1. Functions in the Coordinate Plane
Linear Functions

2. Slope
The rate of change

3. Methods to Determine Slope
From graph, count

4. Methods to Determine Slope
Rise Run
From graph, count

5. Methods to Determine Slope
Rise Run
From graph, count
If it rises from left to right, it is positive

6. Methods to Determine Slope
Rise Run
From graph, count
If it rises from left to right, it is positive
If it goes down from left to right, it is negative

7. Methods to Determine Slope
Rise Run
From graph, count
Run
Rise

8. Methods to Determine Slope
From a table of data, find the average.

9. Methods to Determine Slope
From a table of data, find the average.
y or change in y
x change in x

10. Methods to Determine Slope
From two points, find

11. Methods to Determine Slope
From two points, find
y or y2 – y1
x x2 – x1

12. Methods to Determine Slope
From an equation y = mx + b, the slope is the m value

13. Where the line intersects the x- axis or the y-axis
Intercepts
Where the line intersects the x- axis or the y-axis
Y axis
X axis

14. y-intercept (0,y)
Where the line hits the y axis
(0,y)

15. The y-intercept is the b value
y-intercept (0,y)
Where the line hits the y axis
From the equation:
y = mx + b
The y-intercept is the b value
(0,y)

16. The y-intercept is the b value
y-intercept (0,y)
Where the line hits the y axis
From the equation:
y = mx + b
The y-intercept is the b value
(0,y) 3 2 1
From the equation:
y = x+3
The y-intercept is at 3

17. x-intercept (x,0)
Where the line hits the x axis
(x,0)

18. Plug in 0 for y and solve for x
x-intercept (x,0)
Where the line hits the x axis
From the equation:
y = mx + b
Plug in 0 for y and solve for x
(x,0)

19. Plug in 0 for y and solve for x
x-intercept (x,0)
Where the line hits the x axis
From the equation:
y = mx + b
Plug in 0 for y and solve for x
(x,0)
From the equation:
y = x + 2
0 = x + 2 1 2

20. x-intercept (x,0) x-intercept (x,0) (x,0) (x,0) From the equation:
Where the line hits the x axis
Where the line hits the x axis
From the equation:
y = mx + b
Plug in 0 for y and solve for x
From the equation:
y = mx + b
Plug in 0 for y and solve for x
(x,0)
(x,0)
From the equation:
y = x + 2
0 = x + 2
From the equation:
y = x + 2
0 = x + 2 1 2

21. x-intercept (x,0) x-intercept (x,0) (x,0) (x,0) From the equation:
Where the line hits the x axis
Where the line hits the x axis
From the equation:
y = mx + b
Plug in 0 for y and solve for x
From the equation:
y = mx + b
Plug in 0 for y and solve for x
(x,0)
(x,0)
From the equation:
y = x + 2
0 = x + 2
-2 = x
From the equation:
y = x + 2
0 = x + 2 1 2

22. Example 1 For the given graph:
A. set up a table of values for at least four points
B. find the slope
C. determine the equation of the line
D. identify the x- intercept and y intercept

23. Example 1 X Y For the given graph:
A. set up a table of values for at least four points X Y

24. Example 1 X Y For the given graph:
A. set up a table of values for at least four points X Y -1 -7 -5 1 -3 2

25. Example 1 Rise Run Pick two points For the given graph:
B. find the slope
Pick two points

26. Example 1 Rise Run Pick two points Rise: up or down: COUNT Up + Down -
For the given graph:
B. find the slope
Pick two points
Rise: up or down: COUNT
Up +
Down -
Rise =

27. Example 1 Rise Run Pick two points Rise: up or down: COUNT Up + Down -
For the given graph:
B. find the slope
Pick two points
Rise: up or down: COUNT
Up +
Down -
Rise = 2
Run: right or left
Right +
Left –
Run = 2

28. Example 1 Rise Run Pick two points Rise: up or down: COUNT Up + Down -
For the given graph:
B. find the slope 1
Pick two points
Rise: up or down: COUNT
Up +
Down -
Rise = 2
Run: right or left
Right +
Left –
Run = 1 2

29. Example 1 Rise Run Rise Run 2 1 Slope = = = 2 Pick two points
For the given graph:
B. find the slope 1
Pick two points
Rise: up or down: COUNT
Up +
Down -
Rise = 2
Run: right or left
Right +
Left –
Run = 1 2
Rise Run
2 1
Slope = =
= 2

