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

• 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)

These properties allow algebra to

work!

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

8m a 9 -7j² -4a 8

2cd x/8 7g ¼ b 3xy j 9b

5x –y 2d

4g m 6y 6a³ -9a³

Coefficients

Term• 2b • 1/8c²• K• -5t³ • 2x 3 • 9• -c

Coefficient• 2• 1/8• 1• -5• 2/3• 9• -1

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

Ways to Represent Relations

• 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

• 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

Function Notation

• Equation• y= 3x - 8

• 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)