Download - Slope (Algebra 2)

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.


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



2) Go straight up.



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.


We measure the “change in height” requiredand divide that by the “horizontal change” required.



y

x



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


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


rise

runm


1 1( , )x y

2 2( , )x y

1 1( , )x y

2 2( , )x y

SlopeSlope

x

y


rise

runm


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


rise

runm


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


y

x

(1, 1)

(3, 6)

slope = =rise y

mrun x

SlopeSlope


y

x

(1, 1)

(3, 6)

rise = 6 - 1 = 5 units

slope = =rise y

mrun x

SlopeSlope


y

x

(1, 1)

(3, 6)

run = 3 - 1 = 2 units


slope = =rise y

mrun x

SlopeSlope


y

x

(1, 1)

(3, 6)

run = 3 - 1 = 2 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


(2, 3)

(8, 8)

riseslope =

run

y

x

SlopeSlope


(2, 3)

(8, 8)

run = 8 - 2 = 6 units


riseslope =

run

y

x

SlopeSlope


(2, 3)

(8, 8)

run = 8 - 2 = 6 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


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


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


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


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


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


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


1) Graph the point (1, 1).

SlopeSlope

y

x



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

2

SlopeSlope

y

x



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.

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.

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

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

SlopeSlope