The document discusses slope and how to calculate it. Slope is defined as the ratio of vertical distance change to horizontal distance change between two points on a line. The formula for slope is provided as m=(y2-y1)/(x2-x1). Several examples are worked through to demonstrate calculating slope for different lines by using points on each line in the formula. Horizontal and vertical lines are also discussed, with horizontal lines having a slope of 0 and vertical lines having an undefined slope.
2. Essential Questions
How do you find the slope of a line?
How do you identify horizontal and
vertical lines?
Where you’ll see it:
Business, science, transportation
5. Vocabulary
1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
6. Vocabulary
1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
1. Slope:
7. Vocabulary
1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
1. Slope: How steep a line is, measured in “rise over run”
8. Vocabulary
1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
1. Slope: How steep a line is, measured in “rise over run”
Formula:
9. Vocabulary
1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
1. Slope: How steep a line is, measured in “rise over run”
Formula:
y 2 − y1
m= , for points ( x 1 , y 1 ) and ( x 2 , y 2 )
x 2 − x1
11. Example 1
Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)
D = (4, 4)
12. Example 1
Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0) C
D = (4, 4)
13. Example 1
Graph the line the goes through the given points,
then find the slope of the line.
D
C = (−4,0) C
D = (4, 4)
14. Example 1
Graph the line the goes through the given points,
then find the slope of the line.
D
C = (−4,0) C
D = (4, 4)
15. Example 1
Graph the line the goes through the given points,
then find the slope of the line.
y 2 − y1
D m=
x 2 − x1
C = (−4,0) C
D = (4, 4)
16. Example 1
Graph the line the goes through the given points,
then find the slope of the line.
y 2 − y1
D m=
x 2 − x1
C = (−4,0) C 4−0
=
D = (4, 4) 4 − (−4)
17. Example 1
Graph the line the goes through the given points,
then find the slope of the line.
y 2 − y1
D m=
x 2 − x1
C = (−4,0) C 4−0
=
D = (4, 4) 4 − (−4)
4
=
8
18. Example 1
Graph the line the goes through the given points,
then find the slope of the line.
y 2 − y1
D m=
x 2 − x1
C = (−4,0) C 4−0
=
D = (4, 4) 4 − (−4)
4 1
= =
8 2
19. Example 1
Graph the line the goes through the given points,
then find the slope of the line.
y 2 − y1
D m=
x 2 − x1
C = (−4,0) C 4−0
=
D = (4, 4) 4 − (−4)
4 1
= =
8 2
Here, the slope tells us “Up 1, Right 2”
20. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
21. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1
22. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1
−2 − (−2)
=
3 −9
23. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1
−2 − (−2)
=
3 −9
0
=
−6
24. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1
−2 − (−2)
=
3 −9
0
= =0
−6
25. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1
−2 − (−2)
=
3 −9
0
= =0
−6
Horizontal
26. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2)
=
3 −9
0
= =0
−6
Horizontal
27. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2) −4 − 1 2
= =
3 −9 3 −3
0
= =0
−6
Horizontal
28. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2) −4 − 1 2
= =
3 −9 3 −3
0 −1 6
= =0 =
−6 0
Horizontal
29. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2) −4 − 1 2
= =
3 −9 3 −3
0 −1 6
= =0 = Undefined
−6 0
Horizontal
30. Example 2
Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2) −4 − 1 2
= =
3 −9 3 −3
0 −1 6
= =0 = Undefined
−6 0
Horizontal Vertical
35. Horizontal vs. Vertical
Horizontal lines have slopes of zero
(Think “horizon”)
Vertical lines have a slope that is
36. Horizontal vs. Vertical
Horizontal lines have slopes of zero
(Think “horizon”)
Vertical lines have a slope that is undefined
37. Horizontal vs. Vertical
Horizontal lines have slopes of zero
(Think “horizon”)
Vertical lines have a slope that is undefined
(It’s neither uphill, downhill, or level)
38. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
39. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
40. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
Down 2, right 1
41. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
42. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
43. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
44. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
45. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
46. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
47. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
48. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
49. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
50. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
51. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
52. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
53. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
54. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
55. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
56. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
57. Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2
−2 =
1
P
Down 2, right 1
58. Example 4
a. Find the slope of AB and CD for the given points.
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
59. Example 4
a. Find the slope of AB and CD for the given points.
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
y 2 − y1
m (AB ) =
x 2 − x1
60. Example 4
a. Find the slope of AB and CD for the given points.
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
y 2 − y 1 2 − (−1 )
m (AB ) = =
x 2 − x1 2−0
61. Example 4
a. Find the slope of AB and CD for the given points.
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
y 2 − y 1 2 − (−1 ) 3
m (AB ) = = =
x 2 − x1 2−0 2
62. Example 4
a. Find the slope of AB and CD for the given points.
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
y 2 − y 1 2 − (−1 ) 3
m (AB ) = = =
x 2 − x1 2−0 2
y 2 − y1
m (CD ) =
x 2 − x1
63. Example 4
a. Find the slope of AB and CD for the given points.
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
y 2 − y 1 2 − (−1 ) 3
m (AB ) = = =
x 2 − x1 2−0 2
y 2 − y1 4 −1
m (CD ) = =
x 2 − x 1 −1 − (−3)
64. Example 4
a. Find the slope of AB and CD for the given points.
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
y 2 − y 1 2 − (−1 ) 3
m (AB ) = = =
x 2 − x1 2−0 2
y 2 − y1 4 −1 3
m (CD ) = = =
x 2 − x 1 −1 − (−3) 2
65. Example 4
b. Graph the t wo lines. What do you notice?
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
66. Example 4
b. Graph the t wo lines. What do you notice?
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
A
67. Example 4
b. Graph the t wo lines. What do you notice?
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
B
A
68. Example 4
b. Graph the t wo lines. What do you notice?
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
B
C
A
69. Example 4
b. Graph the t wo lines. What do you notice?
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
D
B
C
A
70. Example 4
b. Graph the t wo lines. What do you notice?
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
D
B
C
A
71. Example 4
b. Graph the t wo lines. What do you notice?
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
D
B
C
A
72. Example 4
b. Graph the t wo lines. What do you notice?
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
D
B The lines are parallel.
C
A
73. Example 4
b. Graph the t wo lines. What do you notice?
A = (0 , −1 ), B = (2, 2), C = (−3 ,1 ), D = (−1 , 4)
D
B The lines are parallel.
C
A They have the same slope.
