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?