Expressions, Equations, and Functions Chapter 1

Introductory terms and symbols: Variable – A letter or symbol to represent an unknown – Examples: Algebraic expression – One or more numbers or variables along with one or more arithmetic operations – Examples: – You may evaluate and simplify expressions, but you cannot solve expressions…you solve equations!

Verbal Translations

Translate verbal expressions to algebraic expressions Examples: 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

Patterns and Sequences 7, 13, 19, 25, … 51, 43, 35, 27, … 343, 81, 27, 9, … 2, 3.5, 5, 6.5, … 1, 4, 16, 64, …. Find the next three terms in each sequence Group the sequences according to similarities

Patterns and Sequences Arithmetic 7, 13, 19, 25, … 51, 43, 35, 27, … 2, 3.5, 5, 6.5, … 3, 6, 9, 12, … Geometric 343, 81, 27, 9, … 1, 4, 16, 64, …. 3, 9, 27, 81, ….

Which Changes Faster?? Geometric! How does it change when: – multiply by a whole number – multiply by a fraction – WHY?

Writing expressions for patterns Draw a picture Make a table and look for a pattern Model using manipulatives

Open Sentences Vocabulary Set Element Replacement set Solution set Solution Equation inequality

Symbols = < > < > 0

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

Algebraic Properties Additive Identity – For any number a, a + 0 = 0 + a = a Multiplicative Identity – For any number a, (a)(1) = 1a = a Multiplicative Inverse Property (reciprocal) For any non-zero number a/b, where a, b don’t = 0, (a/b)(b/a) = 1

More Algebraic Properties 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 Commutative Property – For any numbers a and b, a + b = b + a and ab = ba Associative Property – For any numbers a, b, c, ( a + b ) + c = a = ( b + c ) and (ab)c = a(bc)

These properties along with some others 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

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

Coefficients Term 2b 8c² K -5t³ 9 Coefficient

A Preview to Functions A function is a relationship between input and output values With a function, there is exactly one output for each input! A function (relation) can be expressed as ordered pairs

Relation Input Independent variable X - coordinate domain Output Dependent variable Y-coordinate range