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Algebra 2 Chapter 2 Algebra 2 Chapter 2 1 2.1 Relations and Functions Relation – Any set of inputs and outputs. Maybe represented as a Table Ordered.

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Presentation on theme: "Algebra 2 Chapter 2 Algebra 2 Chapter 2 1 2.1 Relations and Functions Relation – Any set of inputs and outputs. Maybe represented as a Table Ordered."— Presentation transcript:

1

2 Algebra 2 Chapter 2 Algebra 2 Chapter 2 1

3 2.1 Relations and Functions Relation – Any set of inputs and outputs. Maybe represented as a Table Ordered pairs Mapping Graph 2

4 2.1 Relations and Functions Example 1: The monthly average water temperature of the Gulf of Mexico in Key West, Florida is as follows: January69  F February70  F March75  F April 78  F Represent this relation in the 4 ways. 3

5 2.1 Relations and Functions Table MonthTemp 1 2 3 4 69 º F 70 º F 75 º F 78 º F 4

6 2.1 Relations and Functions Ordered Pairs {( ), ( ), ( ), ( )} 1,69 2,70 3,75 4,78 5

7 2.1 Relations and Functions Mapping 1 2 3 4 69º F 70º F 75º F 78º F 6

8 2.1 Relations and Functions 3 68 70 72 74 76 78 1 2 4 Graph 7

9 2.1 Relations and Functions Domain ‒ the set of inputs of a relation the x-coordinates of the ordered pairs Range ‒ the set of outputs of a relation the y-coordinates of the ordered pairs 8

10 2.1 Relations and Functions Example 2: Write the domain and range from example 1. Domain: { } Range: { } 9 1, 2, 3, 4 69, 70, 75, 78

11 2.1 Relations and Functions Function ‒ a relation where no input (x) repeats. 10

12 2.1 Relations and Functions Example 3a Is the relation a function? {( ‒ 3, 5), (5, 4), (4, ‒ 6), (0, ‒ 6)} YES! 11

13 2.1 Relations and Functions Example 3b Is the relation a function? 12 x y 5 4 3 5 ‒9‒9 100 20 ‒ 10 NO!

14 2.1 Relations and Functions 13 4 5 6 7 8 2 4 6 8 Example 3c Is the relation a function? YES!

15 2.1 Relations and Functions 14 Example 4a – Use the vertical line test to determine if the relation is a function. NO!

16 2.1 Relations and Functions 15 Example 4b – Use the vertical line test to determine if the relation is a function. NO!

17 2.1 Relations and Functions 16 Example 4c – Use the vertical line test to determine if the relation is a function. NO!

18 2.1 Relations and Functions 17 Function Rule ‒ An equation that represents an output value in terms of an input value Function Notation ‒ f(x) f(x) is read “ f of x”. On a graph, f(x) is y.

19 2.1 Relations and Functions 18 Example 5 Evaluate the function for the given values of x, and write the input x and output as an ordered pair. a. x = 9 b. x = – 4

20 2.1 Relations and Functions 19 Example 5 (continued) (9,1)

21 2.1 Relations and Functions 20 Example 5 (continued)

22 2.1 Relations and Functions 21 Assignment: p.65 (#9 – 16 all, 18 – 24 evens)

23 2.1 Relations and Functions 22 Independent Variable ‒ Usually x, represents the input value of the function Dependent Variable ‒ Usually f(x), represents the output value of the function (The value of this variable depends on the input value.)

24 2.1 Relations and Functions 23 Example 6 To wash her brother’s clothes Jennifer charges him a base rate of $15 plus $3.50 per hour. Write a function rule to model the cost of washing her brother’s clothes.

25 2.1 Relations and Functions C(x) = ____ + _____ x Then evaluate the function if it takes Jennifer 2½ hours to wash his clothes. C(2.5) = 15 + 3.50(2.5) C(2.5) = 23.75 Jennifer will charge $23.75. 24 153.50

26 2.1 Relations and Functions 25 Example 7 – Find the domain and range of each relation.

27 2.1 Relations and Functions 26 Example 7a – Domain: x > 0 Range: ARN

28 2.1 Relations and Functions 27 Example 7b – Domain: – 4 < x < 4 Range: – 4 < y < 4

29 2.1 Relations and Functions 28 Example 8 – The relationship between your weekly salary S and the number of hours worked h is described by the following function.

30 2.1 Relations and Functions 29 Example 8 (continued) – In the following pairs, the input is the number of hours worked and the output is your weekly salary. Find the unknown measure in each ordered pair.

31 2.1 Relations and Functions 30 Example 8 (continued) – a.)

32 2.1 Relations and Functions 31 Example 8 (continued) – b.) (h, 135.20)

33 2.1 Relations and Functions 32 Assignment: p.65-66 (#25, 26, 29 – 33, 39 – 44, 48)

34 2.2 Direct Variation A function where the ratio of output to input is called direct variation. 33

35 2.2 Direct Variation 34 output input Constant of variation

36 2.2 Direct Variation For each of the following tables, determine whether y varies directly as x. If so, find the constant of variation and the equation of variation. 35

37 2.2 Direct Variation Example 1 36 xy 1 3 7 21 9 3 YES! k = 3 So y = kx would mean y = 3x.

