1. Section 8.1 The Slope of a Line

2. Definition- Linear Equation
Any equation that can be put into the from ax + by = c, where a, b, and c are real numbers and a and b are not both 0, is called a linear equation in two variables.
The graph will be a straight line.
The form ax + by = c is called standard form.

3. Standard Form y = 3x + 4 5 – x = 2y -3x + y = 4 -x – 2y = -5 or

4. Slope describes the direction of a line.

5. Guard against 0 in the denominator
Why is this needed?
Slope
If x1  x2, the slope of the line through the distinct points (x1, y1) and (x2, y2) is
Guard against 0 in the denominator

6. Find the slope between (-3, 6) and (5, 2)
x-axis
y-axis
Rise -4 -1 = =
Run 8 2

7. Find the slope between (-3, 6) and (5, 2)

8. Find the Slope
Yellow
(3, 9)
Blue
(11, 2)
Red
(5, -2)

9. Find the slope between (5, 4) and (5, 2).
STOP
This slope is undefined.

10. Find the slope between (5, 4) and (5, 2).
x-axis
y-axis
Rise -2
Undefined = =
Run

11. Find the slope between (5, 4) and (-3, 4).
This slope is zero.

12. Find the slope between (5, 4) and (-3, 4).
x-axis
y-axis
Rise
Zero = =
Run -8

13. From these results we can see...
The slope of a vertical line is undefined.
The slope of a horizontal line is 0.

14. Find the slope of the line 4x - y = 8
Let x = 0 to find the
y-intercept.
Let y = 0 to find the
x-intercept.
(0, -8)
(2, 0)

15. Find the slope of the line 4x - y = 8
Here is an easier way
Solve for y.
When the equation is solved for y the coefficient of the x is the slope. m
m = 4

16. Graph the line that goes through (1, -3) with
(1,-3)
x-axis
y-axis

17. Sign of the Slope Which have a negative slope? Which have a
positive slope?
Red
Light Blue
White
Undefined
Green
Blue
Zero
Slope

18. Slope of Parallel Lines
Two nonvertical lines with the same slope are parallel.
Two nonvertical parallel lines have the same slope.

19. Are the two lines L1, through (-2, 1) and (4, 5) and L2, through (3, 0) and (0, -2), parallel?

20. Perpendicular Slopes 4 3 y-axis x-axis What can we say
about the intersection
of the two white lines?

21. Slopes of Perpendicular Lines
If neither line is vertical then the slopes of perpendicular lines are negative reciprocals.
Lines with slopes that are negative reciprocals are perpendicular.
If the product of the slopes of two lines is -1 then the lines are perpendicular.
Horizontal lines are perpendicular to vertical lines.

22. Write parallel perpendicular or neither for the pair of lines that passes through (5, -9) and (3, 7) and the line through (0, 2) and (8, 3).

23. Section 8.2 The Equation of a Line
Page 482

24. Objectives
Write the equation of a line, given its slope and a point on the line.
Write the equation of a line, given two points on the line.
Write the equation of a line given its slope and y-intercept.

25. Objectives
Find the slope and the y-intercept of a line, given its equation.
Write the equation of a line parallel or perpendicular to a given line through a given point.
Apply concepts of linear equations to realistic examples.

26. Point-slope Form Objective
Write the equation of a line, given its slope
and a point on the line.

27. Margin 1b Through (-2, 7); m = 3
These loose there subscripts and become generic variables.
These variables represent specific values. This is where you substitute.

28. Margin 1a Through (-2, 7); m = 3

29. Horizontal and Vertical Lines
If k is a constant, the vertical line though (k, y) has equation x = k.
If k is a constant, the horizontal line though ( x, k,) has equation y = k.

30. Write the equation of the line through (8, -2); m = 0

31. Write an equation in standard form through (-1, 2) and (5, 7).
First calculate the slope.
Now plug into point slope.

32. Slope-intercept Form (0, b) Suppose we have a line with slope m.
Then the intercept occurs at some point
(0, b)

33. Write an equation in standard form with m = 2 and passing through (0, -3).

34. Find the slope and the y-intercept of 2x - 5y = 1
Solve for y and then we will be able to read the answer.

35. standard form
Write an equation in standard form for the line through (5, 7) parallel to 2x - 5y = 15.
For lines to be parallel they must have the same slope.
Solve for y
Plug into point slope.

36. Write an equation in standard form for the line through (5, 7) parallel to 2x - 5y = 15.

37. We know the slope and we know a point.
Write an equation in standard form for the line through (5, 7) parallel to 2x - 5y = 15.
We know the slope and we know a point.

38. Now we have to change to standard form.

39. Write an equation in standard form for the line through (5, 7) parallel to 2x - 5y = 15.

40. standard form
Write an equation in standard form for the line through (-8, 3) perpendicular to 2x - 3y = 10.
For lines to be perpendicular they must have negative reciprocals of each other.
Solve for y to find m1
Identify m2
Plug into point slope.

41. Write an equation in standard form for the line through (-8, 3) perpendicular to 2x - 3y = 10.

42. Identify the slope of the perpendicular.
The slope of the perpendicular line is the negative reciprocal of m1
Take the negative reciprocal of m1
Flip it over and change the sign.

43. We know the perpendicular slope and we know a point.
Write an equation in standard form for the line through (-8, 3) perpendicular to 2x - 3y = 10.
We know the perpendicular slope and we know a point.