30. Example 1 Equation of a Line: y = mx + b m = slope = b = y int =
For the given graph:
C. determine the equation of the line
Equation of a Line:
y = mx + b
m = slope =
b = y int =

31. Example 1 Equation of a Line: y = mx + b m = slope = 2
For the given graph:
C. determine the equation of the line
Equation of a Line:
y = mx + b
m = slope = 2
b = y int = (0, )

32. Example 1 Equation of a Line: y = mx + b m = slope = 2
For the given graph:
C. determine the equation of the line
Equation of a Line:
y = mx + b
m = slope = 2
b = y int = (0, -5)

33. Example 1 Equation of a Line: y = mx + b m = slope = 2
For the given graph:
C. determine the equation of the line
Equation of a Line:
y = mx + b
m = slope = 2
b = y int = (0, -5)
SO,
y = 2x -5

34. Example 1 y int = x int = For the given graph:
D. identify the x- intercept and y intercept
y int =
x int =

35. Example 1 y int = (0, -5) x int = For the given graph:
D. identify the x- intercept and y intercept
y int = (0, -5)
x int =

36. x int = (x, 0) since we have our equation, plug in 0 for y
Example 1
For the given graph:
D. identify the x- intercept and y intercept
y int = (0, -5)
x int =
x int = (x, 0) since we have our equation, plug in 0 for y

37. Example 1 y int = (0, -5) x int = y = 2x -5 For the given graph:
D. identify the x- intercept and y intercept
y int = (0, -5)
x int =
x int = (x, 0) since we have our equation, plug in 0 for y
y = 2x -5

38. Example 1 y int = (0, -5) x int = y = 2x -5 0 = 2x-5
For the given graph:
D. identify the x- intercept and y intercept
y int = (0, -5)
x int =
x int = (x, 0) since we have our equation, plug in 0 for y
y = 2x -5
0 = 2x-5

39. Example 1 y int = (0, -5) x int = y = 2x -5 0 = 2x-5 5=2x
For the given graph:
D. identify the x- intercept and y intercept
y int = (0, -5)
x int =
x int = (x, 0) since we have our equation, plug in 0 for y
y = 2x -5
0 = 2x-5
5=2x

40. Example 1 y int = (0, -5) x int = y = 2x -5 0 = 2x-5 5=2x
For the given graph:
D. identify the x- intercept and y intercept
y int = (0, -5)
x int =
x int = (x, 0) since we have our equation, plug in 0 for y
y = 2x -5
0 = 2x-5
5=2x
X = 5/2 = 2.5

41. Example 1 y int = (0, -5) x int = (2.5, 0) y = 2x -5 0 = 2x-5 5=2x
For the given graph:
D. identify the x- intercept and y intercept
y int = (0, -5)
x int = (2.5, 0)
x int = (x, 0) since we have our equation, plug in 0 for y
y = 2x -5
0 = 2x-5
5=2x
X = 5/2 = 2.5

42. Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values:
A. find the slope
B. determine the equation of the line
C. Graph the function
D. identify the x- intercept and y intercept

43. y or change in y x change in x y2 – y1 x2 – x1 = Example 2 x y -2 13 14 -5 8
-17
For the given table of values:
A. find the slope
y or change in y
x change in x
y2 – y1
x2 – x1 =
Choose two points:

44. y or change in y x change in x y2 – y1 x2 – x1 = Example 2 x y -2 13 14 -5 8
-17
For the given table of values:
A. find the slope
y or change in y
x change in x
y2 – y1
x2 – x1 =
(x2, y2)
(x1, y1)
Choose two points: (1,4) and (4,-5), plug into equation

45. y or change in y x change in x y2 – y1 x2 – x1 = Example 2 y2 – y1-2 13 1 4 -5 8
-17
For the given table of values:
A. find the slope
y or change in y
x change in x
y2 – y1
x2 – x1 =
Choose two points: (1,4) and (4,-5), plug into equation
y2 – y1
x2 – x1
-5 – 4
4 – 1 =

46. y or change in y x change in x y2 – y1 x2 – x1 = Example 2 y2 – y1-2 13 1 4 -5 8
-17
For the given table of values:
A. find the slope
y or change in y
x change in x
y2 – y1
x2 – x1 =
Choose two points: (1,4) and (4,-5), plug into equation
y2 – y1
x2 – x1
-5 – 4
4 – 1 -9 3 = =

