The SLOPE of a Line

Section 4.4

In the real world, the roofs of houses are “pitched” differently

Some have a shallow, flat tilt

Others have a steep, large tilt

In algebra, graphs of lines have a special name for the “tilt” associated with them

-2-4 2 4-2

-4

2

4

The SLOPE (m) of a line is defined as the following:

)()(

12

12

xxyy

xy

m

*the triangle is the “DELTA” function, which means “the change in” or “the difference between”

Today we want to be able to calculate the slope of a line

WHAT DOES THAT MEAN ???

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-4

2

4

It means we only need TWO POINTS on the line to find the slope.

Let’s use (-2,-3) and (0,1) as our points

(-2,-3) and (0,1)

The formula for slope requires us to break apart the x and y values of their coordinates in order to substitute and solve.I like to use the “taller” point as the “subscript 2’s” and the “lower” point asThe “subscript 1’s”. The subscripts simply help identify the coordinates but have no mathematical value themselves

(x2, y2)(x1, y1)

)()(

12

12

xxyy

xy

m

Write formula 1st !

Sub-in the components

For y2 , y1 ,x2 , and x1: )20(

)31(

m

simplify: 212

24 m

So what does a slope of 2 mean ?

-2-4 2 4-2

-4

2

4

It means the line followsA pattern of rising up 2Units for every 1 unit ittravels to the right.

Sometimes we refer to the slope as the

“RISE”

“RUN”

What would a slope of mean ?

32

-2-4 2 4-2

-4

2

4

It means the line followsA pattern of falling down2 units for every 3 units ittravels to the right.

Looking left to right, negative slopes travel

DOWNWARD

Remember, the negative moves to the numerator, making it32

32

How about a slope of -5 ?

32

-2-4 2 4-2

-4

2

4

It means the line followsA pattern of falling down5 units for every 1 unit ittravels to the right.

Integers can always be placed over a “1”

15

How about a horizontal line ?

32

-2-4 2 4-2

-4

2

4

Although the points have different x components,the “y” values NEVERchange…so since thegraph doesn’t “RISE”…

Like the equation y = -3 …

The slope of a horizontal line is ZERO (m = 0)

How about a vertical line ?

32

-2-4 2 4-2

-4

2

4

Although the points have different y components,the “x” values NEVERchange…so since thegraph doesn’t “RUN” side to side…and our formula would have zero in the denominator…

Like the equation x = 4 …

The slope of a vertical line is UNDEFINED

In general…

32

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-4

2

4

As lines rotate counterclockwise thru quadrant I, theirslopes increase until theyAre vertical and undefined.

As lines rotate clockwise down thru quadrant IV, their slope becomes more NEGATIVE until they are vertical and undefined.

Lastly, find the missing piece…

• A line travels thru (-2, 1) and (4, k). Find the value of “k” if the line’s slope is -2/3.

12

12

xxyy

xy

m

)2(4

1

32

k

6

1

32

k

)1(312 k

3312 k

k39

So, k = -3. The true

Coordinate was (4, -3)