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 z/3 Pq 2(x + 5) 3x² 5x³ u² - 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 – 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 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 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/ /3 9

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 -3a 3

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! DiscreteContinuous

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) = 1/4x 2 k(x) = 2(12x 2 – 6x + 1)