47. y or change in y x change in x y2 – y1 x2 – x1 = Example 2 y2 – y1-2 13 1 4 -5 8
-17
For the given table of values:
A. find the slope
y or change in y
x change in x
y2 – y1
x2 – x1 =
Choose two points: (1,4) and (4,-5), plug into equation
y2 – y1
x2 – x1
-5 – 4
4 – 1 -9 3 = = = -3

48. Example 2 y– y1 = m ( x – x1) x y -2 13 1 4 -5 8 -17
For the given table of values:
B. determine the equation of the line
y– y1 = m ( x – x1)

49. Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1)-2 13 1 4 -5 8
-17
For the given table of values:
B. determine the equation of the line
When we have our slope, we plug in one point into x1 and y1
y– y1 = m ( x – x1)
y – y1 = -3 ( x – x1)
y – 4 = -3 ( x – 1)

50. Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1)-2 13 1 4 -5 8
-17
For the given table of values:
B. determine the equation of the line
Now we solve for y
y– y1 = m ( x – x1)
y – y1 = -3 ( x – x1)
y – 4 = -3 ( x – 1)

51. Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1)-2 13 1 4 -5 8
-17
For the given table of values:
B. determine the equation of the line
Now we solve for y
y– y1 = m ( x – x1)
y – y1 = -3 ( x – x1)
y – 4 = -3 ( x – 1)
y – 4 = -3x +3

52. Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1)-2 13 1 4 -5 8
-17
For the given table of values:
B. determine the equation of the line
Now we solve for y
y– y1 = m ( x – x1)
y – y1 = -3 ( x – x1)
y – 4 = -3 ( x – 1)
y – 4 = -3x +3
y = -3x +3 +4

53. Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1)-2 13 1 4 -5 8
-17
For the given table of values:
B. determine the equation of the line
Now we solve for y
y– y1 = m ( x – x1)
y – y1 = -3 ( x – x1)
y – 4 = -3 ( x – 1)
y – 4 = -3x +3
y = -3x +7

54. Plot your points Example 2 x y -2 13 1 4 -5 8 -17
For the given table of values:
C. Graph the function x y -2 13 1 4 -5 8
-17
Plot your points

55. Plot your points Example 2 x y -2 13 1 4 -5 8 -17
For the given table of values:
C. Graph the function x y -2 13 1 4 -5 8
-17
Plot your points

56. Connect Points Example 2 x y -2 13 1 4 -5 8 -17
For the given table of values:
C. Graph the function x y -2 13 1 4 -5 8
-17
Connect Points

57. Connect Points Example 2 x y -2 13 1 4 -5 8 -17
For the given table of values:
C. Graph the function x y -2 13 1 4 -5 8
-17
Connect Points

58. Example 2 y int = x int = x y -2 13 1 4 -5 8 -17
For the given table of values:
D. identify the x- intercept and y intercept
y int =
x int =

59. Since we have our equation, y int = b
Example 2 x y -2 13 1 4 -5 8
-17
For the given table of values:
D. identify the x- intercept and y intercept
y int =
x int =
y = -3x +7
Since we have our equation, y int = b

60. We plug in 0 for y to solve for the x int
Example 2 x y -2 13 1 4 -5 8
-17
For the given table of values:
D. identify the x- intercept and y intercept
y int = (0,7)
x int =
y = -3x +7
We plug in 0 for y to solve for the x int

61. Example 2 y int = (0,7) x int = y = -3x +7 0 = -3x +7 x y -2 13 1 4 -58
-17
For the given table of values:
D. identify the x- intercept and y intercept
y int = (0,7)
x int =
y = -3x +7
0 = -3x +7

62. Example 2 y int = (0,7) x int = y = -3x +7 0 = -3x +7 3x = 7 x y -2 134 -5 8
-17
For the given table of values:
D. identify the x- intercept and y intercept
y int = (0,7)
x int =
y = -3x +7
0 = -3x +7
3x = 7

63. Example 2 y int = (0,7) x int = y = -3x +7 0 = -3x +7 3x = 7 x = 7/3 x-2 13 1 4 -5 8
-17
For the given table of values:
D. identify the x- intercept and y intercept
y int = (0,7)
x int =
y = -3x +7
0 = -3x +7
3x = 7
x = 7/3