38 2.2 Direct Variation Example 2 37 xy – 2 2 10 15 3 NO! – 3

39 2.2 Direct Variation Example 3 If y varies directly as x, and y = – 4 when x = 25. What is x when y = 10? 38 – 4x = 250 x = – 62.5

40 2.2 Direct Variation Example 4 If y varies directly as x, and x = – 8 when y = 10, find y when x = 30. 39 300 = – 8y – 37.5 = y

41 2.2 Direct Variation Example 5 The cost buying sirloin steak is directly proportional with the weight in pounds. If 8.5 lbs of steak cost $47.60, how much does 20 lbs cost? 40 = d = $112

42 2.2 Direct Variation Assignment: p.71(#7 – 10, 19 – 26) 41

43 2.3 Linear Functions & Slope Intercept Form 42

44 2.3 Linear Functions & Slope Intercept Form Example 1 – What is the slope of the line that passes through the given points? 43

45 2.3 Linear Functions & Slope Intercept Form Example 1a – ( ‒ 10, 2) and (4, ‒ 5) 44

46 2.3 Linear Functions & Slope Intercept Form Example 1b – (6, ‒ 1) and (5, ‒ 1) 45 0 in numerator

47 2.3 Linear Functions & Slope Intercept Form Example 1c – ( ‒ 2, 5) and ( ‒ 2, 1) 46 0 in denominator The slope isUNDEFINED! 0 in denominator

48 2.3 Linear Functions & Slope Intercept Form 47 Assignment: p.78 (#9-15)

49 2.3 Linear Functions & Slope Intercept Form Slope-intercept Form 48 where m is the slope of the line and (0, b ) is the y-intercept.

50 2.3 Linear Functions & Slope Intercept Form Example 2 – What is an equation of each line in slope-intercept form? 49

51 2.3 Linear Functions & Slope Intercept form Example 2a – 50 Slope = – 3 y-intercept is (0,5)

52 2.3 Linear Functions & Slope Intercept Form Example 2b – Slope = y-intercept = 51 up 2 over 3

53 2.3 Linear Functions & Slope Intercept Form Example 3 – Write the equation in slope- intercept form. What are the slope and y-intercept? 52

54 2.3 Linear Functions & Slope Intercept Form Example 3a 2x + 3y – 15 = 0 – 2x 3y – 15 = – 2x + 15 + 15 3y = – 2x + 15 3 3 3 53

55 2.3 Linear Functions & Slope Intercept Form 54

56 2.3 Linear Functions & Slope Intercept Form Example 3b – 12 = 10y – 3x + 3x 12 + 3x = 10y 10 10 10 55 Slope = y-intercept =

57 2.3 Linear Functions & Slope Intercept Form Example 4 – What is the graph of 24 = 4x + 3y? 56 24 = 4x + 3y –4x –4x – 4x + 24 = 3y 3 3 3

58 2.3 Linear Functions & Slope Intercept Form Example 4 (continued) – 57 8

59 Assignment: 58 p.78(#17-31 odds) 2.3 Linear Functions & Slope Intercept Form

60 Example 5 – A horizontal line has slope 0. Graph y = – 5. m = 0 b = – 5 59 2.3 Linear Functions & Slope Intercept Form

61 Example 6 – The slope of a vertical line is UNDEFINED. Graph x = 3. 60 2.3 Linear Functions & Slope Intercept Form Slope = undefined y-intercept = NONE

62 Assignment: 61 p.78(#32 – 52 evens) 2.3 Linear Functions & Slope Intercept form

63 2.4 More About Linear Equations Point-slope form 62

64 2.4 More About Linear Equations Example 1 Use the given information to write an equation in point-slope form. 63

65 2.4 More About Linear Equations a. slope = through (– 1, 3) 64

66 2.4 More About Linear Equations b.) slope = 0 through (22, – 1) 65

67 2.4 More About Linear Equations 66 c.) passing through (5, 1) and (7, 1)

68 2.4 More About Linear Equations d.) passing through (4, – 1) and (– 6, 5) 67

69 2.4 More About Linear Equations Slopes of parallel lines are equal / the same. Slopes of perpendicular lines are opposite reciprocals. 68

70 2.4 More About Linear Equations Example 2 Use the given information to write the equation of the line described in slope-intercept form. 69

71 2.4 More About Linear Equations a.) parallel to y = x + 2 through (– 5, 3) 70

72 2.4 More About Linear Equations b.) perpendicular to y = – 2x + 3 71

73 2.4 More About Linear Equations Assignment: p.86-88 (#10 – 18, 32, 33, 65, 72, 74) 72

74 2.4 More About Linear Equations Example 3 Use the given information to write the equation of the line described in slope- intercept form. 73

75 2.4 More About Linear Equations a.) parallel to 3x – 2y = 6 through (– 3, 5) 74

76 2.4 More About Linear Equations b.) perpendicular to 4x + y = 1 through (2,1) 75

77 2.4 More About Linear Equations Example 4 Find the intercepts, and graph the line. 76

78 2.4 More About Linear Equations a.) 4x + 3y = 12 77

79 2.4 More About Linear Equations b.) 4x – 5y = 10 78

80 2.4 More About Linear Equations Example 5 The cost of a taxi ride depends on the distance traveled. You paid $8.50 for a 3-mile ride, and your friend paid $18.50 for an 8-mile ride 79

81 2.4 More About Linear Equations 80 Example 5 A.) Sketch a graph that models this situation.

82 2.4 More About Linear Equations Example 5 (continued) B.) Write the equation in slope-intercept form for this situation. C.) How much would a 6 mile taxi ride cost? 81

83 2.4 More About Linear Equations Assignment: p.86-87(#26-31,34-41) 82


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