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PROPERTIES OF NUMBERS

PROPERTIES OF NUMBERS. PROPERTIES OF EQUALITY. Reflexive Property Any quantity is equal to itself. For any number , . Symmetric Property If one quantity equals a second quantity, then the second quantity equals the first. For any numbers and , if , then. PROPERTIES OF EQUALITY.

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PROPERTIES OF NUMBERS

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  1. PROPERTIES OF NUMBERS

  2. PROPERTIES OF EQUALITY • Reflexive Property • Any quantity is equal to itself. • For any number , . • Symmetric Property • If one quantity equals a second quantity, then the second quantity equals the first. • For any numbers and , if , then .

  3. PROPERTIES OF EQUALITY • Transitive Property • If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity. • For any numbers , and , if and , then . • Substitution Property • A quantity may be substituted for its equal in any expression. • If , then may be replaced by in any expression.

  4. ADDITION PROPERTIES • Additive Identity • For any number , the sum of and is . • Additive Inverse • A number and its opposite are additive inverses of each other.

  5. MULTIPLICATION PROPERTIES • Multiplicative Identity • For any number a, the product of a and 1 is a. • or • Multiplicative Property of Zero • For any number a, the product of a and 0 is 0. • or

  6. MULTIPLICATION PROPERTIES • Multiplicative Inverse • For every number , where , there is exactly one number such that the product of and is . • or

  7. COMMUTATIVE PROPERTY • The order in which you add or multiply numbers does not change their sum or product. • For any numbers and , and

  8. ASSOCIATIVE PROPERTY • The way you group three or more numbers when adding or multiplying does not change their sum or product. • For any numbers , and , and .

  9. DISTRIBUTIVE PROPERTY

  10. DISTRIBUTIVE PROPERTY • For any numbers and , • and and and .

  11. EXAMPLE 1 • (a.) A group of 7 adults and 6 children are going to a University of South Florida Bulls baseball game. Use the Distributive Property to write and evaluate an expression for the total ticket cost.

  12. EXAMPLE 1 CONTINUED... • (b.) A group of 3 adults, an 11-year old, and 2 children under 10 years old are going to a baseball game. Write and evaluate an expression to determine the cost of tickets for the group.

  13. EXAMPLE 2 • Rewrite each expression using the distributive property, then simplify. • (a.) • (b.)

  14. EXAMPLE 2 CONTINUED... • Rewrite each expression using the distributive property, then simplify. • (c.) • (d.)

  15. LIKE TERMS AND SIMPLEST FORM • Like terms: terms that contain the same variables, with corresponding variables having the same power • EX: and are like terms since they both have and both have an exponent of 1 • Simplest form: an expression that contains no like terms or parentheses • EX: 12 + 6 is in simplest form since all like terms have been combined

  16. EXAMPLE 3 • Simplify each expression • (a.) • (b.) 14

  17. WRITE AND SIMPLIFY EXPRESSIONS • 1. Use the expression twice the difference of and increased by five times the sum of and . • (a.) Write an algebraic expression for the verbal expression.

  18. (b.) Simplify the expression, and indicate the properties used. Original expression Distributive property Substitution Commutative property Substitution

  19. 2. Use the expression 5 times the difference of squared and plus 8 times the sum of and . • (a.) Write an algebraic expression for the verbal expression.

  20. (b.) Simplify the expression, and indicate the properties used. Original expression Distributive property Substitution Commutative property Substitution

  21. RELATIONS

  22. VOCABULARY • Coordinate system: formed by the intersection of two number lines, the horizontal axis (x-axis) and the vertical axis (y-axis) • Ordered pair: a set of numbers, or coordinates, written in the form • Relation: a set of ordered pairs, can be represented in several different ways: equation, graph, table, mapping

  23. MORE VOCABULARY... • Mapping: illustrates how each element of the domain is paired with an element in the range • Domain: the set of the first numbers of the ordered pairs, x-values • Range: the set of the second numbers of the ordered pairs, y-values

  24. WAYS TO REPRESENT RELATIONS

  25. EXAMPLE 1 • (a.) Express as a table, a graph, and a mapping. INPUTS OUTPUTS -1 2 1 -4 3 2

  26. EXAMPLE 1 CONTINUED... • (b.) Determine the domain and range. • Domain: 4, 3, -2 • Range: -1, 2, 1

  27. MORE VOCABULARY • Independent variable: the value of the variable that determines the output, the x-value • Dependent variable: the value of the variable that is dependent upon the value of the independent variable, the y-value

  28. IDENTIFY THE INDEPENDENT AND DEPENDENT VARIABLES • (a.) The air pressure inside a tire increases with the temperature. • Independent: the temperature • Dependent: air pressure of a tire • (b.) As the amount of rain decreases, so does the water level of the river. • Independent: amount of rain • Dependent: water level of the river

  29. ANALYZE GRAPHS • (a.) (b.) (c.) The farther you travel to school, the longer your drive As time increases, the more income you acquire The longer you ride the bike, the farther you travel

  30. FUNCTION NOTATION

  31. Function notation: equations that are functions can be represented using different notation Equation: Function Notation:

  32. In a function, represents the elements of the domain, and represents the elements of the . The graph of is the graph of the equation . Suppose you want to find the value in the range that corresponds to the element 5 in the . This is written and is read f of 5. The value is found by substituting 5 for in the equation. range domain

  33. FUNCTION VALUES • For find each value. (c.) (a.) (b.)

  34. FUNCTIONS

  35. VOCABULARY • Function: a relation in which each element of the domain is paired with exactlyone element of the range (a.) (b.)

  36. DETERMINE WHETHER EACH RELATION IS A FUNCTION... (b.) (c.) (a.)

  37. DISCRETE VS. CONTINUOUS • Discrete function: A function where not all input/domain/x-values are represented, only certain values are represented, points are NOT connected by a line or curve • Continuous function: A function where ALL input/domain/x-values are represented, all points are connected by a line or curve

  38. EXAMPLE 1: A bird feeder will hold up to 3 quarts of seed. The feeder weighs 2.3 pounds when empty and 13.4 pounds when full. (a.) Make a table that shows the bird feeder with 0, 1, 2, and 3 quarts of seed in it weighing 2.3, 6, 9.7, and 13.4 pounds, respectively.

  39. EXAMPLE 1 CONTINUED... (c.) Write the data as a set of ordered pairs then graph the data. (d.) State whether the function is discrete or continuous. Explain your reasoning. (b.) Determine the domain and range of the function. Domain: 0, 1, 2, 3 Range: 2.3, 6, 9.7, 13.4 Ordered pairs: {(0, 2.3), (1, 6), (2, 9.7), (3, 13.4)} Graph: The function is discrete since the person is adding WHOLE numbers of seed to the bird feeder.

  40. How can you tell if a graph shows a function? • Vertical Line Test: if a vertical line intersects the graph in more than one place, then the graph (relation) is not a function

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