44. Now we have to change to standard form.

45. Write an equation in standard form for the line through (-8, 3) perpendicular to 2x - 3y = 10.

46. Summary of Forms
x-intercept
y-intercept
Standard form
Slope =

47. Summary of Forms Vertical line Horizontal line Slope is undefined
x-intercept is (k, 0)
Horizontal line
Slope is 0.
y-intercept is (0, k)

48. Summary of Forms
Slope-intercept form
Slope is m.
y-intercept is (0, b)

49. Summary of Forms
Point-slope form
Slope is m.
Line passes through (x1, y1)

50. 8.2 Homework
Page 489
1-51 odd

51. Section 8.4 Introduction to Functions
Page 500

52. Relation A relation is a set of ordered pairs of real numbers.
If I say (2, __ ) , can you fill in the blank?
G = {(3, 3) (4, 1) (2, 1) (1, 3)}
If I say (4, __ ) , can you fill in the blank?

53. Domain F = {(3, 2) (4, 1) (2, 4) (1, 3)}
In a domain the set of all of the values of the independent variable is called the domain.
What is the domain of F?
{3, 4, 2, 1}
Does G = {(3, 3) (4, 1) (2, 1) (1, 3)} have the same domain?

54. Range G = {(3, 3) (4, 1) (2, 1) (1, 3)}
In a domain the set of all of the values of the dependent variable is called the range.
What is the range of G?
{3, 1}
Does F = {(3, 2) (4, 1) (2, 4) (1, 3)} have the same range?

55. (Domain, Range)
Notice the alphabetical characteristic of Domain and Range.
(x, y)
(a, b)
(abscissa, ordinate)
Unfortunately (independent, dependent) breaks the rule.

56. Function H = {(3, 2) (4, 1) (3, 4) (1, 3)}
A function is a relation in which , for each value of the first component there is exactly one value of the second component.
H = {(3, 2) (4, 1) (3, 4) (1, 3)}
K = {(2, 3) (4, 1) (3, 1) (2, 3)}
H is not a function,but K is a function.

57. Function Expressed as a Mapping
Domain
Range
F =
{(A,1)
(C, 2)
(B, 3)} A 1 C 2 3 B

58. Function Expressed as a Mapping
Domain
Range
G =
{(A,1)
(C, 2)
(B, 3)
(A, 4)} 4 A 1 C 2 3 B
Since A goes to two ranges G is not a function.

59. Finding Domains and Ranges from Graphs6
The range runs from -6 to 6
[-6, 6]
The domain runs from -4 to 4
[-4, 4] -4 4 -6

60. Finding Domains and Ranges from Graphs
The range runs from - to 
(-, )
The domain runs from -  to 
(-, )

61. Finding Domains and Ranges from Graphs
The range runs from -3 to 
[-3, )
The domain runs from -  to 
(-, ) -3

62. Identify a Function from an Equation
Domain
(-, )
Over the entire domain there is no choice for x that corresponds to two values of y.
First consider the domain
Exactly what is the domain? (what can x be?)
For any choice of x in the domain there should be exactly one value in the range.

63. Identify a Function from an Equation
Domain
Whenever there is a fraction you must guard against a zero denominator.
Range

64. therefore not a function
Vertical Line Test
If a vertical line intersects the graph of a relation in more than one point, then the relation is not a function.
H = {(3, 2) (4, 1) (3, 4) (1, 3)}
Two intersections
therefore not a function

65. Vertical Line Test

66. Vertical Line Test

67. Vertical Line Test

68. Vertical Line Test

69. Think of the uppercase letters. ABCDEFGHIJKLMNOP...
Vertical Line Test
Think of the uppercase letters. ABCDEFGHIJKLMNOP...
Which of the uppercase letters would pass the vertical line test?

70. It does not mean F × x (multiplication)
Functional Notation
y = F(x)
F(x) read F of x
It does not mean F × x (multiplication)

71. Functional Notation Consider y = 2x + 5
Suppose that you wanted to tell someone to substitute in x = 3 into an equation.
With functional notation y = 2x + 5 becomes f(x) = 2x + 5.
And f(3) means substitute in 3 everyplace you see an x.

72. f(x) = mx + b Linear Function Domain Range
A function that can be written in the form
f(x) = mx + b
is a linear function.
Domain
Range

74. Section 8.5 Function Notation
Page 513

75. It does not mean F × x (multiplication)
Functional Notation
y = F(x)
F(x) read F of x
It does not mean F × x (multiplication)

76. Functional Notation Consider y = 2x + 5
Suppose that you wanted to tell someone to substitute in x = 3 into an equation.
With functional notation y = 2x + 5 becomes f(x) = 2x + 5.
And f(3) means substitute in 3 everyplace you see an x.

77. f(x) = mx + b Linear Function Domain Range
A function that can be written in the form
f(x) = mx + b
is a linear function.
Domain
Range

78. Example 1 If f(x) = 7.5x, find f(0), f(10), f(20).

79. Example 5 Find f(0), f(3), and f(-2)

80. Example 5 Find f(0), f(3), and f(-2)

81. Example 5 Find f(0), f(3), and f(-2)

82. Homework problem 4 g(-2)

83. Homework problem #10 f(2) - g(3)

84. #47 Graph the function Draw x = 4 Draw f(4)
f(x) 2 3 4

85. Homework – 51 Odd or EOO

86. Section 8.6 Algebra with Functions
Page 522

87. Example 1 If
and
then write the formula for the functions for

88. Example 1

89. Example 1

90. Example 1

91. Example 1

92. Composition Example If and
then write the formula for the functions for

93. Composition Example

94. Composition Example

95. Homework 1 – 43 Odd or EOO