Post on 18-May-2015
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Find and use the slope of a line.
Graph parallel and perpendicular lines.
1) slope2) rate of change
SlopeSlope
If the pilot doesn’t change something, he / she will not make it home for Christmas. Would you agree?
Consider the options:
1) Keep the same slope of his / her path.
Consider the options:
1) Keep the same slope of his / her path.
Consider the options:
1) Keep the same slope of his / her path.Not a good choice!
Consider the options:
1) Keep the same slope of his / her path.Not a good choice!
2) Go straight up.
Consider the options:
1) Keep the same slope of his / her path.Not a good choice!
2) Go straight up.
Consider the options:
1) Keep the same slope of his / her path.Not a good choice!
2) Go straight up.
Not possible! This is an airplane, not a helicopter.
Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” requiredand divide that by the “horizontal change” required.
Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” requiredand divide that by the “horizontal change” required.
y
x
Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” requiredand divide that by the “horizontal change” required.
y
x
vertical change
horizontal change
ySlope
x
Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” requiredand divide that by the “horizontal change” required.
y
x
vertical change
horizontal change
ySlope
x
Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” requiredand divide that by the “horizontal change” required.
y
x
vertical change
horizontal change
ySlope
x
Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” requiredand divide that by the “horizontal change” required.
y
x
vertical change
horizontal change
ySlope
x
Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” requiredand divide that by the “horizontal change” required.
y
x
vertical change
horizontal change
ySlope
x
vertical change 4,000 4 2
horizontal change 10,000 10 5
y ftSlope
x ft
y
x10000
10000
00
y
x
x
y
FINDING THE SLOPE OF A LINE
SlopeSlope
x
y
FINDING THE SLOPE OF A LINE
The slope m of the non-vertical linepassing through the pointsand is
1 1( , )x y
2 2( , )x y
1 1( , )x y
2 2( , )x y
SlopeSlope
x
y
FINDING THE SLOPE OF A LINE
rise
runm
The slope m of the non-vertical linepassing through the pointsand is
1 1( , )x y
2 2( , )x y
1 1( , )x y
2 2( , )x y
SlopeSlope
x
y
FINDING THE SLOPE OF A LINE
rise
runm
The slope m of the non-vertical linepassing through the pointsand is
1 1( , )x y
2 2( , )x y
1 1( , )x y
2 2( , )x y
ychange in
change x in
SlopeSlope
x
y
2 1x x
FINDING THE SLOPE OF A LINE
rise
runm
The slope m of the non-vertical linepassing through the pointsand is
1 1( , )x y
2 2( , )x y
1 1( , )x y
2 2( , )x y
2 1y yychange in
change x in 2 1
2 1x x
y y
SlopeSlope
x
y
2 1x x
FINDING THE SLOPE OF A LINE
rise
runm
The slope m of the non-vertical linepassing through the pointsand is
1 1( , )x y
2 2( , )x y
1 1( , )x y
2 2( , )x y
2 1y yychange in
change x in 2 1
2 1x x
y y
SlopeSlope
y
x
The slope m of a non-vertical line is the number of units the line rises or fallsfor each unit of horizontal change from left to right.
(1, 1)
(3, 6)
SlopeSlope
The slope m of a non-vertical line is the number of units the line rises or fallsfor each unit of horizontal change from left to right.
y
x
(1, 1)
(3, 6)
slope = =rise y
mrun x
SlopeSlope
The slope m of a non-vertical line is the number of units the line rises or fallsfor each unit of horizontal change from left to right.
y
x
(1, 1)
(3, 6)
slope = =rise y
mrun x
SlopeSlope
The slope m of a non-vertical line is the number of units the line rises or fallsfor each unit of horizontal change from left to right.
y
x
(1, 1)
(3, 6)
rise = 6 - 1 = 5 units
slope = =rise y
mrun x
SlopeSlope
The slope m of a non-vertical line is the number of units the line rises or fallsfor each unit of horizontal change from left to right.
y
x
(1, 1)
(3, 6)
run = 3 - 1 = 2 units
rise = 6 - 1 = 5 units
slope = =rise y
mrun x
SlopeSlope
The slope m of a non-vertical line is the number of units the line rises or fallsfor each unit of horizontal change from left to right.
y
x
(1, 1)
(3, 6)
run = 3 - 1 = 2 units
rise = 6 - 1 = 5 units
slope = =rise y
mrun x
5
=2
SlopeSlope
y
x
SlopeSlope
Find the slope of the line.
(2, 3)
(8, 8)
y
x
SlopeSlope
Find the slope of the line.
(2, 3)
(8, 8)
riseslope =
run
y
x
SlopeSlope
Find the slope of the line.
(2, 3)
(8, 8)
run = 8 - 2 = 6 units
rise = 8 - 3 = 5 units
riseslope =
run
y
x
SlopeSlope
Find the slope of the line.
