what is slope? mountains have slopes…. …and so do lines
TRANSCRIPT
What is Slope?
Mountains have slopes….
…and so do lines.
A skier can go down this slope
When you think of down, you think of negative.
This line, like the mountain, has a
negative slopey = –x + 6
You can also run up the mountain.
When you think of up, you think positive.
This lin
e, like th
e mounta
in, has a
posit
ive
slope.
y = x – 4
Does this mountain have a slope??
Slope?? This hasSlope?? This haszero slope!zero slope!
Horizontal lines have a slope of zero.
0
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
Y = 3
What do you think the slope of a vertical line would be ??
This is not a slope; it’s a cliff!This is not a slope; it’s a cliff!
The slope of a vertical
line is undefined!
It has NO slope.
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
X = -5
A line that goes up from left to right has what kind of slope???
What kind of line has a slope of zero??
What is the slope of this line??
One more time….
+
-
0
undefined
What is Slope?
Slope measures the “steepness” of a line.
Slope can be Positive or Negative or Zero (0) or Undefined.
Higher slope values have steeper slopes.
y = x + 1 y = x – 4 y = 4x
y = –x + 6 y = 2x + 3 y = –3x + 8
Positive SlopeLines with Positive Slope rise from left to right, appearing to go uphill.
Notice the “m” value is positive. The larger the value, the steeper the line.
y = x + 1 y = 4xy = 2x + 3
Positive Slope
Lines with Positive Slope also have the following relationship between x and y.
As x-values increase, y-values also increase.
x -2 0 3 4 8
y -1 1 4 5 9
y = x + 1
x -4 -1 2 3 6
y -5 1 7 9 15
y = 2x + 3
x -1 0 1 2 10
y -4 0 4 8 40
y = 4x
Negative SlopeLines with Negative Slope fall from left to right, appearing to go downhill.
Notice the “m” value is negative.The larger the absolute value, the steeper the line.
y = –x + 6 y = –3x + 8
Negative Slope
Lines with Negative Slope also have the following relationship between x and y.
As x-values increase, y-values decrease
y = −x + 6 y = −3x + 8
y = − 2x
x − 3 1 3 5 7
y 9 5 3 1 −1
x 0 2 5 6 10
y 8 2 −7 −10 −22
x −4 − 2 0 2 4
y 8 4 0 −4 −8
Zero SlopeLines with a Zero Slope are flat (horizontal).
Notice the “m” value is 0
x -2 -1 0 1 2
y -3 -3 -3 -3 -3
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
Y = -3
Undefined Slope
x 4 4 4 4 4
y -2 -1 0 1 2
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
X = 4
Lines with a Undefined Slope are (vertical).
The equation is an “X” equals equation.There is no “Y” in the equation.
m =riserun =
change in y change in x =
y2 - y1
x2 - x1
If you are given two points…how do you find the slope?
What do you think?
Find the slope of the line that passes through the points (-2,-5) and (4,3).
So what about the slopes of these lines?
Try this one… (3 , 6) and (3 , -7)
What do you notice?
And this one… (-2, 4) and (0, 4)
What do you notice?
Memory Device
Zero on top = 0
Zero on bottom = undefined
Oh No!!!
The Slope Ratio
horizontal run = 5
vertical rise = 10
slope =vertical rise
horizontal run
so
slope = 105
= 21
= 2
Remember to always simplify the ratio.
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.
Don’t touch that button unless you have tried it on your own first!
SClick the green solution button to check your own work
Apply the formula- vertical rise to horizontal run
Vertical rise = 40 ft
Horizontal run = 200 ft
40 ft.200 ft.
40
Find the GCF
= 1 5
The slope of the hill is1 5
back forward
..
(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
The Slope Of A Line
1
3
5
7
y
-1-1 1 3 5 7
..
USING A COORDINATE PLANE TO FIND SLOPE
The coordinates are (2,2) and (6,5)
Use the formula:
rise run
= 5 - 2 6 - 2 =
3 4
1. Positive Slope
2. Negative Slope
3. Zero Slope
4. Undefined Slope
A line that rises from left to right
A line falls from left to right
The line is horizontal
The line is vertical
It’s your turn:
Use a piece of graph paper to help you illustrate the slope
(x , y )1 1 = (1,0) and (x , y ) = (3, 4)
2 2
(x , y )1 1 = (0, 9) and (x , y ) = (4, 7)
2 2
(x , y )1 1 = (1, 2) and (x , y ) = (5, 2)
2 2
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
(x , y )1 1 = (1, 2) and (x , y ) = (5, 2)
2 2
1
3
5
7
y
-1-1 1 3 5 7
m = y
2 y
1-
x 2
x 1-
m = 2 - 25 - 1
The slope of the line is zero which makes it a zero slope.
m = 0 4
or 0
. .
Let’s Review
• Slope is the steepness of a line
• It is described by a ratio of rise to run
• A coordinate plane can be used to find the slope of a line
• There are 4 types of slope– positive slope
– negative slope
– zero slope
– undefined slope
WHAT IS THE FORMULA FOR SLOPE OF A LINE?