64. Example 2 y int = (0,7) x int = (7/3, 0) y = -3x +7 0 = -3x +7 3x = 7-2 13 1 4 -5 8
-17
For the given table of values:
D. identify the x- intercept and y intercept
y int = (0,7)
x int = (7/3, 0)
y = -3x +7
0 = -3x +7
3x = 7
x = 7/3

65. Example 3 For the given function: y = -2/3x +4
A. set up a table of values
B. graph the function
C. find the slope
D. identify the x- intercept and y intercept

66. Example 3
For the given function:
y = -2/3x +4
A. set up a table of values x y
Plug in values for x into the equations to find their corresponding y value

67. Example 3
For the given function:
y = -2/3x +4
A. set up a table of values x y -3 6 4 3 2
Plug in values for x into the equations to find their corresponding y value

68. Example 3
For the given function:
y = -2/3x +4
B. graph the function

69. Example 3
For the given function:
y = -2/3x +4
B. graph the function
You have the equation, y=mx+b, so m is the slope and b is the y int.
Plot your y int and count your slope

70. Example 3
For the given function:
y = -2/3x +4
B. graph the function
You have the equation, y=mx+b, so m is the slope and b is the y int.
Plot your y int and count your slope

71. Example 3
For the given function:
y = -2/3x +4
B. graph the function
You have the equation, y=mx+b, so m is the slope and b is the y int.
Plot your y int and count your slope
Note: you could also plot your points from the table

72. Example 3 Connect the points For the given function: y = -2/3x +4
B. graph the function
Connect the points

73. Example 3
For the given function:
y = -2/3x +4
C. find the slope

74. You have your equation, so your slope is the m value in y=mx+b.
Example 3
For the given function:
y = -2/3x +4
C. find the slope
You have your equation, so your slope is the m value in y=mx+b.
What is the slope?

75. You have your equation, so your slope is the m value in y=mx+b.
Example 3
For the given function:
y = -2/3x +4
C. find the slope
You have your equation, so your slope is the m value in y=mx+b.
What is the slope? m=-2/3

76. Example 3 For the given function: y = -2/3x +4
D. identify the x- intercept and y intercept

77. Example 3
For the given function:
y = -2/3x +4
D. identify the x- intercept and y intercept
We have the equation where b in y=mx+b is the y int. What is the y int?

78. Example 3
For the given function:
y = -2/3x +4
D. identify the x- intercept and y intercept
We have the equation where b in y=mx+b is the y int. What is the y int? (0 , 4)
For the x int, check your table and graph. What is the x int?

79. Example 3
For the given function:
y = -2/3x +4
D. identify the x- intercept and y intercept
We have the equation where b in y=mx+b is the y int. What is the y int? (0 , 4)
For the x int, check your table and graph. What is the x int? (6 , 0)

80. Slope Intercept Form y = mx +b Standard Form Ax + By = C
Given information, find the equation in slope intercept form and in standard form
Slope Intercept Form
y = mx +b
Standard Form
Ax + By = C

81. Example 4: find equation in slope intercept form and in standard form
Contains point (-2, 5)

82. Example 4: find equation in slope intercept form and in standard form
Contains point (-2, 5)
Plug in:
y– y1 = m ( x – x1)

83. Example 4: find equation in slope intercept form and in standard form
Contains point (-2, 5)
Plug in:
y– y1 = m ( x – x1)
y – 5 = (-1/2) (x +2)

84. Example 4: find equation in slope intercept form and in standard form
Contains point (-2, 5)
Plug in:
y– y1 = m ( x – x1)
y – 5 = (-1/2) (x +2)
y – 5 = (-1/2)x -1

85. Example 4: find equation in slope intercept form and in standard form
Contains point (-2, 5)
Plug in:
y– y1 = m ( x – x1)
y – 5 = (-1/2) (x +2)
y – 5 = (-1/2)x -1
y = (-1/2)x + 4

86. Example 4: find equation in slope intercept form and in standard form
Contains point (-2, 5)
Plug in:
y– y1 = m ( x – x1)
y – 5 = (-1/2) (x +2)
y – 5 = (-1/2)x -1
y = (-1/2)x + 4
Slope Intercept

