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Expressions, Equations, and Functions

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:

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Expressions, Equations, and Functions

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  1. Expressions, Equations, and Functions Chapter 1

  2. 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!

  3. Verbal Translations

  4. 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

  5. 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

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

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

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

  9. Open SentencesVocabulary • Set • Element • Replacement set • Solution set • Solution • Equation • inequality

  10. Symbols • = • = • < • > • < • > • 0

  11. 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

  12. 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

  13. 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)

  14. These properties along with some others allow algebra to work!

  15. ExpressionsVocabulary • Equivalent expression • denote the same number • Simplify expressions • Write an expression with the least amount of symbols, numbers, and variables

  16. 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

  17. Terms 8m a 9 -7j² -4a 8 2cd x/8 7g ¼ b 3xy j 9b 5x –y 2d 4g m 6y 6a³ -9a³

  18. Terms Like Terms Non Like Terms a and 9 -4a and 8 2x and 3xy 5j and -7j² 2d and 2cd • 8m and m • 4g and 7g • 9b and ¼ b • 5x and x/8 • 6y and –y • 6a³ and -9a³

  19. Equivalent Expressions Expression Simplified expression • 8m - m • 4g + 7g • 9b + ¼ b • 5x + x/8 • 6y + (–y) • 6a³ - 9a³

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

  21. 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

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

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