
6a³ and -9a³
a and 9
-4a and 8
2x and 3xy
5j and -7j²
2d and 2cd
Equivalent Expressions
Expression
Simplified expression
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8m - m
4g + 7g
9b + ¼ b
5x + x/8
6y + (–y)
6a³ - 9a³
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
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•
=
=
<
>
<
>
0
•
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•
•
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
Ways to Represent Relations
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•
•
•
Ordered pairs
Table
Graph
Mapping… new!
•
Mapping
Domain
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
How can you tell if a Relation is a
Function?
• Input - Output
• Vertical line test
Discrete and Continuous Functions
Discrete
Continuous
• 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!
Function Notation
• Equation
• y= 3x - 8
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Function Notation
f(x) = 3x – 8
Read f of x
Find f(3)
Find f(-4)
Find f(2/3)
Other functions:
g(x) = 1/4x2
k(x) = 2(12x2 – 6x + 1)