87. Example 4: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 5 = (-1/2) (x +2)
y – 5 = (-1/2)x -1
y = (-1/2)x + 4
Slope Intercept
Slope = -1/2
Contains point (-2, 5)
For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive
y=(-1/2)x + 4

88. Example 4: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 5 = (-1/2) (x +2)
y – 5 = (-1/2)x -1
y = (-1/2)x + 4
Slope Intercept
Slope = -1/2
Contains point (-2, 5)
For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive
2[y=(-1/2)x + 4]

89. Example 4: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 5 = (-1/2) (x +2)
y – 5 = (-1/2)x -1
y = (-1/2)x + 4
Slope Intercept
Slope = -1/2
Contains point (-2, 5)
For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive
2[y=(-1/2)x + 4]
2y = -x +8

90. Example 4: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 5 = (-1/2) (x +2)
y – 5 = (-1/2)x -1
y = (-1/2)x + 4
Slope Intercept
Slope = -1/2
Contains point (-2, 5)
For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive
2[y=(-1/2)x + 4]
2y = -x +8
x + 2y = 8

91. Example 4: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 5 = (-1/2) (x +2)
y – 5 = (-1/2)x -1
y = (-1/2)x + 4
Slope Intercept
Slope = -1/2
Contains point (-2, 5)
For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive
2[y=(-1/2)x + 4]
2y = -x +8
x + 2y = 8
Standard Form

92. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)

93. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1

94. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1
1 – -3
-6– 2 =

95. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1
1 – -3
-6– 2 4 -8 = =

96. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1
1 – -3
-6– 2 4 -8 -1 2 = = =

97. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1
1 – -3
-6– 2 4 -8 -1 2 = = =
Now plug in:
y– y1 = m ( x – x1)

98. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1
1 – -3
-6– 2 4 -8 -1 2 = = =
Now plug in:
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x - -6)

99. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1
1 – -3
-6– 2 4 -8 -1 2 = = =
Now plug in:
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x + 6)

100. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1
1 – -3
-6– 2 4 -8 -1 2 = = =
Now plug in:
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x + 6)
y – 1 = (-1/2)x - 3

101. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1
1 – -3
-6– 2 4 -8 -1 2 = = =
Now plug in:
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x + 6)
y – 1 = (-1/2)x – 3
y = (-1/2)x - 2

102. Example 5: find equation in slope intercept form and in standard form
Contains points (2, -3) and (-6, 1)
First find the slope:
y2 – y1
x2 – x1
1 – -3
-6– 2 4 -8 -1 2 = = =
Now plug in:
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x + 6)
y – 1 = (-1/2)x – 3
y = (-1/2)x – 2
Slope Intercept Form

103. Example 5: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x + 6)
y – 1 = (-1/2)x – 3
y = (-1/2)x – 2
Slope Intercept Form
Contains points (2, -3) and (-6, 1)
Now rearrange into standard form
y = (-1/2)x – 2

104. Example 5: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x + 6)
y – 1 = (-1/2)x – 3
y = (-1/2)x – 2
Slope Intercept Form
Contains points (2, -3) and (-6, 1)
Now rearrange into standard form
y = (-1/2)x – 2
2[y = (-1/2)x – 2]

105. Example 5: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x + 6)
y – 1 = (-1/2)x – 3
y = (-1/2)x – 2
Slope Intercept Form
Contains points (2, -3) and (-6, 1)
Now rearrange into standard form
y = (-1/2)x – 2
2[y = (-1/2)x – 2]
2y = -x - 4

106. Example 5: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x + 6)
y – 1 = (-1/2)x – 3
y = (-1/2)x – 2
Slope Intercept Form
Contains points (2, -3) and (-6, 1)
Now rearrange into standard form
y = (-1/2)x – 2
2[y = (-1/2)x – 2]
2y = -x – 4
x + 2y = -4

107. Example 5: find equation in slope intercept form and in standard form
y– y1 = m ( x – x1)
y – 1 = (-1/2) (x + 6)
y – 1 = (-1/2)x – 3
y = (-1/2)x – 2
Slope Intercept Form
Contains points (2, -3) and (-6, 1)
Now rearrange into standard form
y = (-1/2)x – 2
2[y = (-1/2)x – 2]
2y = -x – 4
x + 2y = -4
Standard Form