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Find and use the slope of a line.

Graph parallel and perpendicular lines.

1) slope

2) rate of change

SlopeSlope

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If the pilot doesnt change something, he / she will not make ithome for Christmas. Would you agree?

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Consider the options:

1) Keep the same slope of his / her path.

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Consider the options:

1) Keep the same slope of his / her path.

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Consider the options:

1) Keep the same slope of his / her path.Not a good choice!

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Consider the options:

1) Keep the same slope of his / her path.Not a good choice!

2) Go straight up.

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Consider the options:

1) Keep the same slope of his / her path.Not a good choice!

2) Go straight up.

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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.

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F ortunately, there is a way to measure a proper slope to clear the obstacle.

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F ortunately, 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.

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F ortunately, 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(

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F ortunately, 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 changehorizontal change

ySlope

x

(! !

(

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F ortunately, 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 changehorizontal change

ySlope

x

(! !

(

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F ortunately, 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 changehorizontal change

ySlope

x

(! !

(

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F ortunately, 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 changehorizontal change

ySlope

x

(! !

(

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F ortunately, 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 changehorizontal change

ySlope

x

(! !

(

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vertical change 4,000 4 2horizontal change 10,000 10 5

y ft Slope

x ft

(! ! ! ! !

(

y

x10000

10000

00

y(

x(

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x

y

FINDING THE SLOPE O F A LIN E

SlopeSlope

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x

y

FINDING THE SLOPE O F A LIN E

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

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x

y

FINDING THE SLOPE O F A LIN E

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

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x

y

FINDING THE SLOPE O F A LIN E

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 inchange xin

!

SlopeSlope

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x

y

2 1 x x

FINDING THE SLOPE O F A LIN E

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 1 y yychange inchange xin

! 2 1

2 1 x x y y

!

SlopeSlope

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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

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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 = =

ris e ym run x

(

! (

SlopeSlope

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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 = =

ris e ym run x

(

! (

SlopeSlope

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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 = =ris e y

m run x

(

! (

SlopeSlope

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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 = =ris e y

m run x

(

! (

SlopeSlope

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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 = =ris e y

m run x

(

! (

5= 2

SlopeSlope

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y

x

SlopeSlope

F ind the slope of the line.

(2, 3)

(8, 8)

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y

x

SlopeSlope

F ind the slope of the line.

(2, 3)

(8, 8)

riseslope =

run

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y

x

SlopeSlope

F ind the slope of the line.

(2, 3)

(8, 8)

run = 8 - 2= 6 units

rise = 8 - 3= 5 units

riseslope =

run

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y

x

SlopeSlope

F ind the slope of the line.

(2, 3)

(8, 8)

run = 8 - 2= 6 units

rise = 8 - 3= 5 units

riseslope =

run5

=6

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P lot the points (-4, 7) and (4, -1) and draw a line through them.F ind 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

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P lot the points (-4, 7) and (4, -1) and draw a line through them.F ind 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

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P lot the points (-4, 7) and (4, -1) and draw a line through them.F ind 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

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P lot the points (-4, 7) and (4, -1) and draw a line through them.F ind 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 1 x y

my

x!

1 ( 4,7) P

2 (4, 1) P

SlopeSlope

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P lot the points (-4, 7) and (4, -1) and draw a line through them.F ind 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 1 x y

my

x!

4 (1

)74

!

1 ( 4,7) P

2 (4, 1) P

SlopeSlope

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P lot the points (-4, 7) and (4, -1) and draw a line through them.F ind 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 1 x y

my

x!

4 (1

)74

!

88

!

1 ( 4,7) P

2 (4, 1) P

SlopeSlope

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P lot the points (-4, 7) and (4, -1) and draw a line through them.F ind 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 1 x y

my

x!

4 (1

)74

!

88

!

1!

1 ( 4,7) P

2 (4, 1) P

SlopeSlope

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P lot the points (-4, 7) and (4, -1) and draw a line through them.F ind 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 1 x y

my

x!

4 (1

)74

!

88

!

1!

Negative slope:F alls from left to right

1 ( 4,7) P

2 (4, 1) P

SlopeSlope

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y

x

Graph the line passing through point(1, 1) with a slope of 2.

SlopeSlope

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y

x

Graph the line passing through point(1, 1) with a slope of 2.

1) Graph the point (1, 1).

SlopeSlope

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y

x

Graph the line passing through point(1, 1) with a slope of 2.

1) Graph the point (1, 1).

2) F ollow the slope of to locate another point on the line. 1

2

SlopeSlope

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y

x

Graph the line passing through point(1, 1) with a slope of 2.

1) Graph the point (1, 1).

2) F ollow the slope of to locate another point on the line. 1

2

SlopeSlope

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y

x

Graph the line passing through point(1, 1) with a slope of 2.

1) Graph the point (1, 1).

2) F ollow the slope of to locate another point on the line. 1

2

SlopeSlope

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y

x

Graph the line passing through point(1, 1) with a slope of 2.

1) Graph the point (1, 1).

2) F ollow the slope of to locate another point on the line. 1

2

3) Draw the line, connecting the two points.

SlopeSlope

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y

x

If the line rises to the right,then the slope is positive.

SlopeSlope

ll

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y

x

If the line rises to the right,then the slope is positive.

SlopeSlope

SlSl

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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

SlSl

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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

SlSl

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y

x

If the line is horizontal,then the slope is zero.

SlopeSlope

SlSl

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y

x

If the line is horizontal,then the slope is zero.

SlopeSlope

SlSl

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y

x

If the line is horizontal,then the slope is zero.

y

x

If the line is vertical,then the slope isundefined.

SlopeSlope

SlSl

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y

x

If the line is horizontal,then the slope is zero.

y

x

If the line is vertical,then the slope isundefined.

SlopeSlope

SlSl

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SlopeSlope

y

x

In a plane, nonvertical lines _________________ are parallel .

SlopeSlope

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SlopeSlope

y

x

In a plane, nonvertical lines _________________ are parallel .w ith the same slope

SlopeSlope

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SlopeSlope

y

x

In a plane, nonvertical lines _________________ are parallel .w ith the same slope

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SlopeSlope

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y

x

In a plane, nonvertical lines are perpendicular if and only if their slopes are _________________.

SlopeSlope

SlopeSlope

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y

x

In a plane, nonvertical lines are perpendicular if and only if their slopes are _________________.negative reciprocal

SlopeSlope

SlopeSlope

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y

x

In a plane, nonvertical lines are perpendicular if and only if their slopes are _________________.negative reciprocal

SlopeSlope

SlopeSlope

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y

x

In a plane, nonvertical lines are perpendicular if and only if their slopes are _________________.negative reciprocal

SlopeSlope

SlopeSlope

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SlopeSlope

CreditsCredits

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CreditsCredits

P ower P ointcreated by

Using Glencoes Algebra 2 text, 2005

http://robertfant.com