the slope of a line important vocabulary terms the slope of a line is the ratio of the vertical rise...
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The Slope of a Line
Important Vocabulary Terms
The slope of a line is the ratio of the vertical rise to the horizontal run between any two points on the line.
You can describe steepness by using slope of a line.
The Slope Ratio
horizontal run = 5
vertical rise = 10
slope =vertical rise
horizontal run
so
slope = 105
= 2
Slope does not have units
Let’s Try It Out!
horizontal run = 200 ft
vertical rise = 40 ft
Let’s try and find the slope of a hill that has a vertical rise of 40ft and a horizontal run of 200ft. Let m represent the slope.
SClick the green solution button to check work.
Apply the formula- vertical rise to horizontal run
Vertical rise = 40 ft
Horizontal run = 200 ft
40 ft.200 ft.
=1 5
The slope of the hill is 0.2
back forward
= 0.2
How to find the rise and the run
..
(x2, y2)
(x1, y1)x2 - x1
y2 - y1
The slope m of a line that passes through the points (x1, y1) and (x2, y2) is
m =riserun =
change in y change in x =
y2 - y1
x2 - x1
0, 0
6.3, 2
9.54, 3
3.24, 1
0
0.5
1
1.5
2
2.5
3
3.5
0 2 4 6 8 10 12
time (sec)
dis
tan
ce (
m)
run
rise
slope = 1/3.24
rise
run
slope = 1/3.24
HOW TO DETERMINE SLOPE
1. Positive Slope
2. Negative Slope
3. Zero Slope
A line that rises from left to right
A line falls from left to right
A line is horizontal
Different Types of Slope
1
3
5
7
y
-1-1 1 3 5 7.
(x , y )1 1 = (1, 0) and (x , y ) = (3, 4)
2 2
m = y
2 y
1-
x 2
x 1-
m = 4 - 03 - 1
m = 42
.
The slope of the line is 2
1
3
5
7
y
-1-1 1 3 5 7
. m =
y2
y1-
x 2
x 1-
m = 7 - 94 - 0
.(x , y )
1 1 = (0, 9) and (x , y ) = (4, 7)2 2
The slope of the line is -1 which makes it a negative slope. 2
m = -2 4
or-1 2
m = y
2 y
1-
x 2
x 1-
m = 15 - 103 - 2
m = 5 1
or 5
(1,5), (2,10), (3,15), (4,20), (5,25)
m = y
2 y
1-
x 2
x 1-
m = 25 - 55 - 1
m = 20 5
or 5
m = y
2 y
1-
x 2
x 1-
m = 60 - 405 - 4
m = 20 1
or 20
(1,5), (2,15), (3,25), (4,40), (5,60)
m = y
2 y
1-
x 2
x 1-
m = 15 - 52 - 1
m = 10 2
or 5
The slope of this line is changing. The object is speeding up
Seconds 4-5Seconds 1-2
SLOPE AND SPEED1. Who is walking faster during the first 2 seconds?
Answer: Cory. The slope and thus speed is 6/2 = 3 m/s. Notice Cory’s line is steeper at that segment than the other lines.
rise
run
SLOPE AND SPEED2. Who is walking faster the last 3 seconds?
Answer: Justin. The slope and thus speed is 6/3 = 2 m/s. Notice Justin’s line is steeper at that segment than the other lines. Cory is now the slowest (least steep slope.
rise
run
speed of parts of a coaster
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5 6
time (sec)
dis
tan
ce (
m)
straightaway
decline
incline
SLOPE AND SPEED 1. What is happening to the slope/speed as the coaster cars go up an incline?
Answer: The slope gets less steep, and thus the speed decreases. You know that when you go uphill you become slower because you are going against the force of gravity.
Ver
y st
eep
slop
e: h
igh
spee
d
Less steep slope: lower speed
speed of parts of a coaster
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5 6
time (sec)
dis
tan
ce (
m)
straightaway
decline
incline
SLOPE AND SPEED 2. What is happening to the slope/speed as the coaster cars go down a decline?
Answer: The slope gets more steep, and thus the speed increases. You know that when you go downhill you become faster because you are going with the force of gravity.
Less steep slope: lower speed
mor
e st
eep
slop
e: h
ighe
r spe
ed
speed of parts of a coaster
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5 6
time (sec)
dis
tan
ce (
m)
straightaway
decline
incline
SLOPE AND SPEED 3. What is happening to the slope/speed as the coaster cars goes along a straightaway?
Answer: The slope stays the same, and thus the speed stays the same (constant speed).
Same slope (doesn’t change): c
onstant speed
0
5
10
15
20
25
30
0 1 2 3 4 5 6
time
dis
tan
ceWhat type of motion? constant speed
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5 6
time
dis
tan
ceWhat type of motion? positive acceleration
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5 6
time
dis
tan
ceWhat type of motion? negative acceleration
0
0.5
1
1.5
2
2.5
3
3.5
0 2 4 6 8 10 12
time (sec)
dis
tan
ce
(m
)
What type of motion? -constant speed-0 speed-constant speed
0
5
10
15
20
25
30
0 2 4 6
time (s)
dis
tan
ce (
m)
Fastest Speed?
green, becauseit has the highest slope
Speed vs. Time
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
time (minutes)
spee
d (
mi/h
ou
r)What type of motion? negative acceleration
Speed vs. Time
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
time (minutes)
spee
d (
mi/h
ou
r)What type of motion? positive acceleration