1. Math 10: Foundations and Pre-Calculus E. What is a Mathematical Reation?

2. FP10.6 Expand and apply understanding of relations and functions including: relating data, graphs, and situations analyzing and interpreting distinguishing between relations and functions. FP10.8 Demonstrate understanding of linear relations including: representing in words, ordered pairs, tables of values, graphs, function notation, and equations determining characteristics including intercepts, slope, domain, and range relating different equation forms to each other and to graphs.

3. Key Terms: Find the definition of the following terms: Relation Arrow Diagram Function Domain Range Function Notation Rate of Change Linear Function Vertical (y) intercept Horizontal (x) intercept

4. 1. Ways of Representing Relations FP10.6 Expand and apply understanding of relations and functions including: relating data, graphs, and situations analyzing and interpreting FP10.8 Demonstrate understanding of linear relations including: representing in words, ordered pairs, tables of values, graphs, function notation, and equations

6. A set if a collection of distinct objects (in our case numbers) A element of a set is one object in the set A relation associates the elements of one set with the elements of another set

7. One way to write a set is to list its elements in braces. For example we can write the set of real numbers from 1 to 5 as: {1,2,3,4,5} The order of the elements in the set does not matter

8. Consider the set of fruits and the set of colors We can associate fruits with their colors

10. So the set of ordered pairs is a relation:

11. Here are some other ways to represent this relation: The heading of each column describes each set.

12. Two ovals represent the sets Each arrow associates an element in the 1 st set to the 2 nd set

13. The order of the words in the ordered pairs and which column and which oval the words are in in the table and arrow diagram are important It makes sense to say “an apple may be the color red” but does not make sense to say “red may be the color apple”. That is, a relation has direction from one set to the other set

14. Example

17. Practice Ex. 5.1 (p. 261) #1-11 # 3-13

18. 2. Properties of Functions FP10.6 Expand and apply understanding of relations and functions including: distinguishing between relations and functions. FP10.8 Demonstrate understanding of linear relations including: representing in words, ordered pairs, tables of values, graphs, function notation, and equations determining characteristics including intercepts, slope, domain, and range

20. The set of first elements of a relation is called the domain The set of related second elements of a relation is called the Range A function is a special type of relation when each element in the domain is associated with exactly one element in the range.

27. Example

28. In the workplace, a persons gross pay, P dollars, often depends on the number of hours worked, h. So, we say P is the dependent variable. Since the number of hours worked, h, does not depend on the gross pay, P, we say it’s the independent variable

30. Example

31. We can think of a function as an input/output machine The input can be any number in the domain, and the output depends on the input number So the input is the independent variable, and the output is the dependent variable

32. Consider a machine that calculates the value of quarters. This machine represents a function. When the input is “q” quarters, the output or value, V, in dollars is: 0.25q The equation V=0.25q describes this function

33. Since V is a function of q, we can write this using function notation V(q)=0.25q We say: “V of q is equal to 0.25q”

34. This notation shows that V is the dependent variable and the V depends on q. Example What does V(3) mean?

35. Any function that can be written as an equation in two variables can be written in function notation. Example Write the equation d = 4t+5 in function notation. t is the domain d(t) is the range

36. When we write an equation that is not related to a context, we use “x” as the independent variable and “y” as the dependent Then an equation in two variables such as y = 3x-2 may be written as …..

37. Conversely we may write an equation in function notation as an equation in 2 variables For example, for the equation C(n) = 300+5n, we write …. And for the equation g(x) = -2x+5, we write…..

