1. Algebra 2 Chapter

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

3. 2.1 Relations and Functions
Example 1: The monthly average water temperature of the Gulf of Mexico in Key West, Florida is as follows: January 69F February 70F March 75F April 78F Represent this relation in the 4 ways.

4. 2.1 Relations and Functions
Table
Month
Temp 1
69 º F 2
70 º F 3
75 º F 4
78 º F

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

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

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

8. 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

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

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

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

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

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

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

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

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

17. 2.1 Relations and Functions
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. 17

18. 2.1 Relations and Functions
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 18

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

20. 2.1 Relations and Functions
Example 5 (continued) 20

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

22. 2.1 Relations and Functions
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.) 22

23. 2.1 Relations and Functions
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. 23

24. 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) = (2.5) C(2.5) = Jennifer will charge $23.75. 15
3.50

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

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

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

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

29. 2.1 Relations and Functions
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. 29

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

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

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

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

34. 2.2 Direct Variation
output
input
Constant
of variation

35. 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.

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

37. 2.2 Direct Variation
Example 2 x y
NO!
– 2 3 2
– 3 10 15

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

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

40. 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? =
d = $112

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

42. 2.3 Linear Functions & Slope Intercept Form

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

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

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

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

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

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

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

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

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

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

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

54. 2.3 Linear Functions & Slope Intercept Form

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

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

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

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

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

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

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

62. 2.4 More About Linear Equations
Point-slope form
where m is the slope of the line
passing through the point ( 𝒙 𝟏 , 𝒚 𝟏 ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

79. 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

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

81. 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?

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

83. 2.5 Using Linear Models
Month
Temp 1 2 3 4
69 º F
70 º F
75 º F
78 º F

84. 2.5 Using Linear Models Scatter Plot –
A graph that relates two sets of data by plotting the data as ordered pairs

85. 2.5 Using Linear Models
A scatter plot can be used to determine the strength of the relation or correlation between data sets.
The closer the data points fall along a line with a positive slope,
The stronger the linear relationship and the stronger the positive correlation

86. 2.5 Using Linear Models Age (in years) # of cartoons watched
Height (in feet)

87. STRONG POSITIVE CORRELATION WEAK POSITIVE CORRELATION
2.5 Using Linear Models
STRONG POSITIVE CORRELATION
WEAK POSITIVE CORRELATION

88. STRONG NEGATIVE CORRELATION
2.5 Using Linear Models
STRONG NEGATIVE CORRELATION
NO CORRELATION

89. 2.5 Using Linear Models