(2, 3)
(8, 8)
run = 8 - 2 = 6 units
rise = 8 - 3 = 5 units
riseslope =
run
5=
6
Plot the points (-4, 7) and (4, -1) and draw a line through them.Find the slope of the line passing through the points.
y
x
10
0
-5-5
5
-5
10
-5
0
5
1 1 1( , ) ( 4,7)P x y 2 2 2( , ) (4, 1)P x y
SlopeSlope
Plot the points (-4, 7) and (4, -1) and draw a line through them.Find the slope of the line passing through the points.
y
x
10
0
-5-5
5
-5
10
-5
0
5
1 1 1( , ) ( 4,7)P x y 2 2 2( , ) (4, 1)P x y
1( 4,7)P
2 (4, 1)P
SlopeSlope
Plot the points (-4, 7) and (4, -1) and draw a line through them.Find the slope of the line passing through the points.
y
x
10
0
-5-5
5
-5
10
-5
0
5
1 1 1( , ) ( 4,7)P x y 2 2 2( , ) (4, 1)P x y
1( 4,7)P
2 (4, 1)P
SlopeSlope
Plot the points (-4, 7) and (4, -1) and draw a line through them.Find the slope of the line passing through the points.
y
x
10
0
-5-5
5
-5
10
-5
0
5
1 1 1( , ) ( 4,7)P x y 2 2 2( , ) (4, 1)P x y
2 1
2 1x
ym
y
x
1( 4,7)P
2 (4, 1)P
SlopeSlope
Plot the points (-4, 7) and (4, -1) and draw a line through them.Find the slope of the line passing through the points.
y
x
10
0
-5-5
5
-5
10
-5
0
5
1 1 1( , ) ( 4,7)P x y 2 2 2( , ) (4, 1)P x y
2 1
2 1x
ym
y
x
4 (
1
)
7
4
1( 4,7)P
2 (4, 1)P
SlopeSlope
Plot the points (-4, 7) and (4, -1) and draw a line through them.Find the slope of the line passing through the points.
y
x
10
0
-5-5
5
-5
10
-5
0
5
1 1 1( , ) ( 4,7)P x y 2 2 2( , ) (4, 1)P x y
-8
8
2 1
2 1x
ym
y
x
4 (
1
)
7
4
8
8
1( 4,7)P
2 (4, 1)P
SlopeSlope
Plot the points (-4, 7) and (4, -1) and draw a line through them.Find the slope of the line passing through the points.
y
x
10
0
-5-5
5
-5
10
-5
0
5
1 1 1( , ) ( 4,7)P x y 2 2 2( , ) (4, 1)P x y
-8
8
2 1
2 1x
ym
y
x
4 (
1
)
7
4
8
8
1
1( 4,7)P
2 (4, 1)P
SlopeSlope
Plot the points (-4, 7) and (4, -1) and draw a line through them.Find the slope of the line passing through the points.
y
x
10
0
-5-5
5
-5
10
-5
0
5
1 1 1( , ) ( 4,7)P x y 2 2 2( , ) (4, 1)P x y
-8
8
2 1
2 1x
ym
y
x
4 (
1
)
7
4
8
8
1
Negative slope: Falls from left to right
1( 4,7)P
2 (4, 1)P
SlopeSlope
y
x
Graph the line passing through point(1, 1) with a slope of 2.
SlopeSlope
y
x
Graph the line passing through point(1, 1) with a slope of 2.
1) Graph the point (1, 1).
SlopeSlope
y
x
Graph the line passing through point(1, 1) with a slope of 2.
1) Graph the point (1, 1).
2) Follow the slope of to locate another point on the line. 1
2
SlopeSlope
y
x
Graph the line passing through point(1, 1) with a slope of 2.
1) Graph the point (1, 1).
2) Follow the slope of to locate another point on the line. 1
2
SlopeSlope
y
x
Graph the line passing through point(1, 1) with a slope of 2.
1) Graph the point (1, 1).
2) Follow the slope of to locate another point on the line. 1
2
SlopeSlope
y
x
Graph the line passing through point(1, 1) with a slope of 2.
1) Graph the point (1, 1).
2) Follow the slope of to locate another point on the line. 1
2
SlopeSlope
y
x
Graph the line passing through point(1, 1) with a slope of 2.
1) Graph the point (1, 1).
2) Follow the slope of to locate another point on the line. 1
2
3) Draw the line, connecting the two points.
SlopeSlope
y
x
Graph the line passing through point(1, 1) with a slope of 2.
1) Graph the point (1, 1).
2) Follow the slope of to locate another point on the line. 1
2
3) Draw the line, connecting the two points.
SlopeSlope
y
x
If the line rises to the right,then the slope is positive.
SlopeSlope
y
x
If the line rises to the right,then the slope is positive.
SlopeSlope
y
x
If the line rises to the right,then the slope is positive.
y
x
If the line falls to the right,then the slope is negative.
SlopeSlope
y
x
If the line rises to the right,then the slope is positive.
y
x
If the line falls to the right,then the slope is negative.
SlopeSlope
y
x
If the line is horizontal,then the slope is zero.
SlopeSlope
y
x
If the line is horizontal,then the slope is zero.
SlopeSlope
y
x
If the line is horizontal,then the slope is zero.
y
x
If the line is vertical,then the slope is undefined.
SlopeSlope
y
x
If the line is horizontal,then the slope is zero.
y
x
If the line is vertical,then the slope is undefined.
SlopeSlope
SlopeSlope
y
x
In a plane, nonvertical lines _________________ are parallel.
SlopeSlope
y
x
In a plane, nonvertical lines _________________ are parallel.with the same slope
SlopeSlope
y
x
In a plane, nonvertical lines _________________ are parallel.with the same slope
SlopeSlope
y
x
In a plane, nonvertical lines _________________ are parallel.with the same slope
y
x
In a plane, nonvertical lines are perpendicular if and only if their slopes are _________________.
SlopeSlope
y
x
In a plane, nonvertical lines are perpendicular if and only if their slopes are _________________.negative reciprocal
SlopeSlope
y
x
In a plane, nonvertical lines are perpendicular if and only if their slopes are _________________.negative reciprocal
SlopeSlope
y
x
In a plane, nonvertical lines are perpendicular if and only if their slopes are _________________.negative reciprocal
SlopeSlope
SlopeSlope
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Using Glencoe’s Algebra 2 text,© 2005
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