38. Example

39. Practice Ex. 5.2 (p. 270) #1-19 # 4-23

40. 3. Interpret and Sketch Graphs FP10.6 Expand and apply understanding of relations and functions including: relating data, graphs, and situations analyzing and interpreting

42. Construct Understanding p. 277

43. The properties of a graph can provide info about a given situation:

44. Example

45. Lets use the following graph to describe what is happening:

46. Example

48. Practice Ex. 5.3 (p. 281) #1-15 # 3-18

49. 4. Graphing Data FP10.6 Expand and apply understanding of relations and functions including: relating data, graphs, and situations FP10.8 Demonstrate understanding of linear relations including: representing in words, ordered pairs, tables of values, graphs, function notation, and equations

51. Construct Understanding p. 285

52. Practice Ex. 5.4 (p. 286) #1-2

53. 5. Graphing Relations and Functions FP10.6 Expand and apply understanding of relations and functions including: relating data, graphs, and situations analyzing and interpreting distinguishing between relations and functions. FP10.8 Demonstrate understanding of linear relations including: representing in words, ordered pairs, tables of values, graphs, function notation, and equations determining characteristics including domain, and range

55. Construct Understanding p. 288

56. We can represent the function that associates every whole number with its double in several ways

59. We know the relation y=2x is a function because each value of x associates with exactly one y-value, each ordered pair has a different 1 st element. The domain of a function is a set of values of the independent variable; for the previous graph, the domain is all the x- values. The range of a function is a set of values of the dependent variable; for the previous graph, the range is all the y-values.

60. When the domain is restricted to a set of discrete values, the points are not connected.

64. Example

68. Practice Ex. 5.5 (p. 293) #1-20 # 4-23

69. 6. Properties of Linear Functions FP10.8 Demonstrate understanding of linear relations including: representing in words, ordered pairs, tables of values, graphs, function notation, and equations determining characteristics including intercepts, slope, domain, and range relating different equation forms to each other and to graphs.

71. The cost for a car rental is $60, plus $20 for every 100km driven. The independent variable is the distance driven and the dependent variable is the cost. We can identify that this is a linear relation in different ways

72. A Table of Values:

73. For a linear relation, the constant change in the independent variable results in the constant change in the dependent variable.

74. A set of ordered Pairs:

75. A Graph:

76. We can use each representation above to calculate the Rate of Change The rate of change can be expressed as a fraction. Change in dependent variable = $20. Change is independent variable 100 km

77. The rate of change is $0.20/km; that is, for each additional 1 km driven, the rental cost increases by $0.20. The rate of change is constant for a linear relation.

78. We can determine the rate of change from the equation that represents the linear function. Let the cost be “C” dollars and the distance traveled be “d” km. The equation is:………

79. Example

80. When an equation is written using the variables x and y, x represents the independent variables and y represents the dependent variables.

81. Example

84. Practice Ex. 5.6 (p. 307) #1-20 # 6-22

85. 7. Graphs of Linear Functions FP10.8 Demonstrate understanding of linear relations including: representing in words, ordered pairs, tables of values, graphs, function notation, and equations determining characteristics including intercepts, slope, domain, and range relating different equation forms to each other and to graphs.

87. Each graph below shows the temp, T, in degrees C°, as a function of time, t hours, for two locations.

88. Location A: The point where the graph intersects the horizontal (x) axis has the coordinates (4,0) The horizontal (x) intercept is 4. This represents the time after 4h, when the temp is 0°C.

89. The point the graph intersects the vertical (y) axis is (0,-5). The vertical (y) intercept is -5. This point of intersection represents the initial temp, -5°C. Domain = ….. Range = ….. Rate of Change = ….

90. Location B: The point where the graph intersects the horizontal (x) axis has the coordinates (5,0) The horizontal (x) intercept is 5. This represents the time after 5h, when the temp is 0°C.

91. The point the graph intersects the vertical (y) axis is (0,10). The vertical (y) intercept is 10. This point of intersection represents the initial temp, 10°C. Domain = ….. Range = ….. Rate of Change = ….

92. The rate of change is negative because the temp decreases over time (falls left to right) The rate of change is positive when the temp increases over time (rises left to right)

93. Example

94. We can use the intercepts to graph a linear function written in function notation. To determine the y-intercept (vertical) evaluate f(x) when x=0, that is evaluate f(0) (sub in 0 for x solve for y) To determine the x-intercept (horizontal) determine the value of x when f(x)=0. (sub in 0 for y and solve for x)

95. Example

98. Practice Ex. 5.7 (p. 319) #1-16 # 6-20